Encyclopedia Of Physical Science And Technology - Classical Physics [PDF] [2kfr2drg27c0] (2023)

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Acoustic Chaos Werner Lauterborn Universit¨at G¨ottingen

I. II. III. IV. V. VI. VII. VIII.

The Problem of Acoustic Cavitation Noise The Period-Doubling Noise Sequence A Fractal Noise Attractor Lyapunov Analysis Period-Doubling Bubble Oscillations Theory of Driven Bubbles Other Systems Philosophical Implications

GLOSSARY Bifurcation Qualitative change in the behavior of a system, when a parameter (temperature, pressure, etc.) is altered (e.g., period-doubling bifurcation); related to phase change in thermodynamics. Cavitation Rupture of liquids when subject to tension either in flow fields (hydraulic cavitation) or by an acoustic wave (acoustic cavitation). Chaos Behavior (motion) with all signs of statistics despite an underlying deterministic law (often, deterministic chaos). Fractal Object (set of points) that does not have a smooth structure with an integer dimension (e.g., three dimensional). Instead, a fractal (noninteger) dimension must be ascribed to them. Period doubling Special way of obtaining chaotic (irregular) motion; the period of a periodic motion doubles repeatedly until in the limit of infinite doubling aperiodic motion is obtained. Phase space Space spanned by the dependent variables of

a dynamic system. A point in phase space characterizes a specific state of the system. Strange attractor In dissipative systems, the motion tends to certain limits forms (attractors). When the motion comes to rest, this attractor is called a fixed point. Chaotic motions run on a strange attractor, which has involved properties (e.g., a fractal dimension).

THE PAST FEW years have seen a remarkable development in physics, which may be described as the upsurge of “chaos.” Chaos is a term scientists have adapted from common language to describe the motion or behavior of a system (physical or biological) that, although governed by an underlying deterministic law, is irregular and, in the long term, unpredictable. Chaotic motion seems to appear in any sufficiently complex dynamical system. Acoustics, that part of physics that descibes the vibration of usually larger ensembles of molecules in gases, liquids, and solids, makes no exception. As a main necessary ingredient of chaotic

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118 dynamics is nonlinearity, acoustic chaos is closely related to nonlinear oscillations and waves in gases, liquids, and solids. It is the science of never-repeating sound waves. This property it shares with noise, a term having its origin in acoustics and formerly attributed to every sound signal with a broadband Fourier spectrum. But Fourier analysis is especially adapted to linear oscillatory systems. The standard interpretation of the lines in a Fourier spectrum is that each line corresponds to a (linear) mode of vibration and a degree of freedom of the system. However, as examples from chaos physics show, a broadband spectrum can already be obtained with just three (nonlinear) degrees of freedom (that is, three dependent variables). Chaos physics thus develops a totally new view of the noise problem. It is a deterministic view, but it is still an open question how far the new approach will reach in explaining still unsolved noise problems (e.g., the 1/ f -noise spectrum encountered so often). The detailed relationship between chaos and noise is still an area of active research. An example, where the properties of acoustic noise could be related to chaotic dynamics, is given below for the case of acoustic cavitation noise. Acoustic chaos appears in an experiment when a liquid is irradiated with sound of high intensity. The liquid then ruptures to form bubbles or cavities (almost empty bubbles). The phenomenon is known as acoustic cavitation and is accompanied by intense noise emission—the acoustic cavitation noise. It has its origin in the bubbles set into oscillation in the sound field. Bubbles are nonlinear oscillators, and it can be shown both experimentally and theoretically that they exhibit chaotic oscillations after a series of period doublings. The acoustic emission from these bubbles is then a chaotic sound wave (i.e., irregular and never repeats). This is acoustic chaos.

Acoustic Chaos

to ask what physical mechanisms are known to convert a single frequency to a broadband spectrum? This could not be answered before chaos theory was developed. However, although chaos theory is now well established, a physical (intuitive) understanding is still lacking.

II. THE PERIOD-DOUBLING NOISE SEQUENCE To investigate the sound emission from acoustic cavitation the experimental arrangement as depicted in Fig. 1 is used. To irradiate the liquid (water) a piezoceramic cylinder (PZT-4) of 76-mm length, 76-mm inner diameter, and 5-mm wall thickness is used. When driven at its main resonance, 23.56 kHz, a high-intensity acoustic field is generated in the interior and cavitation is easily achieved. The noise is picked up by a broadband hydrophone and digitized at rates up to 60 MHz after suitable lowpass filtering (for correct analog-to-digital conversion for later processing) and strong filtering of the driving frequency, which would otherwise dominate the noise output. The experiment is fully computer controlled. The amplitude of the driving sound field can be made an arbitrary function of time via a programmable synthesizer. In most cases, linear ramp functions are applied to study the buildup of noise when the driving pressure amplitude in the liquid is increased. From the data stored in the memory of the transient recorder, power spectra are calculated via the fast-Fouriertransform algorithm from usually 4096 samples out of the 128 × 1024 samples stored. This yields about 1000 shorttime spectra when the 4096 samples are shifted by 128 samples from one spectrum to the next. Figure 2 shows four power spectra from one such experiment. Each diagram gives the excitation level at

I. THE PROBLEM OF ACOUSTIC CAVITATION NOISE The projection of high-intensity sound into liquids has been investigated since the application of sound to locate objects under water became used. It was soon noticed that at too high an intensity the liquid may rupture, giving rise to acoustic cavitation. This phenomenon is accompanied by broadband noise emission, which is detrimental to the useful operation of, for instance, a sonar device. The noise emission presents an interesting physical problem that may be formulated in the following way. A sound wave of a single frequency (a pure tone) is transformed into a broadband sound spectrum, consisting of an (almost) infinite number of neighboring frequencies. What is the physical mechanism that causes this transformation? The question may even be shifted in its emphasis

FIGURE 1 Experimental arrangement for measurements on acoustic cavitation noise (chaotic sound).

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FIGURE 2 Power spectra of acoustic cavitation noise at different excitation levels (related to the pressure amplitudes of the driving sound wave). (From Lauterborn, W. (1986). Phys. Today 39, S-4.)

the transducer in volts, the time since the experiment (irradiating the liquid with a linear ramp of increasing excitation) has started in milliseconds, and the power spectrum at this time. At the beginning of the experiment, at low sound intensity, only the driving frequency f 0 shows up. In the upper left diagram of Fig. 2 the third harmonic, 3 f 0 , is present. When comparing both lines it should be remembered that the driving frequency is strongly damped by filtering. In the lower left-hand diagram, many more lines are present. Of special interest is the spectral line at 12 f 0 (and their harmonics). A well-known feature of nonlinear systems is that they produce higher harmonics. Not yet widely known is that subharmonics can also be produced by some nonlinear systems. These then seem to spontaneously divide the applied frequency f 0 to yield, for example, exactly half that frequency (or exactly one-third). This phenomenon has become known as a period-doubling (-tripling) bifurcation. A large class of

systems has been found to show period doubling, among them driven nonlinear oscillators. A peculiar feature of the period-doubling bifurcation is that it occurs in sequences; that is, when one period-doubling bifurcation has occurred, it is likely that further period doubling will occur upon altering a parameter of the system, and so on, often in an infinite series. Acoustic cavitation has been one of the first experimental examples known to exhibit this series. In Fig. 2, the upper right-hand diagram shows the noise spectrum after further period doubling to 14 f 0 . The 1 f 0 up doubling sequence can be observed via 18 f 0 and 16 1 to 32 f 0 (not shown here). It is obvious that the spectrum is rapidly “filled” with lines and gets more and more dense. The limit of the infinite series yields an aperiodic motion, a densely packed power spectrum (not homogeneously), that is, broadband noise (but characteristically colored by lines). One such noise spectrum is shown in Fig. 2 (lower right-hand diagram). Thus, at least one way of turning

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a pure tone into broadband noise has been found—via successive period doubling. This finding has a deeper implication. If a system becomes aperiodic through the phenomenon of repeated period doubling, then this is a strong indication that the irregularity attained in this way is of simple deterministic origin. This implies that acoustic cavitation noise is not a basically statistical phenomenon but a deterministic one. It also implies that a description of the system with usual statistical means may not be appropriate and that a successful description by some deterministic theory may be feasible.

III. A FRACTAL NOISE ATTRACTOR In Section II the sound signal has been treated by Fourier analysis. Fourier analysis is a decomposition of a signal into a sum of simple waves (normal modes) and is said to give the degrees of freedom of the described system. Chaos theory shows that this interpretation must be abandoned. Broadband noise, for instance, is usually thought to be due to a high (nearly infinite) number of degrees of freedom that superposed yield noise. Chaotic systems, however, have the ability to produce noise with only a few (nonlinear) degrees of freedom, that is, with only a few dependent variables. Also, it has been found that continuous systems with only three dependent variables are capable of chaotic motions and thus, producing noise. Chaos theory has developed new methods to cope with this problem. One of these is phase-space analysis, which in conjunction with fractal dimension estimation is capable of yielding the intrinsic degrees of freedom of the system. This method has been applied to inspect acoustic cavitation noise. The answer it may give is the dimension of the dynamical system producing acoustic cavitation noise. See SERIES. The sampled noise data are first used to construct a noise attractor in a suitable phase space. Then the (fractal) dimension of the attractor is determined. The procedure to construct an attractor in a space of chosen dimension n simply consists in combining n samples (not necessarily consecutive ones) to an n-tuple, whose entries are interpreted as the coordinate values of a point in n-dimensional Euclidian space. An example of a noise attractor constructed in this way is given in Fig. 3. The attractor has been obtained from a time series of pressure values { p(kts ); t = 1, . . . , 4096; ts = 1 µsec} taken at a sampling frequency of f s = 1/ts = 1 MHz by forming the threetuples [ p(kts ), p(kts + T ), p(kts + 2T )], k = 1, . . . , 4086, with T = 5 µsec. The frequency of the driving sound field has been 23.56 kHz. The attractor in Fig. 3 is shown from different views to demonstrate its nearly flat structure. It is most remarkable that not an unstructured cluster of points is obtained as is expected for noise, but a quite well-defined

FIGURE 3 Strange attractor of acoustic cavitation noise obtained by phase–space analysis of experimental data (a time series of pressure values sampled at 1 MHz). The attractor is rotated to visualize its three-dimensional structure. (Courtesy of J. Holzfuss. From Lauterborn, W. (1986). In “Frontiers in Physical Acoustics” (D. Sette, ed.), pp. 124–144, North Holland, Amsterdam.)

object. This suggests that the dynamical system producing the noise has only a few nonlinear degrees of freedom. The flat appearance of the attractor in a three-dimensional phase space (Fig. 3) suggests that only three essential degrees are needed for the system. This is confirmed by a fractal dimension analysis, which yields a dimension of d = 2.5 for this attractor. Unfortunately, a method has not yet been conceived of how to construct the equations of motion from the data.

IV. LYAPUNOV ANALYSIS Chaotic systems exhibit what is called sensitive dependence on initial conditions. This expression has been introduced to denote the property of a chaotic system that small differences in the initial conditions, however small, are persistently magnified because of the dynamics of the system. This property is captured mathematically by the notion of Lyapunov exponents and Lyapunov spectra. Their definition can be illustrated by the deformation of a small

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FIGURE 4 Idea for defining Lyapunov exponents. A small sphere in phase space is deformed to an ellipsoid, indicating expansion or contraction of neighboring trajectories.

sphere of initial conditions along a fiducial trajectory (see Fig. 4). The expansion or contraction is used to define the Lyapunov exponents λi , i = 1, 2, . . . , m, where m is the dimension of the phase space of the system. When, on the average, for example, r1 (t) is larger than r1 (0), then λ1 > 0 and there is a persistent magnification in the system. The set {λi , i = 1, . . . , m}, whereby the λi usually are ordered λ1 ≥ λ2 ≥ · · · ≥ λm , is called the Lyapunov spectrum.

FIGURE 5 Acoustic cavitation bubble field in water inside a cylindrical piezoelectric transducer of about 7 cm in diameter. Two planes in depth are shown about 5 mm apart. The pictures are obtained by photographs from the reconstructed three-dimensional image of a hologram taken with a ruby laser.

In dissipative systems, the final motion takes place on attractors. Besides the fractal dimension, as discussed in the previous section, the Lyapunov spectrum may serve to characterize these attractors. When at least one Lyapunov exponent is greater than zero, the attractor is said to be chaotic. Progress in the field of nonlinear dynamics has made possible the calculation of the Lyapunov spectrum from a time series. It could be shown that acoustic cavitation in the region of broadband noise emission is characterized by one positive Lyapunov exponent.

V. PERIOD-DOUBLING BUBBLE OSCILLATIONS Thus far, only the acoustic signal has been investigated. An optic inspection of the liquid inside the piezoelectric cylinder (see Fig. 1) reveals that a highly structured cloud of bubbles or cavities is present (Fig. 5) oscillating and moving in the sound field. It is obviously these bubbles that produce the noise. If this is the case, the bubbles must

FIGURE 6 Reconstructed images from (a) a holographic series taken at 23.100 holograms per second of bubbles inside a piezoelectric cylinder driven at 23.100 Hz and (b) the corresponding power spectrum of the noise emitted. Two period-doublings have taken place.

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FIGURE 7 Period-doubling route to chaos for a driven bubble oscillator. Left column: radius-time solution curves; middle left column: trajectories in phase space; middle right column: Poincare´ section plots: right column: power spectra. Rn is the radius of the bubble at rest, Ps and v are the pressure amplitude and frequency of the driving sound field, respectively. (From Lauterborn, W., and Parlitz, U. (1988). J. Acoust. Soc. Am. 84, 1975.)

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FIGURE 7 (Continued )

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124 move chaotically and should show the period-doubling sequence encountered in the noise output. This has been confirmed by holographic investigations where once per period of the driving sound field a hologram of the bubble field has been taken. Holograms have been taken because the bubbles move in three dimensions, and it is difficult to photograph them at high resolution when an extended depth of view is needed. In one experiment the driving frequency was 23,100 Hz, which means 23,100 holograms per second have been taken. The total number of holograms, however, was limited to a few hundred. Figure 6a gives an example of a series of photographs taken from a holographic series. In this case, two perioddoubling bifurcations have already taken place since the oscillations only repeat after four cycles of the driving sound wave. The first period doubling is strongly visible; the second one can only be seen by careful inspection. Figure 6b gives the noise power spectrum taken simultaneously with the holograms. The acoustic measurements show both period doublings more clearly than the optical measurement (documented in Fig. 6a) as the 14 f 0 ( f 0 = 23.1 kHz) spectral line is strongly present together with its harmonics.

Acoustic Chaos

The third column shows so-called Poincar´e section plots. Here, only the dots after the lapse of one full period of the driving sound field are plotted in the radius–velocity plane of the bubble motion. Period doubling is seen most easily here and also the evolution of a strange (or chaotic) attractor. The rightmost column gives the power spectra of the radial bubble motion. The filling of the spectrum with successive lines in between the old lines is evident, as is the ultimate filling when the chaotic motion is reached. A compact way to show the period-doubling route to chaos is by plotting the radius of the bubble as a function of a parameter of the system that can be varied, e.g., the frequency of the driving sound field. Figure 8a gives an example for a bubble of radius at rest of Rn = 10 µm, driven by a sound field of frequency ν between 390 kHz and 510 kHz at a pressure amplitude of Ps = 290 kPa. The period-doubling cascade to chaos is clearly visible. In the chaotic region, “windows” of periodicity show

VI. THEORY OF DRIVEN BUBBLES A theory has not yet been developed that can account for the dynamics of a bubble field as shown in Fig. 5. The most advanced theory is only able to describe the motion of a single spherical bubble in a sound field. Even with suitable neglections the model is a highly nonlinear ordinary differential equation of second order for the radius R of the bubble as a function of time. With a sinusoidal driving term (sound wave) the phase space is three dimensional, just sufficient for a dynamical system to show irregular (chaotic) motion. The model is an example of a driven nonlinear oscillator for which chaotic solutions in certain parameter regions are by now standard. However, period doubling and irregular motion were found in the late 1960s in numerical calculations when chaos theory was not yet available and thus the interpretation of the results difficult. The surprising fact is that already this simple model of a purely spherically oscillating bubble set into oscillation by a sound wave yields successive period doubling up to chaotic oscillations. Figure 7 demonstrates the perioddoubling route to chaos in four ways. The leftmost column gives the radius of the bubble in the sound field as a function of time, where the dot on the curve indicates the lapse of a full period of the driving sound field. The next column shows the corresponding trajectories in the plane spanned by the radius of the bubble and its velocity. The dots again mark the lapse of a full period of the driving sound field.

FIGURE 8 (a) A period-doubling cascade as seen in the bifurcation diagram. (b) The corresponding largest Lyapunov exponent λmax . (c) The winding number w. (From Parlitz, U. et al. (1990). J. Acoust. Soc. Am. 88, 1061.)

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up as regularly experienced with other chaotic systems. In Fig. 8b the largest Lyapunov exponent λmax is plotted. It is seen that λmax > 0 when the chaotic region is reached. Figure 8c gives a further characterization of the system by the winding number w. The winding number describes the winding of a neighboring trajectory around the given one per period of the bubble oscillation. It can be seen that this quantity changes quite regularly in the period-doubling sequence, and rules can be given for this change. The driven bubble system shows resonances at various frequencies that can be labeled by the ratio of the linear resonance frequency of the bubble to the driving frequency of the sound wave. Figure 9 gives an example of the complicated response characteristic of a driven bubble. At somewhat higher driving than given in the figure the oscillations start to become chaotic. A chaotic bubble attractor is shown in Fig. 10. To better reveal its structure, it is not the total trajectory that is plotted but only the points in the velocity–radius plane of the bubble wall at a fixed phase of the driving. These points hop around on the attractor in an irregular fashion. These chaotic bubble oscillations must be considered as the source of the chaotic sound output observed in acoustic cavitation.

VII. OTHER SYSTEMS Are there other systems in acoustics with chaotic dynamics? The answer is surely yes, although the subtleties of chaotic dynamics make it difficult to easily locate them. When looking for chaotic acoustic systems, the question arises as to what ingredients an oscillatory system, as an acoustic one, must possess to be susceptible to chaos. The full answer is not yet known, but some understanding is emerging. A necessary, but unfortunately not sufficient, ingredient is nonlinearity. Next, period doubling is known to be a precursor of chaos. It is a peculiar fact that, when one period doubling has occurred, another one is likely to appear, and indeed a whole series with slight alterations of parameters. Further, the appearance of oscillations when a parameter is altered points to an intrinsic instability of a system and thus to the possibility of becoming a chaotic one. After all, two distinct classes can be formulated: (1) periodically driven passive nonlinear systems (oscillators) and (2) self-excited systems (oscillators). Passive means that in the absence of any external driving the system stays at rest as, for instance, a pendulum does. But a pendulum has the potential to oscillate chaotically when being driven periodically, for instance by a sinusoidally

FIGURE 9 Frequency response curves (resonance curves) for a bubble in water with a radius at rest of Rn = 10 µm for different sound pressure amplitudes pA of 0.4, 0.5, 0.6, 0.7, and 0.8 bar. (From Lauterborn, W. (1976). J. Acoust. Soc. Am. 59, 283.)

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as it is a three-dimensional (space), nonlinear, dynamical (time) system; that is, it requires three space coordinates and one time coordinate to be followed. This is at the border of present-day technology both numerically and experimentally. The latest measurements have singled out mode competition as the mechanism underlying the complex dynamics. Figure 11 gives two examples of oscillatory patterns: a periodic hexagonal structure (Fig. 11a) and

a FIGURE 10 A numerically calculated strange bubble attractor (Ps = 300 kPa, v = 600 kHz). (Courtesy of U. Parlitz.)

varying torque. This is easily shown experimentally by the repeated period doubling that soon appears at higher periodic driving. Self-excited systems develop sustained oscillations from seemingly constant exterior conditions. One example is the Rayleigh-B´enard convection, where a liquid layer is heated from below in a gravitational field. The system goes chaotic at a high enough temperature difference between the bottom and surface of the liquid layer. Self-excited systems may also be driven, giving an important subclass of this type. The simplest model in this class is the driven van der Pol oscillator. A real physical system of this category is the weather (the atmosphere). It is periodically driven by solar radiation with the low period of 24 hr, and it is a self-excited system, as already constant heating by the sun may lead to Rayleigh-B´enard convection as observed on a faster time scale. The first reported period-doubled oscillation from a periodically driven passive system dates back to Faraday in 1831. Starting with the investigation of sound-emitting, vibrating surfaces with the help of Chladni figures, Faraday used water instead of sand, resulting in vibrating a layer of liquid vertically. He was very astonished about the result: regular spatial patterns of a different kinds appeared and, above all, these patterns were oscillating at half the frequency of the vertical motion of the plate. Photography was not yet invented to catch the motion, but Faraday may well have seen chaotic motion without knowing it. It is interesting to note that there is a connection to the oscillation of bubbles as considered before. Besides purely spherical oscillations, bubbles are susceptible to surface oscillations as are drops of liquid. The Faraday case of a vibrating flat surface of a liquid may be considered as the limiting case of either a bubble of larger and larger size or a drop of larger and larger size, when the surface is bent around up or down. Today, the Faraday patterns and Faraday oscillations can be observed better, albeit still with difficulties

b

FIGURE 11 Two patterns appearing on the surface of a liquid layer vibrated vertically in a cylindrical container: (a) regular hexagonal pattern at low amplitude, and (b) pattern when approaching chaotic vibration. (Courtesy of Ch. Merkwirth.)

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its dissolution on the way to chaotic motion (Fig. 11b) at the higher vertical driving oscillation amplitude of a thin liquid layer. The other class of self-excited systems in acoustics is quite large. It comprises (1) musical instruments, (2) thermoacoustic oscillators as used today for cooling with sound waves, and (3) speech production via the vocal folds. Period doubling could be observed in most of these systems; however, very few investigations have been done so far concerning their chaotic properties.

VIII. PHILOSOPHICAL IMPLICATIONS The results of chaos physics have shed new light on the relation between determinism and predictability and on how seemingly random (irregular) motion is produced. It has been found that deterministic laws do not imply predictability. The reason is that there are deterministic laws which persistently show a sensitive dependence on initial conditions. This means that in a finite, mostly short time any significant digit of a measurement has been lost, and another measurement after that time yields a value that appears to come from a random process. Chaos physics has thus shown a way of how random (seemingly random, one must say) motion is produced out of determinism and has developed convincing methods (some of them exemplified in the preceding sections on acoustic chaos) to classify such motion. Random motion is thereby replaced by chaotic motion. Chaos physics suggests that one should not resort too quickly to statistical methods when faced with irregular data but instead should try a deterministic approach. Thus, chaos physics has sharpened our view considerably on how nature operates.

But, as always in physics, when progress has been made on one problem other problems pile up. Quantum mechanics is thought to be the correct theory to describe nature. It contains “true” randomness. But, what then about the relationship between classical deterministic physics and quantum mechanics? Chaos physics has revived interest in these questions and formulated new specific ones, for instance, on how chaotic motion crosses the border to quantum mechanics. What is the quantum mechanical equivalent to sensitive dependence on initial conditions? The exploration of chaos physics, including its relation to quantum mechanics, is therefore thought to be one of the big scientific enterprises of the new century. It is hoped that acoustic chaos will accompany this enterprise further as an experimental testing ground.

SEE ALSO THE FOLLOWING ARTICLES ACOUSTICAL MEASUREMENT • CHAOS • FOURIER SERIES • FRACTALS • QUANTUM MECHANICS

BIBLIOGRAPHY Lauterborn, W., and Holzfuss, J. (1991). “Acoustic chaos.” Int. J. Bifurcation and Chaos 1, 13–26. Lauterborn, W., and Parlitz, U. (1988). Methods of chaos physics and their application to acoustics. J. Acoust. Soc. Am. 84, 1975–1993. Parlitz, U., Englisch, V., Scheffezyk, C., and Lauterborn, W. (1990). “Bifurcation structure of bubble oscillators.” J. Acoust. Soc. Am. 88, 1061–1077. Ruelle, D. (1991). “Chance and Chaos,” Princeton Univ. Press, Princeton, NJ. Schuster, H. G. (1995). “Deterministic Chaos: An Introduction,” WileyVCH, Weinheim.

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Acoustical Measurement Allan J. Zuckerwar NASA Langley Research Center

I. Instruments for Measuring the Properties of Sound II. Instruments for Processing Acoustical Data III. Examples of Acoustical Measurements

GLOSSARY Anechoic Having no reflections or echoes. Audio Pertaining to sound within the frequency range of human hearing, nominally 20 Hz to 20 kHz. Coupler Small leak-tight enclosure into which acoustic devices are inserted for the purpose of calibration, measurement, or testing. Diffuse field Region of uniform acoustic energy density. Free field Region where sound propagation is unaffected by boundaries. Harmonic Pertaining to a pure tone, that is, a sinusoidal wave at a single frequency: an integral multiple of a fundamental tone. Infrasonic Pertaining to sound at frequencies below the limit of human hearing, nominally 20 Hz. Reverberant Highly reflecting. Ultrasonic Pertaining to sound at frequencies above the limit of human hearing, nominally 20 kHz. A SOUND WAVE propagating through a medium produces deviations in pressure and density about their mean or static values. The deviation in pressure is called the acoustic or sound pressure, which has standard international (SI) units of pascal (Pa) or newton per square meter (N/m2 ). Because of the vast range of amplitude covered

in acoustic measurements, the sound pressure is conveniently represented on a logarithmic scale as the sound pressure level (SPL). The SPL unit is the decibel (dB), defined as SPL(dB) = 20 log( p/ p0 ) in which p is the root mean square (rms) sound pressure amplitude and p0 the reference pressure of 20 × 10−6 Pa. The equivalent SPLs of some common units are the following: pascal (Pa) atmosphere (atm) bar

93.98 dB 194.09 193.98

psi (lb/in.2 ) torr (mm Hg) dyne/cm2

170.75 dB 136.48 73.98

The levels of some familiar sound sources and environments are listed in Table I. The displacement per unit time of a fluid particle due to the sound wave, superimposed on that due to its thermal motion, is called the acoustic particle velocity, in units of meters per second. Determination of the sound pressure and acoustic particle velocity at every point completely specifies an acoustic field, just as the voltages and currents completely specify an electrical network. Thus, acoustical instrumentation serves to measure one of these quantities or both. Since in most cases the relationship between

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Acoustical Measurement TABLE I Representative Sound Pressure Levels of Familiar Sound Sources and Environments Source or environment

Level (dB)

Concentrated sources: re 1 m Four-jet airliner Pipe organ, loudest Auto horn, loud Power lawnmower Conversation Whisper Diffuse environments Concert hall, loud orchestra Subway interior Street corner, average traffic Business office Library Bedroom at night Threshold levels Of pain Of hearing impairment, continous exposure Of hearing Of detection, good microphone

155 125 115 100 60 20 105 95 80 60 40 30 130 90 0 −2

sound pressure and particle velocity is known, it is sufficient to measure only one quantity, usually the sound pressure. The scope of this article is to describe instrumentation for measuring the properties of sound in fluids, primarily in air and water, and in the audio (20 Hz–20 kHz) and infrasonic ( 106 , this is the only feasible method for measuring sound attenuation. In a steady-state experiment, the attenuation can be determined by the halfwidth of the resonance curve, shown in Fig. 13c. The sound pressure amplitude is measured as the frequency is incremented or swept through the resonant frequency f 0 . The halfwidth f is defined as the frequency √ interval between the two points where p = pmax / 2; that is, where the amplitude is 3 dB down from the peak pmax . This method cannot be used when the halfwidth is too sharp, due to instability of f 0 , or when the halfwidth is too broad, due to poor peak definition. Furthermore, any losses inherent in the transmitter itself will contribute to the halfwidth. With an efficient transmitter this method can be used effectively over a range of Q from about two to several hundred. If the medium is very lossy (Q < 2), the most effective measure of attenuation is the loss tangent tan δ. Such low Q’s are found in some polymeric liquids, seldom in gases. Because of the high loss the sample is made the lossy element of a composite resonator, in which independent measurements of force and displacement yield the phase angle δ. The measured attenuation in a resonator contains three principal components: wall absorption, absorption due to fluid–structure interaction, and the constituent absorption that is to be measured. The wall and structural components, called the background absorption, can be determined through measurements on a background fluid having negligible constituent absorption over the range of measurement parameters. In gases, argon and nitrogen are frequently used for this purpose.

5. Free-Field Measurements of Attenuation In the free field, corrections must be made for spreading. For a spherical source, the sound pressure falls as 1/r . At sufficiently large distances from the source, a spherical wave can be approximated by a plane wave, and the correction is not needed. 6. Optoacoustical Method: Laser-Induced Thermal Acoustics The passage of laser light through a fluid can induce a strain either thermally (resonant) or electrostrictively (nonresonant). A typical laser-induced thermal acoustics (LITA) arrangement is shown in Fig. 14. Typical component specifications are shown in parentheses. Light from a pulsed pump laser (λpump = 532 nm) is split into two beams which intersect at a small angle (2θ = 0.9◦ ). Optical interference fringes of spatial period = λpump /(2 sin θ) generate electrostrictively counterpropagating ultrasonic waves of fixed wavelength to form a Bragg grating, shown in the insert. A long-pulsed probe laser (750 nm) illuminates the grating, which diffracts a small fraction of the probe beam at an angle φ to a photomultiplier. The diffracted signal is normalized to the direct probe signal measured at the photodetector. Since the acoustical wavelength is known from the intersection angle 2θ and

FIGURE 14 Laser-induced thermal acoustics.

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pump laser wavelength, and the frequency is known from the photomultiplier signal, the speed of sound of the fluid medium can be measured. A “referenced” version of LITA, implemented to avoid the large error associated with the intersection angle measurement, splits the pump and probe beams and directs them to a second LITA cell containing a fluid of known sound speed. E. Measurement of Acoustic Impedance The relationship between sound pressure and acoustic flow velocity plays a central role in the analysis of acoustic devices, such as mufflers and musical instruments, and in the determination of a sound field in the presence of a boundary. Quantitatively, this relationship is described by one of three types of acoustic impedance. The acoustic impedance—the ratio of sound pressure to volume velocity Z = p/U —is used in analyzing acoustic circuits, where devices are represented by equivalent lumped elements. It is a property of the medium, frequency, and geometry and has units of N · sec/m5 = kg/sec · m4 = Rayl/m2 . The interaction between a sound wave and a boundary depends on the specific acoustic impedance of the boundary relative to that of the propagation medium. The specific acoustic impedance is the ratio of sound pressure to acoustic particle velocity z = p/u. For a plane wave z = ρc, and it is basically a property of the medium, although it can have complex frequency-dependent parts. It is related to Z through z = Z S, where S is the cross-sectional area and has units of N · sec/m3 = kg/sec · m2 = Rayl. Thus measurement of z readily leads to the determination of Z and vice versa. The mechanical or radiation impedance, the ratio of force to particle velocity, is of interest in systems containing both discrete and continuous components but is not discussed here. Methods of measuring acoustic impedance fall into three broad categories: (1) impedance tube and waveguide methods in general, (2) free-field methods, and (3) direct measurement of sound pressure and volume velocity. 1. Impedance Tube The test specimen is located at one end of a rigid tube and a transmitter at the other end (Fig. 15a). It is important to distinguish between materials of local reaction and those of extended reaction. In the former, the behavior of one point on the surface depends only on excitation at that point and not on events taking place elsewhere in the material. In the latter, acoustic excitation at a point on the surface generates waves that propagate laterally throughout the material. Generally, a material is locally reacting if normal acoustic penetration does not exceed a wavelength.

FIGURE 15 Measurement of acoustic impedance with an impedance tube. (a) Impedance tube, and (b) standing wave pattern and its envelopes.

For a locally reacting material, a thin test specimen (thickness λ/4) is backed by a λ/4 air gap sandwiched between the specimen and a massive reflector. For a material of extended reaction, a specimen of approximate thickness λ/4 is backed by the massive reflector directly against its surface. The transmitter is tuned to establish a standing wave pattern, which is probed by a microphone located either within the tube or at the end of a probe tube. The observer slides the probe along the impedance tube axis and records the standing wave pattern L(x) in decibels (Fig. 15b). Here L(x) = 20 log[ p(x)/( p0 )], where the reference pressure p0 is immaterial. The impedance is evaluated from the pressure standing wave ratio L 0 at the specimen surface, which cannot be measured directly but is computed from a best fit to the trend of L max and L min shown in the figure. After computing the antilog of the pressure standing wave ratio K 0 and related quantities, K 0 = 10 L 0 /20 x1 1 φ= − × 360◦ x2 − x1 2 M = 12 K 0 + K 0−1 N = 12 K 0 − K 0−1

(41) (42) (43) (44)

we determine the real z and imaginary z

parts of the specific acoustic impedance relative to that of air, ρc: z

1 = ρc M − N cos φ

(45)

z

N sin φ = ρc M − N cos φ

(46)

This method is capable of yielding measurements of high precision, to within a few percent based on repeatability.

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FIGURE 16 Measurement of acoustic impedance by a free-field method.

A major source of error lies in the determination of x, which may be illdefined for a rough or fibrous specimen surface. To improve surface definition, a face sheet composed of a fine-meshed gauze of low acoustic resistance can be used. A disadvantage is the time required to take the number of measurements needed to establish the standing wave pattern. More modern methods based on the transfer function between two microphone stations reduces the measurement time considerably. 2. Free-Field Methods A transmitter sends an incident wave at an angle ψ toward the test specimen, from which it is reflected, also at an angle ψ, toward a receiver (Fig. 16). The specific acoustic impedance is evaluated from the reflection coefficient R p = pr / pi : 1 + Rp z 1 (47) = ρc sin ψ 1 − Rp A variety of techniques, both transient and steady state, have been devised to determine the three wave components pr , pi , and pd . One steady-state method utilizes three separate measurements at each frequency: (1) with the specimen in place, yielding p1 = pr + pd ; (2) with the specimen replaced by a reflector of high impedance, yielding p2 = pr + pd ≈ pi + pd ; and (3) with the reflector removed, yielding p3 = pd alone. Thus, pr p1 − p3 Rp = = (48) pi p2 − p3 Free-field methods are used for testing materials at short wavelengths and are popular for outdoor measurements of the earth’s ground surface. 3. Direct Measurement of Sound Pressure and Volume Velocity For measurement of the acoustic impedance within an acoustic device, the sound pressure can be measured with the aid of a probe tube (Section I.A.8), but measurement of the acoustic particle or volume velocity is difficult (Section I.C). The most common method of attacking the latter problem is to control the volume velocity at the transmitter.

This can be achieved in several ways: (1) by mounting a displacement sensor on the driver; (2) by using a dual driver, directing one side to the test region and the other side to a known impedance Z k and using U = p/Z k and (3) by exciting a driving piston with a cam so that the generated volume velocity will be independent of acoustic load. The first two methods rely on the integrity of the velocity measurement technique: the third is limited to relatively low frequencies. To measure the specific acoustic impedance of a material, a transmitter, receiver, and test specimen are mounted in a coupler; the impedance of the latter must be taken into account.

II. INSTRUMENTS FOR PROCESSING ACOUSTICAL DATA A. Filters The representation of an acoustic time history in the domain of an integral (or discrete) transform has two advantages. First, it transforms an integrodifferential (or difference) equation into a more tractable algebraic equation. Second, it often separates relevant signal from irrelevant signal and random noise. The two most common transforms used in acoustics are the Fourier transform for continuous time histories and the z transform for discrete (sampled) time histories. The Fourier transform represents a time history f (t) in the frequency domain, ∞ 1 F(ω) = f (t) exp(− jωt) dt (49) 2π −∞ with ω = 2π f . The z transform represents the sampled values f (nTs ) in the z domain: F(z) =

∞

f (nTs )z −n

(50)

n=0

where Ts is the sample interval and n the sample number. The representation of a time history in the transformed domain is called a spectrum. We shall be concerned with the frequency spectrum. Filters fulfill three major functions in acoustics: spectral selection, analysis, and shaping. It is assumed that the reader is familiar with the general characteristics of filters and with filter terminology. 1. Spectral Selection We shall present two examples of spectral selection. The first is antialiasing. In sampled systems, it is essential that all frequency components above half the sampling frequency f s be suppressed to avoid “aliasing,” that is, the appearance of components of frequency f s − f in the observed spectrum. This is a consequence of the Nyquist sampling theorem. The maximum frequency for which

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108 the spectrum is uncorrupted by aliaising is called the Nyquist frequency. The second example is signal-to-noise improvement. The observed signal is often a pure tone, for which narrow-band filtering will produce a considerable improvement in signal-to-noise (S/N) ratio. If the noise is “white,” that is, has a uniform spectral power density, then a reduction in bandwidth from BW to BN improves the S/N ratio by 20 log(BW /BN ) decibels.

2. Spectral Analysis The role of the filter here is to permit observation of a narrow portion of a wideband spectrum. The selected band is specified by a center frequency f 0 and a bandwidth B, defined as the frequency interval about f 0 where the output/input ratio remains within 3 dB of that at the center. The bandwidth of the filter may be constant (i.e., independent of f 0 ) or a constant percentage of f 0 . The constant-bandwidth filter is advantageous in cases where the measured spectrum is rich in detail over a limited frequency range, for example, where a series of harmonics appears as the result of nonlinear distortion or where a number of sharp resonances are generated from a complex sound source. The constant-percentagebandwidth filter is more appropriate in cases where the measured spectrum encompasses a large number of decades, say two or more; where the source is unstable, constantly shifting its prominent frequencies; or where the power transmitted over a band of frequencies is of interest, as in noise control engineering. Popular choices for the constant-percentage bandwidth 1 1 are the octave (factor of 2), 13 , 16 , 12 , and 24 octave. 1 The bandwidth of a 3 -octave filter, for example, is 21/6 f 0 − 2−1/6 f 0 = 0.231 f 0 . The 13 -octave filter, in fact, is the most widely used in acoustic spectral analysis. The reason is rooted in a property of human auditory response. Consider an experiment in which a human subject is exposed to a 60-dB narrow-band tone at 10 kHz. If the amplitude and center frequency of the tone remain fixed but the bandwidth increases, the subject will perceive no change in loudness until the bandwidth reaches 2.3 kHz, and then the loudness begins to increase. This is called the critical bandwidth and has a value of ∼ 13 octave. If the test is repeated at other, sufficiently high center frequencies, the resulting critical bandwidth remains at about 13 octave. For sound measurements geared to human response, then, a narrower bandwidth does not influence loudness and a greater bandwidth yields a false measurement of loudness—hence, the choice of 13 -octave spectral resolution. A list of preferred 13 -octave center frequencies is given in Table IV. The audible spectrum, 20 Hz to 20 kHz, encompasses thirty-one 13 -octave bands.

Acoustical Measurement TABLE IV Preferred 13 -Octave Center Frequenciesa 16 63

20 80 a

25 100

31.5 125

40 160

50 200

In hertz (also ×10 or ×100).

3. Spectral Shaping The perceived loudness of a tone of constant amplitude is a strong function of frequency and amplitude. Many acoustic instruments feature not only a linear response, an objective measurement of sound pressure, but also a weighted response, which conforms to the frequency response of the human ear. The function of a weighting filter is to shape an acoustic spectrum to match the response of the ear. Three standard frequency response curves, called A, B, and C curves, conform to equal loudness curves at 40, 70, and 100 phons, respectively. A phon is a unit of loudness, usually specified in decibels; it is the same as the SPL at 1 kHz but differs at most other frequencies. The D weight has been proposed for applications involving aircraft noise measurement. The filter response curves for the A, B, C, and D weighting are shown in Fig. 17. B. Spectrum Analyzers A spectrum analyzer enables an observer to view the frequency spectrum of an acoustic time history on an output device such as a television monitor, chart recorder, or digital printer. A real-time analyzer produces a complete, continuously updated spectrum without interruption. The first real-time analyzers were analog in nature, based on either of two principles: (1) time compression, which used a frequency transformation to speed up processing time, or (2) a parallel bank of analog filters and detectors. The advent of VLSI (very large-scale integration) in the semiconductor industry made the all-digital, real-time analyzer a reality, offering competitive cost and enhanced stability, linearity, and flexibility.

FIGURE 17 Response curves of A, B, C, and D weighting filters.

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The spectrum analyzer performs the basic functions of preamplification, analog filtering, detection, analog-todigital (A/D) conversion, logic control, computation, and output presentation. The frequency range usually covers the audio band but may exceed it at both ends. Digital real-time analyzers operate on either of two principles: the digital filter or the fast Fourier transform (FFT). 1. Digital Filter The transfer function of a two-pole analog filter is written: H (s) =

(s + r1 )(s + r2 ) (s + p1 )(s + p2 )

(51)

where s is the Laplace operator, r1.2 the zeros, and p1.2 the poles. The filter characteristics—gain and cutoff frequencies—are fixed and can be changed only by changing the components making up the filter. The frequency response can be found by replacing s by jω. The digital filter accepts samples f (nTs ) of the time history from an A/D converter, where Ts is the sample interval, and yields an output in the form of a sequence of numbers. The transfer function is represented in the z domain: H (z) =

A0 + A1 z −1 + A2 z −2 1 − B1 z −1 − B2 z −2

(52)

where z −1 = exp(−sTs ) is called the unit delay operator, since multiplication by z −1 is equivalent to delaying the sequence by one sample number. Synthesis of H (z) requires a system that performs the basic operations of multiplying, summing, and delaying. Noteworthy is the fact that once the filter characteristics are set by choice of coefficients A0 . . . B2 , the frequency response parameters (center frequency f 0 and bandwidth B for a bandpass filter) are controlled by the sample rate f s = 1/Ts . For example, doubling f s doubles f 0 and B. Thus, the digital filter is a constant-percentage-bandwidth filter and is appropriate for those applications where such is required (Section II.A.2). Typically, the filters are six-pole Butterworth or Chebycheff filters of 13 -octave bandwidth. Several twopole filters can be cascaded to produce filters of higher poles, or the data can be recirculated through the same filter several times. 2. Fast Fourier Transform First consider the discrete Fourier transform (DFT), the digital version of Eq. (49), F(k) =

N −1 1

f (n) exp(− j2π kn/N ) N n=0

(53)

where f (n) is the value of the nth time sample, k the frequency component number, N the block size, or number of time samples. The time resolution depends on the time window t = T /N , and the frequency resolution depends on the sampling frequency f = f max /N . Obviously, the filter is a constant-bandwidth filter and again is suited to the appropriate applications (Section II.A.2). In contrast to the digital filtering technique, the data throughput is not continuous but is segmented into data blocks. Thus, for real-time analysis, the analyzer must be capable of processing one block of data while simultaneously acquiring a new block. The FFT exploits the symmetry properties of the DFT to reduce the number of computations. The DFT requires N 2 multiplications to transform a data block of N samples from the time domain to the frequency domain: the FFT requires only N log2 N multiplications. For a block size of N = 1024 samples, the reduction is over a factor of 100. The DFT of Eq. (53) differs from the continuous Fourier transform in three ways, each presenting a data-processing problem that must be addressed by man/machine. First, the transformed function is a sampled time history. The sampling frequency must exceed twice the Nyquist frequency, as explained in Section II.A.1. In fact, it is beneficial to choose an even higher sampling frequency. For example, in a six-pole low-pass filter, the signal is down ∼18 dB at 12 octave past the cutoff frequency f c . A strong component at this√frequency will “fold over” as a component of frequency 2 f c − f c ≈ 0.4 f c , attenuated only 18 dB, and may have a level comparable to the true signal. Increasing the sampling frequency to f s = 2.5 f c will relieve the problem in this case. Second, the filter time window yields the well-known sin x/x transform. In the frequency domain, the window spectrum is convolved with the signal spectrum and introduces ripples in the latter. The sidelobes of the sin x/x spectrum introduce leakage of power from a spectral component to its neighbors. A countermeasure to this effect is to use a Hanning window, a weighting time function that is maximum at the center of the window and zero at its edges. The Hanning window improves the sidelobe suppression at the expense of increased bandwidth. However, the Hanning window may not be needed if the signal is small at the edges of the window. Finally, the digitally computed transform of the sampled time history must itself be presented as a sampled frequency spectrum. This fact is responsible for the socalled picket fence effect, whereby we do not observe the complete spectrum but only samples. Thus, we may miss a sharp peak and observe only the slopes. A Hanning window also helps to compensate for this effect. Examples of acoustic signals necessitating analysis in real time are signals in the form of a sequence of transients,

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as speech; aircraft flyover noise, as required by the Federal Aviation Administration (FAA); and measurements where the analyzer is an element in a control loop. For other types of signals, such as stationary or quasi-stationary signals, or transients shorter than the time window, the time history can be stored and analyzed at a later time. 3. Correlation Many spectrum analyzers provide the capability of computing the autocorrelation and cross-correlation functions and their Fourier transforms, namely, the spectral and cross-spectral density functions. These operations are used to compare the data at one test station with that at another station. The cross-correlation of the time-varying functions f 1 (t) and f 2 (t) is expressed in terms of a time delay τ : 1 T g12 (τ ) = lim f 1 (t) f 2 (t + τ ) dt (54) T →∞ T 0 The Fourier transform of this function is the cross-power spectral density function: ∞ G 12 ( f ) = g12 (τ ) exp(− j2π f τ ) dτ (55) −∞

If f 1 (t) and f 2 (t) are the same signal, say at station 1, then Eqs. (54) and (55) yield g11 (τ ) and G 11 ( f ), the autocorrelation function and the spectral power density function. Two important acoustic applications of the crossfunctions are transfer function determination and time delay estimation. Let us consider the transfer function. Suppose a noise or vibration source produces responses at two stations f 1 (t) and f 2 (t), having Fourier transforms F1 ( f ) and F2 ( f ). The transfer function H12 ( f ) = F2 ( f )/F1 ( f ) is related to the power spectra as follows: H12 ( f ) = G 12 ( f )/G 11 ( f )

(56)

Thus, Eq. (56) permits the determination of H12 ( f ), while the source is operating in its natural condition. Now consider time delay estimation. Suppose an acoustic signal propagates from station 1 to station 2 in time τ0 . Then g12 (t) will show a peak at τ = τ0 , and G 12 ( f ) will have a phase angle φ12 = 2π f τ0 . If the signal is a pure tone, say a cosine wave, then g12 (τ ) will also be a cosine wave of the same frequency but shifted by φ12 ; that is, the maximum will be displaced by an angle φ12 . If the time delay τ0 exceeds the period 1/ f of the wave, then g12 (τ ) will reveal two maxima and thus a twofold ambiguity in τ0 . Consequently, the maximum delay that can be uniquely determined is τmax < 1/ f . If the signal is a mixture of two tones of frequencies f 1 and f 2 , then the maximum delay will be determined by the beat frequency, τmax < ( f 2 − f 1 )−1 . Formal analysis leads to the criterion: τmax < 0.3/( f 2 − f 1 )

(57)

where f 2 − f 1 is the bandwidth of the signal. In these two cases, the cross-spectral density is strongly peaked at a few prominent frequencies. If, on the other hand, the time signal is strongly peaked as in the case of a narrow pulse, comprising a broad spectrum of frequencies, there is a criterion on minimum system bandwidth to measure a given delay similar to Eq. (57), with the inequality reversed. The autocorrelation function reveals the presence of periodic signals in the presence of noise. An important function in acoustic signal processing is the coherence function, C12 ( f ) =

|G 12 ( f )|2 G 11 ( f )G 22 ( f )

(58)

which has a value between 0 and 1. This function serves as a criterion as to whether the signals received at stations 1 and 2 have the same cause. It should have a reasonably high value even in measurement systems subject to noise and random events. C. Sound Level Meters A sound level meter is a compact portable instrument, usually battery-operated, for measuring SPL at a selected location. The microphone signal is preamplified (attenuated), weighted, again amplified (attenuated), detected, and displayed on an analog meter. The detector is a squarelaw detector followed by an averaging (mean or rms) network. There are a variety of additional features such as calibration, overload indication, and external connectors for filters and output signal. The directional response of the microphone affects the accuracy of the measurement. In a free field, corrections are based on curves such as those in Fig. 7 if the angle of incidence is known. In a diffuse field, the random response curve must be relied on: The smaller the microphone, the more accurate are the results. Two switch selections available to the user are weighting and time constant. The weighting networks are linear (unweighted), A, B, C, and sometimes D (Section II.A.3). For stationary or quasi-stationary signals, a “fast” or “slow” time constant, based on the response to a 200- or 500-msec signal, respectively, is used. The fast response follows time-varying sound pressures more closely at the expense of accuracy; the slow response offers a higher confidence level for the rms sound pressure measurement. Impulsive signals present something of a problem. Current standards specify a time constant of 35 msec, in an attempt to simulate the response of the human ear, plus the capability of storing the peak or rms value of the applied signal. To prevent saturation resulting from high peak amplitudes, the detector circuit must be capable of sustaining a crest factor, the ratio of peak to rms signal, of at least 5.

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D. Storage of Acoustical Data Up to the mid-1970s the workhorse of acoustical data storage was the magnetic tape recorder in both am and fm versions. The major limitation was the limited dynamic range, amounting to less than 40 dB for am tape and 50– 55 dB for fm tape. This was followed by 7- to 9-track digital tape, which improved the dynamic range but in the 1980s yielded to VHS (video home systems) cassettes having greater storage density. Typical specifications for VHS cassette recorders, which are still on the market today, are 70 dB dynamic range, dc to 80-kHz frequency response, and recording time ranging from 50 min to 426.7 hr at sample rates of 1280 and 2.5 thousands of samples per second, respectively. With the explosive development of personal computers, the development of digital storage systems has proceeded at a comparable pace. These are classified as either random access or sequential access devices. Random-access devices include hard drives, CD (compact disc) writers, and DVD (digital versatile disc) RAM (random-access memory) devices. The hard drive typically has a storage capacity of 20 gigabytes (GB) and a data transfer rate of over 10 megabytes (MB) per second. Traditional hard drives are not meant for archiving data nor for removal from one system to another. Now more options with removable hard-disk systems are available, such as the “Jaz” and “Orb,” which have the disk in a removable cartridge. These cartridge-based hard drives have capacities of up to 2 GB and sustained data rates of over 8 MB/sec. The removable DVD-RAM has shown capacities of 5.2 GB and transfer rates of up to 1 MB per second. While sequential access times are significantly greater than random-access devices, sequential access provides the highest storage capacities (up to 50 GB per tape) and very high sustained data transfer rates of over 6 MB/sec. The advent of advanced intelligent tape has an electronic memory device on each tape that speeds up the search process. In addition, 8-mm, digital linear tape, and 4-mm tapes are among forms of storage that allow up to 20 terabytes of information to be stored and accessed in a costeffective manner. High-quality digital storage devices conform to the Small Computer Systems Interface (SCSI) standard. The advantages are far-reaching. The conforming devices (including those mentioned above) are easily upgraded, mutually compatible, and interchangeable from one system to another. A single SCSI controller can control up to 15 independent SCSI devices. An option available to users of digital storage devices is data compression, whereby data density is compressed by a two-to-one ratio. Most compression schemes are very robust and, combined with error detection and correction, produce error rates on the order of 10−15 .

E. The Computer as an Instrument in Acoustical Measurements The integration of a digital computer into an acoustic measurement system offers many practical advantages in addition to improved specifications regarding dynamic range, data storage density, flexibility, and cost effectiveness. Many acoustic measurements require inordinately complex evaluation procedures. The capability of performing an on-line evaluation during a test provides the user with an immediate readout of the evaluated data: this may aid in the making of decisions regarding further data acquisition and the choice of test parameters. The decisionmaking procedure can even be automated. The digital data can readily be telecommunicated over ordinary telephone lines. Most digital systems accommodate a great variety of peripheral equipment. Figure 18 shows examples of a computer integrated into an acoustical measurement system: 1. Active noise cancellation (Fig. 18a). The computer implements real-time digital filters 1 and 2, which serve as adaptive controllers to produce the required responses of noise-cancelling speakers 1 and 2. 2. Spatial transformation of sound fields (Fig. 18b). A cross spectrum analyzer yields a cross spectral representation of a sound field, based on acoustical measurements over a selected scan plane; then, a computer predicts the near field from the scan data using near field acoustic holography and the far field from the Helmholtz integral equation. 3. Computer-steered microphone arrays (Fig. 18c). In a large room, such as an auditorium or conference hall, the computer introduces a preprogrammed time delay in each microphone of a rectangular array, thus steering the array to the direction of high selectivity; coordinating more than one array, it controls the location from which the received sound is especially sensitive.

III. EXAMPLES OF ACOUSTICAL MEASUREMENTS A. Measurement of Reverberation Time Reverberation time (RT) is the time required for the sound in a room to decay over a specific dynamic range, usually taken to be 60 dB, when a source is suddenly interrupted. The Sabine formula relates the RT to the properties of the room. T = 0.161V /αS (59) where V is the volume of the room, S the area of its surfaces, and α the absorption coefficient due to losses

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in the air and at the surfaces. Recommended values for a 500-Hz tone in a 1000-m3 room are about 1.6 sec for a church, 1.2 sec for a concert hall, 1.0 sec for a broadcasting studio, and 0.8 sec for a motion picture theater, the values increasing slightly with room size. The room constant R, appearing in Eq. (32), is related to α through: R = Sα/(1 − α)

(60)

A typical measuring arrangement is shown in Fig. 19a. A sound source is placed at a propitious location and the response is averaged over several microphone locations about the room. If the source is excited into a pure tone, the measurement is beset with two basic difficulties. The act of switching generates additional tones, which establish beat frequencies and irregularities on the decay curves; furthermore, the excitation of room resonances can produce a break in the slope of the decay curve (Fig. 19b). The smoothness of the decay curve can be improved by widening the bandwidth of the source. Three types of excitation are used for this purpose: random noise, an impulse, or a warble tone, in which the center frequency is FM-modulated. The 13 -octave analyzer performs two function: It permits the frequency dependence of the RT to be determined, and it provides a logarithmic output to linearize the free decay curve. The output device can be a recorder (logarithmic if the analyzer provides a linear output) or a digital data acquisition system. The microphones can be multiplexed or measured individually. A typical decay curve obtained by this method is shown in Fig. 19c. Because many

FIGURE 18 Measurement systems using computers. (a) Active noise cancellation. (b) Spatial transformation of sound fields. (c) Computer-steered microphone arrays. (Courtesy of NASA, B&K Instruments, and J. Acoust. Soc. Am.)

FIGURE 19 Measurement of reverberation time. (a) Experimental arrangement showing microphones (circles) positioned at suitable locations about the room. (b) Response curve showing a break in slope due to simultaneous room resonances. (c) Response curve showing unambiguous reverberation.

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measurements are averaged to enhance confidence level, the method is time-consuming. If 20 averages are taken over each 13 -octave band from 125 Hz to 10 kHz, then 400 decay curves would have to be evaluated.

B. Measurement of Impulsive Noises The measurement of noise from impulsive sources such as gunshots, explosives, punch presses, and impact hammers, as well as short transients in general, requires considerable care on the part of the observer. Sometimes these sources are under the observer’s control, but on some occasions their occurrence is unpredictable, often affording but a single opportunity to make the measurement. By nature such sources are of large amplitude and short duration, requiring instruments capable of handling high crest factors and extended frequency content. Measurement of the peak pressure does not give information on duration. Of greater interest is the measurement of rms pressure, from which loudness and energy content can be inferred. Such a measurement can be made with simple analog equipment, such as an impulse sound level meter. The pressure signal is squared and time-averaged, the square root extracted, and the result presented on a meter. The averaging time will affect the measurement. By convention, an averaging time constant of 35 msec is recommended in an effort to simulate the response of the human ear. A description of an impulsive source in the frequency domain has several advantages. First, sources can be identified by their characteristic spectral signatures. Second, those components bearing a large amount of energy can be identified, as for noise control purposes. Finally, the response of an acoustic device to the signal is more readily analyzed in the frequency domain than in the time domain. Consider the Fourier spectrum of a pulse of constant amplitude A and duration T , shown in Fig. 20a: F( f ) = AT sin (π f T )/π f T

(61)

Suppose the pulse is applied to an ideal, unity-gain filter of bandwidth B, center frequency f 0 , and phase slope tL = d φ/d ω. The spectrum of the pulse and transfer function of the filter are shown in Fig. 20b. The filter output will exhibit a characteristic ringing response; if T 1/B, this can be approximated as: sin(π f 0 T ) sin π B(t − tL ) × cos(2π f 0 t) π f0 T π B(t − tL ) (62) shown in Fig. 20c. The spectral component F( f 0 ) is intimately related both to the peak response of the envelope, occurring at t = tL , and to the integrated-squared response: υ0 (t) ≈ 2ABT

FIGURE 20 Measurement of impulsive noise. (a) Time history of a single pulse and (b) its amplitude–frequency spectrum together with that of an ideal narrow-band filter. (c) Time history of the filter response to the single pulse. (d) Reconstruction of the pulse spectrum from the outputs of several adjacent narrowband filters. (e) Time history of a periodic sequence of pulses and (f) its Fourier series amplitude spectrum (with envelope).

sin(π f 0 T ) (63) = 2B F( f 0 ) π f0 T ∞ sin(π f 0 T ) 2 E= υ02 (t) dt ≈ 2A2 BT 2 π f0 T −∞

υ0 peak ≈ 2ABT

= 2B F 2 ( f 0 )

(64)

Thus, the Fourier spectrum can be reconstructed from the measurements corresponding to Eq. (63) or (64), using narrow-band filters of different center frequencies (Fig. 20d). If the condition T 1/B is not fulfilled, the filter response shows two bursts, each similar to that of Fig. 20c and separated by the pulse duration T , and Eqs. (63) and (64) are no longer valid. If the impulsive noise is repetitive or if a single pulse can be reproduced repetitively, the pulse sequence can be represented as a Fourier series. The Fourier coefficients Fn are given by Eq. (61) if f is replaced by n/Tr , where Tr is the pulse repetition interval. The time history and Fourier spectrum are shown in Figs. 20e,f. The number of components per spectral lobe depends on the ratio T /Tr . Too large a ratio will yield too few components for

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114 accurate representation of the spectrum, but too small a ratio must be avoided due to crest factor limitations in the analyzing equipment. A reasonable compromise is a ratio between 0.2 and 0.5. The reconstruction of short-transient spectra is a fine application of constant-bandwidth filtering. Some final notes are pertinent to the measurement of transients: 1. The principles discussed here for the constant-amplitude pulse apply to short pulses of other shapes. The features of the spectrum of Fig. 20b are generally retained. In the case of a tone burst, the main lobe is displaced from the origin. 2. If the occurrence of the transient event cannot be predicted, a digital event recorder will prove useful. A time history is continuously sampled and transferred to a buffer. The buffer content is transferred to a storage register only when the signal exceeds a threshold. In this manner the pre- and postevent background signals are included on both sides of the event. 3. For long transients, such as sonic booms, a 13 -octave real-time analyzer may prove advantageous because the condition T 1/B may be difficult to fulfill, and energy content may spread over a wide band of frequencies. The rms response, √ υ0 rms = E/τ (65) depends on the averaging (or integrating) time τ . For a fixed value of τ there will be a low-frequency rolloff due to the long response times, tL , of the filters, which causes part of the signal to be excluded from the averaging. At high frequencies some error will occur because of high crest factors. C. Measurement of Aircraft Noise

Acoustical Measurement

2. Aircraft Flyover Testing for Certification Federal Aviation Regulations, “Part 36—Noise Standards: Aircraft Type and Airworthiness Certification,” define instrumentation requirements and test procedures for aircraft noise certification. The instrumentation system consists of microphones and their mounting, recording and reproducing equipment, calibrators, analysis equipment, and attenuators. For subsonic transports and turbojet-powered airplanes, microphones are located on the extended centerline of the runway, 6500 m from the start of takeoff or 2000 m from the threshold of approach, and on the sideline 450 m from the runway. The microphones are of the capacitive type, either pressure or free field, with a minimum frequency response from 44–11,200 Hz. If the wind exceeds 6 knots, a windscreen is used. If the recording and reproducing instrument is a magnetic tape recorder, it has a minimum dynamic range of 45 dB (noise floor to 3% distortion level), with a standard reading level 10 dB below the maximum and a frequency response comparable to that of the microphone. The analyzer is a 13 -octave, real-time analyzer, having 24 bands in the frequency interval from 50–10,000 Hz. It has a minimum crest factor of 3, a minimum dynamic range of 60 dB, and a specified response time and provides an rms output from each filter every 500 msec. Field calibrations are performed immediately before and after each day’s testing. The microphone–preamplifier system is calibrated with a normal incidence pressure calibrator, the electronic system with “pink” noise (constant power in each 13 -octave band), and the magnetic tape recorder with the aid of a pistonphone. After the recorded data are corrected to reference atmospheric conditions and reference flight conditions, an effective perceived noise level—a measure of subjective response—is evaluated. The noise spectrum from a noncertification flyover of a Boeing 747 aircraft is shown in Fig. 21. The aircraft had

Aircraft noise measurements can be organized into two broad categories: aircraft noise monitoring and aircraft flyover testing, the latter for both engineering applications and noise certification. 1. Aircraft Noise Monitoring Aircraft noise is measured routinely at numerous airports around the world to evaluate noise exposure in adjacent communities and to compare noise sources. The instruments are basically weather-protected sound level meters, covering an SPL range from about 60 to 120 dB at frequencies up to 10 kHz, and are generally installed near the airport boundaries.

FIGURE 21 Noncertification flyover noise spectrum, in 13 octaves, of a Boeing 747 aircraft. The reference level of 0 dB is arbitrary. (Courtesy of NASA.)

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a speed of 130 m/sec, an altitude of 60 m, and a position directly over the microphone at the time the noise was recorded. The microphone was located 1.2 m above the ground, and the averaging time of the analyzer was 0.9 sec.

SEE ALSO THE FOLLOWING ARTICLES ACOUSTIC CHAOS • ACOUSTICS, LINEAR • ACOUSTIC WAVE DEVICES • ANALOG SIGNAL ELECTRONIC CIRCUITS • SIGNAL PROCESSING, ACOUSTIC • ULTRASONICS AND ACOUSTICS • UNDERWATER ACOUSTICS

BIBLIOGRAPHY Acoustical Society of America, Standards Secretariat, 120 Wall Street, 32nd Floor, New York, NY 10005-3993. Crocker, M.J., ed.-in-chief (1997). “Encyclopedia of Acoustics,” John Wiley & Sons, New York. Hassall, J.R., and Zavari, K., (1979). “Acoustic Noise Measurement,” 4th ed., Bruel & Kjaer Instruments, Marlborough, MA. International Organization for Standardization (ISO), Case Postale 56, CH-1211, Geneve, Switzerland. Kundert, W.R. (1978). Sound and Vibration 12, 10–23. Wong, G.S.K., and Embleton, T.F.W., eds. (1995). “AIP Handbook of Condenser Microphones,” American Institute of Physics Press, New York.

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Acoustics, Linear Joshua E. Greenspon J. G. Engineering Research Associates

I. Introduction II. Physical Phenomena in Linear Acoustics III. Basic Assumptions and Equations in Linear Acoustics IV. Free Sound Propagation V. Sound Propagation with Obstacles VI. Free and Confined Waves VII. Sound Radiation and Vibration VIII. Coupling of Structure/Medium (Interactions) IX. Random Linear Acoustics

GLOSSARY Attenuation Reduction in amplitude of a wave as it travels. Condensation Ratio of density change to static density. Coupling Mutual interaction between two wave fields. Diffraction Bending of waves around corners and over barriers. Dispersion Dependence of velocity on frequency, manifested by distortion in the shape of a disturbance. Elastic waves Traveling disturbances in solid materials. Ergodic Statistical process in which each record is statistically equivalent to every other record. Ensemble averages over a large number of records at fixed times can be replaced by corresponding time averages on a single representative record. Impedance Pressure per unit velocity. Interaction Effect of two media on each other. Medium Material through which a wave propagates.

Nondispersive medium Medium in which the velocity is independent of frequency and the shape of the disturbance remains undistorted. Normal mode Shape function of a wave pattern in transmission. Propagation Motion of a disturbance characteristic of radiaton or other phenomena governed by wave equations. Ray Line drawn along the path that the sound travels, perpendicular to the wave front. Reflection Process of a disturbance bouncing off an obstacle. Refraction Change in propagation direction of a wave with change in medium density. Reverberation Wave pattern set up in an enclosed space. Scattering Property of waves in which a sound pattern is formed around an obstacle enveloped by an incoming wave.

29

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130 Sommerfeld radiation condition Equation stating that waves must go out from their source towards infinity and not come in from infinity. Standing waves Stationary wave pattern. Wave guide Structure or channel along which the wave is confined. ACOUSTICS is the science of sound—its generation, transmission, reception, and effects. Linear acoustics is the study of the physical phenomena of sound in which the ratio of density change to static density is small, typically much less than 0.1. A sound wave is a disturbance that produces vibrations of the medium in which it propagates.

I. INTRODUCTION A unified treatment of the principles of linear acoustics must begin with the well-known phenomena of singlefrequency acoustics. A second essential topic is random linear acoustics, a relatively new field, which is given a tutorial treatment in the final section of this article. The objective is to present the elementary principles of linear acoustics and then to use straightforward mathematical development to describe some advanced concepts. Section II gives a physical description of phenomena in acoustics. Section III starts with the difference between linear and nonlinear acoustics and leads to the derivation of the basic wave equation of linear acoustics. Section IV discusses the fundamentals of normal-mode and ray acoustics, which is used extensively in studies of underwater sound propagation. In Section V, details are given on sound propagation as it is affected by barriers and obstacles. Sections VI–VIII deal with waves in confined spaces; sound radiation, with methods of solution to determine the sound radiated by structures; and the coupling of sound with its surroundings. Section IX discusses the fundamentals of radom systems as applied to structural acoustics.

II. PHYSICAL PHENOMENA IN LINEAR ACOUSTICS A. Sound Propagation in Air, Water, and Solids Many practical problems are associated with the propagation of sound waves in air or water. Sound does not propagate in free space but must have a dense medium to propagate. Thus, for example, when a sound wave is produced by a voice, the air particles in front of the mouth are vibrated, and this vibration, in turn, produces a disturbance in the adjacent air particles, and so on. [See ACOUSTICAL MEASUREMENT.]

Acoustics, Linear

If the wave travels in the same direction as the particles are being moved, it is called a longitudinal wave. This same phenomenon occurs whether the medium is air, water, or a solid. If the wave is moving perpendicular to the moving particles, it is called a transverse wave. The rate at which a sound wave thins out, or attenuates, depends to a large extent on the medium through which it is propagating. For example, sound attenuates more rapidly in air than in water, which is the reason that sonar is used more extensively under water than in air. Conversely, radar (electromagnetic energy) attenuates much less in air than in water, so that it is more useful as a communication tool in air. Sound waves travel in solid or fluid materials by elastic deformation of the material, which is called an elastic wave. In air (below a frequency of 20 kHz) and in water, a sound wave travels at constant speed without its shape being distorted. In solid material, the velocity of the wave changes, and the disturbance changes shape as it travels. This phenomenon in solids is called dispersion. Air and water are for the most part nondispersive media, whereas most solids are dispersive media. B. Reflection, Refraction, Diffraction, Interference, and Scattering Sound propagates undisturbed in a nondispersive medium until it reaches some obstacle. The obstacle, which can be a density change in the medium or a physical object, distorts the sound wave in various ways. (It is interesting to note that sound and light have many propagation characteristics in common: The phenomena of reflection, refraction, diffraction, interference, and scattering for sound are very similar to the phenomena for light.) [See WAVE PHENOMENA.] 1. Reflection When sound impinges on a rigid or elastic obstacle, part of it bounces off the obstacle, a characteristic that is called reflection. The reflection of sound back toward its source is called an echo. Echoes are used in sonar to locate objects under water. Most people have experienced echoes in air by calling out in an empty hall and hearing their words repeated as the sound bounces off the walls. 2. Refraction and Transmission Refraction is the change of direction of a wave when it travels from a medium in which it has one velocity to a medium in which it has a different velocity. Refraction of sound occurs in the ocean because the temperature or the water changes with depth, which causes the velocity of

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sound also to change with depth. For simple ocean models, the layers of water at different temperatures act as though they are layers of different media. The following example explains refraction: Imagine a sound wave that is constant over a plane (i.e., a plane wave) in a given medium and a line drawn perpendicular to this plane (i.e., the normal to the plane) which indicates the travel direction of the wave. When the wave travels to a different medium, the normal bends, thus changing the direction of the sound wave. This normal line is called a ray and is discussed later with ray acoustics in Section IV.A. When a sound wave impinges on a plate, part of the wave reflects and part goes through the plate. The part that goes through the plate is the transmitted wave. Reflection and transmission are related phenomena that are used extensively to describe the characteristics of sound baffles and absorbers.

When sound waves propagate in an enclosed space, they reflect from the walls of the enclosure and travel in a different direction until they hit another wall. In a regular enclosure, such as a rectangular room, the waves reflect back and forth between the sound source and the wall, setting up a constant wave pattern that no longer shows the characteristics of a traveling wave. This wave pattern, called a standing wave, results from the superposition of two traveling waves propagating in opposite directions. The standing wave pattern exists as long as the source continues to emit sound waves. The continuous rebounding of the sound waves causes a reverberant field to be set up in the enclosure. If the walls of the enclosure are absorbent, the reverberant field is decreased. If the sound source stops emitting the waves, the reverberant standing wave field dies out because of the absorptive character of the walls. The time it takes for the reverberant field to decay is sometimes called the time constant of the room.

3. Diffraction Diffraction is associated with the bending of sound waves around or over barriers. A sound wave can often be heard on the other side of a barrier even if the listener cannot see the source of the sound. However, the barrier projects a shadow, called the shadow zone, within which the sound cannot be heard. This phenomenon is similar to that of a light that is blocked by a barrier.

D. Sound Radiation The interaction of a vibrating structure with a medium produces disturbances in the medium that propagate out from the structure. The sound field set up by these propagating disturbances is known as the sound radiation field. Whenever there is a disturbance in a sound medium, the waves propagate out from the disturbance, forming a radiation field.

4. Interference Interference is the phenomenon that occurs when two sound waves converge. In linear acoustics the sound waves can be superimposed. When this occurs, the waves interfere with each other, and the resultant sound is the sum of the two waves, taking into consideration the magnitude and the phase of each wave. 5. Scattering Sound scattering is related closely to reflection and transmission. It is the phenomenon that occurs when a sound wave envelops an obstacle and breaks up, producing a sound pattern around the obstacle. The sound travels off in all directions around the obstacle. The sound that travels back toward the source is called the backscattered sound, and the sound that travels away from the source is known as the forwardscattered field. C. Standing Waves, Propagating Waves, and Reverberation When a sound wave travels freely in a medium without obstacles, it continues to propagate unless it is attentuated by some characteristic of the medium, such as absorption.

E. Coupling and Interaction between Structures and the Surrounding Medium A structure vibrating in air produces sound waves, which propagate out into the air. If the same vibrating structure is put into a vacuum, no sound is produced. However, whether the vibrating body is in a vacuum or air makes little difference in the vibration patterns, and the reaction of the structure to the medium is small. If the same vibrating body is put into water, the high density of water compared with air produces marked changes in the vibration and consequent radiation from the structure. The water, or any heavy liquid, produces two main effects on the structure. The first is an added mass effect, and the second is a damping effect known as radiation damping. The same type of phenomenon also occurs in air, but to a much smaller degree unless the body is traveling at high speed. The coupling phenomenon in air at these speeds is associated with flutter. F. Deterministic (Single-Frequency) Versus Random Linear Acoustics When the vibrations are not single frequency but are random, new concepts must be introduced. Instead of dealing

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with ordinary parameters such as pressure and velocity, it is necessary to use statistical concepts such as auto- and cross-correlation of pressure in the time domain and autoand cross-spectrum of pressure in the frequency domain. Frequency is a continuous variable in random systems, as opposed to a discrete variable in single-frequency systems. In some acoustic problems there is randomness in both space and time. Thus, statistical concepts have to be applied to both time and spatial variables.

ρ = ρo (1 + S)

(2)

where S is called the condensation. By substituting Eq. (2) into (1) we obtain: (1 + S)(1 + ∂ξ/∂ x) = 1

(3)

If p is the sound pressure at x, then p + ∂ p/∂ x d x is the sound pressure at x + d x (by expanding p into a Taylor series in x and neglecting higher-order terms in d x). Applying Newton’s law to the differential element, we find that:

III. BASIC ASSUMPTIONS AND EQUATIONS IN LINEAR ACOUSTICS A. Linear Versus Nonlinear Acoustics The basic difference between linear and nonlinear acoustics is determined by the amplitude of the sound. The amplitude is dependent on a parameter, called the condensation, that describes how much the medium is compressed as the sound wave moves. When the condensation reaches certain levels, the sound becomes nonlinear. The major difference between linear and nonlinear acoustics can best be understood by deriving the one-dimensional wave equation for sound waves and studying the parameters involved in the derivation. Consider a plane sound wave traveling down a tube, as shown in Fig. 1. Let the cross-sectional area of the tube be A and let ξ be the particle displacement along the x axis from the equilibrium position. Applying the principle of conservation of mass to the volume A d x before and after it is displaced, the following equation is obtained: ρ A d x(1 + ∂ξ/∂ x) = ρo A d x

where ρ is the new density of the disturbed medium. This disturbed density ρ can be defined in terms of the original density ρo by the following relation:

∂p ∂ 2ξ (4) = ρo 2 ∂x ∂t If it is assumed that the process of sound propagation is adiabatic (i.e., there is no change of heat during the process), then the pressure and density are related by the following equation: γ P ρ = (5) po ρo −

where P = total pressure = p + po , p is the disturbance sound pressure, and γ is the adiabatic constant, which has a value of about 1.4 for air. Using Eqs. (2) and (3) gives: ρo ρ= 1 + ∂ξ/∂ x Thus,

(1)

The mass of the element before the disturbance arrives is ρo A d x where ρo is the original density of the medium. The mass of this element as the disturbance passes is changed to: ρ A d x(1 + ∂ξ/∂ x)

∂p ∂ξ −1−γ ∂ 2 ξ ∂P = = −γ po 1 + ∂x ∂x ∂x ∂x2

(6)

Substituting into Eq. (4) gives: γ po

∂ 2 ξ/∂ x 2 ∂ 2ξ = ρo 2 1+γ (1 + ∂ξ/∂ x) ∂t

(7)

∂ 2 ξ/∂ x 2 ∂ 2ξ = 2 1+γ (1 + ∂ξ/∂ x) ∂t

(8)

or finally, c2

where c2 = po γ /ρo (c is the sound speed in the medium). If ∂ξ/∂ x is small compared with 1, then Eq. (3) gives: S=−

∂ξ ∂x

(9)

and (8) gives: c2 FIGURE 1 Propagation of a plane one-dimensional sound wave. A = cross sectional area of tube; ξ = particle displacement along the x axis; p = acoustic pressure.

∂ 2ξ ∂ 2ξ = ∂x2 ∂t 2

(10)

Thus, ξ = f 1 (x − ct) + f 2 (x − ct)

(10a)

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Equations (9) and (10) are the linear acoustic approximations. The first term in Eq. (10a) is an undistorted traveling wave that is moving at speed c in the +x direction, and the second term is an undistorted traveling wave moving with speed c in the −x direction. Condensation values S of the order of 1 are characteristic of sound waves with amplitudes approaching 200 db rel. 0.0002 dyne/cm2 . The threshold of pain is about 130 db rel. 0.0002 dyne/cm2 . This is the sound pressure level that results in a feeling of pain in the ear. The condensation value for this pressure is about S = 0.0001. For a condensation value S = 0.1, we are in the nonlinear region. This condensation value corresponds to a sound pressure level of about 177 db rel. 0.0002 dyne/cm2 . All the ordinary sounds that we hear such as speech and music (even very loud music) are usually well below 120 db rel. 0.0002 dyne/cm2 . A person who is very close to an explosion or is exposed to sonar transducer sounds underwater would suffer permanent damage to his hearing because the sounds are usually well above 130 db rel. 0.0002 dyne/cm2 . B. Derivation of Basic Equations It is now necessary to derive the general three-dimensional acoustic wave equations. In Section III.A, the onedimensional wave equation was derived for both the linear and nonlinear cases. If there is a fluid particle in the medium with coordinates x, y, z, the fluid particle can move in three dimensions. Let the displacement vector of the particle be b having components ξ , η, ζ , as shown in Fig. 2. The velocity vector q is q = ∂b/∂t

(11)

Let this velocity vector have components u, v, w where u = ∂ξ/∂t

v = ∂η/∂t

w = ∂ζ /∂t

(12)

As the sound wave passes an element of volume, V = d x d y dz, the element changes volume because of the displacement ξ , η, ζ . The change in length of the element in the x, y, z directions, respectively, is (∂ξ/∂ x) d x, (∂η/∂ y) dy, (∂ζ /∂z) dz; so the new volume is V + V where: ∂ξ ∂η ∂ρ V + V = d x 1 + dy 1 + dz 1 + ∂x ∂y ∂z (13) The density of the medium before displacement is ρo and the density during displacement is ρo (1+ S), as in the onedimensional case developed in the last section. Applying the principle of conservation of mass to the element before and after displacement, we find that: (1 + S)(1 + ∂ξ/∂ x)(1 + ∂η/∂ y)(1 + ∂ζ /∂z) = 1 (14) Now we make the linear acoustic approximation that ∂ξ/∂ x, ∂η/∂ y, and ∂ζ /∂z are small compared with 1. So Eq. (14) becomes the counterpart of Eq. (9) in one dimension: S = −(∂ξ/∂ x + ∂η/∂ y + ∂ζ /∂z)

(15)

This equation is called the equation of continuity for linear acoustics. The equations of motion for the element d x d y dz are merely three equations in the three coordinate directions that parallel the one-dimensional Eq. (4); thus, the three equations of motion are ∂p ∂ 2ξ = ρo 2 ∂x ∂t

∂ 2ρ ∂p = ρo 2 ∂z ∂t (16) If one differentiates the first of these equations with respect to x, the second with respect to y, and the third with respect to z, and adds them, then letting

−

−

∂2 y ∂p = ρo 2 ∂y ∂t

−

∇ 2 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 + ∂ 2 /∂z 2 one obtains: ∂2S (17) ∂t 2 Now we introduce the adiabatic assumption in Eq. (17); that is, γ P ρ = (18) po ρo where P = total pressure = p + po and p is the sound pressure due to the disturbance. Since ∇ 2 p = ρo

FIGURE 2 The fluid particle. x, y, z = rectangular coordinates of fluid particle; b = displacement vector of the particle (components of b are ξ , η, ζ ).

ρ = ρo (1 + S), P/ po = (1 + S)γ

(19)

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FIGURE 3 Temperatures, velocities, and refraction angles of the sound. T1 , T2 , . . . , Tn = temperatures of the n layers of the model medium; V1 , V2 , . . . , Vn = sound velocities in the n layers of the model medium.

For small S, the binomial theorem applied to Eq. (19) gives: P − po = Sc2 ρo

A. Ray Acoustics (20)

(c being the adiabatic sound velocity, as discussed for the one-dimensional case). Thus, p = ρo Sc2 Substituting into Eq. (17) we obtain: c2 ∇ 2 p = ∂ 2 p/∂t 2

(21)

C. Intensity and Energy The one-dimensional equation for place waves is given by Eq. (10). The displacement for a harmonic wave can be written: ξ = Aei(ωt+kx) The pressure is given by Eq. (20); that is, p = ρo Co2 S, where S = ∂ξ/∂ x for the one-dimensional wave. Then, p = ρo co2

∂ξ ∂x

The velocity is given by u = ∂ξ/∂t, so, for onedimensional harmonic waves, p = ρo co2 (ik) and u = iωξ , but k = ω/co . Thus, p = ρo co u. The intensity is defined as the power flow per unit area (or the rate at which energy is transmitted per unit area). Thus, I = pξ . The energy per unit area is the work done on the medium by the pressure in going through displacement ξ , that is, E f = p ξ˙. And by the above, I = p 2 ρ o co

IV. FREE SOUND PROPAGATION

Characteristics of sound waves can be studied by the same theory regardless of whether the sound is propagating in air or water. The simplest of the sound-propagation theories is known as ray acoustics. A sound ray is a line drawn normal to the wave front of the sound wave as the sound travels. In Section II. B. 2, refraction was described as the bending of sound waves when going from one medium to another. When a medium such as air or water has a temperature gradient, then it can be thought of as having layers that act as different media. The objective of this theory, or any other transmission theory, is to describe the sound field at a distance from the source. There are two main equations of ray theory. The first is Snell’s law of refraction, which states that V1 V2 V3 Vn = = = ··· = cos θ1 cos θ2 cos θ3 cos θn

(22)

where V1 , V2 , . . . Vn are the velocities of sound through the various layers of the medium, which are at different temperatures as shown in Fig. 3. The second relation is that the power flow remains constant along a ray tube (i.e., there is conservation of energy along a ray tube). A ray tube is a closed surface formed by adjacent rays, as shown in Fig. 4. If the power flow remains constant along a ray tube, then, p12 A1 p 2 A2 p 2 An = 2 = ··· n ρo c1 ρo c2 ρ o cn

(23)

FIGURE 4 Ray tube, A1 , A2 , . . . , An = cross section area of the ray tube at the n stations along the tube.

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where p refers to the sound pressure, A is the crosssectional area of the ray tube, ρo the mass density of the medium, and C the sound velocity in the medium. But, p 2 /ρo c = I , the sound intensity. Thus, I1 A 1 = I2 A 2 = · · · In A n

(24)

The intensity can therefore be found at any point if the initial intensity I1 and the areas of the ray tube A1 , . . . , An are known. The ray tube and the consequent areas can be determined by tracing the rays. The tracing is done by using Snell’s law (Eq. 22). The velocities of propagation V1 , V2 , . . . , Vn are determined from the temperature and salinity of the layer. One such equation for sound velocity is V = 1449 + 4.6T − 0.055T 2 + 0.0003T 3 + (1.39 − 0.012T )(s − 35) + .017d

The δ functions describe the point source. The boundary conditions are

(25)

where V is the velocity of sound in meters per second, T the temperature in degrees centigrade, s the salinity in parts per thousand, and d the depth in meters. The smaller the ray tubes, that is, the closer together the rays, the more accurate are the results. Simple ray-acoustics theory is good only at high frequencies (usually in the kilohertz region). For low frequencies (e.g., less than 100 Hz), another theory, the normal mode theory, has to be used to compute transmission characteristics. B. Normal Mode Theory The normal mode theory consists of forming a solution of the acoustic wave equation that satisfies specific boundary conditions. Consider the sound velocity C(z) as a function of the depth z, and let h be the depth of the medium. The medium is bounded by a free surface at z = 0 and a rigid bottom at z = h. Let a point source be located at z = z 1 , r = 0 (in cylindrical coordinates) as shown in Fig. 5. The pressure p is given by the Helmholtz equation: ∂2 p 1 ∂ p ∂2 p 2 + 2 + k 2 (z) p = − δ(z − z 1 )δ(r ) + 2 ∂r r ∂r ∂z r (26) ω2 2 k (z) = c(z)2

p(r, o) = 0

(free surface)

∂p (r, h) = 0 ∂z

(rigid bottom)

(27)

Equations (26) and (27) essentially constitute all of the physics of the solution. The rest is mathematics. The solution of the homogeneous form of Eq. (26) is first found by separation of variables. Since the wave has to be an outgoing wave, this solution is p(r, z) = Ho(1) (ξr )ψ(z, ξ )

(28)

where Ho(1) is the Hankel function of the first kind of order zero. The function ψ(z, ξ ) satisfies the equation: d 2ψ + [k 2 (z) − ξ 2 ]ψ = 0 dz 2

(29)

with boundary conditions: ψ(o, ξ ) = 0

dψ(h, ξ ) =0 dz

(30)

Since Eq. (30) is a second-order linear differential equation, let the two linearly independent solutions be ψ1 (z, ξ ) and ψ2 (z, ξ ). Thus, the complete solution is ψ(z, ξ ) = B1 ψ1 (z, ξ ) + B2 ψ2 (z, ξ )

(31)

where B1 and B2 are constants. Substitution of Eq. (31) into (30) leads to an equation from which the allowable values of ξ (the eigenvalues) can be obtained, that is, ψ1 (o, ξ )

dψ2 (h, ξ ) dψ1 (h, ξ ) − ψ2 (o, ξ ) =0 dz dz

(32)

The nth root of this equation is called ξn . The ratio of the constants B1 and B2 is B1 ψ2 (o, ξn ) =− B2 ψ1 (o, ξn )

(33)

The Ho(1) (ξn r )ψ(z, ξn ) are known as the normal mode functions, and the solution of the original inhomogeneous equation can be expanded in terms of these normal mode functions as follows: p(r, z) = An Ho(1) (ξn r )ψ(z, ξn ) (34) n

FIGURE 5 Geometry for normal mode propagation. r , z = cylindrical coordinates of a point in the medium; h = depth of medium; z1 = z coordinate of the source.

with unknown coefficients An , which will be determined next. Substituting Eq. (34) into (26) and employing the relation for the Hankel function, 2 d 1 d 2i 2 (1) + + ξ δ(r ) (35) n Ho (ξn r ) = dr 2 r dr πr

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leads to:

An ψn (z) = iπδ(z − z 1 )

(36)

n

Next one must multiply Eq. (36) by ψm (z) and integrate over the depth 0 to h. Using the orthogonality of mode assumption, which states: h ψn (z)ψm (z) dz = 0 if m = n (37) 0

we find that: iπψn (z 1 ) An = h 2 0 ψn (z) dz

(38)

Concorde sonic boom has been studied at about 300 km, and signals of about 0.6 N/m2 were received at frequencies of the order of 0.4 Hz. The same phenomenon occurs for thunder and explosions on the ground. The same principles hold in the atmosphere as in water for the bending of rays in areas of changing temperature. Because of the large attenuation of higher frequency sound in air as opposed to water, sound energy is not used for communication in air. For example, considering the various mechanisms of absorption in the atmosphere, the total attenuation is about 24 db per kiloyard at 100 Hz, whereas, for underwater, the sound attenuates at 100 Hz at about 0.001 db per kiloyard.

So, p(r, z) = πi

n

H (1) (ξn r ) ψn (z 1 )ψn (z) h o 2 0 ψn (z) dz

(39)

If the medium consists of a single layer with constant velocity, Co , it is found that: ψn (z) = cosh bn z ξn = bn2 + k 2 (40)

bn = i n + 12 π/ h k = ω/co C. Underwater Sound Propagation Ray theory and normal mode theory are used extensively in studying the transmission of sound in the ocean. At frequencies below 100 Hz, the normal mode theory is necessary. Two types of sonar are used underwater: active and passive. Active sonar produces sound waves that are sent out to locate objects by receiving echos. Passive sonar listens for sounds. Since the sound rays are bent by refraction (as discussed in Section IV.B), there are shadow zones in which the sound does not travel. Thus a submarine located in a shadow zone has very little chance of being detected by sonar. Since sound is the principal means of detection under water, there has been extensive research in various aspects of this field. The research has been divided essentially into three areas: generation, propagation, and signal processing. Generation deals with the mechanisms of producing the sound, propagation deals with the transmission from the source to the receiver, and signal processing deals with analyzing the signal to extract information.

V. SOUND PROPAGATION WITH OBSTACLES A. Refraction Refraction is the bending of sound waves. (Section II.B.2.) The transmission of sound through the water with various temperature layers and the application of Snell’s law have already been treated in this article. Transmission of sound through water is probably the most extensive practical application of refraction in acoustics. B. Reflection and Transmission There are many practical problems related to reflection and transmission of sound waves. One example is used here to acquaint the reader with the concepts involved in the reflection and transmission of sound. Consider a sound wave coming from one medium and hitting another, as shown in Fig. 6. What happens in layered media, such as the temperature layers described in connection with ray acoustics and underwater sound, can now be noted. When ray acoustics and underwater sound were discussed, only refraction and transmission were described. The entire process for one transition layer can now be explained. The mass density and sound velocity in the upper medium is ρ, c and in the lower medium is ρ1 , c1 . The pressure in the incident wave, pinc , can be written: pinc = po eik(x sin θ −z cos θ )

k = ω/c

(41)

D. Atmospheric Sound Propagation

(i.e., assuming a plane wave front). From Snell’s law, it is found that the refracted wave angle θ1 is determined by the relation:

It has been shown that large amplitude sounds such as sonic booms from supersonic aircraft can be detected at very low frequencies (called infrasonic frequencies) at distances above 100 km from the source. In particular, the

sin θ c = (42) sin θ1 c1 There is also a reflected wave that goes off at angle θ , as shown in the figure. The magnitude of the reflected wave

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The second boundary condition at z = 0 is, therefore, 1 ∂ plower 1 ∂ pupper = ρ ∂z ρ1 ∂z

(48)

The total field in the upper medium consists of the reflected and incident waves combined, so: pupper = pinc + prefl = po eikx sin θ (e−ikz cos θ + V eikz cos θ )

(49)

Substituting into the boundary conditions, we find that: po eikx sin θ (1 + V ) = W po eik1 x sin θ1 so 1 + V = W eik1 x sin θ1 −ikx sin θ Since 1 + V is independent of x, then e must also be independent of x. Thus, k1 sin θ1 = k sin θ

FIGURE 6 Reflection and transmission from an interface. ρ, c = mass density and sound velocity in upper medium; ρ1 , c1 = mass density and sound velocity in lower medium; θ = angle of incidence and reflection; θ1 = angle of refraction.

prefl = V eik(x sin θ+z cos θ )

1 po eikx sin θ (−ik cos θ + V ik cos θ ) ρ 1 W po eik1 x sin θ1 (−ik1 cos θ1 ) ρ1

prefrac = W po e

ik1 (x sin θ1 −z cos θ1 )

pupper = plower

(vz )upper = (vz )lower

(acoustic pressure is continuous across ) the boundary) (particle velocity normal to boundary is continuous across the boundary)

(45)

For harmonic motion v ∼ e

iωt

(46)

, so:

∂ p/∂z = iωρvz

(47)

(53)

and substituting Eq. (53) into (52) gives: 1 ikx sin θ e (−ik cos θ + V ik cos θ) ρ =

1 (1 + V )eik1 x sin θ1 (−ik1 cos θ1 ) ρ1

(54)

or, ρ1 (−ik cos θ + V ik cos θ) = (1 + V )(−ik1 cos θ1 ) ρ So V =

The velocity is related to the pressure by the expression: ∂p ∂vz =ρ ∂z ∂t

1+V = W

(44)

where k1 = ω/c1 and W is the transmission coefficient. The boundary conditions at z = 0 are

(52)

Substituting Eq. (51) into (50) gives:

(43)

where V is the reflection coefficient. Similarly, the refracted wave can be written in the form

(51)

which is Snell’s law. Thus, the first boundary condition leads to Snell’s law. The second boundary condition leads to the equation:

= is not known, but since its direction θ is known, it can be written in the form:

(50)

ik1 x sin θ1 −ikx sin θ

=

(ρ1 /ρ)k cos θ − k1 cos θ1 (ρ1 /ρ)k cos θ + k1 cos θ1 ρ1 ρ ρ1 ρ

cos θ − cos θ +

k1 k k1 k

cos θ1 cos θ1

(55) c k1 = k c1

(56)

Equations (51), (53), and (56) give the unknowns θ1 , V , and W as functions of the known quantities ρ1 , ρ, c1 , c, and θ . Note that if the two media are the same then V = 0 and W = 1. Thus, there is no reflection, and the transmission is

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FIGURE 7 Diffraction over a wide barrier. Plot of the ratio of square of diffracted sound pressure amplitude p Diffr to the square of the amplitude pATL expected at an equivalent distance L from the source in the absence of the barrier. Source and listener locations are as indicated in the sketch with zs = zL on opposite sides of a rectangular three-sided barrier. Computations based on the Maekawa approximation and on the double-edge diffraction theory are presented for listener angle θ between 0◦ and 90◦ . Here, L represents a distance of 30 wavelengths (10λ + 10λ + 10λ). (Reprinted by permission from Pierce, A. D. (1974). J. Acoust. Soc. Am. 55 (5), 953.)

100%; that is, the incident wave continues to move along the original path. As θ → π/2 (grazing incidence) then V → −1 and W → 0. This says that there is no wave transmitted to the second medium at grazing incidence of the incident wave. For θ such that (ρ1 /ρ) cos θ = (k1 /k) cos θ , the reflection coefficient vanishes, and there is complete transmission similar to the case in which the two media are the same.

D. Interference If the pressure in two different acoustic waves is p1 , p2 and the velocity of the waves is u 1 , u 2 , respectively, then the intensity I for each of the waves is I 1 = p1 u 1

I 2 = p2 u 2

When the waves are combined, the pressure p in the combined wave is p = p1 + p2

C. Diffraction One of the most interesting practical applications of diffraction is in barriers. Figure 7 shows results of diffraction over a wide barrier. This plot illustrates how sound bends around corners. As the listener gets closer to the barrier (i.e., as θ → 0), the sound is reduced by almost 40 db for the case shown. When the listener is at the top of the barrier (θ → 90◦ ), the reduction is only 20 db. In the first case (θ → 0), the sound has to bend around the two corners. However, for the second case, it has to bend only around the first corner. Such barriers are often used to block the noise from superhighways to housing developments. As can be seen from the curve, the listener has to be well in the shadow zone to achieve maximum benefits.

(57)

(58)

and the velocity u in the combined wave is u = u1 + u2

(59)

The intensity of the combined wave is I = pu = ( p1 + p2 )(u 1 + u 2 ) = p1 u 1 + p2 u 2 + p2 u 1 + p1 u 2 = I 1 + I 2 + ( p2 u 1 + p1 u 2 )

(60)

Equation (60) states that the sum of the intensities of the two waves is not merely the sum of the intensities of each of the waves, but that there is an extra term. This term is called the interference term. The phenomena that the superposition principle does not hold for intensity in linear acoustics is known as interference. If both u 1 and

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u 2 are positive, then what results is called constructive interference. If u 1 = −u 2 , then I = 0 and what results is called destructive interference. E. Scattering The discussion of reflection, refraction, and transmission was limited to waves that impinged on a flat infinite surface such as the interface between two fluids. In those cases, the phenomena of reflection, refraction, and transmission were clear cut, and the various phenomena could be separated. If the acoustic wave encounters a finite object, the processes are not so clear cut and cannot be separated. The process of scattering actually involves reflection, refraction, and transmission combined, but it is called scattering because the wave scatters from the object in all directions. Consider the classical two-dimensional problem of a plane wave impinging on a rigid cylinder, as shown in Fig. 8. The intensity of the scattered wave can be written: 2Io a Is ≈ (61) |ψs (θ )|2 πr where Io is the intensity of the incident wave (Io = Po2 / 2ρo co where Po is the pressure in the incident wave, ρo is the density of the medium, and co is the sound velocity in the medium), and ψs (θ ) is a distribution function. Figure 9 shows the scattered power and distribution in intensity for various values of ka. Several interesting cases can be noted. If ka → 0, then the wavelength of the sound is very large compared with the radius of the cylinder, and the scattered power goes to zero. This means that the sound travels as if the object were not present at all. If the wavelength is very small

compared with the cylinder radius, it can be shown that most of the scattering is in the backward direction in the form of an echo or reflection, in the same manner as would occur at normal incidence of a plane wave on an infinite plane. Thus, for small ka (low frequency), there is mostly forward scattering, and for large ka (high frequency), there is mostly backscattering. Consider now the contrast between scattering from elastic bodies compared with rigid bodies. Let a plane wave of magnitude p and frequency ω impinge broadside on a cylinder as shown in Fig. 10(a). Let f ∞ (θ ) be defined as follows: 1/2 ps (θ ) 2r f ∞ (θ ) = = form function a po where r = radial distance to the point where the scattered pressure is being measured; a = outside radius of the cylinder; b = inside radius of a shell whose outside radius is a; ps (θ) = amplitude of scattered pressure; po = amplitude of incident wave; ka = ωa/co ; ω = 2π f ; f = frequency of incoming wave; co = sound velocity in the medium. Figure 10(b) shows the form function for a rigid cylinder as a function of ka. Figure 10(c) shows this function for a rigid sphere of outside radius a. Contrast this with Fig. 10(d), which gives the form function for a solid aluminum cylinder in water, and with Fig. 10(e), which shows the function for elastic aluminum shells of various thicknesses. As one can see, the elasticity of the body has a dominant effect on the acoustic scattering from the body.

VI. FREE AND CONFINED WAVES A. Propagating Waves The acoustic wave equation states that the pressure satisfies the equation: c2 ∇ 2 p =

∂2 p ∂t 2

(62)

For illustrative purposes, consider the one-dimensional case in which the waves are traveling in the x direction. The equation satisfied in this case is c2

∂2 p ∂2 p = ∂x2 ∂t 2

(63)

The most general solution to this equation can be written in the form: FIGURE 8 Plane wave impinging on a rigid cylinder. I o = intensity of incident plane wave; a = radius of cylinder; r , θ = cylindrical coordinates of field point.

p = f 1 (x + ct) + f 2 (x − ct)

(64)

This solution consists of two free traveling waves moving in opposite directions.

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FIGURE 9 Scattered power and distribution in intensity for a rigid cylinder. ψS (θ ) = angular distribution function; a = radius of cylinder, I o = incident intensity of plane wave; k = ω/c o ; ω = frequency of wave; c o = sound velocity in the medium. (Reprinted by permission from Lindsay, R. B. (1960). “Mechanical Radiation,” McGraw-Hill, New York.)

B. Standing Waves

C. Reverberation

Consider the waves described immediately above, but limit the discussion to harmonic motion of frequency ω. One of the waves takes the form:

When traveling waves are sent out in an enclosed space, they reflect from the walls and form a standing wave pattern in the space. This is a very simple description of a very complicated process in which waves impinge on the walls from various angles, reflect, and impinge again on another wall, and so on. The process of reflection takes place continually, and the sound is built up into a sound field known as a reverberant field. If the source of the sound is cut off, the field decays. The amount of time that it takes for the sound energy density to decay by a factor of 106 (i.e., 60 db) is called the reverberation time of the room. The sound energy density concept was used by Sabine in his fundamental discoveries on reverberation. He found that sound fills a reverberant room in such a way that the average energy per unit volume (i.e., the energy density) in any region is nearly the same as in any other region. The amount of reverberation depends on how much sound is absorbed by the walls in each reflection and in the air. The study of room reverberation and the answering of questions such as how much the sound is absorbed by people in the room, and other absorbers placed in the room, are included in the field of architectural acoustics. The acoustical design of concert halls or any structures in which sound and reverberation are of importance is a specialized and intricate art.

p = A cos(kx − ωt)

(65)

where k = ω/c = 2π/λ and λ = wavelength of the sound. This equation can also be written in the form: p = A cos k(x − ct)

(66)

If this wave hits a rigid barrier, another wave of equal magnitude is reflected back toward the source of the wave motion. The reflected wave is of the form: p = A cos k(x + ct)

(67)

If the source continues to emit waves of the form of Eq. (66), and reflections of the form of Eq. (67) come back, then the resulting pattern is a superposition of the waves, that is, p = A cos(kx + ωt) + A cos(kx − ωt)

(68)

or p = 2A cos kx cos ωt The resultant wave pattern no longer has the characteristics of a traveling wave. The pattern is stationary and is known as a standing wave.

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propagation, the boundary conditions were stipulated on the surface and bottom. The problem thus became one of propagation in a wave guide. One wave guide application that leads to interesting implications when coupled and uncoupled systems are considered, is the propagation of axially symmetric waves in a fluid-filled elastic pipe. If axially symmetric pressure waves of magnitude po and frequency ω are sent out from a plane wave source in a fluid-filled circular pipe, the pressure at any time t and at any location x from the source can be written as follows: 2 r p ≈ po 1 + iω e−(xκt /a)+i[ω/c+(σt /a)]x−iωt 2acζt (69) where r is the radial distance from the center of the pipe to any point in the fluid, a the mean radius of the pipe, c the sound velocity in the fluid inside the pipe, ω the radian frequency of the sound, x the longitudinal distance from the disturbance to any point in the fluid, and z t the impedance of the pipe such that: 1 1 1 = = (κt − iσt ) zt ρcζt ρc

(70)

where κt /ρc is the conductance of the pipe and σt /ρc the susceptance of the pipe. The approximate value of the wave velocity υ down the tube is υ = c[1 − σt (λ/2πa)]

FIGURE 10 (a) The geometry used in the description of the scattering of a plane wave by an infinitely long cylinder. (b) The form function for a rigid cylinder. (c) The form function vs. ka for a rigid sphere.

D. Wave Guides and Ducts When a wave is confined in a pipe or duct, the duct is known as a wave guide because it prevents the wave from moving freely into the surrounding medium. In discussing normal mode theory in connection with underwater sound

(71)

If the tube wall were perfectly rigid, then the tube impedance would be infinite (σt → 0) and the velocity would be c. The attenuation is given by the factor eiκt x/a . If the tube were perfectly rigid (κt = 0), then the attenuation would be zero. If the tube is flexible, then energy gradually leaks out as the wave travels and the wave in the fluid attenuates. This phenomenon is used extensively in trying to reduce sound in tubes and ducts by using acoustic liners. These acoustic liners are flexible and absorb energy as the wave travels down the tube. One critical item must be mentioned at this point. It has been assumed that the tube impedance (or conductance and susceptance) can be calculated independently. This is an assumption that can lead to gross errors in certain cases. It will be discussed further when coupled systems are considered. The equation for an axisymmetric wave propagating in a rigid circular duct or pipe is as follows: p(r, z) = pm0 J0 (αom r/a)ei(γom z−ωt) pmo is the amplitude of the pressure wave, r is the radial distance from the center of the pipe to any point, a is the radius of the pipe, z is the distance along the pipe, ω is the radian frequency, and t is time.

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FIGURE 10 (continued ). (d) Top, the form function for an aluminum cylinder in water, Bottom, comparison of theory (——) and experimental observation (the points) for an aluminum cylinder in water.

γom and αom are related by the following formula: 1/2

γom = k 2 − (αom /a)2 J0 is the Bessel Function of order 0.

propagated down the tube. Propagation takes place only for frequencies in which k > αom /a. Since α00 = 0, propagation always takes place for this mode. The frequency at which γom is 0 is called the cutoff frequency and is as follows:

k = 2π f /ci

f om = ci αom /2πa

In the above relation, f is the frequency of the wave and ci is the sound velocity in the fluid inside the pipe (for water this sound velocity is about 5000 ft/sec and for air it is about 1100 ft/ sec). The values of αom for the first few m are m=0

α00 = 0

m=1

α01 = 3.83

m=2

α02 = 7.02

p(r, z) = Pm0 J0 (αom r/a)e

p(r, z, φ) = Pnm Jn (αnm r/a)ei(γnm z−ωt) cos nφ where γnm = [k 2 − (αnm /a)2 ]1/2 , f nm = ci αnm /2πa. A few values for αnm for n > 0 are as follows:

If k < αom /a then γom is a pure imaginary number and the pressure takes the form: −γ

For frequencies below the cutoff frequency, no propagation takes place. For general asymmetric waves the pressure takes the form:

z −iωt

om e

which is the equation for a decaying wave in the z direction. For frequencies which give k < αom /a, no wave is

α10 = 1.84

α11 = 5.31

α12 = 8.53

α20 = 3.05

α21 = 6.71

α22 = 9.97

It is seen that only α00 = 0, and this is the only mode that propagates at all frequencies regardless of the size of the duct. Consider a 10-in.-diameter pipe containing water. The lowest cutoff frequency greater than the 00 mode is

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axisymmetric mode (00 mode) will propagate in the duct.

VII. SOUND RADIATION AND VIBRATION A. Helmholtz Integral, Sommerfeld Radiation Condition, and Green’s Functions In this presentation, acoustic fields that satisfy the wave equation of linear acoustics are of interest. c2 ∇ 2 p = ∂ 2 p/∂t 2

(72)

where p is the pressure in the field at point P and at time t. For sinusoidal time-varying fields, p(P, t) = p(P)eiωt

(73)

so that p satisfies the Helmholtz equation: ∇2 p + k2 p = 0

k 2 = ω2 /c2

(74)

This Helmholtz equation can be solved in general form by introducing an auxiliary function called the Green’s function. First, let ϕ and ψ be any two functions of space variables that have first and second derivatives on S and outside S (see Fig. 11). Let Vo be the volume between So and the boundary at ∞. Green’s theorem states that within the volume Vo , ∂ψ ∂ϕ ϕ (ϕ∇ 2 ψ −ψ∇ 2 ϕ) d Vo (75) −ψ dS = ∂n ∂n S Vo where S denotes the entire boundary surface and Vo the entire volume of the region outside So . In Eq. (75) ∂/∂n denotes the normal derivative at the boundary surface. Rearrange the terms in Eq. (75) and subtract Vo k 2 ϕψ d Vo from each side, and the result is ∂ψ ∂ϕ 2 2 ϕ ϕ(∇ ψ + k ψ) d Vo = ψ dS − dS ∂n ∂n S Vo S − ψ(∇ 2 ϕ + k 2 ϕ) d Vo (76) Vo

FIGURE 10 (continued ). (e) The form function vs. ka over the range of 0.2 ≤ ka ≤ 20 for aluminum shells with b/a values of (a) 0.85, (b) 0.90, (c) 0.92, (d) 0.94, (e) 0.96, (f) 0.98, and (g) 0.99. (Figs. 10(a–e) reprinted by permission from Neubauer, W. G., (June 1986). “Acoustic Reflection from Surfaces and Shapes,” Chapter 4, Eq. (6) and Figs. 1, 2, 7(a), 13, and 27, Naval Research Laboratory, Washington, D.C.)

3516 Hz. Thus, nothing will propagate below 3516 Hz except the lowest axisymmetric mode (i.e., the 00 mode). If the pipe were 2 in. in diameter then nothing would propagate in the pipe below 17,580 Hz except the 00 mode. This means that in a great many practical cases no matter what is exciting the sound inside the duct, only the lowest

Now choose ϕ as the pressure p in the region Vo ; thus, ∇ 2 p + k2 p = ∇ 2ϕ + k2ϕ = 0

(77)

and choose ψ as a function that satisfies: ∇ 2 ψ + k 2 ψ = δ(P − P )

(78)

where δ(P − P ) is a δ function of field points P and P . Choose another symbol for ψ, that is, ψ = g(P, P , ω)

(79)

By virtue of the definition of the δ function, the following is obtained: ϕ(P )δ(P − P ) d P = ϕ(P) (80) Vo

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FIGURE 11 The volume and boundary surface. S ∞ = surface at ∞; S o = radiating surface; V0 = volume between S ∞ and S o ; S 1 , S 2 = surfaces connecting S o to S ∞ .

Thus, Eq. (76) becomes: ∂g(P, S, ω) p(S, ω) − p(P, ω) ∂n S ∂ p(S, ω) = g(P, S, ω) dS ∂n S or

Green’s function as follows: ∂ eikr eikr ∂ p(S, ω) 1 ρ(S, ω) − dS p(P, ω) = 4π S ∂n r r ∂n (86) (81)

p(P, ω) =

∂g(P, S, ω) ∂n S ∂ p(S, ω) − g(P, S, ω) dS ∂n p(S, ω)

(82)

It is now clear that the arbitrary function ψ was chosen so that the volume integral would reduce to the pressure at P. The function g is the Green’s function, which thus far is a completely arbitrary solution of: ∇ 2 g(P, P , ω) + k 2 g(P, P , ω) = δ(P − P )

(83)

For this Green’s function to be a possible solution, it must satisfy the condition that there are only outgoing traveling waves from the surface So to ∞, and no waves are coming in from ∞. Sommerfeld formulated this condition as follows: ∂g lim r − ikg = 0 (84) r →∞ ∂r

Several useful alternative forms of Eqs. (82) and (86) can be derived. If a Green’s function can be found whose normal derivative vanishes on the boundary So , then: ∂g (P, S, ω) = 0 on So ∂n and Eq. (82) becomes: ∂p p(P, ω) = − g(P, S, ω) dS ∂n S

(87)

Alternatively, if a Green’s function can be found that itself vanishes on the boundary So , then g(P, S, ω) = 0 on So and Eq. (82) becomes: ∂g(P, S, ω) p(P, ω) = dS (88) p(S, ω) ∂n S From Newton’s law, ∂ p/∂n = −ρ w ¨n

(89)

where w ¨ n is the normal acceleration of the surface. Thus, Eq. (87) can be written as: p(P, ω) = ρ g(P, S, ω)w ¨ n dS (90) S

where r is the distance from any point on the surface to any point in the field. A solution that satisfies Eqs. (83) and (84) can be written as follows: g=

1 eikr 4π r

(85)

This function is known as the free-space Green’s function. Thus, Eq. (82) can be written in terms of this free-space

If γ is the angle between the outward normal to the surface at S and the line drawn from S to P, then Eq. (86) can be written in the form (assuming harmonic motion): 1 eikr p(P, ω) = d S (91) (ρ w ¨ n (s) − ikp(s) cos γ ) 4π S r Since p(s) is unknown, then Eq. (86) or, alternatively, Eq. (91) is an integral equation for p.

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An interesting limiting case can be studied. Assume a low-frequency oscillation of the body enclosed by So . For this case, k is very small and Eq. (91) reduces to: 1 p(P, ω) = ρw ¨ n ds (92) 4πr S or p(P, ω) = ρ V¨ /4π R

(93)

where V¨ is the volume acceleration of the body. For some cases of slender bodies vibrating at low frequency, it can be argued that the term involving p in Eq. (91) is small compared with the acceleration term. For these cases, 1 eikr p(P, ω) ≈ ρw ¨ n (S) dS (94) 4π S r

FIGURE 12 Geometry of the vibrating cylinder. R S 1 = radius vector to point S 1 on the cylinder surface; R1 = radius vector to the far-field point; aRi = unit vector in the direction of R1 ; P1 = farfield point (with spherical coordinates R 1 , ϕ1 , θ1 ). (Reprinted by permission from Greenspon, J. E. (1967). J. Acoust. Soc. Am. 41 (5), 1203.)

B. Rayleigh’s Formula for Planar Sources It was stated in the last section that if a Green’s function could be found whose normal derivative vanishes at the boundary, then the sound pressure radiated from the surface could be written as Eq. (87) or, alternatively, using Eq. (89): p(P, ω) = ρ g(P, S, ω)w ¨ n (S) ds (95) S

It can be shown that such a Green’s function for an infinite plane is exactly twice the free-space Green’s function, that is, g(P, S, ω) = 2 ×

1 eikr 4π r

(96)

The argument here is that the pressure field generated by two identical point sources in free space displays a zero derivative in the direction normal to the plane of symmetry of the two sources. Substituting Eq. (96) into Eq. (95) gives: ikr ρ e p= (97) w ¨ n (S) ds 2π S r Equation (97) is known as Rayleigh’s formula for planar sources.

C. Vibrating Structures and Radiation: Multipole Contributions To make it clear how the above relations can be applied, consider the case of a slender vibrating cylindrical surface

at low frequency such that Eq. (94) applies. The geometry of the problem is shown in Fig. 12. eikr eik R1 −ik (a R ·RS ) 1 1 e = r R1

(98) a R1 · R S1 = z o cos θ1 + xo sin θ1 cos ϕ1 + yo sin θ1 sin ϕ1 where xo , yo , z o are the rectangular coordinates of a point on the vibrating surface of the structure, R S1 is the radius vector to point S1 on the surface, R1 , θ, ϕ1 are the spherical coordinates of point P1 in the far field, any a R1 is a unit vector in the direction of R1 (the radius vector from the origin to the far field point). Thus, a R1 · R S1 is the projection of R S1 on R1 making R1 − a R1 · R S1 the distance from the far field point to the surface point. Assume the acceleration distribution of the cylindrical surface to be that of a freely supported shell subdivided into its modes, that is, w ¨ 2 (S) =

∞ ∞ m=1 q=0

(Amq cos qϕ1 +Bmq sin qϕ) sin

mπ z l (99)

Expression (99) is a half-range Fourier expansion in the longitudinal z direction (which is chosen as the distance along the generator) between bulkheads, which are distance 1 apart. The expression (99) is a full-range Fourier expansion in the peripheral ϕ direction. It is known that such an expression does approximately satisfy the differential equations of shell vibration and practical end conditions at the edges of the compartment. Substitution of Eqs. (98) and (99) into (94) and integrating results in:

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p(P1 , ω) =

∞ ∞ ρo eik R1 {Amq cos qϕ1 + Bmq sin qϕ1 } 4π R1 m=1 q=0

1 2(mπ − kl cos θ1 ) 1 cos(mπ − kl cos θ1 ) + − 2(mπ + kl cos θ1 ) 2(mπ + kl cos θ1 ) cos(mπ − kl cos θ1 ) sin(mπ − kl cos ϑ1 ) + −i 2(mπ + kl cos θ1 ) 2(mπ + kl cos θ1 ) sin(mπ − kl cos ϑ1 ) − 2(mπ + kl cos θ1 ) × Jq (ka sin θ1 )2πali q

Consider the directivity pattern in the horizontal plane of the cylindrical structure, that is, at ϕ1 = π/2, and let us examine the Amq term in the series above. After some algebraic manipulation, the amplitude of the pressure can be written as follows: For mπ = kl cos θ1 , Imq =

pmq /2πal Amq ρo eik R1 cos qθ1 4π R1 √

=

D. Vibrations of Flat Plates in Water Consider a simply supported flat rectangular elastic plate placed in an infinite plane baffle and excited by a point force as shown in Fig. 14. The plate is made of aluminum and is square with each side being 13.8 in. long. Its thickness is 14 in., and it has a damping loss factor of 0.05. The force is equal to 1 lb, has a frequency of 3000 cps, and is located at x0 = 9 in., y0 = 7 in. If the plate is stopped at the instant when the deflection is a maximum, the velocity pattern would be as shown in Fig. 14. Since the velocity is just the frequency multiplied by the deflection, this is an instantaneous picture of the plate.

VIII. COUPLING OF STRUCTURE/MEDIUM (INTERACTIONS) A. Coupled Versus Uncoupled Systems

√ 2mπ Jq (ka sin θ1 ) 1 − (−1)m cos(kl cos θ1 ) (mπ )2 − (kl cos θ1 )2

For mπ = kl cos θ1 , Imq = 12 Jq (ka sin θ1 )

the directivity pattern for two point sources at the edges of the compartment modified by Jq (ka sin θ1 ). If the longitudinal mode number m is even, then the sources are 180 degrees out of phase, and if m is odd, the sources are in phase. Such modes are called edge modes.

(100)

where Jq is the Bessel function of order q. Figure 13 shows the patterns of the far-field pressure for various values of ka, m, q. A source pattern is defined as one that is uniform in all directions. A dipole pattern has two lobes, a quadrupole pattern has four lobes, and so on. Note that for ka = 0.1, q = 1, m = 1, 3, 5; all show dipole-type radiation. In general. Fig. 13 shows how the multipole contributions depend upon the spatial pattern of the acceleration of the structure. For low frequencies where kl mπ, it is seen that (noting that πl/λ = kl/2) for m even,

cos θ1 2Jq (ka sin θ1 ) sin πl λ Imq ≈ (101) mπ for m odd,

cos θ1 2Jq (ka sin θ1 ) cos πl λ Imq ≈ mπ Thus, at low frequencies (i.e., for kl mπ ), the structural modes radiate as though there were two sources at the edges of the compartment (i.e., l apart). What results is

When a problem involving the interaction between two media can be solved without involving the solution in both media simultaneously, the problem is said to be uncoupled. One example of a coupled system is a vibrating structure submerged in a fluid. Usually the amplitude of vibration depends on the dynamic fluid pressure, and the dynamic pressure in the fluid depends on the amplitude of vibration. In certain limiting cases, the pressure on the structure can be written as an explicit function of the velocity of the structure. In these cases, the system is said to be uncoupled. Another example is a pipe containing an acoustic liner. Sometimes it is possible to represent the effect of the liner by an acoustic impedance, as described in a previous section of this presentation. Such a theory was offered by Morse. As Scott indicated, implicit in Morse’s theory is the assumption that the motion of the surface between the liner and the fluid depends only on the acoustic impedance and the local acoustic pressure, and not on the acoustic pressure elsewhere. This is associated with the concept of “local” and “extended” reaction. In a truly coupled system, the motion of the surface depends on the distribution of acoustic pressure, and, conversely, the acoustic pressure depends on the distribution of motion. Thus, the reaction of the surface and the pressure produced are interrelated at all points. The motion of the surface at point A is not only governed by the pressure at point A. There is motion at A due to pressure at B and, conversely, motion at B due

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FIGURE 13 Horizontal directivity patterns for a cylinder in which L/a = 4, where L is length of cylinder and a is radius of cylinder. The plots show l mq as a function of θ1 at ϕ1 = π/2 for various values of ka, m, and q. (a) ka = 0.1, q = 0; (b) ka = 0.1, q = 1; (c) ka = 0.1, q = 2; (d) ka = 1.0, q = 0; (e) ka = 1.0, q = 1; (f) ka = 3.0, q = 0; (g) ka = 3.0, q = 1; (h) ka = 3.0, q = 5. The numbers shown on the figure are the values of m. (Reprinted by permission from Greenspon, J. E. (1967). J. Acoust. Soc. Am. 41 (5), 1205.)

to pressure at A. Figure 15 illustrates how this assumption can lead to errors in the phase velocity and attenuation in lined ducts. The alternative is to solve the completely coupled problem of the liner and fluid as outlined by Scott.

In aeroelastic or hydroelastic problems, it is necessary to solve the coupled problem of structure and fluid because the stability of the system usually depends on the feeding of energy from the fluid to the structure. Similarly, in acoustoelastic problems such as soft duct liners

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FIGURE 14 Real part of velocity of plate.

in pipes, the attenuation of the acoustic wave in the pipe is dependent upon the coupling of this acoustic wave with the wave in the liner. Similar problems were encountered with viscoelastic liners in water-filled pipes. A recent solution of the coupled problem gave realistic results, whereas the uncoupled problem gave attenuations that were much too high in much the same manner as that shown in Fig. 15.

B. Methods for Simplifying Coupled Systems 1. Impedance Considerations In cases of acoustic waves propagating in pipes or ducts that are not quite rigid but have some elastic deformation as the wave passes, it is satisfactory to use an acoustic impedance to represent the effect of the pipe on the acoustic wave. Only when the acoustic liner or pipe is rather soft and undergoes considerable deformation during the passage of the acoustic wave is it necessary to solve the coupled problem. It is interesting to contrast a typical uncoupled problem with a typical coupled problem. Consider a plane acoustic wave incident on a plane surface, where ζ is the dimen-

sionless specific impedance of the surface (Fig. 16). The impedance is given by: p Zn ζ = = ∂ p/∂n ikρc ik

k = w/c

(102)

In the above equation, p is the pressure, ∂ p/∂n is the normal derivative of the pressure, Z n is the normal impedance of the surface, ρ is the density of the medium, and c is the sound velocity in the medium. If pi is the magnitude of the incident pressure and pr is the magnitude of the reflected pressure from the surface, then the reflection coefficient R can be written as: R=

pr ζ cos ϕi − 1 = pi ζ cos ϕi + 1

(103)

Contrast this result with the coupled problem of the reflection coefficient of an incident wave on an elastic boundary (Fig. 17). In this case, the reflection coefficient, R, and the transmission coefficient, S, are R=

Z− + Zm − Z+ Z− + Zm + Z+

(104)

S=

2Z − Z+ + Zm + Z−

(104a)

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FIGURE 15 Comparison of the results of experiment and Morse’s theory for a narrow lined duct. (Reprinted by permission from Scott, R. A. (1946). Proc. Phys. Soc. 58, 358.)

where

Z m = Z p = iωδ

Z − = ρ− c− / cos ϕt

Z m = iωδ

cm c+

2

4 sin ϕi − 1 4

if the elastic surface is a plate.

Z + = ρ+ c+ / cos ϕi c− sin2 ϕi cos ϕt = 1 − c+

cp c+

sin ϕi − 1 4

if the elastic surface is a membrane

Where δ is the mass per unit area of surface, ρ− the density of the lower medium, ρ+ the density of the upper medium, c− the sound velocity in the lower medium, √ c+ the sound velocity in the upper medium, cm = T /δ the sound velocity in the √ membrane, T the tension in the membrane, cρ = c+ ω/ν+ the sound velocity in the 4 plate, ν+2 = 12δ(12 − ν¯ 2 )c+ /Eh 3 , ν¯ is Poisson’s ratio for the plate material, and h the thickness of the plate.

FIGURE 16 Plane wave incident on impedance surface. ζ = impedance of surface; ρ, c = density and sound velocity in medium; ϕi = angle of incidence.

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FIGURE 17 Plane wave incident on elastic boundary. ρ+ , c+ = density and sound velocity in the upper medium; ρ− , c− = density and sound velocity in the lower medium; ϕi = angle of incidence.

It is seen that the reflection coefficient for the coupled problem depends on both the media and the characteristics of the surface in between the media. In the uncoupled problem, the reflection coefficient depended only on the impedance of the surface. Figure 18 illustrates the reflection and transmission coefficients between 0 and 15,000 Hz for 38 -in., 12 -in., and 1-in. steel plates with water on both sides and with water on one side and air on the other. Note that, with water on both sides, most of the sound gets transmitted through the plate at low frequency (below 2000 Hz), whereas most of the sound gets reflected at 15,000 Hz. For plates with air on one side and water on the incoming wave side, the plate acts like a perfect baffle— i.e., all energy gets reflected back into the water and no transmission to the air takes place.

The equation of the elastic structure can be written in operator form as: L(w) = p(s) (109) where L is a differential operator. Equation (108) is an integral equation for the pressure. Equations (106–109) constitute the set of equations needed to solve for the scattered pressure. The integral equation (108) is solved by coupling Eqs. (106), (107), and (109) with Eq. (108), dividing the surface into many parts, and solving the resulting system of equations. An alternative method of solution is offered by the asymptotic approximations which give the scattered pressure explicitly in terms of the motion of the surface. First, write the equation of motion of the elastic structure in matrix form as follows: Ms x¨ + Cs x¨ + K s x = f int + f ext

2. Asymptotic Approximations

f ext = −G Af ( pI + psc )

In classical scattering problems, solutions to the Helmholtz equation are sought, ∇2 p + k2 p = 0

k = ω/c

(105)

that satisfy the boundary condition at the fluid structure interface: ∂ p/∂n = ρω2 w

(106)

where p is the fluid pressure, ρ is the density of the medium, and w is the normal component of the displacement of the surface of the structure. The pressure in the field can be written as p = pI = psc

(107)

where p is the total field pressure, pI is the pressure in the incident wave, and psc is the pressure in the scattered wave. For points p on the structural surface S, the scattered pressure can be written in terms of the Helmholtz relation: ∂ eikr eikr ∂ p(s) 1 psc (P) = p(s) − ds 2π s ∂n r r ∂n (108)

(110)

G T x = u I + u sc where x is the structural displacement vector; Ms , Cs , and K s are the structural mass, damping, and stiffness matrices, respectively; Af is a diagonal area matrix for the fluid-structure interface; G is a transformation matrix that relates the forces on the structure to those on the interface; and f int is the known internal force vector. The terms pI and u I are the (known) free-field pressure and fluid particle velocity associated with the incident wave, and psc and u sc are the pressure and fluid particle velocity for the scattered wave. The dots denote differentiation with respect to time. The following fluid-structure interaction equations are then used to relate the pressure and motion on the fluid-structure interface. 1. First doubly asymptotic approximation (DAA1):

Mf p˙ s + ρc Af ps = ρc M˙ f G T x¨ − u˙ I (111) 2. Second doubly asymptotic approximation (DAA2):

Mf p¨ s + ρc Af p˙ s = ρcf Af ps = ρc Mf G T˙˙˙ x + u¨ I

+ f Mf G T x¨ − u˙ I (112)

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FIGURE 18 (a) Reflection (R ) and transmission (S ) coefficients for 38 -in., 12 -in., and 1-in. steel plates with water on both sides (φi = 30◦ ). (b) Reflection (R ) and transmission (S ) coefficients for 38 -in., 12 -in., and 1-in. steel plates with water on the incident wave side and air on the other side (φi = 30◦ ). (Note that ordinates are in thousandths; thus, R ≈ 1 and S is very small.)

151

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The Mf and f are the fluid added mass and frequency matrices pertaining to the fluid-structure interface, and G T is the transpose of G. In essence, the doubly asymptotic approximations uncouple the fluid–structure interaction by giving an explicit relation between pressure and motion of the surface. An illustration of the uncoupling procedure of the asymptotic approximations can be formulated by taking the very simplest case of a plane wave. The pressure and velocity are related by (this holds for high frequency): p = ρo cu

(113)

where p is the pressure and u is the velocity. If this pressure were applied to a simple elastic plate, the resulting differential equation would be (noting that u = w): ˙ D∇ 4 w + ρh w ¨ = −ρo cw ˙

(114)

Thus, for this simple case, the entire fluid-structure interaction can be solved by one equation instead of having

to solve the Helmholtz integral equation coupled with the elastic plate equation.

IX. RANDOM LINEAR ACOUSTICS A. Linear System Equations 1. Impulse Response Consider a system with n inputs and m outputs as shown in Fig. 19. Each of the inputs and outputs is a function of time. The central problem lies in trying to determine the outputs or some function of the outputs in terms of the inputs or some function of them. Let any input x(t) be divided into a succession of impulses as shown in Fig. 20. Let h(t − τ ) be the response at time t due to a unit impulse at time τ . A unit impulse is defined as one in which the area under the input versus time curve is unity. Thus, if

FIGURE 19 Block diagram for the system. x1 (t), x2 (t) . . . xn (t) = n inputs; h1 , h2 , . . . hn = n transfer functions; y1 (t)1 t2 (t) . . . ym(t) = m outputs.

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FIGURE 20 Input divided into infinitesimal impulses. t = time; τ = the value of time at which x(τ ) is taken; τ = time width of impulse; x(τ ) = value of the impulse at time τ .

the base is τ , the height of the unit impulse is 1/ τ . Thus, h(t − τ ) is the response per unit area (or per unit impulse at t = τ ). The area (or impulse) is x(τ ) τ . The response at time t is the sum of the responses due to all the unit impulses for all time up to t, that is, from −∞ to t. But it is physically impossible for a system to respond to anything but past inputs; therefore, h(t − τ ) = 0

for τ > t

(115)

function Hi j (ω) is Hi j (ω) =

y¯ j (ω) x¯ i (ω)

(120)

For sinusoidal input and output, Eq. (117) becomes: +∞ y¯ i (ω) h i j (τ )e−iωτ dτ (121) = x¯ i (ω) −∞

Thus, the upper limit of integration can be changed to +∞. By a simple change of variable (θ = t − τ ) it can be demonstrated that: +∞ y(t) = h(θ )x(t − θ) dθ (116)

It is therefore proven that the frequency response function is the Fourier transform of the unit impulse function.

Since there are n inputs and m outputs, there has to be one of these equations for each input and output. Thus, for the ith input and jth output, +∞ yi j (t) = h i j (τ )xi (t − τ ) dτ (117)

Since the linear process is assumed to be random, the results are based on statistical operations on the process. In this section, the pertinent statistical parameters will be derived. Referring back to Eq. (117), we see that the total response y j is the sum over all inputs. Thus,

−∞

3. Statistics of the Response

−∞

y j (t) =

2. Frequency Response Function

i=1

The frequency response function or the transfer function (the system function, as it is sometimes known) is defined as the ratio of the complex output amplitude to the complex input amplitude for a steady-state sinusoidal input. (The frequency response function is the output per unit sinusoidal input at frequency ω.) Thus, the input is xi (t) = x¯ i (ω)eiωt

where x¯ i (ω) and y¯ j (ω) are the complex amplitudes of the input and output respectively. Then the frequency response

−∞

h i j (τ )xi (t − τ ) dτ

From the definition of C jk (τ ) it is seen that:

(118)

(119)

+∞

(122)

The cross correlation between outputs y j (t) and yk (t) is defined as follows: +T 1 C jk (τ ) = lim y j (t)yk (t + τ ) dt (123) T →∞ 2T −T

Ck j (τ ) = yk (t)y j (t + τ )

and the corresponding output is y j = y¯ j (ω)eiωt

n

where: 1 T →∞ 2T

( ) = lim

+T

( ) dt −T

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Substituting Eq. (122) and rearranging, +∞ m n +∞ C jk (τ ) = du dv h js (u)h kr (υ) s=1 r =1

×

−∞

−∞

1 lim Z →∞ 2T

+T −T

xs (t − u)xr (t − υ + τ ) dt (124)

By definition of the cross correlation, +T 1 lim xs (t − u)xr (t − υ + τ ) dt T →∞ 2T −T = Cr s (u − υ + τ ) m n

(125)

+∞

−∞

s=1 r =1

+∞

du

dυ −∞

× [h js (u)h kr (υ)Cr s (u − υ + τ )]

(126)

The cross spectrum G jk (ω) is defined as the Fourier transform of the cross correlation. The inverse Fourier transform relation is, then, +∞ 1 Ck j (τ ) = G jk (ω)eiωτ dω 2π −∞ Thus,

G jk (ω) =

Note that: G k j (ω) =

+∞ −∞

=

+∞

−∞

+∞

−∞

Ck j (τ )e−iωτ dτ

Ck j (τ )eiωτ dτ =

+∞ −∞

4. Important Quantities Derivable from the Cross Spectrum C jk (−τ )e−iωτ dτ

C jk (θ )eiωθ dθ = G ∗jk (ω)

n n

∗ H js (ω)Hkr (ω)G r s (ω)

(128)

s=1 r =1 ∗ H js

in which is the complex conjugate of H js . Equation (128) gives the cross spectrum of the outputs G jk (ω) in terms of the cross spectrum of the inputs G r s (ω). In matrix notation, Eq. (128) can be written as: ∗

G (ω) = H G H o

i

T

(131) G i is also Hermitian of order r × r = s × s = r × s. The transfer function matrix is a k × r complex matrix (not Hermitian): H11 (ω) H12 (ω) · · · H1r (ω) H21 (ω) H (ω) = . (132) .. ··· Hkr (ω) Hk1 (ω)

(127)

where G ∗jk is the complex conjugate of G jk . Substituting Eq. (126), changing variables θ = u − υ + τ , and using definition (127) and relation (121), G jk (ω) =

(130) By virtue of the fact that the above matrix is a square Hermitian matrix. The input matrix G i is i G 11 (ω) G i12 (ω) G i13 (ω) · · · G i15 (ω) i G 21 (ω) i G (ω) = . .. i i ··· G rr (ω) G r 1 (ω) G i j (ω) = G ∗ji (ω),

Thus, C jk (τ ) =

fer functions. H T denotes the transpose matrix of H , and H ∗ is the complex conjugate matrix of H . Thus, o G 11 (ω) G o12 (ω) G o13 (ω) · · · G o1k (ω) G o (ω) 21 G o (ω) = .. . G o (ω) ··· G oj j (ω) j1

(129)

where G o is the output matrix of cross spectra, G i is the input matrix of cross spectra, and H is the matrix of trans-

The cross spectrum can be used as a starting point to derive several important quantities. The spectrum of the response at point j is obtained by letting k = j in Eq. (128). The autocorrelation is obtained by letting k = j in Eq. (123). The mean-square response is further obtained from the autocorrelation by letting τ = 0. If the Fourier inverse of Eq. (127) is used to determine mean square, then: +∞ 1 C j j (0) = G j j (ω) dω 2π −∞ = meam square = M 2j +T 1 = lim y 2j (t) dt T →∞ 2T −T

(133)

If the mean square is desired over a frequency band

= 2 − 1 , then it is given by: 2 2

1 M j = G j j (ω) dω (134) 2π 2

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The mean value of y j (t) is defined as: +T 1 ¯ M j = lim y j (t) dt T →∞ 2T −T

(135)

The variance σ j2 is defined as the mean square value about the mean: +T 1 σ j2 = lim [y j (t) − M¯ j ]2 dt (136) T →∞ 2T −T The square root of the variance is known as the standard deviation. By using Eqs. (133) and (135), Eq. (136) can be written: σ j2 = M 2j − ( M¯ j )2

(137)

The mean, the variance, and the standard deviation are three important parameters involved in probability distributions. Note that if the process is one with zero mean value, then the variance is equal to the mean square, and the standard deviation is the root mean square. The above quantities are associated with the ordinary spectrum rather than the cross spectrum. An important physical quantity associated with the cross spectrum is the coherence, which is defined as: 2 γ jk (ω) =

|G jk (ω)|2 G j j (ω)G kk (ω)

(138)

2 The lower limit of γ jk must be zero since the lower limit of G jk (ω) is zero. This corresponds to no correlation between 2 the signals at j and k. In addition, γ jk ≤ 1. Going back to Eq. (128), we see that if there is only one input, then: ∗ G jk (ω) = H js Hkr G rr

(139)

Thus, 2 γ jk =

∗ 2 Hkr H js Hkr∗ G rr H js ∗ H js H js G rr Hkr∗ Hkr G rr

=1

1 T →∞ 2T

C jk (τ ) = lim

(140)

So the field is completely coherent for a single input to the system. In an acoustic field, sound emanating from a single source is coherent. If the coherence is less than unity, then the field is partially coherent. The partial coherence effect is sometimes due to the fact that the source is of finite extent. It is also sometimes due to the fact that there are several sources causing the radiation and these sources are correlated in some way with each other. 5. The Cross Spectrum in Terms of Fourier Transforms The cross spectrum can also be expressed in terms of Fourier transforms alone. To see this, start with the basic definition of cross spectrum as given by Eq. (127), where:

Thus, G jk (ω) =

+∞

+T −T

y j (t)yk (t + τ ) dt

(141)

1 T →∞ 2T lim

−∞

×

+T

−T

y j (t)yk (t + τ ) dt e−iωτ dτ

(142)

Letting t = u and t + τ = υ, we have: +T +∞ 1 G jk = lim y j (u)yk (υ) du e−iω(υ−u) dυ T →∞ 2T −T −∞ =

+∞

1 T →∞ 2T

+T

lim

−∞

−T

(143)

y j (u)eiωu du yk (υ)e−iωυ dυ (144)

The next step is true only under the condition that the process is ergodic. In this case, the last equation can be written as: +T 1 G jk (ω) = lim y j (u)eiωu du T →∞ 2T −T ×

+T

−T

yk (υ)e−iωυ dυ

This last relation can then be written: y¯ ∗j (T, ω) y¯ k (T, ω) G jk (ω) = lim T →∞ 2T where: +T y¯ j (T, ω) = y j (t)eiωt dt

(145)

(146)

(147)

−T

and y ∗j is the complex conjugate of y j . +T y¯ k (T, ω) = yk (t)e−iωt dt

(148)

−T

Equation (146) expresses the cross spectrum in terms of the limit of the product of truncated Fourier transforms. 6. The Conceptual Meaning of Cross Correlation, Cross Spectrum, and Coherence Given two functions of time x(t) and y(t), the cross correlation between these two functions is defined mathematically by the formula: +T 1 C(x, y, τ ) = lim x(t)y(t + τ ) dt (149) T →∞ 2T −T

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This formula states that we take x at any time t, multiply it by y at a time t + τ (i.e., at a time τ later than t), and sum the product over all values (− T < t < + T ). The result is then divided by 2T . In real systems, naturally, T is finite, and the meaning of ∞ in the formula is that various values of T must be tried to make sure that the same answer results independent of T . For two arbitrary functions of time, formula (149) has no real meaning. It is only when the two signals have something to do with each other that the cross correlation tells us something. To see this point clearly, consider an arbitrary random wave train moving in space. (It could be an acoustic wave, an elastic wave, an electromagnetic wave, etc.) Let x(t) = x1 (t) be the response (pressure, stress, etc.) at one point, and y(t) = x2 (t) be the response at another point. Now form the cross correlation between x1 and x2 (the limit is eliminated, it being understood): +T 1 C(x1 , x2 , τ ) = x1 (t)x2 (t + τ ) dt (150) 2T −T When the points coincide (i.e., x1 = x2 ), the relation becomes: +T 1 C(x1 , τ ) = x1 (t)x1 (t + τ ) dt (151) 2T −T and if τ = 0, 1 C(x1 , 0) = 2T

+T −T

x12 (t) dt

(152)

2, and τ is the time of travel from 1 to 2. Forming the cross correlation C(x1 , x2 , τ ) gives +T 1 C(x1 , x2 , τ ) = x1 (t)Ax1 (t + τ − τ1 ) dt 2T −T = AC(x1 , x1 , τ − τ1 )

(154)

Thus, the cross correlation of a random wave train in a nondispersive system is exactly the same form as the autocorrelation of the wave at the starting point, except that the peak occurs at a time delay corresponding to the time necessary for the wave to travel between the points. In the absence of attenuation, the wave is transmitted undisturbed in the medium. However, in most cases it is probable that the peak is attenuated somewhat as the wave progresses. It is thus seen that cross correlation is an extremely useful concept for measuring the time delay of a propagating random signal. In the above case, it had to be assumed that the signal was propagating in a nondispersive medium and that when the cross correlation was done, the signal was actually being traced as it moved through the system. Consider the meaning of cross correlation if the system was dispersive (i.e., if the velocity was a function of frequency). White has addressed himself to this question and has demonstrated that time delays in the cross correlation can still be measured with confidence if the signal that is traveling is band-limited noise. For dispersive systems where the velocity is a function of frequency, it has been pointed out in the literature that time delays can also be obtained. For this case, the following cross spectrum is formed: +∞ S12 (ω) = C12 (τ )e−iωτ dτ (155)

which is, by definition, the mean square value of the response at point x1 . For other values of τ , Eq. (151) defines the autocorrelation at point 1. It is the mean value between the response at one time and the response at another time τ later than t. Thus, Eq. (150) is the mean product between the response at point 1 and the response at point 2 at a time τ later. Now, going back to the random wave train, let us assume that it is traveling in a nondispersive medium (i.e., with velocity independent of frequency). It is seen that if the wave train leaves point 1 at time t (see Fig. 21) and travels through the system with no distortion, then:

The phase angle θ12 (ω) is actually the phase between input and output at frequency ω. The time delay from input to output is then:

y(t) = x2 (t) = Ax1 (t − τ1 )

τ (ω) = θ12 (ω)/ω

(153)

where A is some decay constant giving the amount that the wave has decreased in amplitude from point 1 to point

−∞

The cross spectrum is a complex number and can be written in terms of amplitude and phase angle θ12 (ω) as follows: S12 (ω) = |S12 (ω)|e−iθ12 (ω)

(156)

(157)

Suppose that the signal has lost its propagating properties in that it has reflected many times and set up

FIGURE 21 Input and output in a linear system. x(t) = input; y (t ) + output.

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a reverberant field in the system. Consider the physical meaning of cross correlation in this case. To answer this question partially, examine an optical field. In optical systems, extensive use has been made of the concept of partial coherence. At the beginning of this section, two functions x(t) and y(t) were chosen, and the cross correlation between them was formed. It was pointed out that if the two functions are completely arbitrary, then there is no real physical meaning to the cross correlation. However, if the two functions are descriptions of response at points of a field, then there is a common ground to interpret cross correlation. Thus, the cross correlation and any function that is derived therefrom give some measure of the dependence of the vibrations at one point on the vibrations at the other point. This is a general statement, and to tie it down, the concept coherence has been used. In the optical case, suppose light is coming into two points in a field. If the light comes from a small, single source of narrow spectral range (i.e., almost single frequency), then the light at the two field points is dependent. If each point in the field receives light from a different source, then the light at the two field points is independent. The first case is termed coherent, and the field at the points is highly correlated (or dependent). The second case is termed incoherent, and the field between the two points is uncorrelated (independent). These are the two extreme cases, and between them there are degrees of coherence (i.e., partial coherence). Just as in everyday usage, coherence is analogous to clarity or sharpness of the field, whereas incoherence is tantamount to haziness or “jumbledness.” The same idea is used when speaking about someone’s speech or written article. If it is concise and presented clearly, it is coherent. If the ideas and presentation are jumbled, they can be called incoherent. Single-frequency radiation is coherent radiation; radiation with a finite bandwidth is not. The partial coherence associated with finite spectral bandwidth is called the temporal (or timewise) coherence. On the other hand, light or sound emanating from a single source gives coherent radiation, but a point source is never actually obtained. The partial coherence effect, due to the fact that the source is of finite extent, is termed space coherence. The point source gives unit coherence in a system, whereas an extended source gives coherence somewhat less than unity. The square of coherence γ12 (ω) between signals at points 1 and 2 at frequency ω is defined as:

2 γ12 (ω) =

|S12 (ω)|2 S11 (ω)S22 (ω)

(158)

where:

S12 (ω) =

+∞ −∞

C12 (τ )e−iωτ dτ

(159)

in which C12 (τ ) is the cross correlation between signals at points 1 and 2. The function S12 (ω) is the cross spectrum between signals at points 1 and 2, and S11 (ω) and S22 (ω) are the autospectra of the signals at points 1 and 2, respectively. Wolf has other ways of defining coherence by functions called complex degree of coherence or mutual coherence function, but it all amounts conceptually to the same cross spectrum as given by Eq. (158). Although there are formal proofs that γ12 (ω) is always between 0 and 1, one can reason this out nonrigorously by going back to the basic physical ideas associated with correlation and coherence. If the signals at two points are uncorrelated, and therefore incoherent, the cross correlation is zero, thus γ12 (ω) is zero. If the signals are perfectly correlated, then this is tantamount to saying that the signals in the field are a result of input to the system from a single source, as shown in Fig. 22. As seen before, the cross spectrum Syz (ω) can be written in terms of the input spectrum Sx (ω) and the transfer functions Yz (iω) and Y y (iω) as follows: Syz (iω) = Y y (iω)Yz∗ (iω)Sx (ω)

(160)

Thus, 2 γ yz (ω) =

=

|Syz (ω)|2 Syy (ω)Szz (ω)

(161)

Y y (iω)Yz∗ (iω)Y y∗ (iω)Yz (iω)Sx2 (ω) |Y y (iω)|2 Sx (ω)|Yz (iω)|2 Sx (ω)

= 1 (162)

So that: 0 ≤ γ12 (ω) ≤ 1

(complete coherence)

(163)

Between the cases of complete coherence and complete incoherence there are many degrees of partial coherence. B. Statistical Acoustics 1. Physical Concept of Transfer Function In Section IX.A.2 it was shown that Hi j was the transfer function that gave the output at j per unit sinusoidal input at i. Suppose there is an acoustic field which is generated by a group of sound sources and these sources are surrounded by an imaginery surface S o as shown in Fig. 23. Through each element of So , sound passes into the field. Thus, each element of So , denoted by ds, can be considered a source that radiates sound into the field. Consider the

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FIGURE 22 Single input or coherent system. x(t) = input; S x (ω) = spectrum of input; Yz (i ω) = transfer function for z output; Yy (i ω) = transfer function for y output; z (t), y (t) = outputs.

pressure dp(P, ω) at field point P at frequency ω due to radiation out of element ds, dp(P, ω) = H p (P, S, ω) p(S, ω) ds

(164)

where H p (P, S, ω) is the pressure at field point P per unit area of S due to a unit sinusoidal input pressure on S. The total pressure in the field at point P is p(P, ω) = H p (P, S, ω) p(S, ω) ds (165) So

If motion (e.g., acceleration) of the surface S is considered instead of pressure, the counterpart to Eq. (165) is p(P, ω) = Ha (P, S, ω)a(S, ω) ds (166) So

where Ha (P, S, ω) is the transfer function associated with acceleration; that is, it is the pressure at field point P per unit area of S due to a unit sinusoidal input acceleration of S. Applying these ideas to Eq. (128), it is seen that the cross spectrum of the field pressure can immediately be written

in terms of the cross spectrum of the surface pressure or the cross spectrum of the surface acceleration. For surface pressure, Eq. (128) becomes: G(P, Q, ω) = H p∗ (P, Si , ω)H p (Q, Sr , ω) Si

Sr

× G(Si , Sr , ω) dsi dsr

(167)

Comparing Eq. (167) with Eq. (128) we find that the points j, k become field points P, Q. G(P, Q, ω) is the cross spectrum of pressure at field points P, Q. The trans∗ fer functions H js (ω) become H p∗ (P, Si , ω) in which Si is the surface point or ith input point. Hkr (ω) becomes H p (Q, Sr , ω) where Sr is the other surface point or r th input point. G(Si , Sr , ω) is the input cross spectrum, which is the cross spectrum of surface pressure. The summations over r and s become integrals over Si and Sr . For acceleration, G(P, Q, ω) = Ha∗ (P, Si , ω)Ha (Q, Sr , ω) Si

Sr

× A(Si , Sr , ω) dsi dsr

(168)

The transfer functions are those for acceleration, and A(Si , Sr , ω) is the cross spectrum of the surface acceleration. The relation can be written for any other surface input such as velocity. 2. Response in Terms of Green’s Functions The Green’s functions for single-frequency systems were taken up in a previous section. The transfer function for pressure is associated with the Green’s function that vanishes over the surface. Thus, FIGURE 23 Surface surrounding the sources. S o = surrounding surface; Vo = volume outside the surface; P, Q = field points.

H p (P, S, ω) =

∂g1 (P, S, ω) ∂n

(169)

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FIGURE 24 Surface surrounding main sources with presence of other field sources. S o = surrounding surface; Qk , Q j = strengths of other sources not within S o .

The transfer function for acceleration is associated with the Green’s function whose normal derivative vanishes over So . Thus, Ha (P, S, ω) = ρg2 (P, S, ω)

(169a)

The statistical relations for the field pressure can therefore immediately be written in terms of the cross spectrum of pressure or acceleration over the surface surrounding the sources, ∂g1∗ (P, Si , ω) ∂g1 (Q, Sr , ω) G(P, Q, ω) = ∂n i ∂n r Si Sr × G(Si , Sr , ω) dsi dsr or

G(P, Q, ω) = Si

Sr

(170)

ρ 2 g2∗ (P, Si , ω)g2 (Q, Sr , ω)

× A(Si , Sr , ω) dsi dsr

(171)

These relations give the cross spectrum of the field pressure as a function of either the cross spectrum of the surface pressure G(Si , Sr , ω) or the cross spectrum of the surface acceleration A(Si , Sr , ω). Equation (170) was derived by Parrent using a different approach. At this point, one should review the relationship between Eqs. (170) and (171) for acoustic systems and (128) for general linear systems. It is evident that the inputs in Eqs. (170) and (171) are G(Si , Sr , ω), A(Si , Sr , ω), respectively, and the output is G(P, Q, ω) in both cases. The frequency response functions are the transfer functions described by Eqs. (169) and (169a). 3. Statistical Differential Equations Governing the Sound Field Consider the general case where there are source terms present in the field equation. This is tantamount to saying that outside the series of main radiating sources which

have been surrounded by a surface (see Fig. 24) there are other sources arbitrarily located in the field. For example, in the case of turbulence surrounding a moving structure, the turbulent volume constitutes such a source, whereas the surface of the structure surrounds all the other vibrating sources. The equation governing the propagation of sound waves in the medium is ∇ 2 p(P, t) −

1 ∂ 2 p(P, t) = V (Q, t) ∂t 2 c02

where: V (Q, t) =

Vi (Q i , t)

(172)

(173)

i

In the above equation, V is a general source term that may consist of a series of sources at various points in the medium. Actually, the medium being considered is bounded internally by So so that the sources inside So are not in the medium. The sources at Q i , however, are in the medium. The various types of source terms that can enter acoustical fields arise from the injection of mass, momentum, heat energy, or vorticity into the field. These are discussed by Morse and Ingard and will not be treated here. It is assumed that the source term V (Q, t) is a known function of space and time, or if it is a random function, then some statistical information is known about it such as its cross correlation or cross spectrum. In cases where the field is random, a statistical description has to be used. The cross correlation function (P1 , P2 , τ ) between pressures at field points P1 and P2 are defined as: +T 1 p(P1 , t) p(P2 , t + τ ) dt (P1 , P2 , τ ) = lim T →∞ 2T −T (174) (To the author’s knowledge, one of the first pieces of work on correlation in wavefields was the paper of Marsh.)

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The Fourier transform, U (P, ω) of the pressure p(P, t) is

U (P, ω) =

+∞

p(P, t)e−iωt dt

where ∇22 stands for operations performed in the P2 (x2 , y2 , z 2 ) coordinates. Also, ∂ 2 (P1 , P2 , τ ) 1 = lim T →∞ 2T ∂τ 2

(175)

−∞

Taking the inverse, we can write the above equation +∞ 1 ∇2 p = ∇ 2 U (P, ω)eiωt dω (176) 2π −∞ Also from the Fourier transform of V (Q, t): +∞ W (Q, ω) = V (Q, t)e−iωt dt (177)

×

−T

1 × 2π

−∞

+T

1 2π

−∞

+∞

−∞

+∞

U (P1 , ω)e−iωt dω

(−ω )U (P2 , ω)e 2

−iω(t+τ )

dω dt

and its inverse:

+∞ 1 V (Q, t) = W (Q, ω)eiωt dω (178) 2π −∞ Substitution into the original nonhomogeneous wave equation (172) gives: +∞ 1 ω2 2 ∇ U + 2 U − W dω = 0 (179) 2π −∞ c0 Thus, for this relation to hold for all P and all ω, there must be ∇ 2 U (P, ω) + k 2 U (P, ω) = W (Q, ω)

−∞

(P1 , P2 , τ ) =

1 2π

+∞

−∞

G(P1 , P2 , ω)e−iωτ dω

Thus, ∇22 (P1 , P2 , τ ) −

T →∞

×

1 2T +T

−T

1 × 2π

∇12 (P2 , P1 , τ ) −

1 ∂ 2 (P2 , P1 , τ ) ∂τ 2 c02

= p(P2 , t)V (Q 1 , t + τ )

(187)

This set of Eqs. (186) and (187) is an extension of the equation obtained by Eckart and Wolf. If the source term were zero, then:

(182)

∇22 (P1 , P2 , τ ) −

1 ∂ 2 (P1 , P2 , τ ) =0 ∂τ 2 c02

∇12 (P2 , P1 , τ ) −

1 ∂ 2 (P2 , P1 , τ ) =0 ∂τ 2 c02

(188)

However, since:

1 2π +∞

−∞

+∞

−∞

U (P1 , ω)e−iωt dω

(P2 , P1 , τ ) = (P1 , P2 , −τ )

(189)

and

∂2 ∂2 = ∂τ 2 ∂(−τ )2

U (P2 , ω)e−iω(t+τ ) dω dt (183)

So

(186)

It should be clear from an analysis similar to that given above that the following relation also holds:

Thus, (P1 , P2 , τ ) can be written as: (P1 , P2 , τ ) = lim

1 ∂ 2 (P1 , P2 , τ ) ∂τ 2 c02

= p(P1 , t)V (Q 2 , t + τ )

(180)

The cross spectrum G(P1 , P2 , ω) between the pressures at P1 and P2 at frequency ω is defined in terms of the cross correlation (P1 , P2 , τ ), by: +∞ G(P1 , P2 , ω) = (P1 , P2 , τ )eiωτ dτ (181) and the inverse is

(185)

1 T →∞ 2T +T +∞ 1 × U (P1 , ω)e−iωt dω 2π −∞ −T +∞ 1 × ∇22U (P2 , ω)e−iω(t+τ ) dω dt 2π −∞

∇22 (P1 , P2 , τ ) = lim

(184)

(190)

Then, 2 ∇1,2 (P1 , P2 , τ ) −

1 ∂ 2 (P1 , P2 , τ ) =0 ∂τ 2 c02

(191)

From the above relations, it is seen that the cross correlation is propagated in the same way that the original pressure wave propagates, except that real time t is replaced by correlation time τ . The nonhomogeneous counterparts given by Eqs. (186) and (187) state that the source term takes the statistical

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FIGURE 25 The loaded structure. f (r o , t) = force component at location ro and time t; w (r, t) = deflection component at r at time t.

form of the cross correlation between the pressure p at a reference point and the source function V . Taking the Fourier transform of Eqs. (186) and (187), we see that the cross spectrum satisfies: ∇22 G(P1 , P2 , ω) + k 2 G(P1 , P2 , ω) = !2 (P1 , Q 2 , ω) ∇12 G(P2 , P1 , ω) + k 2 G(P2 , P1 , ω) = !1 (P2 , Q 1 , ω) (192) where !1 and !2 are the Fourier transforms of the cross correlation between the reference pressure and source function, that is +∞ !2 (P1 , Q 2 , ω) = p(P1 , t)V (Q 2 , t + τ )e−iωτ dτ −∞

!1 (P2 , Q 1 , ω) =

+∞ −∞

p(P2 , t)V (Q 1 , t + τ )e−iωτ dτ (193)

Thus, !1 and !2 are cross-spectrum functions between the pressure and source term. In Eqs. (186), (187), (192), and (193), it is important to note that one point is being used as a reference point and the other is the actual variable. For example, in Eq. (187), the varying is being done in the P1 (x1 , y1 , z 1 ) coordinates; thus, all cross correlations are performed with P2 fixed. Conversely, in Eq. (186) all the operations are being carried out in the P2 space with P1 remaining fixed.

C. Statistics of Structures 1. Integral Relation for the Response Let the loading (per unit area) on the structure be represented by the function f (r0 , t) where r0 is the position vector of a loaded point on the body with respect to a fixed system of axes, as shown in Fig. 25. Let the unit impulse response be h(r, r0 , t − θ); this is the output at r corresponding to a unit impulse at t = 0 and at location r0 . The response at r at time t due to an arbitrary distributed excitation f (r0 , t) can then be written: t w(r, t) = dr0 f (r0 , θ )h(r, r0 , t − θ) dθ (194) r0

−∞

The integration is taken over the whole loaded surface denoted by r0 . Since the loading is usually random in nature, only the statistics of the response (that is, the mean square values, the power spectral density, and so on) are determinable. Thus, let U = t − θ and form the cross correlation of the response at two points r1 and r2 . This cross correlation is denoted by Rw and is +∞ +T 1 Rw (r1 , r2 , τ ) = lim dr0 T →∞ 2T −T −∞ r0 × f (r0 , t − U1 )h(r1 , r0 , U1 ) dU1 × r0

dr0

+∞

−∞

f (r0 , t − U2 + τ )

× h(r2 , r0 , U2 ) dU2 dt

(195)

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or

Rw (r1 , r2 , τ ) =

dr0 dr0

r0

r0

+∞

−∞

S" (r0 , r0 , ω) =

+∞

−∞

× h(r1 , r0 , U1 )h(r2 , r0 , U2 ) dU1

×

1 T →∞ 2T

+T

lim

−T

dU2

f (r0 , t − U1 )

× f (r0 , t − U2 + τ ) dt

Rw (r1 , r2 , τ ) =

r0

r0

dr0 dr0

−∞

+∞

−∞

× R" (r0 , r0 , τ3 ) dU1 dU2

where R" (r0 , r0 , τ3 ) is the cross-correlation function of the loading. Now form the cross spectrum of the response: +∞ Sw (r1 , r2 , ω) = Rw (r1 , r2 , τ3 )e−iωτ dτ e Thus,

=e

Sw (r1 , r2 , ω) =

r0 r0

×

+∞ −∞

×

+∞

−∞

h(r1 , r0 , U1)eiωU1 dU1

R" (r0 , r0 , τ3 )e−iωτ3 dτ3

−∞

−∞

r0

(201) h(r2 , r0 , U2 )e−iωU2

dU2 =

G(r2 , r0 , ω)

where G ∗ denotes the complex conjugate of G. The Green’s function G(r2 , r0 , ω) is the response at r2 due to a unit sinusoidal load at r0 , and G ∗ (r1 , r0 , ω) is the complex conjugate of the response at r1 due to a unit sinusoidal load at r0 . The bracket can be written:

(205)

r0

× G(r2 , r0 , ω)S(ξ, ω)

(206)

White has shown that by applying Parseval’s theorem and letting r1 = r2 , the above equation can be written: ˜ ω)ψ(k, ω) dk (207) Sw (r1 , r1 , ω) = (2π )2 S(k, k

+∞

−∞

but the Fourier transform of the impulse function is the Green’s function. Thus, +∞ h(r1 , r0 , U1 )eiωU1 dU1 = G ∗ (r1 , r0 , ω) +∞

(204)

Let r0 − r0 = ξ . Equation (203) now becomes: Sw (r1 , r2 , ω) = dr0 dr0 G(r1 , r0 , ω)

1 ψ(k, ω) = (2π )2

h(r2 , r0 , U2 )e−iω U2 dU2

(200)

S" (r0 , r0 , ω) = S(r0 − r0 , ω)

where:

+∞ −∞

r0

× G(r1 , r0 , ω)S" (r0 , r0 , ω)

(199)

iω(τ3 −U1 +U2 )

dr0 dr0

The spectrum at any point r1 is obtained by setting r1 = r2 ; thus, Sw (r1 , ω) = dr0 dr0 G ∗ (r1 , r0 , ω)

(198)

−∞

(203)

Note the equivalence between Eq. (203) and the general Eq. (128) for linear systems. In many practical cases, especially in turbulence excitation, the cross spectrum of the loading takes a homogeneous form as follows:

× h(r1 , r0 , U1 )(r2 , r0 , U2 )

−iωτ

r0

× G(r2 , r0 , ω)S" (r0 , r0 , ω)

(197) +∞

(202)

where S" is the cross spectrum of the load. Thus, the expression for the cross spectrum of the response becomes: Sw (r1 , r2 , ω) = dr0 dr0 G ∗ (r1 , r0 , ω)

r0

= U1 − U2 + τ

R" (r0 , r0 , τ3 )e−iωτ3 dτ3

−∞

(196)

τ3 = (t − U2 + τ ) − (t − U1 )

+∞

r0

We assume a stationary process so that the loading is only a function of the difference of the times t − U2 + τ and t − U1 . Let

then,

G(r1 , r, ω)e

ik(r−r1 )

2 dr

(208)

˜ ω) is the spectrum of the In the above equations, S(k, excitation field in wave number space, and ψ(k, ω) is the square of the Fourier transform of the Green’s function, which can be obtained very quickly on a computer by application of the fast Fourier transform technique. The Green’s functions take on the true spatial character of an influence function. They represent the response at one point due to a unit sinuosidal load at another point. The inputs are loads, the outputs are deflections, and the linear black boxes are pieces of the structure as used in the first section of this chapter. A few very interesting results can immediately be written from Eq. (204). Supposing a body is loaded by a single random force at point p, the loading "(r, t) can be written: "(r, t) = P(t)δ(r − r p )

(209)

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The δ function signifies that " is 0 except when r = r p . Thus, S" (r0 , r0 , ω) = S p (ω)δ(r0 − rp )δ(r0 − r p )

(210)

The spectrum of the response is, therefore, Sw (r1 , ω) = dr0 dr0 G ∗ (r1 , r0 , ω) r0

2. Computation of the Response in Terms of Modes The general variational equation of motion for any elastic structure can be written as: ¨ + v¨ δv + wδw) [ρ(uδu ¨ + δW ] d V V

r0

× G(r1 , r0 , ω)S p (ω)δ(r0 − r p )δ(r0 − r p ) (211) G ∗ (r1 , r0 , ω)

= S p (ω) r0

× δ(r0 − r p ) dr0

r0

G(r1 , r0 , ω)

× δ(r0 − r p ) dr0

(212)

= S p (ω)|G(r1 , r p , ω)|

2

(213)

The spectrum of the response is the square absolute value of the Green’s function multiplied by the spectrum of the force. The Green’s function in this case is the response at r1 due to unit sinusoidal force of frequency ω at r p ( p being the loading point). Suppose there is a group of independent forces on the structure. The cross correlation between them is 0, so: S" (r0 , r0 , ω) = S(r0 , ω)δ(r0 − r0 )

(214)

−

(X ν δu + Yν δυ + Z ν δw) d S = 0 S

where ρ is mass density of body; u, v, w are displacements at any point; δu, δv, δw are variations of the displacements; X ν , Yν , Z ν are surface forces; ds is the elemental surface area; d V is the elemental volume; and δW is the variation of potential energy. In accordance with Love’s analysis, let the displacements in the normal modes be described by: u = u r ϕr ,

r

r0

r0

× S(r0 , ω)δ(r0 − = =

r0

r0

r0 ) dr0

(217)

r

where u r , vr , wr are the mode shapes, and ϕr is a function of time. In accordance with Love, let u = u r ϕr

δu = u s ϕs

v = vr ϕr

δv = vs ϕs

w = wr ϕr

δw = ws ϕs

(219)

Substituting into the variational equation of motion, we obtain the following: ρ(u r ϕ¨ r u s ϕs + vr ϕ¨ r vs ϕs + wr ϕ¨ r ws ϕs ) d V

dr0

V

+

G(r1 , r0 , ω)G ∗ (r1 , r0 , ω)S(r0 , ω)] dr0 |G(r1 , r0 , ω)|2 S(r0 , ω) dr0

w = wr ϕr

r

r0

× S(r0 , ω)δ(r0 − r0 ) dr0 dr0 = G(r1 , r0 , ω) G ∗ (r1 , r0 , ω)

v = vr ϕr ,

where ϕr = Ar cos pr t, pr being the natural frequency of the r th mode. Now let the forced motion of the system be described by: u= u r ϕr , v = vr ϕr , w = wr ϕr (218)

That is, S" = 0 except when r0 = r0 , so: Sw (r1 , ω) = G ∗ (r1 , r0 , ω)G(r1 , r0 , ω) r0

(216)

δW d V =

V

(X ν u s ϕs + Yν vs ϕs S

+ Z ν ws ϕs ) d S

(220)

(215)

If there are n forces, each with spectrum S(rn , ω), Sw (r1 , ω) = |G(r1 , rn , ω)|2 S(rn , ω) (215a) n

The response is just the sum of the spectra for each force acting separately.

However, since the modal functions satisfy the equation for free vibration: δW d V = ρ pr2 u r ϕr u s ϕs V

V

+ pr2 vr ϕr vs ϕs + pr2 wr ϕr ws ϕs d V (221)

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and Love shows that: ρ(u r u s + vr vs + wr ws ) d V = 0

r = s

(222)

and the Fourier transform of ϕr is +∞ Sϕr (ω) = ϕr (t)e−iωr dt

(231)

−∞

V

Now, the final equation of motion becomes: ϕ¨ r (t) + = Fr (t) (223) where Mr = V ρ(u r2 + vr2 + wr2 ) d V (the generalized mass for the r th mode): 1 Fr (t) = [X ν (t)u r + Yν (t)vr + Z ν (t)wr ] d S Mr pr2 ϕ(t)

ϕ˙ r (t) =

ϕ¨ r + ψr ϕ¨ r + pr2 ϕr = Fr

(226)

u r2

+

vr2

+

wr2

−∞

−ω2 Sϕr (ω)eiωt dω Sϕ¨ r (ω) = −ω2 Sϕr (ω)

SF (rs , ω) · qr (rs ) ds

(227)

where i, j, k are the unit vectors in the x, y, z directions, respectively. Let F(s, t) = X ν i + Yνj + Z ν k

(228)

Thus, ρqr · qr d V,

qr = qr (V )

(229) F · qr ds,

F = F(S, t)

The Fourier transform of F(S, t) is

−∞

F(S, t)e−iωr dt

(234)

(235)

(236)

Sϕr (ω) =

SF (rs , ω) · qr (rs ) ds

Mr pr2 − ω2 + iωβr s

In dealing with statistical averaging, the cross correlation function is used. The cross correlation between the displacement at two points in any direction (the direction can be different at the two points) is +T 1 (r , r , τ ) = lim q(r1 , t)q(r2 , t + τ ) dt q 1 2 T →∞ 2T −T (237) We are picking a given direction at each point, so the two quantities are scalar (no longer vector). Then, q= qr ϕr r

S

+∞

(233)

S

So,

+∞

(232)

dV

qr = u r i + vr j + wr k

SF (S, ω) =

−ω2 Sϕr (ω) + iωβr Sϕr (ω) + pr2 Sϕr (ω) = S Fr (ω) 1 S Fr (ω) = Mr

It is convenient to employ the vector notation; thus, let the displacement functions in the r th mode be written as:

1 Fr (t) = Mr

−∞

iωSϕr (ω) eiωt dω

which is

where:

V

V

+∞

Let βr be the damping constant for the r th mode. Now take the Fourier transform of the equation of motion:

where:

Mr =

Sϕr (ω)eiωt dω

Sϕ¨ r + βr Sϕ˙ r + pr2 Sϕr = S Fr

where κ is the damping force per unit volume per unit velocity. Finally, the equation of motion becomes:

1 2π

Sϕ˙ r (ω) = iωSϕr (ω),

V

κ ψr = Mr

+∞

−∞

1 ϕ¨ r (t) = 2π

S

(224) If structural damping is taken into account, it can be written as another generalized force that opposes the motion:

2 (Fr )damping = −κ ϕ¨ r u r + vr2 + wr2 d V (225)

1 ϕr (t) = 2π

(230)

q(r2 , t + τ ) =

1 2π

q(r1 , t) =

1 2π

+∞

−∞

+∞

−∞

Sq (r2 , ω)eiω(t+τ ) dω Sq (r1 , ω)eiωt dω

(238)

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So,

q (r1 , r2 , τ )

+T 1 q(r1 , t) T →∞ 2π −T +T 1 iω(t+τ ) × Sq (r2 , ω)e dω dt 2π −T +T 1 = lim Sq (r2 , ω)eiωτ T →∞ 2T −T +T 1 × q(r1 , t)eiωt dt dω 2π −T +T 1 = lim SqT (r2 , ω)Sq∗ (r1 , ω)eiωτ dω T →∞ 2T −T

= lim

Note the tensor properties of the last expression involving each component of loading. Note that in the general formula involving the dot product, the component of the modal vector in the direction of the loading function at the two points has to be taken. Now, assuming that the loading is normal to the surface of the structure, our concern is with the cross spectral density of the normal acceleration at two points (or cross spectral density between normal acceleration at two points r1s and r2s ) on the surface: [qr (r1s )qk (r2s )]n an (r1s , r2s , ω) = ω4 Yr∗ (iω)Yk (iω) r q

× G p r Su , r Sv , ω Su

× qr r Su qk r Sv dsu dsv

(239) Now, the power spectral density of the displacement is defined in terms of the cross correlation as: +∞ 1 G q (r1 , r2 , ω)eiωτ dω (240) q (r1 , r2 , τ ) = 2π −∞ Then, G q (r1 , r2 , ω) = lim

T →∞

1 ∗T S (r1 , ω)Sq∗T (r2 , ω) (241) 2T q

Now, SqT (r2 , ω) =

qr (r2 )SϕTr (ω)

(242)

Sv

(246)

where G p (r Su , r Sv , ω) is the cross spectral density of the loading normal to the surface at points r Su and r Sv . The mean square acceleration over a frequency band 1 to 2 at point r1S is given by 2 1 2 an (r1s ) = an (r1S , r1s , ω) dω (247) 2π 1 Equation (246) is nothing other than Eq. (203) with the integrand expanded in terms of modes of the structure, and Eq. (203) in turn is nothing other than Eq. (128) written for a continuous structure instead of just a linear black box system.

r

Thus,

D. Coupled Structural Acoustic Systems

1 G q (r1 , r2 , ω) = lim T →∞ 2T r k × qr (r1 )SϕTr (ω)qk (r2 )SϕTk (ω) (243) qr (r1 )qk (r2 ) G q (r1 , r2 , ω) = Yr∗ (iω)Yk (I ω) r k Su

Sv

1 ∗T S F r Su , ω · qr r Su T →∞ 2T

T

× S F r Sv , ω · qk r Sv dsu dsv

× lim

(244) Now, if the integrand is written in the double surface integral, it is S X∗T S XT u r u k + S X∗T SYT u r vk + S X∗T S ZT u r wk + SY∗T S XT vr u k + SY∗T SYT vr vk + SY∗T S ZT vr wk + S Z∗T S X wr u k + S Z∗T SYT wr vk + S Z∗T S ZT wr wk

Equation (171) stated that the cross spectral density of the field pressure in tems of acceleration spectra on the surface is 1 G(P1 , P2 , ω) = ρ 2 an (S1 , S2 , ω)g¯ (P1 , S1 , ω)g¯ ∗ (4π)2 S1 S2 0 × (P2 , S2 , ω) d S1 d S2

(248)

Furthermore, it was found in the last section that the crossspectral density of the normal acceleration for a structure in which the loading is normal to the surface can then be written: qr n (S1 )qmn (S2 ) an (S1 , S2 , ω) = ω4 Cr m (ω) ∗ r m Yr (iω)Ym (iω) (249) in which: Cr m (ω) = G(S1 , S2 , ω)qr n (S1 )qmn (S2 ) d S1 d S2 S1 S2

(245)

where G(S1 , S2 , ω) is the cross-spectral density of the pressure that excites the structure, and qr n (S1 ) is the

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normal component of the r th mode evaluated at point S1 of the surface. If the damping in the structure is relatively low, then in accordance with the analysis of Powell, Hurty, and Rubenstein, the cross-product terms can be neglected and qr n (S1 )qr n (S2 ) an (S1 , S2 , ω) ≈ ω4 Crr (ω) (250) |Yr (iω)|2 r where:

Crr (ω) =

G(S1 , S2 , ω)qr n (S1 )qr n (S2 ) d S1 d S2 S1

S2

To carry the analysis further, a Green’s function must be obtained. Using the analysis of Strasberg and Morse and Ingard as a guide, we assume the use of a free-field Green’s function. The analysis, although approximate, then comes out in general form instead of being limited to a particular surface. Therefore, let g¯ (P1 , S1 , ω) =

eik R1 −ik (a R ·RS ) 1 1 dS e 2 R1

a R1 · R S1 = z o cos θ1 + xo sin θ1 cos ϕ1 + yo sin θ1 sin ϕ1 where xo , yo , z o are the rectangular coordinates of the point on the vibrating surface of the structure; R S1 is the radius vector to point S1 on the surface; R1 , θ1 , ϕ1 are the spherical coordinates of point P1 in the far field; a R1 is a unit vector in the direction of R1 (the radius vector from the origin to the far-field point). Thus, a R1 · R S1 is the projection of R S1 on R1 , making R1 − a R · R S1 the distance from the far-field point to the surface point. Therefore, e−ik R2 ik (a R ·RS ) 2 2 e R2

(253)

Combining Eqs. (248), (249), (251), and (253) gives the following expression for far-field cross spectrum of farfield pressure: G(P1 , P2 , ω) = ω4 ρo2

eik(R1 −R2 ) Ir (θ1 , ϕ1 , ω)Im∗ (4π)2 R1 R2 r m

× (θ2 , ϕ2 , ω) × where:

Im∗ =

(254)

S1

(255) qmn (S2 )e−ik (a R2·RS2 ) d S2

S2

×

ρo ω 4 (4π )2 R12

|Ir (θ1 , ϕ1 , ω)|2 r

|Yr (iω)|2

Crr (ω)

(256)

The far-field mean square pressure in a frequency band

= 2 − can be written: 2 1 p(P1 ) 2 = G(P1 , P2 , ω) dw (257) 2π 1 Thus, p(P1 ) 2 =

1 ρo2 2 2π 2 (4π ) R1 2 4 ω Crr (ω) 2 × |I (θ , ϕ1 , ω)| dw 2 r 1 1 r |Yr (iω)| (258)

In cases in which the structure is lightly damped, the following can be written: 2 4 1 ω Crr (ω) dω pr Crr ( pr ) ≈ (259) 2 2π 1 |Yr (iω)| 8ζr Mr2 Where Crr ( pr ) is defined as the joint acceptance evaluated at the natural frequency pr ( pr consists of those natural frequencies between 1 and 2 ), Mr is the total generalized mass of the r th mode (including virtual mass), and ζr = C¯ r /(C¯ c )r

(260)

where C¯ r is the damping constant for the r th mode (including radiation damping) and (C¯ c )r is the critical damping constant for that mode. Thus, the mean square pressure at the far-field point P1 in the frequency band is p(P1 ) 2 ≈

pr Crr ( pr ) ρo2 1 |Ir (θ1 , ϕ1 , pr )|2 2 (4π )2 2 8ζ M R1 r r r in (261)

In Eq. (261), Crr ( pr ) describes the characteristics of the generalized force of the random loading, pr /8ζr Mr2 describes the characteristics of the structure, and Ir describes the directivity of the noise field. The sum is taken over those modes that resonate in the band.

SEE ALSO THE FOLLOWING ARTICLES

qr n (S1 )e−ik (a R1·RS1 ) d S1

Ir =

Cr m Yr (iω)Ym∗ (iω)

G(P1 , P1 , ω) ≈

(251)

in which (see Fig. 11):

g¯ ∗ (P2 , S2 , ω) =

With the low damping approximation given by Eq. (250), the far-field auto spectrum at point P1 is

ACOUSTICAL MEASUREMENT • ACOUSTIC CHAOS • ACOUSTIC WAVE DEVICES • SIGNAL PROCESSING, ACOUSTIC • WAVE PHENOMENA

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Acoustics, Linear

BIBLIOGRAPHY Ando, Y. (1998). “Architectural Acoustics: Blending Sound Sources, Sound Fields, and Listener,” Modern Acoustics and Signal Processing, Springer-Verlag. Brekhovskikh, L. M., and Godin, O. A. (1998). “Acoustics of Layered Media I: Plane and Quasi-Plane Waves,” Springer Series on Wave Phenomena, Vol. 5, Springer-Verlag. Brekhovskikh, L. M., and Godin, O. A. (1999). “Acoustics of Layered Media II: Point Sources and Bounded Beams,” Springer Series on

167 Wave Phenomena, Vol. 14, Springer-Verlag. Howe, M. S. (1998). “Acoustics of Fluid-Structure Interactions,” Cambridge University Press. Kishi, T., Ohtsu, M., and Yuyama, S., eds. (2000). “Acoustic Emission— Beyond the Millennium,” Elsevier. Munk, W., Worcester, P., and Wunsch, C. (1995). “Ocean Acoustic Tomography,” Cambridge University Press. Ohayon, R., and Soize, C. (1998). “Structural Acoustics and Vibration,” Academic Press. Tohyama, M., Suzuki, H., and Ando, Y. (1996). “The Nature and Technology of Acoustic Space,” Academic Press, London.

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Chaos Joshua Socolar Duke University

I. II. III. IV. V.

Introduction Classical Chaos Dissipative Dynamical Systems Hamiltonian Systems Quantum Chaos

GLOSSARY Cantor set Simple example of a fractal set of points with a noninteger dimension. Chaos Technical term referring to the irregular, unpredictable, and apparently random behavior of deterministic dynamical systems. Deterministic equations Equations of motion with no random elements for which formal existence and uniqueness theorems guarantee that once the necessary initial and boundary conditions are specified the solutions in the past and future are uniquely determined. Dissipative system Dynamical system in which frictional or dissipative effects cause volumes in the phase space to contract and the long-time motion to approach an attractor consisting of a fixed point, a periodic cycle, or a strange attractor. Dynamical system System of equations describing the time evolution of one or more dependent variables. The equations of motion may be difference equations if the time is measured in discrete units, a set of ordinary differential equations, or a set of partial differential equations. Ergodic theory Branch of mathematics that introduces

statistical concepts to describe average properties of deterministic dynamical systems. Extreme sensitivity to initial conditions Refers to the rapid, exponential divergence of nearby trajectories in chaotic dynamical systems. Fractal Geometrical structure with self-similar structure on all scales that may have a noninteger dimension, such as the outline of a cloud, a coastline, or a snowflake. Hamiltonian system Dynamical system that conserves volumes in phase space, such as a mechanical oscillator moving without friction, the motion of a planet, or a particle in an accelerator. KAM theorem The Kolmogorov-Arnold–Moser theorem proves that when a small, nonlinear perturbation is applied to an integrable Hamiltonian system it remains nearly integrable if the perturbation is sufficiently small. Kicked rotor Simple model of a Hamiltonian dynamical system that is exactly described by the classical standard map and the quantum standard map. Kolmogorov–Sinai entropy Measure of the rate of mixing in a chaotic dynamical system that is closely related

637

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638 to the average Lyapunov exponent, which measures the exponential rate of divergence of nearby trajectories. Localization Quantum interference effect, introduced by Anderson in solid-state physics, which inhibits the transport of electrons in disordered or chaotic dynamical systems as in the conduction of electronics in disordered media or the microwave excitation and ionization of highly excited hydrogen atoms. Lyapunov exponent A real number λ specifying the average exponential rate at which nearby trajectories in phase space diverge or converge. Mixing Technical term from ergodic theory that refers to dynamical behavior that resembles the evolution of cream poured in a stirred cup of coffee. Period-doubling bifurcations Refers to a common route from regularity to chaos in chaotic dynamical systems in which a sequence of periodic cycles appears in which the period increases by a factor of two as a control parameter is varied. Phase space Mathematical space spanned by the dependent variables of the dynamical system. For example, a mechanical oscillator moving in one dimension has a two-dimensional phase space spanned by the position and momentum variables. Poincar´e section Stroboscopic picture of the evolution of a dynamical system in which the values of two dependent variables are plotted as points in a plane each time the other dependent variables assume a specified set of values. Random matrix theory Theory introduced to describe the statistical fluctuations of the spacings of nuclear energy levels based on the statistical properties of the eigenvalues of matrices with random elements. Resonance overlap criterion Simple analytical estimate of the conditions for breakup of KAM surfaces leading to widespread, global chaos. Strange attractor Aperiodic attracting set with a fractal structure that often characterizes the longtime dynamics of chaotic dissipative systems. Trajectory A path in the phase space of a dynamical system that is traced out by a system starting from a particular set of initial values of the dependent variables. Universality Refers to the detailed quantitative similarity of the transition from regular behavior to chaos in a broad class of disparate dynamical systems.

A WIDE VARIETY of natural phenomena exhibit complex, irregular behavior. In the past, many of these phenomena were considered to be too difficult to analyze; however, the advent of high-speed digital computers coupled with new mathematical and physical insight has led to the development of a new interdisciplinary field of science

Chaos

called nonlinear dynamics, which has been very successful in finding some underlying order concealed in nature’s complexity. In particular, research in the latter half on the 20th century has revealed how very simple, diterministic mathematical models of physical and biological systems can exhibit surprisingly complex behavior. The apparently random behavior of these deterministic, nonlinear dynamical systems is called chaos. Since many different fields of science and engineering are confronted with difficult problems involving nonlinear equations, the field of nonlinear dynamics has evolved in a highly interdisciplinary manner, with important contributions coming from biologists, mathematicians, engineers, and physicists. In the physical sciences, important advances have been made in our understanding of complex processes and patterns in dissipative systems, such as damped, driven, nonlinear oscillators and turbulent fluids, and in the derivation of statistical descriptions of Hamiltonian systems, such as the motion of celestial bodies and the motion of charged particles in accelerators and plasmas. Moreover, the predictions of chaotic behavior in simple mechanical systems have led to the investigation of the manifestations of chaos in the corresponding quantum systems, such as atoms and molecules in very strong fields. This article attempts to describe some of the fundamental ideas; to highlight a few of the important advances in the study of chaos in classical, dissipative, and Hamiltonian systems; and to indicate some of the implications for quantum systems.

I. INTRODUCTION In the last 25 years, the word chaos has emerged as a technical term to refer to the complex, irregular, and apparently random behavior of a wide variety of physical phenomena, such as turbulent fluid flow, oscillating chemical reactions, vibrating structures, the behavior of nonlinear electrical circuits, the motion of charged particles in accelerators and fusion devices, the orbits of asteroids, and the dynamics of atoms and molecules in strong fields. In the past, these complex phenomena were often referred to as random or stochastic, which meant that researchers gave up all hope of providing a detailed microscopic description of these phenomena and restricted themselves to statistical descriptions alone. What distinguishes chaos from these older terms is the recognition that many complex physcial phenomena are actually described by deterministic equations, such as the Navier–Stokes equations of fluid mechanics, Newton’s equations of classical mechanics, or Schr¨odinger’s equation of quantum mechanics, and the important discovery that even very simple, deterministic equations of motion can exhibit exceedingly

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Chaos

complex behavior and structure that is indistinguishable from an idealized random process. Consequently, a new term was required to describe the irregular behavior of these deterministic dynamical systems that reflected the new found hope for a deeper understanding of these various physical phenomena. These realizations also led to the rapid development of a new, highly interdisciplinary field of scientific research called nonlinear dynamics, which is devoted to the description of complex, but deterministic, behavior and to the search for “order in chaos.” The rise of nonlinear dynamics was stimulated by the combination of some old and often obscure mathematics from the early part of the 20th century that were preserved and developed by isolated mathematicians in the United States, the Soviet Union, and Europe; the deep natural insight of a number of pioneering researchers in meteorology, biology, and physics; and by the widespread availability of high-speed digital computers with high-resolution computer graphics. The mathematicians constructed simple, but abstract, dynamical systems that could generate complex behavior and geometrical patterns. Then, early researchers studying the nonlinear evolution of weather patterns and the fluctuations of biological populations realized that their crude approximations to the full mathematical equations, in the form of a single difference equation or a few ordinary differential equations, could also exhibit behavior as complex and seemingly random as the natural phenomena. Finally, high-speed computers provided a means for detailed computer experiments on these simple mathematical models with complex behavior. In particular, high-resolution computer graphics have enabled experimental mathematicians to search for order in chaos that would otherwise be buried in reams of computer output. This rich interplay of mathematical theory, physical insight, and computer experimentation, which characterizes the study of chaos and the field of nonlinear dynamics, will be clearly illustrated in each of the examples discussed in this article. Chaos research in the physical sciences and engineering can be divided into three distinct areas relating to the study of nonlinear dynamical systems that correspond to (1) classical dissipative systems, such as turbulent flows or mechanical, electrical, and chemical oscillators; (2) classical Hamiltonian systems, where dissipative processes can be neglected, such as charged particles in accelerators and magnetic confinement fusion devices or the orbits of asteroids and planets; and (3) quantum systems, such as atoms and molecules in strong static or intense electromagnetic fields or electrons confined to submicron-scale cavities. The study of chaos in classical systems (both dissipative and Hamiltonian) is now a fairly well-developed field that has been described in great detail in a number of pop-

20:28

639 ular and technical books. In particular, the term chaos has a very precise mathematical definition for classical nonlinear systems, and many of the characteristic features of chaotic motion, such as the extreme sensitivity to initial conditions, the appearance of strange attractors with noninteger fractal dimensions, and the period-doubling route to chaos, have been cataloged in a large number of examples and applications, and new discoveries continue to fill technical journals. In Section II, we will begin with a precise definition of chaos for classical systems and present a very simple mathematical example that illustrates the origin of this complex, apparently random motion in simple deterministic dynamical systems. In Section II, we will also consider additional examples to illustrate some of the other important general features of chaotic classical systems, such as the notion of geometric structures with noninteger dimensions. Some of the principal accomplishments of the application of these new ideas to dissipative systems include the discovery of a universal theory for the transition from regular, periodic behavior to chaos via a sequence of perioddoubling bifurcations, which provides quantitative predictions for a wide variety of physical systems, and the discoveries that mathematical models of turbulence with as few as three nonlinear differential equations can exhibit chaotic behavior that is governed by a strange attractor. The ideas and analytical methods introduced by the simple models of nonlinear dynamics have provided important analogies and metaphors for describing complex natural phenomena that should ultimately pave the way for a better theoretical understanding. Section III will be devoted to a detailed discussion of several models of dissipative systems with important applications in the description of turbulence and the onset of chaotic behavior in a variety of nonlinear oscillators. The latter portion of Section III introduces concepts associated with the description of dissipative dynamical systems with many degrees of freedom and briefly discusses some issues that have been central to chaos research in the last decade. In the realm of Hamiltonian systems, the exact nonlinear equations for the motion of particles in accelerators and fusion devices and of celestial bodies are simple enough to be analyzed using the analytical and numerical methods of nonlinear dynamics without any gross approximations. Consequently, accurate quantitative predictions of the conditions for the onset of chaotic behavior that play significant roles in the design of accelerators and fusion devices and in understanding the irregular dynamics of asteroids can be made. Moreover, the important realization that only a few interacting particles, representing a small number of degrees of freedom, can exhibit motion that is

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640 sufficiently chaotic to permit a statistical description has greatly enhanced our understanding of the microscopic foundations of statistical mechanics, which have also remained an outstanding problem of theoretical physics for over a century. Section IV will examine several simple mathematical models of Hamiltonian systems with applications to the motion of particles in accelerators and fusion devices and to the motion of celestial bodies. Finally, in Section V, we will discuss the more recent and more controversial studies of the quantum behavior of strongly coupled and strongly perturbed Hamiltonian systems, which are classically chaotic. In contrast to the theory of classical chaos, there is not yet a consensus on the definition of quantum chaos because the Schrodinger equation is a linear equation for the deterministic evolution of the quantum wave function, which is incapable of exhibiting the strong dynamical instability that defines chaos in nonlinear classical systems. Nevertheless, both numerical studies of model problems and real experiments on atoms and molecules reveal that quantum systems can exhibit behavior that resembles classical chaos for long times. In addition, considerable research has been devoted to identifying the distinct signatures or symptoms of the quantum behavior of classically chaotic systems. At present, the principal contributions of these studies has been the demonstration that atomic and molecular physics of strongly perturbed and strongly coupled systems can be very different from that predicted by the traditional perturbative methods of quantum mechanics. For example, experiments with highly excited hydrogen atoms in strong microwave fields have revealed a novel ionization mechanism that depends strongly on the intensity of the radiation but only weakly on the frequency. This dependence is just the opposite of the quantum photoelectric effect, but the sharp onset of ionization in the experiments is very well described by the onset of chaos in the corresponding classical system.

II. CLASSICAL CHAOS This section provides a summary of the fundamental ideas that underlie the discussion of chaos in all classical dynamical systems. It begins with a precise definition of chaos and illustrates the important features of the definition using some very simple mathematical models. These examples are also used to exhibit some important properties of chaotic dynamical systems, such as extreme sensitivity to initial conditions, the unpredictability of the longtime dynamics, and the possibility of geometric structures corresponding to strange attractors with noninteger, fractal dimensions. The manifestations of these fundamental concepts in more realistic examples of dissipative

Chaos

and Hamiltonian systems will be provided in Sections III and IV. A. The Definition of Chaos The word chaos describes the irregular, unpredictable, and apparently random behavior of nonlinear dynamical systems that are described mathematically by the deterministic iteration of nonlinear difference equations or the evolution of systems of nonlinear ordinary or partial differential equations. The precise mathematical definition of chaos requires that the dynamical system exhibit mixing behavior with positive Kolmogorov–Sinai entropy (or positive average Lyapunov exponent). This definition of chaos invokes a number of concepts from ergodic theory, which is a branch of mathematics that arose in response to attempts to reconcile statistical mechanics with the deterministic equations of classical mechanics. Although the equations that describe the evolution of chaotic dynamical systems are fully deterministic (no averages over random forces or initial conditions are involved), the complexity of the dynamics invites a statistical description. Consequently, the statistical concepts of ergodic theory provide a natural language to define and characterize chaotic behavior. 1. Ergodicity A central concept familiar to physicists because of its importance to the foundations of statistical mechanics the notion of ergodicity. Roughly speaking, a dynamical system is ergodic if the system comes arbitrarily close to every possible point (or state) in the accessible phase space over time. In this case, the celebrated ergodic theorem guarantees that long-time averages of any physical quantity can be determined by performing averages over phase space with respect to a probability distribution. However, although there has been considerable confusion in the physical literature, ergodicity alone is not sufficiently irregular to account for the complex behavior of turbulent flows or interacting many-body systems. A simple mathematical example clearly reveals these limitations. Consider the dynamical system described by the difference equation, xn+1 = xn + a,

Mod 1

(1)

which takes a real number xn between 0 and 1, adds another real number a, and subtracts the integer part of the sum (Mod 1) to return a value of xn+1 on the unit interval [0, 1]. The sequence of numbers, {xn }n=0,1,2,3,... , generated by iterating this one-dimensional map describes the time history of the dynamical variable xn (where time is measured in discrete units labeled by n). If a = p/q is a rational number (where p and q are integers), then starting with any initial x0 , this dynamical system generates a time

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sequence of {xn } that returns to x0 after q iterations since xq = x0 + p (Mod1) = x0 (Mod 1). In this case, the longtime behavior is described by a periodic cycle of period q that visits only q different values of x on the unit interval [0, 1]. Since this time sequence does not come arbitrarily close to every point in the unit interval (which is the phase or state space of this dynamical system), this map is not ergodic for rational values of a. However, if a is an irrational number, the time sequence never repeats and xn will come arbitrarily close to every point in the unit interval. Moreover, since the time sequence visits every region of the unit interval with equal probability, the long-time averages of any functions of the dynamical variable x can be replaced by spatial averages with respect to the uniform probability distribution P(x) = 1 for x in [0, 1]. Therefore, for irrational values of a, this dynamical system, described by a single, deterministic difference equation, is an ergodic system. Unfortunately, the time sequence generated by this map is much too regular to be chaotic. For example, if we initially colored all the points in the phase space between 0 and 14 red and iterated the map, then the red points would remain clumped together in a continuous interval (Mod 1) for all time. But, if we pour a little cream in a stirred cup of coffee or release a dyed gas in the corner of the room, the different particles of the cream or the colored gas quickly spread uniformly over the accessible phase space. 2. Mixing A stronger notion of statistical behavior is required to describe turbulent flows and the approach to equilibrium in many-body systems. In ergodic theory, this property is naturally called mixing. Roughly speaking, a dynamical system described by deterministic difference or differential equations is said to be a mixing system if sets of initial conditions that cover limited regions of the phase space spread throughout the accessible phase space and evolve in time like the particles of cream in coffee. Once again a simple difference equations serves to illustrate this concept. Consider the shift map: xn+1 = 2xn ,

Mod 1

(2)

which takes xn on the unit interval, multiplies it by 2, and subtracts the integer part to return a value of xn+1 on the unit interval. If we take almost any initial condition, x0 , then this deterministic map generates a time sequence {xn } that never repeats and for long times is indistinguishable from a random process. Since the successive iterates wander over the entire unit interval and come arbitrarily close to every point in the phase space, this map is ergodic. Moreover, like Eq. (1), the long-time averages of

any function of the {xn } can be replaced by the spatial average with respect to the uniform probability distribution P(x) =1. However, the dynamics of each individual trajectory is much more irregular than that generated by Eq. (1). If we were to start with a set of red initial conditions on the interval [0, 14 ], then it is easy to see that these points would be uniformly dispersed on the unit interval after only two iterations of the map. Therefore, we call this dynamical system a mixing system. (Of course, if we were to choose very special initial conditions, such as x0 = 0 or x0 = p/2m , where p and m are positive integers, then the time sequence would still be periodic. However, in the set of all possible initial conditions, these exceptional initial conditions are very rare. Mathematically, they comprise a set of zero measure, which means the chance of choosing one of these special initial conditions by accident is nil.) It is very easy to see that the time sequences generated by the vast majority of possible initial conditions is as random as the time sequence generated by flipping a coin. Simply write the initial condition in binary representation, that is, x0 = 0.0110011011100011010 . . . . Multiplication by 2 corresponds to a register shift that moves the binary point to the right (just like multiplying a decimal number by 10). Therefore, when we iterate Eq. (2), we read off successive digits in the initial condition. If the leading digit to the left of the binary point is a one, then the Mod 1 replaces it by a 0. Since a theorem by Martin–L¨of guarantees that the binary digits of almost every real number are a random sequence with no apparent order, the time sequence {xn } generated by iterating this map will also be random. In particular, if we call out heads whenever the leading digit is a 1 (which means that xn lies on the interval [ 12 , 1]) and tails whenever the leading digit is a 0 (which means that xn lies on the interval [0, 12 ]), then the time sequence {xn } generated by this deterministic difference equation will jump back and forth between the left and right halves of the unit interval in a process that is indistinguishable from that generated by a series of coin flips. The technical definition of chaos refers to the behavior of the time sequence generated by a mixing system, such as the shift map defined by Eq. (2). This simple, deterministic dynamical system with random behavior is the simplest chaotic system, and it serves as the paradigm for all chaotic systems. 3. Extreme Sensitivity to Initial Conditions One of the essential characteristics of chaotic systems is that they exhibit extreme sensitivity to initial conditions. This means that two trajectories with initial conditions that

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642 are arbitrarily close will diverge at an exponential rate. The exponential rate of divergence in mixing systems is related to a positive Kolmogorov–Sinai entropy. For simple systems, such as the one-dimensional maps defined by Eqs. (1) and (2), this local instability is characterized by the average Lyapunov exponent, which in practice is much easier to evaluate than the Kolmogorov–Sinai entropy. It is easy to see that Eq. (2) exhibits extreme sensitivity to initial conditions with a positive average Lyapunov exponent, while Eq.(1) does not. If we consider two nearby initial conditions x0 and y0 , which are d0 = |x0 − y0 | apart, then after one iteration of a map, xn+1 = F(xn ) of the form of Eqs. (1) or (2), the two trajectories will be approximately separated by a distance d1 = |(d F/d x)(x0 |d0 ). Clearly, if |d F/d x| < 1, the distance between the two points decreases; if |d F/d x| > 1, the two distance increases; while if |d F/d x| = 1, the two trajectories remain approximately the same distance apart. We can easily see by differentiating the map or looking at the slopes of the graphs of the return maps in Figs. 1 and 2 that |d F/d x| = 1 for Eq. (1), while |d F/d x| = 2 for Eq. (2). Therefore, after many iterations of Eq. (1), nearby initial conditions will generate trajectories that stay close together (the red points remained clumped), while the trajectories generated by Eq. (2) diverge at an exponential rate (the red points mix throughout the phase space). Moreover, the average Lyapunov exponent for these one-dimensional maps, defined by:

FIGURE √ 1 A graph of the return map defined by Eq. (1) for a = ( 5 − 1)/2 0.618. The successive values of the time sequence {xn }n = 1,2,3,... are simply determined by taking the old values of xn and reading off the new values xn+1 from the graph.

Chaos

FIGURE 2 A graph of the return map defined by Eq. (2). For values of xn between 0 and 0.5, the map increases linearly with slope 2; but for xn larger than 0.5, the action of the Mod 1 requires that the line reenter the unit square at 0.5 and rise again to 1.0.

λ = lim

N →∞

N dF 1 ln (xn ) N n=0 dx

(3)

provides a direct measure of the exponential rate of divergence of nearby trajectories. Since the slope of the return maps for Eqs. (1) and (2) are the same for almost all values of xn , the average Lyapunov exponents can be easily evaluated. For Eq. (1), we get λ = 0, while Eq. (2) gives λ = log 2 > 0. However, it is important to note that all trajectories generated by Eq. (2) do not diverge exponentially. As mentioned earlier, the set of rational x0’s with even denominators generate regular periodic orbits. Although these points are a set of measure zero compared with all of the real numbers on the unit interval, they are dense, which means that in every subinterval, no matter how small, we can always find one of these periodic orbits. The significance of these special trajectories is that, figuratively speaking, they play the role of rocks or obstructions in a rushing stream around which the other trajectories must wander. If this dense set of periodic points were not present in the phase space, then the extreme sensitivity to initial conditions alone would not be sufficient to guarantee mixing behavior. For example, if we iterated Eq. (2) without the Mod 1, then all trajectories would diverge exponentially, but the red points would never be able to spread throughout the accessible space that would consist of the entire positive real axis. In this case, the dynamical system is simply unstable, not chaotic.

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4. Unpredictability One important consequence of the extreme sensitivity to initial conditions is that long-term prediction of the evolution of chaotic dynamical systems, like predicting the weather, is a practical impossibility. Although chaotic dynamical systems are fully deterministic, which means that once the initial conditions are specified the solution of the differential or difference equations are uniquely determined for all time, this does not mean that it is humanly possible to find the solution for all time. If nearby initial conditions diverge exponentially, then any errors in specifying the initial conditions, no matter how small, will also grow exponentially. For example, if we can only specify the initial condition in Eq. (2) to an accuracy of one part in a thousand, then the uncertainty in predicting xn will double each time step. After only 10 time steps, the uncertainty will be as large as the entire phase space, so that even approximate predictions will be impossible. (If we can specify the initial conditions to double precision accuracy on a digital computer (1 part in 1018 ), then we can only provide approximate predictions of the future values of xn for 60 time steps before the error spans the entire unit interval.) In contrast, if we specify the initial condition for Eq. (1) to an accuracy of 10−3 , then we can always predict the future values of xn to the same accuracy. (Of course, errors in the time evolution can also arise from uncertainties in the parameters in the equations of evolution. However, if we can only specify the parameter a in Eq. (1) to an accuracy of 10−3 , we could still make approximate predictions for the time sequence for as many as 103 iterations before the uncertainty becomes as large as the unit interval.)

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643 simpler no matter how much we increase the magnifications. The term fractal was coined by Benoit Mandelbrot to describe these complex geometrical objects. Like the shapes of snowflakes and clouds and the outlines of coastlines and mountain ranges, these fractal objects are best characterized by a noninteger dimension. The simplest geometrical object with a noninteger dimension is the middle-thirds Cantor set. If we take the unit interval [0, 1] and remove all of the points in the middle third, then we will be left with a set consisting of two pieces [0, 13 ] and [ 23 , 1] each of length 13 . If we remove the middle thirds of these remaining pieces. we get a set consisting of four pieces of length 19 . By repeating this construction ad infinitum, we end up with a strange set of points called a Cantor set. Although it consists of points none is isolated. In fact, if we magnify any interval containing elements of the set—for example, the segment contained on the intern val [0, 13 ] for any positive n—then the magnified interval will look the same as the complete set (see Fig. 3). In order to calculate a dimension for this set, we must first provide a mathematical definition of dimension that agrees with our natural intuition for geometrical objects with integer dimension. Although there are a variety of

B. Fractals Another common feature of chaotic dynamical systems is the natural appearance of geometrical structures with noninteger dimensions. For example, in dissipative dynamical systems, described by systems of differential equations, the presence of dissipation (friction) causes the long-time behavior to converge to a geometrical structure in the phase space called an attractor. The attractor may consist of a single fixed point with dimension 0, a periodic limit cycle described by a closed curve with dimension 1, or, if the long-time dynamics is chaotic, the attracting set may resemble a curve with an infinite number of twists, turns, and folds that never closes on itself. This strange attractor is more than a simple curve with dimension 1, but it may fail to completely cover an area of dimension 2. In addition, strange attractors are found to exhibit the same level of structure on all scales. If we look at the complex structure through a microscope, it does not look any

FIGURE 3 The middle-thirds Cantor set is constructed by first removing the points in the middle third of the unit interval and then successively removing the middle thirds of the remaining intervals ad infinitum. This figure shows the first four stages of Cantor set construction. After the first two steps, the first segment is magnified to illustrate the self-similar structure of the set.

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different definitions of dimension corresponding to different levels of mathematical rigor, one definition that serves our purpose is to define the dimension of a geometrical object in terms of the number of boxes of uniform size required to cover the object. For example, if we consider two-dimensional boxes with sides of length L (for example, L = 1 cm), then the number of boxes required to cover a two-dimensional object, N (L), will be approximately equal to the area measured in units of L 2 (that is, cm2 ). Now, if we decrease the size of the boxes to L , then the number of boxes N (L ) will increase approximately as (L/L 2 . [If L = 1 mm, then N (L ) ≈ 100N (L).] Similarly, if we try to cover a one-dimensional object, such as a closed curve, with these boxes, the number of boxes will increase only as (L/L ). (For L = 1 cm and L = 1 mm, N (L) will be approximately equal to the length of the curve in centimeters, while N (L ) will be approximately the length of the curve in millimeters.) In general, N (L) ∞L +d

(4)

where d is the dimension of the object. Therefore, one natural mathematical definition of dimension is provided by the equation: d = lim log N (L)/ log(1/L) L→0

(5)

obtained by taking the logarithm of both sides of Eq. (4). For common geometrical objects, such as a point, a simple curve, an area, or a volume, this definition yields the usual integer dimensions 0, 1, 2, and 3, respectively. However, for the strange fractal sets associated with many chaotic dynamical systems, this definition allows for the possibility of noninteger values. For example, if we count the number of boxes required to cover the middle-thirds Cantor set at each level of construction, we find that we can always cover every point in the set using 2n boxes 1 n of length 3 (that is, 2 boxes of length 13 , 4 boxes of length 19 , etc.), Therefore, Eq. (5) yields a dimension of d = log 2/ log 3 = 0.63093 . . . , which reflects the intricate self-similar structure of this set.

III. DISSIPATIVE DYNAMICAL SYSTEMS In this section, we will examine three important examples of simple mathematical models that can exhibit chaotic behavior and that arise in applications to problems in science and engineering. Each represents a dynamical system with dissipation so that the long-time behavior converges to an attractor in the phase space. The examples increase in complexity from a single difference equation, such as Eqs. (1) and (2), to a system of two coupled difference equations and then to a system of three coupled ordinary differential equations. Each example illustrates the characteris-

tic properties of chaos in dissipative dynamical systems with irregular, unpredictable behavior that exhibits extreme sensitivity to initial conditions and fractal attractors. A. The Logistic Map The first example is a one-dimensional difference equation, like Eqs. (1) and (2), called the logistic map, which is defined by: X n+1 = axn (1 − xn ) ≡ F(X n )

(6)

For values of the control parameter a between 0 and 4, this nonlinear difference equation also takes values of xn between 0 and 1 and returns a value xn+1 on the unit interval. However, as a is varied, the time sequences {xn } generated by this map exhibit extraordinary transitions from regular behavior, such as that generated by Eq. (1), to chaos, such as that generated by Eq. (2). Although this mathematical model is too simple to be directly applicable to problems in physics and engineering, which are usually described by differential equations, Mitchell Feigenbaum has shown that the transition from order to chaos in dissipative dynamical systems exhibits universal characteristics (to be discussed later), so that the logistic map is representative of a large class of dissipative dynamical systems. Moreover, since the analysis of this deceptively simple difference equation involves a number of standard techniques used in the study of nonlinear dynamical systems, we will examine it in considerable detail. As noted in the seminal review article in 1974 by Robert May, a biologist who considered the logistic map as a model for annual variations of insect populations, the time evolution generated by the map can be easily studied using a graphical analysis of the return maps displayed in Fig. 4. Equation (6) describes an inverted parabola that intercepts the xn+1 = 0 axis at xn = 0 and 1, with a maximum of xn+1 = a/4 at xn = 0.5. Although this map can be easily iterated using a short computer program, the qualitative behavior of the time sequence {xn } generated by any initial x0 can be examined by simply tracing lines on the graph of the return map with a pencil as illustrated in Fig. 4. For values of a < 1, almost every initial condition is attracted to x = 0 as shown in Fig. 4 for a = 0.95. Clearly, x = 0 is a fixed point of the nonlinear map. If we start with x0 = 0, then the logistic map returns the value xn = 0 for all future iterations. Moreover, a simple linear analysis, such as that used to define the Lyapunov exponent in Section II, shows that for a < 1 this fixed point is stable. (Initial conditions that are slightly displaced from the origin will be attracted back since |(d F/d x)(0)| = a < 1.) However, when the control parameter is increased to a > 1, this fixed point becomes unstable and the long-time behavior is attracted to a new fixed point, as shown in Fig. 4 for a = 2.9, which lies at the other intersection of

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FIGURE 4 Return maps for the logistic map, Eq. (6), are shown for four different values of the control parameter a. These figures illustrate how pencil and paper can be used to compute the time evolution of the map. For example, if we start our pencil at an initial value of x0 = 0.6 for a = 0.95, then the new value of x1 is determined by tracing vertically to the graph of the inverted parabola. Then, to get x2 , we could return to the horizontal axis and repeat this procedure, but it is easier to simply reflect off the 45◦ line and return to the parabola. Successive iterations of this procedure give rapid convergence to the stable, fixed point at x = 0. However, if we start at x0 = 0.1 for a = 2.9, our pencil computer diverges from x = 0 and eventually settles down to a stable, fixed point at the intersection of the parabola and the 45◦ line. Then, when we increase a > 3, this fixed point repels the trace of the trajectory, which settles into either a periodic cycle, such as the period-2 cycle for a = 3.2, or a chaotic orbit, such as that for a = 4.0.

the 45◦ line and the graph of the return map. In this case, the dynamical system approaches an equilibrium with a nonzero value of the dependent variable x. Elementary algebra shows that this point corresponds to the nonzero root of the quadratic equation x = ax(1 − x) given by x ∗ = (a − 1)/a. Again, a simple linear analysis of small displacements from this fixed point reveals that it remains stable for values of a between 1 and 3. When a becomes larger than 3, this fixed point also becomes unstable and the long-time behavior becomes more complicated, as shown in Fig. 4.

1. Period Doubling For values of a slightly bigger than 3, empirical observations of the time sequences for this nonlinear dynamical

system generated by using a hand calculator, a digital computer, or our “pencil computer” reveals that the long-time behavior approaches a periodic cycle of period 2, which alternates between two different values of x. Because of the large nonlinearity in the difference equation, this periodic behavior could not be deduced from any analytical arguments based on exact solutions or from perturbation theory. However, as typically occurs in the field of nonlinear dynamics, the empirical observations provide us with clues to new analytical procedures for describing and understanding the dynamics. Once again, the graphical analysis provides an easy way of understanding the origin of the period-2 cycle. Consider a new map. xn+2 = F (2) (xn ) = F[F(xn )] = a 2 xn − xn2 − a 3 xn2 − 2xn3 + xn4

(7)

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FIGURE 5 The return maps are shown for the second iterate of the logistic map, F (2) , defined by Eq. (7). The fixed points at the intersection of the 45◦ line and the map correspond to values of x that repeat every two periods. For a = 2.9, the two intersections are just the period-1 fixed points at 0 and a*, which repeat every period and therefore every other period, as well. However, when a is increased to 3.2, the peaks and valleys of the return map become more pronounced and pass through the 45◦ line and two new fixed points appear. Both of the old, fixed points are now unstable because the absolute value of the slope of the return map is larger than 1, but the new points are stable, and they correspond to the two elements of the period-2 cycle displayed in Fig. 4. Moreover, because the portion of the return map contained in the dashed box resembles an inverted image of the original logistic map, one might expect that the same bifurcation process will be repeated for each of these period-2 points as a is increased further.

constructed by composing the logistic map with itself. The graph of the corresponding return map, which gives the values of xn every other iteration of the logistic map, is displayed in Fig. 5. If we use the same methods of analysis as we applied to Eq. (6), we find that there can be at most four fixed points that correspond to the intersection of the graph of the quartic return map with the 45◦ line. Because the fixed points of Eq. (4) are values of x that return every other iteration, these points must be members of the period-2 cycles of the original logistic map. However, since the period-1 fixed points of the logistic map at x = 0 and x ∗ are automatically period-2 points, two of the fixed points of Eq. (7) must be x = 0, x ∗ . When 1 < a < 3, these are the only two fixed points of Eq. (7), as shown in Fig. 5 for a = 2.9. However, when a is increased above 3, two new fixed points of Eq. (7) appear, as shown in Fig. 5 for a = 3.2, on either side of the fixed point at x = x ∗ , which has just become unstable. Therefore, when the stable period-1 point at x ∗ becomes unstable, it gives birth to a pair of fixed points, x (1) , x (2) of Eq. (7), which form the elements of the period-2 cycle found empirically for the logistic map. This process is called a pitchfork bifurcation. For values of a just above

3, these new fixed points are stable and the long-time dynamics of the second iterate of the logistic map, F (2) , is attracted to one or the other of these fixed points. However, as a increases, the new fixed points move away from x ∗ , the graphs of the return maps for Eq. (7) get steeper and steeper, and when |d F (2) /d x|x (1) ,x (2) > 1 the period-2 cycle also becomes unstable. (A simple application of the chain rule of differential calculus shows that both periodic points destabilize at the same value of a, since F(x (1),(2) ) = x (2),(1) and (d F (2) /d x)(x (1) ) = (d F/d x)(x (2) )(d F/d x)(x (1) ) = (d F (2) /d x)(x (1) ).) Once again, empirical observations of the long-time behavior of the iterates of the map reveal that when the period-2 cycle becomes unstable it gives birth to a stable period-4 cycle. Then, as a increases, the period-4 cycle becomes unstable and undergoes a pitchfork bifurcation to a period-16 cycle, then a period-32 cycle, and so on. Since the successive period-doubling bifurcations require smaller and smaller changes in the control parameter, this bifurcation sequence rapidly accumulates to a period cycle of infinite period at a∞ = 3.57 . . . . This sequence of pitchfork bifurcations is clearly displayed in the bifurcation diagram shown in Fig. 6. This

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FIGURE 6 A bifurcation diagram illustrates the variety of longtime behavior exhibited by the logistic map as the control parameter a is increased from 3.5 to 4.0. The sequences of perioddoubling bifurcations from period-4 to period-8 to period-16 are clearly visible in addition to ranges of a in which the orbits appear to wander over continuous intervals and ranges of a in which periodic orbits, including odd periods, appear to emerge from the chaos.

graph is generated by iterating the map for several hundred time steps for successive values of a. For each value of a, we plot only the last hundred values of xn to display the long-time behavior. For a < 3, all of these points land close to the fixed point at a ∗ ; for a > 3, these points alternate between the two period-2 points, then between the four period-4 points, and so on. The origin of each of these new periodic cycles can be qualitatively understood by applying the same analysis that we used to explain the birth of the period-2 cycle from period 1. For the period-4 cycle, we consider the second iterate of the period-2 map: xn+4 = F (4) (xn ) = F (2) F (2) (xn ) = F{F[F(F(xn )]}

(8)

In this case, the return map is described by a polynomial of degree 16 that can have as many as 16 fixed points that correspond to intersections of the 45◦ line with the graph of the return map. Two of these period-4 points correspond to the period-1 fixed points at 0 and x ∗ , and for a > 3, two correspond to the period-2 points at x (1) and x (2) . The remaining 12 period-4 points can form three different period-4 cycles that appear for different values of a. Figure 7 shows a graph of F (4) (xn ) for a = 3.2, where the period-2 cycle is still stable, and for a = 3.5, where the unstable period-2 cycle has bifurcated into a period4 cycle. (The other two period-4 cycles are only briefly stable for other values of a > a∞· )

We could repeat the same arguments to describe the origin of period 8; however, now the graph of the return map of the corresponding polynomial of degree 32 would begin to tax the abilities of our graphics display terminal as well as our eyes. Fortunately, the “slaving” of the stability properties of each periodic point via the chain-rule argument (described previously for the period-2 cycle) means that we only have to focus on the behavior of the successive iterates of the map in the vicinity of the periodic point closest to x = 0.5. In fact, a close examination of Figs. 4, 5, and 7 reveals that the bifurcation process for each F (n) is simply a miniature replica of the original period-doubling bifurcation from the period-1 cycle to the period-2 cycle. In each case, the return map is locally described by a parabolic curve (although it is not exactly a parabola beyond the first iteration and the curve is flipped over for every other F (N ) . Because each successive period-doubling bifurcation is described by the fixed points of a return map xn+N = F (N ) (X n ) with ever greater oscillations on the unit interval, the amount the parameter a must increase before the next bifurcation decreases rapidly, as shown in the bifurcation diagram in Fig. 6. The differences in the changes in the control parameter for each succeeding bifurcation, an+1 − an , decreases at a geometric rate that is found to rapidly converge to a value of: an − an−1 δ= = 4.6692016 . . . (9) an+1 − an In addition, the maximum separation of the stable daughter cycles of each pitchfork bifurcation also decreases rapidly, as shown in Fig. 6, by a geometric factor that rapidly converges to: α = 2.502907875 . . .

(10)

2. Universality The fact that each successive period doubling is controlled by the behavior of the iterates of the map, F (N ) (x), near x = 0.5, lies at the root of a very significant property of nonlinear dynamical systems that exhibit sequences of period-doubling bifurcations called universality. In the process of developing a quantitative description of period doubling in the logistic map, Feigenbaum discovered that the precise functional form of the map did not seem to matter. For example, he found that a map on the unit interval described by F(x) = a sin π x gave a similar sequence of period-doubling bifurcations. Although the values of the control parameter a at which each period-doubling bifurcation occurs are different, he found that both the ratios of the changes in the control parameter and the separations of the stable daughter cycles decreased at the same geometrical rates δ and α as the logistic map.

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FIGURE 7 The appearance of the period-4 cycle as a is increased from 3.2 to 3.5 is illustrated by these graphs of the return maps for the fourth iterate of the logistic map, F (4) . For a = 3.2, there are only four period-4 fixed points that correspond to the two unstable period-1 points and the two stable period-2 points. However, when a is increased to 3.5, the same process that led to the birth of the period-2 fixed points is repeated again in miniature. Moreover, the similarity of the portion of the map near xn = 0.5 to the original map indicates how this same bifurcation process occurs again as a is increased.

This observation ultimately led to a rigorous proof, using the mathematical methods of the renormalization group borrowed from the theory of critical phenomena, that these geometrical ratios were universal numbers that would apply to the quantitative description of any perioddoubling sequence generated by nonlinear maps with a single quadratic extremum. The logistic map and the sine map are just two examples of this large universality class. The great significance of this result is that the global details of the dynamical system do not matter. A thorough understanding of the simple logistic map is sufficient for describing both qualitatively and, to a large extent, quantitatively the period-doubling route to chaos in a wide variety of nonlinear dynamical systems. In fact, we will see that this universality class extends beyond one-dimensional maps to nonlinear dynamical systems described by more realistic physical models corresponding to two-dimensional maps, systems of ordinary differential equations, and even partial differential equations.

age Lyapunov exponent and therefore satisfy the definition of chaos. Figure 8 plots the average Lyapunov exponent computed numerically using Eq. (3) for the same range of values of a, as displayed in the bifurcation diagram in Fig. 6.

3. Chaos Of course, these stable periodic cycles, described by Feigenbaum’s universal theory, are not chaotic. Even the cycle with an infinite period at the period-doubling accumulation point a has a zero average Lyapunov exponent. However, for many values of a above a∞ , the time sequences generated by the logistic map have a positive aver-

FIGURE 8 The values of the average Lyapunov exponent, computed numerically using Eq. (3), are displayed for the same values of a shown in Fig. 6. Positive values of λ correspond to chaotic dynamics, while negative values represent regular, periodic motion.

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Wherever the trajectory appears to wander chaotically over continuous intervals, the average Lyapunov exponent is positive. However, embedded in the chaos for a∞ < a < 4 we see stable period attractors in the bifurcation diagram with sharply negative average Lyapunov exponents. The most prominent periodic cycle is the period-3 cycle, which appears near a3 = 3.83. In fact, between a∞ and a3 , there is a range of values of a for which cycles of every odd and even period are stable. However, the intervals for the longer cycles are too small to discern in Fig. 6. The period-5 cycle near a = 3.74 and the period6 cycles near a = 3.63 and a = 3.85 are the most readily apparent in both the bifurcation diagram and the graph of the average Lyapunov exponent. Although these stable periodic cycles are mathematically dense over this range of control parameters, the values of a where the dynamics are truly chaotic can be mathematically proven to be a significant set with nonzero measure. The proof of the positivity of the average Lyapunov exponent is much more difficult for the logistic map than for Eq. (2) since log |(d F/d x)(xn )| can take on both negative and positive values depending on whether xn is close to 12 or to 0 or 1. However, one simple case for which the logistic map is easily proven to be chaotic is for a = 4. In this case, the time sequence appears to wander over the entire unit interval in the bifurcation diagram, and the numerically computed average Lyapunov exponent is positive. If we simply change variables from xn √ to yn = (2/π) sin−1 xn , then the logistic map for a = 4 transforms to the tent map: 2yn 0 ≤ yn ≤ 0.5 yn+1 = (11) 2(1 − yn ) 0.5 ≤ yn ≤ 1 which is closely related to the shift map, Eq. (2). In particular, since |d F/dy| = 2, the average Lyapunov exponent is found to be λ = log 2 ≈ 0.693, which is the same as the numerical value for the logistic map. B. The Henon ´ Map Most nonlinear dynamical systems that arise in physical applications involve more than one dependent variable. For example, the dynamical description of any mechanical oscillator requires at least two variables—a position and a momentum variable. One of the simplest dissipative dynamical systems that describes the coupled evolution of two variables was introduced by Michel H´enon in 1976. It is defined by taking a one-dimensional quadratic map for xn+1 similar to the logistic map and coupling it to a second linear map for yn+1 : xn+1 = 1 − axn2 + yn

(12a)

yn+1 = bxn

(12b)

This pair of difference equations takes points in the x–y plane with coordinates (xn , yn ) and maps them to new points (xn+1 , yn+1 ). The behavior of the sequence of points generated by successive iterates of this two-dimensional map from an initial point (x0 , y0 ) is determined by the values of two control parameters a and b. If a and b are both 0, then Eq. (12) maps every point in the plane to the attracting fixed point at (1, 0) after at most two iterations. If b = 0 but a is nonzero, then the H´enon map reduces to a one-dimensional quadratic map that can be transformed into the logistic map by shifting the variable x. Therefore, for b = 0 and even for b small, the behavior of the time sequence of points generated by the H´enon map closely resembles the behavior of the logistic map. For small values of a, the long-time iterates are attracted to stable periodic orbits that exhibit a sequence of period-doubling bifurcations to chaos as the nonlinear control parameter a is increased. For small but nonzero b, the main difference from the one-dimensional maps is that these regular orbits of period N are described by N points in the (x–y) plane rather than points on the unit interval. (In addition, the basin of attraction for these periodic cycles consists of a finite region in the plane rather than the unit interval alone. Just as in the one-dimensional logistic map, if a point lies outside this basin of attraction, then the successive iterates diverge to ∞.) The H´enon map remains a dissipative map with time sequences that converge to a finite attractor as long as b is less than 1. This is easy to understand if we think of the action of the map as a coordinate transformation in the plane from the variables (xn , yn ) to (xn+1 , yn+1 ). From elementary calculus, we know that the Jacobian of this coordinate transformation, which is given by the determinant of the matrix,

−2ax 1 M= (13) b 0 describes how the area covered by any set of points increases or decreases under the coordinate transformation. In this case, J = Det M = −b. When |J | > 1, areas grow larger and sets of initial conditions disperse throughout the x–y plane under the iteration of the map. But, when |J | < 1, the areas decrease under each iteration, so areas must contract to sets of points that correspond to the attractors. 1. Strange Attractors However, these attracting sets need not be a simple fixed point or a finite number of points forming a periodic cycle. In fact, when the parameters a and b have values that give

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than for one-dimensional maps) indicate that the dynamics on this strange attractor are indeed chaotic. C. The Lorenz Attractor The study of chaos is not restricted to nonlinear difference equations such as the logistic map and the H´enon map. Systems of coupled nonlinear differential equations also exhibit the rich variety of behavior that we have already seen in the simplest nonlinear dynamical systems described by maps. A classic example is provided by the Lorenz model described by three coupled nonlinear differential equations:

FIGURE 9 The first 10,000 iterates of the two-dimensional Henon ´ map trace the outlines of a strange attractor in the xn −yn plane. The parameters were chosen to be a = 1.4 and b = 0.3 and the initial point was (0, 0).

rise to chaotic dynamics, the attractors can be exceedingly complex, composed of an uncountable set of points that form intricate patterns in the plane. These strange attractors are best characterized as fractal objects with noninteger dimensions. Figure 9 displays 10,000 iterates of the H´enon map for a = 1.4 and b = 0.3. (In this case, the initial point was chosen to be (x0 , y0 ) = (0, 0), but any initial point in the basin of attraction would give similar results.) Because b < 1, the successive iterates rapidly converge to an intricate geometrical structure that looks like a line that is folded on itself an infinite number of times. The magnifications of the sections of the attractor shown in Fig. 10 display the detailed self-similar structure. The cross sections of the folded line resemble the Cantor set described in Section II.B. Therefore, since the attractor is more than a line but less than an area (since there are always gaps between the strands at every magnification), we might expect it to be characterized by a fractal dimension that lies between 1 and 2. In fact, an application of the box-counting definition of fractal dimension given by Eq. (5) yields a fractal dimension of d = 1.26 . . . . Moreover, if you were to watch a computer screen while these points are plotted, you would see that they wander about the screen in a very irregular manner, slowly revealing this complex structure. Numerical measurements of the sensitivity to initial conditions and of the average Lyapunov exponents (which are more difficult to compute

d x/dt = −σ x + σ y

(14a)

dy/dt = −x z + r x − y

(14b)

dz/dt = x y − bz

(14c)

These equations were introduced in 1963 by Edward Lorenz, a meteorologist, as a severe truncation of the Navier–Stokes equations describing Rayleigh–Benard convection in a fluid (like Earth’s atmosphere), which is heated from below in a gravitational field. The dependent variable x represents a single Fourier mode of the stream function for the velocity flow, the variables y and z represent two Fourier components of the temperature field; and the constants r , σ , and b are the Rayleigh number, the Prandtl number, and a geometrical factor, respectively. The Lorenz equations provide our first example of a model dynamical system that is reasonably close to a real physical system. (The same equations provide an even better description of optical instabilities in lasers, and similar equations have been introduced to describe chemical oscillators.) Numerical studies of the solutions of these equations, starting with Lorenz’s own pioneering work using primitive digital computers in 1963, have revealed the same complexity as the H´enon map. In fact, H´enon originally introduced Eq. (12) as a simple model that exhibits the essential properties of the Lorenz equations. A linear analysis of the evolution of small volumes in the three-dimensional phase space spanned by the dependent variables x, y, and z shows that this dissipative dynamical system rapidly contracts sets of initial conditions to an attractor. When the Rayleigh number r is less than 1, the point (x, y, z) = (0, 0, 0) is an attracting fixed point. But, when r > 1, a wide variety of different attractors that depend in a complicated way on all three parameters r , σ , and b are possible. Like the H´enon map, the long-time behavior of the solutions of these differential equations can be attracted to fixed points; to periodic cycles, which are described by limit cycles consisting of closed curves in the

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FIGURE 10 To see how strange the attractor displayed in Fig. 9 really is, we show two successive magnifications of a strand of the attractor contained in the box in Fig. 9. Here, (a) shows that the single strand in Fig. 9 breaks up into several distinct bands, which shows even finer structure in (b) when the map is iterated 10,000,000 time steps.

three-dimensional phase space; and to strange attractors, which are described by a fractal structure in phase space. In the first two cases, the dynamics is regular and predictable, but the dynamics on strange attractors is chaotic and unpredictable (as unpredictable as the weather). The possibility of strange attractors for three or more autonomous differential equations, such as the Lorenz model, was established mathematically by Ruelle and Takens. Figure 11 shows a three-dimensional graph of the famous strange attractor for the Lorenz equations corresponding to the values of the parameters r = 28, σ = 10, and b = 83 , which provides a graphic illustration of the consequences of their theorem. The initial conditions were chosen to be (1, 1, 1). The trajectory appears to loop around on two surfaces that resemble the wings of a butterfly, jumping from one wing to the other in an irregular manner. However, a close inspection of these surfaces reveals that under successive magnification they exhibit the same kind of intricate, self-similar structure as the striations of the H´enon attractor. This detailed structure is best revealed by a so-called Poincar´e section of continuous dynamics, shown in Fig. 12, which was generated by plotting a point in the x–z plane every time the orbit passes from negative y to positive y. Since we could imagine that this Poincar´e section was generated by iterating a pair of nonlinear difference equations, such as the H´enon map, it is easy to understand, by analogy with the analysis described in Section III.B, how the time evolution can be chaotic with extreme sensitivity to initial conditions and how this cross section of the

Lorenz attractor, as well as the Lorenz attractor itself, can have a noninteger, fractal dimension. D. Applications Perhaps the most significant conclusion that can be drawn from these three examples of dissipative dynamical systems that exhibit chaotic behavior is that the essential features of the behavior of the more realistic Lorenz model are well described by the properties of the much simpler H´enon map and to a large extent by the logistic map. These observations provide strong motivation to hope that simple nonlinear systems will also capture the essential properties of even more complex dynamical systems that describe a wide variety of physical phenomena with irregular behavior. In fact, the great advances of nonlinear dynamics and the study of chaos in the last [20] years can be attributed to the fulfillment of this hope in both numerical studies of more complicated mathematical models and experimental studies of a variety of complicated natural phenomena. The successes of this program of reducing the essential features of complicated dynamical processes to simple nonlinear maps or to a few coupled, nonlinear differential equations have been well documented in a number of conference proceedings and textbooks. For example, the universality of the period-doubling route to chaos and the appearance of strange attractors have been demonstrated in numerical studies of a wide variety of nonlinear maps, systems of nonlinear, ordinary, and partial differential equations. Even more importantly, Feigenbaum’s

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FIGURE 11 The solution of the Lorenz equations for the parameters r = 28, σ = 10, and b = 83 rapidly converges to a strange attractor. This figure shows a projection of this three-dimensional attractor onto the x−z plane, which is traced out by approximately 100 turns of the orbit.

universal constants δ and α, which characterize the quantitative scaling properties of the period-doubling sequence, have been measured to good accuracy in a number of careful experiments on Rayleigh–Benard convection and nonlinear electrical circuits and in oscillating chemical reactions (such as the Belousov–Zabotinsky reaction), laser oscillators, acoustical oscillators, and even the response of heart cells to electrical stimuli. In addition, a large number of papers have been devoted to the measurement of the fractal dimensions of strange attractors that may govern the irregular, chaotic behavior of chemical reactions, turbulent flows, climatic changes, and brainwave patterns. Perhaps the most important lesson of nonlinear dynamics has been the realization that complex behavior need not have complex causes and that many aspects of irregular, unpredictable phenomena may be understood in terms of simple nonlinear models. However, the study of chaos also teaches us that despite an underlying simplicity and order we will never be able to describe the precise behav-

ior of chaotic systems analytically nor will we succeed in making accurate long-term predictions no matter how much computer power is available. At the very best, we may hope to discern some of this underlying order in an effort to develop reliable statistical methods for making predictions for average properties of chaotic systems. E. Hyperchaos The notion of Lyapunov exponent can be extended to systems of differential equations or higher dimensional maps. In general, a system has a set of Lyapunov exponents, each characterizing the average stretching or shrinking of phase space in a particular direction. The logistic map discussed above has only one Lyapunov exponent because it has only one dependent variable that can be displaced. In the Lorenz model, the system has only one positive Lyapunov exponent, but has three altogether. Consider an initial point in phase space that is on the strange attractor and

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FIGURE 12 A Poincare´ section of the Lorenz attractor is constructed by plotting a cross section of the butterfly wings. This graph is generated by plotting the point in the x −z plane each time the orbit displayed in Fig. 11 passes through y = 0. This view of the strange attractor is analogous to that displayed in Fig. 9 for the Henon ´ map. This figure appears to consist of only a single strand, but this is because of the large contraction rate of the Lorenz model. Successive magnifications would reveal a finescale structure similar to that shown in Fig. 10 for the Henon ´ map.

another point displaced infinitesimally from it. The trajectories followed from these two points may remain the same distance apart (on average), diverge exponentially, or converge exponentially. The first case corresponds to a Lyapunov exponent of zero and is realized when the displacement lies along the trajectory of the initial point. The two points then follow the same trajectory, but displaced in time. The second case corresponds to a positive Lyapunov exponent, the third to a negative one. In the Lorenz system, the fact that the attractor has a planar structure locally indicates that trajectories converge in the direction transverse to the plane, hence one of the Lyapunov exponents is negative. An arbitrary, infinitesimal perturbation will almost certainly have some projection on each of the directions corresponding to the different Lyapunov exponents. Since the growth of the perturbation in the direction associated with the largest Lyapunov exponent is exponentially faster than that in any other direction, the observed trajectory divergence will occur in that direction. In numerical models, one can measure the n largest Lyapunov exponents by integrating the linearized equations for the deviations from a given trajectory for n different initial conditions. One must repeatedly rescale the deviations to avoid both expo-

nential growth that causes overflow errors and problems associated with the convergence of all the deviations to the direction associated with the largest exponent. It is possible to have an attractor with two or more Lyapunov exponents greater than zero. This is sometimes referred to as “hyperchaos” and is common in systems with many degrees of freedom. The problem of distinguishing between hyperchaos and stochastic fluctuations in interpreting experimental data has received substantial attention. We are typically presented with an experimental trace of the time variation of a single variable and wish to determine whether the system that generated it was essentially deterministic or stochastic. The distinction here is quantitative rather than qualitative. If the observed fluctuations involve so many degrees of freedom that it appears hopeless to model them with a simple set of deterministic equations, we label it stochastic and introduce noise terms into the equations. “Time-series analysis” algorithms have been developed to identify underlying deterministic dynamics in apparently random systems. The central idea behind these algorithms is the construction of a representation of the strange attractor (if it exists) via delay coordinates. Given a time series for a single variable x(t), the n-dimensional vector X(t) = (x(t), x(t − τ ), x(t − 2τ ), . . . , x(t − (n − 1)τ )) is formed, where τ is a fixed delay time comparable to the scale on which x(t) fluctuates. For sufficiently large n, the topological structure of the attractor for X (t) will generically be identical to that of the dynamical system that generated the data. This allows for mesasures of the geometric structure of the trajectory in the space of delay coordinates, called the embedding space, to provide an upper bound on the dimension of the true attractor. As a practical matter, hyperchaos with more than about 10 positive Lyapunov exponents is extremely difficult to identify unambiguously. F. Spatiotemporal Chaos The term spatiotemporal chaos has been used to refer to any system in which some variable exhibits chaotic motion in time and the spatial structure of the system varies with time as well. Real systems are always composed of spatially extended materials for which a fully detailed mathematical model would require either the use of partial differential equations or an enormous number of ordinary differential equations. In many cases, the dynamics of the vast majority of degrees of freedom represented in these equations need not be solved explicitly. All but a few dependent variables exhibit trivial behavior, decaying exponentially quickly to a steady state, or else oscillate at amplitudes negligible for the problem at hand. The remaining variables are described by a few ordinary

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654 differential equations (or maps) of the type discussed in the previous section. In some cases, the relevant variables are easily identified. For example, it is not difficult to guess that the motion of a pendulum can be described by solving coupled equations for the position and velocity of the bob. One generally does not have to worry about the elastic deformations of the bar. In other cases, the relevant variable are amplitudes of modes of oscillation that can have a nontrivial spatial structure. A system of ordinary differential equations for the amplitudes of a few modes may correspond to an complicated pattern of activity in real space. For example, a violin string can vibrate in different harmonic modes, each of which corresponds to a particular shape of the string that oscillates sinusoidally in time. A model of large-amplitude vibrations might be cast in the form of coupled, nonlinear equations for the amplitudes of a few of the lowest frequency modes. If those equations yielded chaos, the spatial shape of the string would fluctuate in complicated, unpredictable ways. This complex motion in space and time is sometimes referred to as spatiotemporal chaos, though it is a rather simple version since the dynamics simplifies greatly when the correct modes are identified. In general, models of smaller systems require fewer variables in the following sense. What determines the number of modes necessary for an accurate description is the smallest scale spatial variation that has an appreciable probability of occuring at a noticeable amplitude. Since large amplitude variations over very short-length scales generally require large amounts of energy, there will be an effective small-scale cutoff determined by the strength with which the system is driven. Systems whose size is comparable to the cutoff scale will require the analysis of only a few modes; in systems much larger than this scale many modes may be involved and the dynamics can be considerably more complex. The term “spatiotemporal chaos” is sometimes reserved for this regime. Interest in the general subject of turbulence and its statistical description has led to a number of studies of deterministic systems that exhibit spatiotemporal chaos with a level of complexity proportional to the volume of the system. By analogy with thermodynamic properties that are proportional to the volume, such systems are said to exhibit “extensive chaos.” A well-studied example is the irregular pattern of activity known as Benard convection, where a fluid confined to a thin, horizontal layer is heated from below. As the temperature difference between the bottom and top surface of the fluid is increased, the fluid begins to move, arranging itself in a pattern of roughly cylindrical rolls in which warm fluid rises on one side and falls on the other. At the onset of convection, the rolls form straight stripes (apart from boundary effects). As the temperature difference is increased further, the rolls may form

Chaos

more complicated patterns of spirals and defects that continually move around, never settling into a periodic pattern or steady state. The question of whether the “spiral defect chaos” state is an example of extensive chaos is not easy to answer directly, but numerical simulations of models exhibiting similar behavior can be analyzed in detail. To establish the fact that a numerical model exhibits extensive chaos, one must define an appropriate quantity that characterizes the complexity of the chaotic attractor. A quantity that has proven useful is the Lyapunov dimension, Dλ . Let λ1 be the largest Lyapunov exponent, λ2 the second largest, etc. Note that in most extended systems the exponents with higher indices become increasingly strongly negative. Let N be the largest integer for which N λ > 0. We define the Lyapunov dimension as i=1 i Dλ = N +

1

N

|λ N +1 |

i=1

λi .

Numerical studies of systems of partial differential equations such as the complex Ginzburg-Landau equation in two dimensions have demonstrated the existence of attractors for which Dλ does indeed grow proportionally to the system volume; that is, extensive chaos does exist in simple, spatially extended systems, and the spiral defect chaos state is a real example of this phenomenon. G. Control and Synchronization of Chaos Over the past 10 years, mathematicians, physicists, and engineers have become increasingly interested in the possibilities of using the unique properties of chaotic systems for novel applications. A key feature of strange attractors that spurred much of this effort was that they have embedded within them an infinite set of perfectly periodic trajectories. These trajectories, called unstable periodic orbits (UPOs) lie on the attractor but are not normally observed because they are unstable. In 1990, Ott, Grebogi, and Yorke pointed out that UPOs could form the basis of a switching system. Using standard techniques of control theory for feedback stabilize, we can arrange for an intrinsically chaotic system to follow a selected UPO. By turning off that feedback and turning on a different one, we can stabilize a different UPO. The beauty of the scheme is that we are guaranteed, due to the nature of the strange attractor, that the system will come very close to the desired UPO in a relatively short time. Thus, our feedback system need only be capable of applying tiny perturbations to the system. The chaotic dynamics does the hard work of switching from the vicinity of one orbit to the vicinity of the other. The notion that chaos can be suppressed using small feedback perturbations has generated a great deal of

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interest even independent of the possibility of switching between UPOs. At the time of this writing, applications of “controlling chaos” (or simply suppressing it) are being actively pursued in systems as diverse as semiconductor lasers, mechanical systems, fluid flows, the electrodynamics of cardiac tissue. A development closely related to controlling chaos has been the use of simple coupling between two nearly identical chaotic systems to synchronize their chaotic behaviors. Given two identical chaotic systems that are uncoupled, their behaviors deviate wildly from each other because of the exponetial divergence of nearby initial conditions. It is possible, however, to couple the systems together in a simple way such that the orginal strange attractor is not altered, but the two systems follow the same trajectory on that attractor. The coupling must be based on the differences between the values of corresponding variables in the two systems. When the systems are synchronized, those differences vanish and the two systems follow the chaotic attractor. If the two systems begin to diverge, however, a feedback is generated via the coupling. An appropriately chosen coupling scheme can maintain the synchronized motion. Synchronization is currently being pursued as a novel means for efficient transmission of information through an electronic or optical channel.

IV. HAMILTONIAN SYSTEMS Although most physics textbooks on classical mechanics are largely devoted to the description of Hamiltonian systems in which dissipative, frictional forces can be neglected, such systems are rare in nature. The most important examples arise in celestial mechanics, which describes the motions of planets and stars; accelerator design, which deals with tenuous beams of high-energy charged particles moving in guiding magnetic fields; and the physics of magnetically confined plasmas, which is primarily concerned with the dynamics of trapped electrons and ions in high-temperature fusion devices. Although few in number, these examples are very important. In this section, we will examine three simple examples of classical Hamiltonian systems that exhibit chaotic behavior. The first example is the well-known baker’s transformation, which clearly illustrates the fundamental concepts of chaotic behavior in Hamiltonian systems. Although it has no direct applications to physical problems, the baker’s transformation, like the logistic map, serves as a paradigm for all chaotic Hamiltonian systems. The second example is the standard map, which has direct applications in the description of the behavior of a wide variety of periodically perturbed nonlinear oscillators ranging from particle motion in accelerators and plasma fusion devices

to the irregular rotation of Hyperion, one of the moons of Saturn. Finally, we will consider the H´enon–Heiles model, which corresponds to an autonomous Hamiltonian system with two degrees of freedom, describing, for example, the motion of a particle in a nonaxisymmetric, two-dimensional potential well or the interaction of three nonlinear oscillators (the three-body problem). A. The Baker’s Transformation The description of a Hamiltonian system, like a frictionless mechanical oscillator, requires at least two dependent variables that usually correspond to a generalized position variable and a generalized momentum variable. These variables define a phase space for the mechanical system, and the solutions of the equations of motion describe the motion of a point in the phase space. Starting from the initial conditions specified by an initial point in the 2–d plane, the time evolution generated by the equations of motion trace out a trajectory or orbit. The distinctive feature of Hamiltonian systems is that the areas or volumes of small sets of initial conditions are preserved under the time evolution, in contrast to the dissipative systems, such as the H´enon map or the Lorenz model, where phase-space volumes are contracted. Therefore, Hamiltonian systems are not characterized by attractors, either regular or strange, but the dynamics can nevertheless exhibit the same rich variety of behavior with regular periodic and quasi-periodic cycles and chaos. The simplest Hamiltonian systems correspond to areapreserving maps on the x–y plane. One well-studied example is the so-called baker’s transformation, defined by a pair of difference equations: xn+1 = 2xn , Mod 1 0.5yn yn+1 = 0.5(yn + 1)

(15a) 0 ≤ xn ≤ 0.5 0.5 ≤ xn ≤ 1

(15b)

The action of this map is easy to describe by using the analogy of how a baker kneads dough (hence, the origin of the name of the map). If we take a set of points (xn , yn ) covering the unit square (0 ≤ xn ≤ 1 and 0 ≤ yn ≤ 1), Eq. (15a) requires that each value of xn be doubled so that the square (or dough) is stretched out in the x direction to twice its original length. Then, Eq. (15b) reduces the values of yn by a factor of two and simultaneously cuts the resulting rectangular set of points (or dough) in half at x = 1 and places one piece on top of the other, which returns the dough to its original shape, as shown in Fig. 13. Then, this dynamical process (or kneading) is repeated over and over again. Since area is preserved under each iteration, this dynamical system is Hamiltonian. This can be easily seen

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FIGURE 13 The baker’s transformation takes all of the points in the unit square (the dough), compresses them vertically by a factor of 12 , and stretches them out horizontally by a factor of 2. Then, this rectangular set of points is cut at x = 1, the two resulting rectangles are stacked one on top of the other to return the shape to the unit square, and the transformation is repeated over again. In the process, a “raisin,” indicated schematically by the black dot, wanders chaotically around the unit square.

mathematically if we think of the successive iterates of the baker’s transformation as changes of coordinates from xn , yn to xn−1 , yn+1 . As in the case of the H´enon map, we can analyze the effects of this transformation by evaluating the Jacobian of the coordinate transformation that is the determinant of the matrix:

2 0 M= (16) 0 12 Since J = Det M = 1, we know from elementary integral calculus that volumes are preserved by this change of variables. 1. Chaotic Mixing Starting from a single initial condition (x0 , y0 ), the time evolution will be described by a sequence of points in the plane. (To return to the baking analogy we could imagine

Chaos

that (x0 , y0 ) specifies the initial coordinate of a raisin in the dough.) For almost all initial conditions, the trajectories generated by this simple, deterministic map will be chaotic. Because the evolution of the x coordinate is completely determined by the one-dimensional, chaotic shift map, Eq. (2), the trajectory will move from the right half to the left half of the unit square in a sequence that is indistinguishable from the sequence of heads and tails generated by flipping a coin. Moreover, since the location of the orbit in the upper half or lower half of the unit square is determined by the same random sequence, the successive iterates of the initial point (the raisin) will wander around the unit square in a chaotic fashion. In this simple model, it is easy to see that the mechanism responsible for the chaotic dynamics is the process of stretching and folding of the phase space. In fact, this same stretching and folding lies at the root of all chaotic behavior in both dissipative and Hamiltonian systems. The stretching is responsible for the exponential divergence of nearby trajectories, which is the cause of the extreme sensitivity to initial conditions that characterizes chaotic dynamics. The folding ensures that trajectories return to the initial region of phase space so that the unstable system does not simply explode. Since the stretching only occurs in the x direction for the baker’s transformation, we can easily compute the value of the exponential divergence of nearby trajectories, which is simply the logarithm of the largest eigenvalue of the matrix M. Therefore, the baker’s transformation has a positive Kolmogorov–Sinai entropy, λ = log 2, so that the dynamics satisfy out definition of chaos. B. The Standard Map Our second example of a simple Hamiltonian system that exhibits chaotic behavior is the standard map described by the pair of nonlinear difference equations: xn+1 = xn + yn+1 ,

Mod 2π

(17a)

and yn+1 = yn + k sin xn

Mod 2π

(17b)

Starting from an initial point (x0 , y0 ) on the “2π square,” Eq. (17b) determines the new value of y1 and Eq. (17a) gives the new value of x1 . The behavior of the trajectory generated by successive iterates is determined by the control parameter k, which measures the strength of the nonlinearity. The standard map provides a remarkably good model of a wide variety of physical phenomena that are properly described by systems of nonlinear differential equations (hence, the name standard map). In particular, it serves as a paradigm for the response of all nonlinear oscillators

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FIGURE 14 Successive iterates of the two-dimensional standard map for a number of different initial conditions are displayed for four values of the control parameter k. For small values of k, the orbits trace out smooth, regular curves in the two-dimensional phase space that become more distorted by resonance effects as k increases. For k = 1, the interaction of these resonances has generated visible regions of chaos in which individual trajectories wander over large regions of the phase space. However, for k = 1 and k = 2, the chaotic regions coexist with regular islets of stability associated with strong nonlinear resonances. The boundaries of the chaotic regions are defined by residual KAM curves. For still larger values of k (not shown), these regular islands shrink until they are no longer visible in these figures.

to periodic perturbations. For example, it provides an approximate description of a particle interacting with a broad spectrum of traveling waves, an electron moving in the imperfect magnetic fields of magnetic bottles used to confine fusion plasmas, and the motion of an electron in a highly excited hydrogen atom in the presence of intense electromagnetic fields. In each case, xn and yn correspond to the values of the generalized position and momentum variables, respectively, at discrete times n. Since this model exhibits most of the generic features of Hamiltonian systems that exhibit a transition from regular behavior to chaos, we will examine this example in detail. The standard map actually provides the exact mathematical description for one physical system called the “kicked rotor.” Consider a rigid rotor in the absence of any gravitational or frictional forces that is subject to periodic kicks every unit of time n = 1, 2, 3, . . . . Then, xn and yn describe the angle and the angular velocity (an-

gular momentum) just before the nth kick. The rotor can be kicked either forward or backward depending on the sign of sin xn , and the strengths of the kicks are determined by the value of k. As the nonlinear parameter k is increased, the trajectories generated by this map exhibit a dramatic transition from regular, ordered behavior to chaos. This remarkable transformation is illustrated in Fig. 14, where a number of trajectories are plotted for four different values of k. When k = 0, the value of y remains constant at y0 and the value of xn increases each iteration by the amount y0 (Mod 2π , which means that if xn does not lie on the interval [0, 2π ] we add or subtract 2π until it does). In this case, the motion is regular and the trajectories trace out straight lines in the phase space. The rotor rotates continuously at the constant angular velocity y0 . If y0 is a rational multiple of 2π, then Eq. (17a), like Eq. (1), exhibits a periodic cycle. However, if y0 is an irrational

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658 multiple of 2π, then the dynamics is quasi-periodic for almost all initial values of x0 and the points describing the orbit gradually trace out a solid horizontal line in the phase space. 1. Resonance Islands As k is increased to k = 0.5, most of the orbits remain regular and lie on smooth curves in the phase space; however, elliptical islands begin to appear around the point (π, 0) = (π, 2π ). (Remember, the intrinsic periodicity of the map implies that the top of the 2π square is connected to the bottom and the right-hand side to the left.) These islands correspond to a resonance between the weak periodic kicks and the rotational frequency of the rotor. Consequently, when the kicks and the rotations are synchronous, the rotor is accelerated. However, because it is a nonlinear oscillator (as opposed to a linear, harmonic oscillator), the rotation frequency changes as the velocity increases so that the motion goes out of resonance and therefore the kicks retard the motion and the velocity decreases until the rotation velocity returns to resonance, then this pattern is repeated. The orbits associated with these quasi-periodic cycles of increasing and decreasing angular velocity trace out elliptical paths in the phase space, as shown in Fig. 14. The center of the island, (π, 0), corresponds to a period-1 point of the standard map. (This is easy to check by simply plugging (π, 0) into the right-hand side of Eq. (17).) Figure 14 also shows indications of a smaller island centered at (π, π). Again, it is easy to verify that this point is a member of a period-2 cycle (the other element is the point (2π, π) = (0, π)). In fact, there are resonance islands described by chains of ellipses throughout the phase space associated with periodic orbits of all orders. However, most of these islands are much too small to show up in the graphs displayed in Fig. 14. As the strength of the kicks increases, these islands increase in size and become more prominent. For k = 1, several different resonance island chains are clearly visible corresponding to the period-1, period-2, period-3, and period-7 cycles. However, as the resonance regions increase in size and they begin to overlap, individual trajectories between the resonance regions become confused, and the motion becomes chaotic. These chaotic orbits no longer lie on smooth curves in the phase space but begin to wander about larger and larger areas of the phase space as k is increased. For k = 2, a single orbit wanders over more than half of the phase space, and for k = 5 (not shown), a single orbit would appear to uniformly cover the entire 2π square (although a microscopic examination would always reveal small regular regions near some periodic points).

Chaos

2. The Kolmogorov–Arnold–Moser Theorem Since the periodic orbits and the associated resonance regions are mathematically dense in the phase space (though a set of measure zero), there are always small regions of chaos for any nonzero value of k. However, for small values of k, an important mathematical theorem, called the Kolmogorov–Arnold–Moser (KAM) theorem, guarantees that if the perturbation applied to the integrable Hamiltonian system is sufficiently small, then most of the trajectories will lie on smooth curves, such as those displayed in Fig. 14 for k = 0 and k = 0.5. However, Fig. 14 clearly shows that some of these so-called KAM curves (also called KAM surfaces or invariant tori in higher dimensions) persist for relatively large values of k ∼ 1. The significance of these KAM surfaces is that they form barriers in the phase space. Although these barriers can be circumvented by the slow process of Arnold diffusion in four or more dimensions, they are strictly confining in the two-dimensional phase space of the standard map. This means that orbits starting on one side cannot cross to the other side, and the chaotic regions will be confined by these curves. However, as resonance regions grow with increasing k and begin to overlap, these KAM curves are destroyed and the chaos spreads, as shown in Fig. 14. The critical kc for this onset of global chaos can be estimated analytically using Chirikov’s resonance overlap criteria, which yields an approximate value of kc ≈ 2. However, a more precise value of kc can be determined by a detailed examination of the breakup of the last confining KAM curve. Since the resonance regions associated with low-order periodic orbits are the largest, the last KAM curve to survive is the one furthest from a periodic orbit. This corresponds to an orbit with the most irrational value of average rotation frequency, which is the golden √ mean = ( 5 − 1)/2. Careful numerical studies of the standard map show that the golden mean KAM curve, which is the last smooth curve to divide the top of the phase space from the bottom, is destroyed for k 1 (more precisely for kc = 0.971635406). For k > kc , MacKay et al. (1987) have shown that this last confining curve breaks up into a so-called cantorus, which is a curve filled with gaps resembling a Cantor set. These gaps allow chaotic trajectories to leak through so that single orbits can wander throughout large regions of the phase space, as shown in Fig. 14 for k = 2. 3. Chaotic Diffusion Because of the intrinsic nonlinearity of Eq. (17b), the restriction of the map to the 2π square was only a graphical convenience that exploited the natural periodicities of the map. However, in reality, both the angle variable and the

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angular velocity of a real physical system described by the standard map can take on all real values. In particular, when the golden mean KAM torus is destroyed, the angular velocity associated with the chaotic orbits can wander to arbitrarily large positive and negative values. Because the chaotic evolution of both the angle and angular velocity appears to execute a random walk in the phase space, it is natural to attempt to describe the dynamics using a statistical description despite the fact that the underlying dynamical equations are fully deterministic. In fact, when k kc , careful numerical studies show that the evolution of an ensemble of initial conditions can be well described by a diffusion equation. Consequently, this simple deterministic dynamical system provides an interesting model for studying the problem of the microscopic foundations of statistical mechanics, which is concerned with the question of how the reversible and deterministic equations of classical mechanics can give rise to the irreversible and statistical equations of classical statistical mechanics and thermodynamics.

C. The Henon–Heiles ´ Model Our third example of a Hamiltonian system that exhibits a transition from regular behavior to chaos is described by a system of four coupled, nonlinear differential equations. It was originally introduced by Michel H´enon and Carl Heiles in 1964 as a model of the motion of a star in a nonaxisymmetric, two-dimensional potential corresponding to the mean gravitational field in a galaxy. The equations of motion for the two components of the position and momentum, d x/dt = px

(18a)

dy/dt = p y

(18b)

dpx /dt = −x − 2x y

(18a)

dp y /dt = −y + y 2 − x 2

(18b)

are generated by the Hamiltonian H (x, y, px , p y ) =

p 2y 1 2 px2 1 + + (x + y 2 ) + x 2 y − y 3 2 2 2 3 (19)

where the mass is taken to be unity. Equation 19 corresponds to the Hamiltonian of two uncoupled harmonic oscillators H0 = ( px2 /2) + ( p 2y /2) + 12 (x 2 + y 2 ) (consisting of the sum of the kinetic and a quadratic potential energy) plus a cubic perturbation H1 = x 2 y − 13 y 3 , which provides a nonlinear coupling for the two linear oscillators. Since the Hamiltonian is independent of time, it is a constant of motion that corresponds to the total energy of

the system E = H (x, y, px , p y ). When E is small, both the values of the momenta ( px , p y ) and the positions (x, y) must remain small. Therefore, in the limit E 1, the cubic perturbation can be neglected and the motion will be approximately described by the equations of motion for the unperturbed Hamiltonian, which are easily integrated analytically. Moreover, the application of the KAM theorem to this problem guarantees that as long as E is sufficiently small the motion will remain regular. However, as E is increased, the solutions of the equations of motion, like the orbits generated by the standard map, will become increasingly complicated. First, nonlinear resonances will appear from the coupling of the motions in the x and the y directions. As the energy increases, the effect of the nonlinear coupling grows, the sizes of the resonances grow, and, when they begin to overlap, the orbits begin to exhibit chaotic motion. 1. Poincare´ Sections Although Eq. (18) can be easily integrated numerically for any value of E, it is difficult to graphically display the transition from regular behavior to chaos because the resulting trajectories move in a four-dimensional phase space spanned by x, y, px , and p y . Although we can use the constancy of the energy to reduce the dimension of the accessible phase space to three, the graphs of the resulting three-dimensional trajectories would be even less revealing than the three-dimensional graphs of the Lorenz attractor since there is no attractor to consolidate the dynamics. However, we can simplify the display of the trajectories by exploiting the same device used to relate the H´enon map to the Lorenz model. If we plot the value of px versus x every time the orbit passes through y = 0, then we can construct a Poincar´e section of the trajectory that provides a very clear display of the transition from regular behavior to chaos. Figure 15 displays these Poincare sections for a number of different initial conditions corresponding to three 1 1 different energies, E = 12 , 8 , and 16 . For very small E, most of the trajectories lie on an ellipsoid in fourdimensional phase space, so the intersection of the orbits with the px –x plane traces out simple ellipses centered 1 at (x, px ) = (0, 0). For E = 12 , these ellipses are distorted and island chains associated with the nonlinear resonances between the coupled motions appear; however, most orbits appear to remain on smooth, regular curves. Finally, as E is increased to 18 and 16 , the Poincar´e sections reveal a transition from ordered motion to chaos, similar to that observed in the standard map. In particular, when E = 16 , a single orbit appears to uniformly cover most of the accessible phase space defined by the surface of constant energy in the full four-dimensional

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phase space. Although the dynamics of individual trajectories is very complicated in this case, the average properties of an ensemble of trajectories generated by this deterministic but chaotic dynamical system should be well described using the standard methods of statistical mechanics. For example, we may not be able to predict when a star will move chaotically into a particular region of the galaxy, but the average time that the star spends in that region can be computed by simply measuring the relative volume of the corresponding region of the phase space. D. Applications

FIGURE 15 Poincare´ sections for a number of different orbits generated by the Henon–Heiles ´ equations are plotted for three different values of the energy E. These figure were created by plotting the position of the orbit in the x– px plane each time the solutions of the Henon–Heiles ´ equations passed through y = 0 1 with positive, py . For E = 12 , the effect of the perturbation is small and the orbits resemble the smooth but distorted curves observed in the standard map for small k, with resonance islands associated with coupling of the x and y oscillations. However, as the energy increases and the effects of the nonlinearities become more pronounced, large regions of chaotic dynamics become visible and grow until most of the accessible phase space appears to be chaotic for E = 16 . (These figures can be compared with the less symmetrical Poincare´ sections plotted in the y– py plane that usually appear in the literature).

The earliest applications of the modern ideas of nonlinear dynamics and chaos to Hamiltonian systems were in the field of accelerator design starting in the late 1950s. In order to maintain a beam of charged particles in an accelerator or storage ring, it is important to understand the dynamics of the corresponding Hamiltonian equations of motion for very long times (in some cases, for more than 108 revolutions). For example, the nonlinear resonances associated with the coupling of the radial and vertical oscillations of the beam can be described by models similar to the H´enon–Heiles equations, and the coupling to field oscillations around the accelerator can be approximated by models related to the standard map. In both cases, if the nonlinear coupling or perturbations are too large, the chaotic orbits can cause the beam to defocus and run into the wall. Similar problems arise in the description of magnetically confined electrons and ions in plasma fusion devices. The densities of these thermonuclear plasmas are sufficiently low that the individual particle motions are effectively collisionless on the time scales of the experiments, so dissipation can be neglected. Again, the nonlinear equations describing the motion of the plasma particles can exhibit chaotic behavior that allows the particles to escape from the confining fields. For example, electrons circulating along the guiding magnetic field lines in a toroidal confinement device called a TOKAMAK will feel a periodic perturbation because of slight variations in magnetic fields, which can be described by a model similar to the standard map. When this perturbation is sufficiently large, electron orbits can become chaotic, which leads to an anomalous loss of plasma confinement that poses a serious impediment to the successful design of a fusion reactor. The fact that a high-temperature plasma is effectively collisionless also raises another problem in which chaos actually plays a beneficial role and which goes right to the root of a fundamental problem of the microscopic foundations of statistical mechanics. The problem is how do you heat a collisionless plasma? How do you make an irreversible transfer of energy from an external source,

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such as the injection of a high-energy particle beam or high-intensity electromagnetic radiation, to a reversible, Hamiltonian system? The answer is chaos. For example, the application of intense radio-frequency radiation induces a strong periodic perturbation on the natural oscillatory motion of the plasma particles. Then, if the perturbation is strong enough, the particle motion will become chaotic. Although the motion remains deterministic and reversible, the chaotic trajectories associated with the ensemble of particles can wander over a large region of the phase space, in particular to higher and lower velocities. Since the temperature is a measure of the range of possible velocities, this process causes the plasma temperature to increase. Progress in the understanding of chaotic behavior has also caused a revival of interest in a number of problems related to celestial mechanics. In addition to H´enon and Heiles’ work on stellar dynamics described previously, Jack Wisdom at MIT has recently solved several old puzzles relating to the origin of meteorites and the presence of gaps in the asteroid belt by invoking chaos. Each time an asteroid that initially lies in an orbit between Mars and Jupiter passes the massive planet Jupiter, it feels a gravitational tug. This periodic perturbation on small orbiting asteroids results in a strong resonant interaction when the two frequencies are related by low-order rational numbers. As in the standard map and the H´enon–Heiles model, if this resonant interaction is sufficiently strong, the asteroid motion can become chaotic. The ideal Kepler ellipses begin to precess and elongate until their orbits cross the orbit of Earth. Then, we see them as meteors and meteorites, and the depletion of the asteroid belts leaves gaps that correspond to the observations. The study of chaotic behavior in Hamiltonian systems has also found many recent applications in physical chemistry. Many models similar to the H´enon–Heiles model have been proposed for the description of the interaction of coupled nonlinear oscillators that correspond to atoms in a molecule. The interesting questions here relate to how energy is transferred from one part of the molecule to the other. If the classical dynamics of the interacting atoms is regular, then the transfer of energy is impeded by KAM surfaces, such as those in Figs. 14 and 15. However, if the classical dynamics is fully chaotic, then the molecule may exhibit equipartition of energy as predicted by statistical theories. Even more interesting is the common case where some regions of the phase space are chaotic and some are regular. Since most realistic, classical models of molecules involve more than two degrees of freedom, the unraveling of this complex phase-space structure in six or more dimensions remains a challenging problem. Finally, most recently there has been considerable interest in the classical Hamiltonian dynamics of electrons

in highly excited atoms in the presence of strong magnetic fields and intense electromagnetic radiation. The studies of the regular and chaotic dynamics of these strongly perturbed systems have provided a new understanding of the atomic physics in a realm in which conventional methods of quantum perturbation theory fail. However, these studies of chaos in microscopic systems, like those of molecules, have also raised profound, new questions relating to whether the effects of classical chaos can survive in the quantum world. These issues will be discussed in Section V.

V. QUANTUM CHAOS The discovery that simple nonlinear models of classical dynamical systems can exhibit behavior that is indistinguishable from a random process has naturally raised the question of whether this behavior persists in the quantum realm where the classical nonlinear equations of motion are replaced by the linear Schrodinger equation. This is currently a lively area of research. Although there is general consensus on the key problems, the solutions remain a subject of controversy. In contrast to the subject of classical chaos, there is not even agreement on the definition of quantum chaos. There is only a list of possible symptoms for this poorly characterized disease. In this section, we will briefly discuss the problem of quantum chaos and describe some of the characteristic features of quantum systems that correspond to classically chaotic Hamiltonian systems. Some of these features will be illustrated using a simple model that corresponds to the quantized description of the kicked rotor described in Section IV.B. Then, we will conclude with a description of the comparison of classical and quantum theory with real experiments on highly excited atoms in strong fields. A. The Problem of Quantum Chaos Guided by Bohr’s correspondence principle, it might be natural to conclude that quantum mechanics should agree with the predictions of classical chaos for macroscopic systems. In addition, because chaos has played a fundamental role in improving our understanding of the microscopic foundations of classical statistical mechanics, one would hope that it would play a similar role in shoring up the foundations of quantum statistical mechanics. Unfortunately, quantum mechanics appears to be incapable of exhibiting the strong local instability that defines classical chaos as a mixing system with positive Kolmogorof–Sinai entropy. One way of seeing this difficulty is to note that the Schrodinger equation is a linear equation for the wave

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662 function, and neither the wave function nor any observable quantities (determined by taking expectation values of self-adjoint operators) can exhibit extreme sensitivity to initial conditions. In fact, if the Hamiltonian system is bounded (like the H´enon–Heiles Model), then the quantum mechanical energy spectrum is discrete and the time evolution of all quantum mechanical quantities is doomed to quasiperiodic behavior, such as that Eq. (1). Although the question of the existence of quantum chaos remains a controversial topic, nearly everyone agrees that the most important questions relate to how quantum systems behave when the corresponding classical Hamiltonian systems exhibit chaotic behavior. For example, how does the wave function behave for strongly perturbed oscillators, such as those modeled by the classical standard map, and what are the characteristics of the energy levels for a system of strongly coupled oscillators, such as those described by the H´enon–Heiles model? B. Symptoms of Quantum Chaos Even though the Schr¨odinger equation is a linear equation, the essential nonintegrability of chaotic Hamiltonian systems carries over to the quantum domain. There are no known examples of chaotic classical systems for which the corresponding wave equations can be solved analytically. Consequently, theoretical searches for quantum chaos have also relied heavily on numerical solutions. These detailed numerical studies by physical chemists and physicists studying the dynamics of molecules and the excitation and ionization of atoms in strong fields have led to the identification of several characteristic features of the quantum wave functions and energy levels that reveal the manifestation of chaos in the corresponding classical systems. One of the most studied characteristics of nonintegrable quantum systems that correspond to classically chaotic Hamiltonian systems is the appearance of irregular energy spectra. The energy levels in the hydrogen atom, described classically by regular, elliptical Kepler orbits, form an orderly sequence, E n = −1/(2n 2 ), where n = 1, 2, 3, . . . is the principal quantum number. However, the energy levels of chaotic systems, such as the quantum H´enon–Heiles model, do not appear to have any simple order at large energies that can be expressed in terms of well-defined quantum numbers. This correspondence makes sense since the quantum numbers that define the energy levels of integrable systems are associated with the classical constants of motion (such as angular momentum), which are destroyed by the nonintegrable perturbation. For example, Fig. 16 displays the calculated energy levels for a hydrogen atom in a magnetic field that shows the transition from the regular spectrum at low magnetic fields to an ir-

Chaos

FIGURE 16 The quantum mechanical energy levels for a highly excited hydrogen atom in a strong magnetic field are highly irregular. This figure shows the numerically calculated energy levels as a function of the square of the magnetic field for a range of energies corresponding to quantum states with principal quantum numbers n ≈ 40–50. Because the magnetic field breaks the natural spherical and Coulomb symmetries of the hydrogen atom, the energy levels and associated quantum states exhibit a jumble of multiple avoided crossings caused by level repulsion, which is a common symptom of quantum systems that are classically chaotic. [From Delande, D. (1988). Ph. D. thesis, Universite´ Pierre & Marie Curie, Paris.]

regular spectrum (“spaghetti”) at high fields in which the magnetic forces are comparable to the Coulomb binding fields. This irregular spacing of the quantum energy levels can be conveniently characterized in terms of the statistics of the energy level spacings. For example, Fig. 17 shows a histogram of the energy level spacings, s = E i+1 ∼ E i , for the hydrogen atom in a magnetic field that is strong enough to make most of the classical electron orbits chaotic. Remarkably, this distribution of energy level spacings, P(s), is identical to that found for a much more complicated quantum system with irregular spectra–compound nuclei. Moreover, both distributions are well described by the predictions of random matrix theory, which simply replaces the nonintegrable (or unknown) quantum Hamiltonian with an ensemble of large matrices with random values for the matrix elements. In particular, this distribution of energy level spacings is expected to be given by the Wigner–Dyson distribution, P(s) ∼ s exp(−s 2 ), displayed in Fig. 17. Although these random matrices cannot predict the location of specific energy levels, they do account for many of the statistical features relating to the fluctuations in the energy level spacings. Despite the apparent statistical character of the quantum energy levels for classically chaotic systems, these level spacings are not completely random. If they were

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FIGURE 17 The repulsion of the quantum mechanical energy levels displayed in Fig. 16 results in a distribution of energy level spacings, P(s), in which accidental degeneracies (s = 0) are extremely rare. This figure displays a histogram of the energy level spacings for 1295 levels, such as those in Fig. 16. This distribution compares very well with the Wigner–Dyson distribution (solid curve), which is predicted for the energy level spacing for random matrices. If the energy levels were uncorrelated random numbers, then they would be expected to have a Poisson distribution indicated by the dashed curve. [From Delande. D., and Gay, J. C. (1986). Phys. Rev. Lett. 57, 2006.]

completely uncorrelated, then the spacings statistics would obey a Poison distribution, P(s ) ∼ exp(s), which would predict a much higher probability of nearly degenerate energy levels. The absence of degeneracies in chaotic systems is easily understood because the interaction of all the quantum states induced by the nonintegrable perturbation leads to a repulsion of nearby levels. In addition, the energy levels exhibit an important long-range correlation called spectral rigidity, which means that fluctuations about the average level spacing are relatively small over a wide energy range. Michael Berry has traced this spectral rigidity in the spectra of simple chaotic Hamiltonians to the persistence of regular (but not necessarily stable) periodic orbits in the classical phase space. Remarkably, these sets of measure-zero classical orbits appear to have a dominant influence on the characteristics of the quantum energy levels and quantum states. Experimental studies of the energy levels of Rydberg atoms in strong magnetic fields by Karl Welge and collaborators at the University of Bielefeld appear to have confirmed many of these theoretical and numerical predictions. Unfortunately, the experiments can only resolve a limited range of energy levels, which makes the confirmation of statistical predictions difficult. However, the experimental observations of this symptom of quantum chaos are very suggestive. In addition, the experiments have provided very striking evidence for the important role of classical regular orbits embedded in the chaotic sea of trajectories in determining gross features in the fluctuations in the irregular spectrum. In particular, there appears

to be a one-to-one correspondence between regular oscillations in the spectrum and the periods of the shortest periodic orbits in the classical Hamiltonian system. Although the corresponding classical dynamics of these simple systems is fully chaotic, the quantum mechanics appears to cling to the remnants of regularity. Another symptom of quantum chaos that is more direct is to simply look for quantum behavior that resembles the predictions of classical chaos. In the cases of atoms or molecules in strong electromagnetic fields where classical chaos predicts ionization or dissociation, this symptom is unambiguous. (The patient dies.) However, quantum systems appear to be only capable of mimicking classical chaotic behavior for finite times determined by the density of quantum states (or the size of the quantum numbers). In the case of as few as 50 interacting particles, this break time may exceed the age of the universe, however, for small quantum systems, such as those described by the simple models of Hamiltonian chaos, this time scale, where the Bohr correspondence principle for chaotic systems breaks down, may be accessible to experimental measurements. C. The Quantum Standard Map One model system that has greatly enhanced our understanding of the quantum behavior of classically chaotic systems is the quantum standard map, which was first introduced by Casati et al. in 1979. The Schrodinger equation for the kicked rotor described in Section IV.B also reduces to a map that describes how the wave function (expressed in terms of the unperturbed quantum eigenstates of the rotor) spreads at each kick. Although this map is formally described by an infinite system of linear difference equations, these equations can be solved numerically to good approximation by truncating the set of equations to a large but finite number (typically, ≤1000 states). The comparison of the results of these quantum calculations with the classical results for the evolution of the standard map over a wide range of parameters has revealed a number of striking features. For short times, the quantum evolution resembles the classical dynamics generated by evolving an ensemble of initial conditions with the same initial energy or angular momenta but different initial angles. In particular, when the classical dynamics is chaotic, the quantum mechanical average of the kinetic energy also increases linearly up to a break time where the classical dynamics continue to diffuse in angular velocity but the quantum evolution freezes and eventually exhibits quasi-periodic recurrences to the initial state. Moreover, when the classical mechanics is regular the quantum wave function is also confined by the KAM surfaces for short times but may eventually “tunnel” or leak through.

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This relatively simple example shows that quantum mechanics is capable of stabilizing the dynamics of the classically chaotic systems and destabilizing the regular classical dynamics, depending on the system parameters. In addition, this dramatic quantum suppression of classical chaos in the quantum standard map has been related to the phenomenon of Anderson localization in solid-state physics where an electron in a disordered lattice will remain localized (will not conduct electricity) through destructive quantum interference effects. Although there is no random disorder in the quantum standard map, the classical chaos appears to play the same role. D. Microwave Ionization of Highly Excited Hydrogen Atoms As a consequence of these suggestive results for the quantum standard map, there has been a considerable effort to see whether the manifestations of classical chaos and its suppression by quantum interference effects could be observed experimentally in a real quantum system consisting of a hydrogen atom prepared in a highly excited state that is then exposed to intense microwave fields. Since the experiments can be performed with atoms prepared in states with principal quantum numbers as high as n = 100, one could hope that the dynamics of this electron with a 0.5-µm Bohr radius would be well described by classical dynamics. In the presence of an intense oscillating field, this classical nonlinear oscillator is expected to exhibit a transition to global chaos such as that exhibited by the classical standard map at k ≈ 1. For example, Fig. 18 shows a Poincar´e section of the classical actionangle phase space for a one-dimensional model of a hydrogen atom in an oscillating field for parameters that correspond closely to those of the experiments. For small values of the classical action I , which correspond to low quantum numbers by the Bohr–Somerfeld quantization rule, the perturbing field is much weaker than the Coulomb binding fields and the orbits lie on smooth curves that are bounded by invariant KAM tori. However, for larger values of I , the relative size of the perturbation increases and the orbits become chaotic, filling large regions of phase space and wandering to arbitrarily large values of the action and ionizing. Since these chaotic orbits ionize, the classical theory predicts an ionization mechanism that depends strongly on the intensity of the radiation and only weakly on the frequency, which is just the opposite of the dependence of the traditional photoelectric effect. In fact, this chaotic ionization mechanism was first experimentally observed in the pioneering experiments of Jim Bayfield and Peter Koch in 1974, who observed the sharp onset of ionization in atoms prepared in the n ≈ 66 state, when a 10-GHz microwave field exceeded a critical threshold. Subsequently, the agreement of the predictions

FIGURE 18 This Poincare´ section of the classical dynamics of a one-dimensional hydrogen atom in a strong oscillating electric field was generated by plotting the value of the classical action I and angle θ once every period of the perturbation with strength I 4 F = 0.03 and frequency I 3 = 1.5. In the absence of the perturbations, the action (which corresponds to principal quantum number n by the Bohr–Sommerfeld quantization rule) is a constant of motion. In this case, different initial conditions (corresponding to different quantum states of the hydrogen atom) would trace out horizontal lines in the phase space, such as those in Fig. 14, for the standard map at k = 0. Since the Coulomb binding field decreases as 1/I 4 (or 1/n4 ), the relative strength of the perturbation increases with I . For a fixed value of the perturbing field F, the classical dynamics is regular for small values of I with a prominent nonlinear resonance below I = 1.0. A prominent pair of islands also appears near I = 1.1, but it is surrounded by a chaotic sea. Since the chaotic orbits can wander to arbitrarily high values of the action, they ultimately led to ionization of the atom.

of classical chaos on the quantum measurements has been confirmed for a wide range of parameters corresponding to principal quantum numbers from n = 32 to 90. Figure 19 shows the comparison of the measured thresholds for the onset of ionization with the theoretical predictions for the onset of classical chaos in a one-dimensional model of the experiment. Moreover, detailed numerical studies of the solution of the Schr¨odinger equation for the one-dimensional model have revealed that the quantum mechanism that mimics the onset of classical chaos is the abrupt delocalization of the evolving wave packet when the perturbation exceeds a critical threshold. However, these quantum calculations also showed that in a parameter range just beyond that studied in the original experiments the threshold fields for quantum delocalization would become larger than the classical predictions for the onset of chaotic ionization. This quantum suppression of the classical chaos would be analogous to that observed in the quantum standard map. Very recently, the experiments in this new regime

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• QUANTUM THEORY • TECTONOPHYSICS • VIMECHANICAL

BRATION,

BIBLIOGRAPHY

FIGURE 19 A comparison of the threshold field strengths for the onset of microwave ionization predicted by the classical theory for the onset of chaos (solid curve) with the results of experimental measurements on real hydrogen atoms with n = 32 to 90 (open squares) and with estimates from the numerical solution of the corresponding Schrodinger ¨ equation (crosses). The threshold field strengths are conveniently plotted in terms of the scaled variable n 4 F = I 4 F, which is the ratio of the perturbing field F to the Coulomb binding field 1/n4 versus the scaled frequency n3 = l 3 , which is the ratio of the microwave frequency to the Kepler orbital frequency 1/n3 . The prominent features near rational values of the scaled frequency, n3 = 1, 12 , 13 , and 14 , which appear in both the classical and quantum calculations as well as the experimental measurements, are associated with the presence of nonlinear resonances in the classical phase space.

have been performed, and the experimental evidence supports the theoretical prediction for quantum suppression of classical chaos, although the detailed mechanisms remain a topic of controversy. These experiments and the associated classical and quantum theories are parts of the exploration of the frontiers of a new regime of atomic and molecular physics for strongly interacting and strongly perturbed systems. As our understanding of the dynamics of the simplest quantum systems improves, these studies promise a number of important applications to problems in atomic and molecular physics, physical chemistry, solid-state physics, and nuclear physics.

SEE ALSO THE FOLLOWING ARTICLES ACOUSTIC CHAOS • ATOMIC AND MOLECULAR COLLISIONS • COLLIDER DETECTORS FOR MULTI-TEV PARTICLES • FLUID DYNAMICS • FRACTALS • MATHEMATICAL MODELING • MECHANICS, CLASSICAL • NONLINEAR DY-

Baker, G. L., and Gollub, J. P. (1990). “Chaotic Dynamics: An Introduction,” Cambridge University Press, New York. Berry, M. V. (1983). “Semi-classical mechanics of regular and irregular motion,” In “Chaotic Behavior of Deterministic Systems” (G. Iooss, R. H. G. Helleman, and R. H. G. Stora, eds.), p. 171. North-Holland, Amsterdam. Berry, M. V. (1985). “Semi-classical theory of spectral rigidity,” Proc. R. Soc. Lond. A 400, 229. Bohr, T., Jensen, M. H., Paladin, G., and Vulpiani, A. (1998). “Dynamical Systems Approach to Turbulence,” Cambridge University Press, New York. Campbell, D., ed. (1983). “Order in Chaos, Physica 7D,” Plenum, New York. Casati, G., ed. (1985). “Chaotic Behavior in Quantum Systems,” Plenum, New York. Casati, G., Chirikov, B. V., Shepelyansky, D. L., and Guarneri, I. (1987). “Relevance of classical chaos in quantum mechanics: the hydrogen atom in a monochromatic field,” Phys. Rep. 154, 77. Crutchfield, J. P., Farmer, J. D., Packard, N. H., and Shaw, R. S. (1986). “Chaos,” Sci. Am. 255, 46. Cvitanovic, P., ed. (1984). “Universality in Chaos,” Adam Hilger, Bristol. (This volume contains a collection of the seminal articles by M. Feigenbaum, E. Lorenz, R. M. May, and D. Ruelle, as well as an excellent review by R. H. G. Helleman.) Ford, J. (1983). “How random is a coin toss?” Phys. Today 36, 40. Giannoni, M.-J., Voros, A., and Zinn-Justin, J., eds. (1990). “Chaos and Quantum Physics,” Elsevier Science, London. Gleick, J. (1987). “Chaos: Making of a New Science,” Viking, New York. Gutzwiller, M. C. (1990). “Choas in Classical and Quantum Mechanics,” Springer-Verlag, New York. (This book treats the correspondence between classical chaos and relevant quantum systems in detail, on a rather formal level.) Jensen, R. V. (1987a). “Classical chaos,” Am. Sci. 75, 166. Jensen, R. V. (1987b). “Chaos in atomic physics,” In “Atomic Physics 10” (H. Narami and I. Shimimura, eds.), p. 319, North-Holland, Amsterdam. Jensen, R. V. (1988). “Chaos in atomic physics,” Phys. Today 41, S-30. Jensen, R. V., Susskind, S. M., and Sanders, M. M. (1991). “Chaotic ionization of highly excited hydrogen atoms: comparison of classical and quantum theory with experiment,” Phys. Rep. 201, 1. Lichtenberg, A. J., and Lieberman, M. A. (1983). “Regular and Stochastic Motion,” Springer-Verlag, New York. MacKay, R. S., and Meiss, J. D., eds. (1987). “Hamiltonian Dynamical Systems,” Adam Hilger, Bristol. Mandelbrot, B. B. (1982). “The Fractal Geometry of Nature,” Freeman, San Francisco. Ott, E. (1981). “Strange attractors and chaotic motions off dynamical systems,” Rev. Mod. Phys. 53, 655. Ott, E. (1993). “Chaos in Dynamical Systems,” Cambridge University Press, New York. (This is a comprehensive, self-contained introduction to the subject of chaos, presented at a level appropriate for graduate students and researchers in the physical sciences, mathematics, and engineering.) Physics Today (1985). “Chaotic orbits and spins in the solar system,” Phys. Today 38, 17. Schuster, H. G. (1984). “Deterministic Chaos,” Physik-Verlag, Weinheim, F. R. G.

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I. II. III. IV.

Introduction Geometric Optics Wave Optics Concluding Remarks

GLOSSARY Aberration A perfect lens would produce an image that was a scaled representation of the object; real lenses suffer from defects known as aberrations and measured by aberration coefficients. Cardinal elements The focusing properties of optical components such as lenses are characterized by a set of quantities known as cardinal elements; the most important are the positions of the foci and of the principal planes and the focal lengths. Conjugate Planes are said to be conjugate if a sharp image is formed in one plane of an object situated in the other. Corresponding points in such pairs of planes are also called conjugates. Electron lens A region of space containing a rotationally symmetric electric or magnetic field created by suitably shaped electrodes or coils and magnetic materials is known as a round (electrostatic or magnetic) lens. Other types of lenses have lower symmetry; quadrupole lenses, for example, have planes of symmetry or antisymmetry. Electron prism A region of space containing a field in which a plane but not a straight optic axis can be defined forms a prism.

Image processing Images can be improved in various ways by manipulation in a digital computer or by optical analog techniques; they may contain latent information, which can similarly be extracted, or they may be so complex that a computer is used to reduce the labor of analyzing them. Image processing is conveniently divided into acquisition and coding; enhancement; restoration; and analysis. Optic axis In the optical as opposed to the ballistic study of particle motion in electric and magnetic fields, the behavior of particles that remain in the neighborhood of a central trajectory is studied. This central trajectory is known as the optic axis. Paraxial Remaining in the close vicinity of the optic axis. In the paraxial approximation, all but the lowest order terms in the general equations of motion are neglected, and the distance from the optic axis and the gradient of the trajectories are assumed to be very small. Scanning electron microscope (SEM) Instrument in which a small probe is scanned in a raster over the surface of a specimen and provokes one or several signals, which are then used to create an image on a cathoderay tube or monitor. These signals may be X-ray intensities or secondary electron or backscattered electron currents, and there are several other possibilities.

667

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668 Scanning transmission electron microscope (STEM) As in the scanning electron microscope, a small probe explores the specimen, but the specimen is thin and the signals used to generate the images are detected downstream. The resolution is comparable with that of the transmission electron microscope. Scattering When electrons strike a solid target or pass through a thin object, they are deflected by the local field. They are said to be scattered, elastically if the change of direction is affected with negligible loss of energy, inelastically when the energy loss is appreciable. Transmission electron microscope (TEM) Instrument closely resembling a light microscope in its general principles. A specimen area is suitably illuminated by means of condenser lenses. An objective close to the specimen provides the first stage of magnification, and intermediate and projector lens magnify the image further. Unlike glass lenses, the lens strength can be varied at will, and the total magnification can hence be varied from a few hundred times to hundreds of thousands of times. Either the object plane or the plane in which the diffraction pattern of the object is formed can be made conjugate to the image plane.

OF THE MANY PROBES used to explore the structure of matter, charged particles are among the most versatile. At high energies they are the only tools available to the nuclear physicist; at lower energies, electrons and ions are used for high-resolution microscopy and many related tasks in the physical and life sciences. The behavior of the associated instruments can often be accurately described in the language of optics. When the wavelength associated with the particles is unimportant, geometric optics are applicable and the geometric optical properties of the principal optical components—round lenses, quadrupoles, and prisms—are therefore discussed in detail. Electron microscopes, however, are operated close to their theoretical limit of resolution, and to understand how the image is formed a knowledge of wave optics is essential. The theory is presented and applied to the two families of high-resolution instruments.

I. INTRODUCTION Charged particles in motion are deflected by electric and magnetic fields, and their behavior is described either by the Lorentz equation, which is Newton’s equation of motion modified to include any relativistic effects, or by Schr¨odinger’s equation when spin is negligible. There

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are many devices in which charged particles travel in a restricted zone in the neighborhood of a curve, or axis, which is frequently a straight line, and in the vast majority of these devices, the electric or magnetic fields exhibit some very simple symmetry. It is then possible to describe the deviations of the particle motion by the fields in the familiar language of optics. If the fields are rotationally symmetric about an axis, for example, their effects are closely analogous to those of round glass lenses on light rays. Focusing can be described by cardinal elements, and the associated defects resemble the geometric and chromatic aberrations of the lenses used in light microscopes, telescopes, and other optical instruments. If the fields are not rotationally symmetric but possess planes of symmetry or antisymmetry that intersect along the optic axis, they have an analog in toric lenses, for example the glass lenses in spectacles that correct astigmatism. The other important field configuration is the analog of the glass prism; here the axis is no longer straight but a plane curve, typically a circle, and such fields separate particles of different energy or wavelength just as glass prisms redistribute white light into a spectrum. In these remarks, we have been regarding charged particles as classical particles, obeying Newton’s laws. The mention of wavelength reminds us that their behavior is also governed by Schr¨odinger’s equation, and the resulting description of the propagation of particle beams is needed to discuss the resolution of electron-optical instruments, notably electron microscopes, and indeed any physical effect involving charged particles in which the wavelength is not negligible. Charged-particle optics is still a young subject. The first experiments on electron diffraction were made in the 1920s, shortly after Louis de Broglie associated the notion of wavelength with particles, and in the same decade Hans Busch showed that the effect of a rotationally symmetric magnetic field acting on a beam of electrons traveling close to the symmetry axis could be described in optical terms. The first approximate formula for the focal length was given by Busch in 1926–1927. The fundamental equations and formulas of the subject were derived during the 1930s, with Walter Glaser and Otto Scherzer contributing many original ideas, and by the end of the decade the German Siemens Company had put the first commercial electron microscope with magnetic lenses on the market. The latter was a direct descendant of the prototypes built by Max Knoll, Ernst Ruska, and Bodo von Borries from 1932 onwards. Comparable work on the development of an electrostatic instrument was being done by the AEG Company. Subsequently, several commercial ventures were launched, and French, British, Dutch, Japanese, Swiss,

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American, Czechoslovakian, and Russian electron microscopes appeared on the market as well as the German instruments. These are not the only devices that depend on charged-particle optics, however. Particle accelerators also use electric and magnetic fields to guide the particles being accelerated, but in many cases these fields are not static but dynamic; frequently the current density in the particle beam is very high. Although the traditional optical concepts need not be completely abandoned, they do not provide an adequate representation of all the properties of “heavy” beams, that is, beams in which the current density is so high that interactions between individual particles are important. The use of very high frequencies likewise requires different methods and a new vocabulary that, although known as “dynamic electron optics,” is far removed from the optics of lenses and prisms. This account is confined to the charged-particle optics of static fields or fields that vary so slowly that the static equations can be employed with negligible error (scanning devices); it is likewise restricted to beams in which the current density is so low that interactions between individual particles can be neglected, except in a few local regions (the crossover of electron guns). New devices that exploit charged-particle optics are constantly being added to the family that began with the transmission electron microscope of Knoll and Ruska. Thus, in 1965, the Cambridge Instrument Co. launched the first commercial scanning electron microscope after many years of development under Charles Oatley in the Cambridge University Engineering Department. Here, the image is formed by generating a signal at the specimen by scanning a small electron probe over the latter in a regular pattern and using this signal to modulate the intensity of a cathode-ray tube. Shortly afterward, Albert Crewe of the Argonne National Laboratory and the University of Chicago developed the first scanning transmission electron microscope, which combines all the attractions of a scanning device with the very high resolution of a “conventional” electron microscope. More recently still, fine electron beams have been used for microlithography, for in the quest for microminiaturization of circuits, the wavelength of light set a lower limit on the dimensions attainable. Finally, there are, many devices in which the charged particles are ions of one or many species. Some of these operate on essentially the same principles as their electron counterparts; in others, such as mass spectrometers, the presence of several ion species is intrinsic. The laws that govern the motion of all charged particles are essentially the same, however, and we shall consider mainly electron optics; the equations are applicable to any charged particle, provided that the appropriate mass and charge are inserted.

II. GEOMETRIC OPTICS A. Paraxial Equations Although it is, strictly speaking, true that any beam of charged particles that remains in the vicinity of an arbitrary curve in space can be described in optical language, this is far too general a starting point for our present purposes. Even for light, the optics of systems in which the axis is a skew curve in space, developed for the study of the eye by Allvar Gullstrand and pursued by Constantin Carath´eodory, are little known and rarely used. The same is true of the corresponding theory for particles, developed by G. A. Grinberg and Peter Sturrock. We shall instead consider the other extreme case, in which the axis is straight and any magnetic and electrostatic fields are rotationally symmetric about this axis. 1. Round Lenses We introduce a Cartesian coordinate system in which the z axis coincides with the symmetry axis, and we provisionally denote the transverse axes X and Y . The motion of a charged particle of rest mass m 0 and charge Q in an electrostatic field E and a magnetic field B is then determined by the differential equation (d /dt)(γ m 0 v) = Q(E + v × B) γ = (1 − v 2 /c2 )−1/2 ,

(1)

which represents Newton’s second law modified for relativistic effects (Lorentz equation); v is the velocity. For electrons, we have e = −Q 1.6 × 10−19 C and e/m 0 176 C/µg. Since we are concerned with static fields, the time of arrival of the particles is often of no interest, and it is then preferable to differentiate not with respect to time but with respect to the axial coordinate z. A fairly lengthy calculation yields the trajectory equations d2 X ρ 2 ∂g ∂g = −X dz 2 g ∂X ∂z Qρ Y (Bz + X B X ) − BY (1 + X 2 ) g 2 2 d Y ρ ∂g ∂g = − Y dz 2 g ∂Y ∂z +

+

Qρ −X (Bz + Y BY ) + B X (1 + Y 2 ) (2) g

in which ρ 2 = 1 + X 2 + Y 2 and g = γ m 0 v. By specializing these equations to the various cases of interest, we obtain equations from which the optical properties can be derived by the “trajectory method.” It is well

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known that equations such as Eq. (1) are identical with the Euler–Lagrange equations of a variational principle of the form t1 W = L(r, v, t) dt = extremum (3) t0

provided that t0 , t1 , r(t0 ), and r(t1 ) are held constant. The Lagrangian L has the form L = m 0 c2 [1 − (1 − v 2 /c2 )1/2 ] + Q(v · A − )

(4)

in which and A are the scalar and vector potentials corresponding to E, E = −grad and to B, B = curl A. For static systems with a straight axis, we can rewrite Eq. (3) in the form z1 S= M(x, y, z, x , y ) dz, (5) z0

where M = (1 + X 2 + Y 2 )1/2 g(r) +Q(X A X + Y AY + A z ). The Euler–Lagrange equations, d ∂M ∂M ∂M d ∂M = = ; dz ∂X ∂X dz ∂Y ∂Y

(6)

(7)

again define trajectory equations. A very powerful method of analyzing optical properties is based on a study of the function M and its integral S; this is known as the method of characteristic functions, or eikonal method. We now consider the special case of rotationally symmetric systems in the paraxial approximation; that is, we examine the behavior of charged particles, specifically electrons, that remain very close to the axis. For such particles, the trajectory equations collapse to a simpler form, namely, X +

ηB γ φ γ φ ηB X + X+ Y + Y =0 2φˆ 4φˆ 2φˆ 1/2 φˆ 1/2

ηB γ φ γ φ ηB Y + Y + Y− X − X =0 2φˆ 4φˆ 2φˆ 1/2 φˆ 1/2

(8)

in which φ(z) denotes the distribution of electrostatic ˆ = potential on the optic axis, φ(z) = (0, 0, z); φ(z) φ(z)[1 + eφ(z)/2m 0 c2 ]. Likewise, B(z) denotes the magnetic field distribution on the axis. These equations are coupled, in the sense that X and Y occur in both, but this can be remedied by introducing new coordinate axes x, y, inclined to X and Y at an angle θ (z) that varies with z; x = 0, y = 0 will therefore define not planes but surfaces. By choosing θ (z) such that dθ/dz = ηB/2φˆ 1/2 ;

η = (e/2m 0 )1/2 ,

(9)

FIGURE 1 Paraxial solutions demonstrating image formation.

we find ˆ x + γ φ x /2φˆ + [(γ φ + η2 B 2 )/4φ]/x =0 ˆ y + γ φ y /2φˆ + [(γ φ + η2 B 2 )/4φ]/y = 0.

(10)

These differential equations are linear, homogeneous, and second order. The general solution of either is a linear combination of any two linearly independent solutions, and this fact is alone sufficient to show that the corresponding fields B(z) and potentials φ(z) have an imaging action, as we now show. Consider the particular solution h(z) of Eq. (10) that intersects the axis at z = z 0 and z = z i (Fig. 1). A pencil of rays that intersects the plane z = z o at some point Po (xo , yo ) can be described by x(z) = xo g(z) + λh(z) y(z) = yo g(z) + µh(z)

(11)

in which g(z) is any solution of Eq. (10) that is linearly independent of h(z) such that g(z o ) = 1 and λ, µ are parameters; each member of the pencil corresponds to a different pair of values of λ, µ. In the plane z = z i , we find x(z i ) = xo g(z i );

y(z i ) = yo g(z i )

(12)

for all λ and µ and hence for all rays passing through Po . This is true for every point in the plane z = z o , and hence the latter will be stigmatically imaged in z = z i . Furthermore, both ratios and x(z i )/xo and y(z i )/yo are equal to the constant g(z i ), which means that any pattern of points in z = z o will be reproduced faithfully in the image plane, magnified by this factor g(z i ), which is hence known as the (transverse) magnification and denoted by M. The form of the paraxial equations has numerous other consequences. We have seen that the coordinate frame x– y–z rotates relative to the fixed frame X –Y –Z about the optic axis, with the result that the image will be rotated with respect to the object if magnetic fields are used. In an instrument such as an electron microscope, the image therefore rotates as the magnification is altered, since the latter is affected by altering the strength of the magnetic field and Eq. (9) shows that the angle of rotation is a function of this quantity. Even more important is the fact that the coefficient of the linear term is strictly positive in the

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case of magnetic fields. This implies that the curvature of any solution x(z) is opposite in sign to x(z), with the result that the field always drives the electrons toward the axis; magnetic electron lenses always have a convergent action. The same is true of the overall effect of electrostatic lenses, although the reasoning is not quite so simple. A particular combination of any two linearly independent solutions of Eq. (10) forms the invariant known as the Wronskian. This quantity is defined by φˆ 1/2 (x1 x2 − x1 x2 );

φˆ 1/2 (y1 y2 − y1 y2 )

(13)

Suppose that we select x1 = h and x2 = g, where h(z o ) = h(z i ) = 0 and g(z o ) = 1 so that g(z i ) = M. Then ˆ 1/2 φˆ 1/2 o ho = φi hi M

(14)

The ratio h i / h o is the angular magnification MA and so MMA = (φˆ o /φˆ i )1/2

(15)

or MMA = 1 if the lens has no overall accelerating effect and hence φˆ o = φˆ i . Identifying φ 1/2 with the refractive index, Eq. (15) is the particle analog of the Smith–Helmholtz formula of light optics. Analogs of all the other optical laws can be established; in particular, we find that the longitudinal magnification Ml is given by. Ml = M/MA = (φˆ i /φˆ o )1/2 M 2

(16)

and that Abbe’s sine condition and Herschel’s condition take their familiar forms. We now show that image formation by electron lenses can be characterized with the aid of cardinal elements: foci, focal lengths, and principal planes. First, however, we must explain the novel notions of real and asymptotic imaging. So far, we have simply spoken of rotationally symmetric fields without specifying their distribution in space. Electron lenses are localized regions in which the magnetic or electrostatic field is strong and outside of which the field is weak but, in theory at least, does not vanish. Some typical lens geometries are shown in Fig. 2. If the object and image are far from the lens, in effectively field-free space, or if the object is not a physical specimen but an intermediate image of the latter, the image formation can be analyzed in terms of the asymptotes to rays entering or emerging from the lens region. If, however, the true object or image is immersed within the lens field, as frequently occurs in the case of magnetic lenses, a different method of characterizing the lens properties must be adopted, and we shall speak of real cardinal elements. We consider the asymptotic case first. It is convenient to introduce the solutions of Eq. (10) that satisfy the boundary conditions lim G(z) = 1;

z→−∞

¯ lim G(z) =1

z→∞

(17)

FIGURE 2 Typical electron lenses: (a–c) electrostatic lenses, of which (c) is an einzel lens; (d–e) magnetic lenses of traditional design.

These are rays that arrive at or leave the lens parallel to the axis (Fig. 3). As usual, the general solution is ¯ x(z) = αG(z) + β G(z), where α and β are constants. We denote the emergent asymptote to G(z) thus: lim G(z) = G i (z − z Fi )

z→∞

(18)

¯ We denote the incident asymptote to G(z) thus: ¯ lim G(z) = G¯ o (z − z Fo )

z→−∞

FIGURE 3 Rays G(z ) and G(z ).

(19)

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the emergent asymptote; x = d x/dz. The matrix that appears in this equation is widely used to study systems with many focusing elements; it is known as the (paraxial) transfer matrix and takes slightly different forms for the various elements in use, quadrupoles in particular. We denote the transfer matrix by T . If the planes z 1 and z 2 are conjugate, the point of arrival of a ray in z 2 will vary with the position coordinates of its point of departure in z 1 but will be independent of the gradient at that point. The transfer matrix element T12 must therefore vanish,

FIGURE 4 Focal and principal planes.

Clearly, all rays incident parallel to the axis have emergent asymptotes that intersect at z = z Fi ; this point is known as the asymptotic image focus. It is not difficult to show that the emergent asymptotes to any family of rays that are parallel to one another but not to the axis intersect at a point in the plane z = z Fi . By applying a similar reasoning to ¯ G(z), we recognize that z Fo is the asymptotic object focus. The incident and emergent asymptotes to G(z) intersect in a plane z Pi , which is known as the image principal plane (Fig. 4). The distance between z Fi and z Pi is the asymptotic image focal length: z Fi − z Pi = −1/G i = f i

(20)

We can likewise define z Po and f o : z Po − z Fo = 1/G¯ o = f o

(21)

¯ is constant ¯ − G G) The Wronskian tells us that φˆ 1/2 (G G and so ¯ ˆ 1/2 G i φˆ 1/2 o G o = −φ i

or

1/2 ˆ f o φˆ 1/2 o = fi φi

(22)

In magnetic lenses and electrostatic lenses, that provide no overall acceleration, φˆ o = φˆ i and so f o = f i ; we drop the subscript when no confusion can arise. The coupling between an object space and an image space is conveniently expressed in terms of z Fo , z Fi , f o , and ¯ we see that f i . From the general solution x = αG + β G, lim x(z) = α + β(z − z Fo )/ f o

z→−∞

lim x(z) = −α(z − z Fi )/ f i + β

(23)

(z o − z Fo )(z i − z Fi ) = − f o f i

in which we have replaced z 1 and z 2 by z o and z i to indicate that these are now conjugates (object and image). This is the familiar lens equation in Newtonian form. Writing z Fi = z Pi + f i and z Fo = z Po − f o , we obtain fo fi + =1 z Po − z o z i − z Pi

(26)

the thick-lens form of the regular lens equation. Between conjugates, the matrix T takes the form   M 0 T = 1 (27) fo 1  − fi fi M in which M denotes the asymptotic magnification, the height of the image asymptote to G(z) in the image plane. If, however, the object is a real physical specimen and not a mere intermediate image, the asymptotic cardinal elements cannot in general be used, because the object may well be situated inside the field region and only a part of the field will then contribute to the image formation. Fortunately, objective lenses, in which this situation arises, are normally operated at high magnification with the specimen close to the real object focus, the point at which the ¯ ray G(z) itself intersects the axis [whereas the asymptotic ¯ object focus is the point at which the asymptote to G(z) in object space intersects the optic axis]. The corresponding ¯ real focal length is then defined by the slope of G(z) at the object focus Fo : f = 1/G (Fo ); see Fig. 5.

z→∞

and likewise for y(z). Eliminating α and β, we find   z 2 − z Fi (z 1 − z Fo )(z 2 − z Fi ) fo +  x1 – x2 fi fi   =  x  z − z 1 o Fo x2 1 – fi fo (24) where x1 denotes x(z) in some plane z = z 1 on the incident asymptote and x2 denotes x(z) in some plane z = z 2 on

(25)

FIGURE 5 Real focus and focal length.

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2. Quadrupoles In the foregoing discussion, we have considered only rotationally symmetric fields and have needed only the axial distributions B(z) and φ(z). The other symmetry of greatest practical interest is that associated with electrostatic and magnetic quadrupoles, widely used in particle accelerators. Here, the symmetry is lower, the fields possessing planes of symmetry and antisymmetry only; these planes intersect in the optic axis, and we shall assume forthwith that electrostatic and magnetic quadrupoles are disposed as shown in Fig. 6. The reason for this is simple: The paraxial equations of motion for charged particles traveling through quadrupoles separate into two uncoupled equations only if this choice is adopted. This is not merely a question of mathematical convenience; if quadrupole fields overlap and the total system does not have the symmetry indicated, the desired imaging will not be achieved.

The paraxial equations are now different in the x–z and y–z planes: d ˆ 1/2 γ φ − 2γ p2 + 4ηQ 2 φˆ 1/2 (φ x ) + x =0 dz 4φˆ 1/2 d ˆ 1/2 γ φ + 2γ p2 − 4ηQ 2 φˆ 1/2 y=0 (φ y ) + dz 4φˆ 1/2

(28)

in which we have retained the possible presence of a round electrostatic lens field φ(z). The functions p2 (z) and Q 2 (z) that also appear characterize the quadrupole fields; their meaning is easily seen from the field expansions [for B(z) = 0]: 1

(x, y, z) = φ(z) − (x 2 + y 2 )φ (z) 4 +

1 2 (x + y 2 )2 φ (4) (z) 64

1 1 + (x 2 − y 2 ) p2 (z) − (x 4 + y 4 ) p2 (z) 2 24 +

1 p4 (z)(x 4 − 6x 2 y 2 + y 4 ) + · · · 24

(29)

1 1

(r, ψ, z) = φ(z) − r 2 φ + r 4 φ (4) 4 64

Ax = − Ay =

+

1 1 4 p2r 2 cos 2ψ − p r cos 2ψ 2 24 2

+

1 p4r 4 cos 4ψ + · · · 24

x 2 (x − 3y 2 )Q 2 (z) 12

y 2 (y − 3x 2 )Q 2 (z) 12

(30)

1 1 A z = (x 2 − y 2 )Q 2 (z) − (x 4 − y 4 )Q 2 (z) 2 24 +

FIGURE 6 (a) Magnetic and (b) electrostatic quadrupoles.

1 4 (x − 6x 2 y 2 + y 4 )Q 4 (z) 24

The terms p4 (z) and Q 4 (z) characterize octopole fields, and we shall refer to them briefly in connection with the aberration correction below. It is now necessary to define separate transfer matrices for the x–z plane and for the y–z plane. These have exactly the same form as Eqs. (24) and (27), but we have to distinguish between two sets of cardinal elements. For arbitrary planes z 1 and z 2 , we have

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

T (x)

T (y)

(x) z − z Fi − 2  f xi =  1 − f xi  (y) z 2 − z Fi −  f yi =  1 − f yi

 (x) (x) z 2 − z Fi z 2 − z Fo + f xi   f xo  (x)  z 1 − z Fo f xi  (y) (y) z 2 − z Fi z 1 − z Fo + f yi   f yo · (y)  z 1 − z Fo f yi

(31) Suppose now that z = z xo and z = z xi and conjugate so that (y) (x) = 0; in general, T12 = 0 and so a point in the object T12 plane z = z xo will be imaged as a line parallel to the y axis. Similarly, if we consider a pair of conjugates z = z yo and z = z yi , we obtain a line parallel to the x axis. The imaging is hence astigmatic, and the astigmatic differences in object and image space can be related to the magnification ∧i := z xi − z yi = ∧Fi − f xi Mx + f yi M y ∧i := z xo − z yo = ∧Fo + f xo /Mx − f yo /M y ,

(32)

where (y)

(x) ∧Fi := z Fi − z Fi = ∧i (Mx = M y = 0) (y)

(x) ∧Fo := z Fo − z Fo = ∧o (Mx = M y → ∞).

(33)

Solving the equations ∧i = ∧o = 0 for Mx and M y , we find that there is a pair of object planes for which the image is stigmatic though not free of distortion. 3. Prisms There is an important class of devices in which the optic axis is not straight but a simple curve, almost invariably lying in a plane. The particles remain in the vicinity of this curve, but they experience different focusing forces in the plane and perpendicular to it. In many cases, the axis is a circular arc terminated by straight lines. We consider the situation in which charged particles travel through a magnetic sector field (Fig. 7); for simplicity, we assume that the field falls abruptly to zero at entrance and exit planes (rather than curved surfaces) and that the latter are normal to the optic axis, which is circular. We regard the plane containing the axis as horizontal. The vertical field at the axis is denoted by Bo , and off the axis, B = Bo (r/R)−n in the horizontal plane. It can then be shown, with the notation of Fig. 7, that paraxial trajectory equations of the form x + kv2 x = 0;

y + kH2 y = 0

(34)

describe the particle motion, with kH2 = (1 − n)/R 2 and kv2 = n/R 2 . Since these are identical in appearance with

FIGURE 7 Passage through a sector magnet.

the quadrupole equations but do not have different signs, the particles will be focused in both directions but not in the same “image” plane unless kH = kv and hence n = 12 . The cases n = 0, for which the magnetic field is homogeneous, and n = 12 have been extensively studied. Since prisms are widely used to separate particles of different energy or momentum, the dispersion is an important quantity, and the transfer matrices are usually extended to include this information. In practice, more complex end faces are employed than the simple planes normal to the axis considered here, and the fringing fields cannot be completely neglected, as they are in the sharp cutoff approximation. Electrostatic prisms can be analyzed in a similar way and will not be discussed separately. B. Aberrations 1. Traditional Method The paraxial approximation describes the dominant focusing in the principal electron-optical devices, but this is inevitably perturbed by higher order effects, or aberrations. There are several kinds of aberrations. By retaining higher order terms in the field or potential expansions, we obtain the family of geometric aberrations. By considering small changes in particle energy and lens strength, we obtain the chromatic aberrations. Finally, by examining the effect of small departures from the assumed symmetry of the field, we obtain the parasitic aberrations. All these types of aberrations are conveniently studied by means of perturbation theory. Suppose that we have obtained the paraxial equations as the Euler–Lagrange equations of the paraxial form of M [Eq. (6)], which we denote

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z where Soi(A) denotes zoi M (4) dz, with a similar expression for y (A) . The quantities with superscript (A) indicate the departure from the paraxial approximation, and we write

M (P) . Apart from a trivial change of scale, we have M (P) = −(1/8φˆ 1/2 )(γ φ + η2 B 2 )(x 2 + y 2 ) 1 + φˆ 1/2 (x 2 + y 2 ) 2

(35)

Suppose now that M (P) is perturbed to M (P) + M (A) . The second term M (A) may represent additional terms, neglected in the paraxial approximation, and will then enable us to calculate the geometric aberrations; alternatively, M (A) may measure the change in M (P) when particle energy and lens strength fluctuate, in which case it tells us the chromatic aberration. Other field terms yield the parasitic aberration. We illustrate the use of perturbation theory by considering the geometric aberrations of round lenses. Here, we have 1 M (A) = M (4) = − L 1 (x 2 + y 2 )2 4 1 − L 2 (x 2 + y 2 )(x 2 + y 2 ) 2 1 − L 3 (x 2 + y 2 )2 4

− Qφ with L1 =

2

(x + y )(x y − x y) (36)

2 φ 1 2γ φ η2 B 2 − γ φ (4) + 32φˆ 1/2 φˆ φˆ η4 B 4 + − 4η2 B B φˆ

1 L2 = (γ φ + η2 B 2 ) 8φˆ 1/2 L3 =

1 ˆ 1/2 φ ; 2

P=

η γφ B η2 B 2 + − B 16φˆ 1/2 φˆ φˆ

Q=

ηB ; 4φˆ 1/2

R=

η2 B 2 8φˆ 1/2

and with S (A) =

z z0

(37)

M (4) dz, we can show that

∂ S (A) = px(A) t(z) − x (A) φˆ 1/2 t (z) ∂ xa ∂ S (A) = px(A) s(z) − x (A) φˆ 1/2 s (z) ∂ ya

(38)

where s(z) and t(z) are the solutions of Eq. (10) for which s(z 0 ) = t(z a ) = 1, s(z a ) = t(z 0 ) = 0, and z = z a denotes some aperture plane. Thus, in the image plane, x (A) = −(M/W )∂ Soi(A) /∂ xa

(40)

The remainder of the calculation is lengthy but straightforward. Into M (4) , the paraxial solutions are substituted and the resulting terms are grouped according to their dependence on xo , yo , xa , and ya . We find that S (A) can be written 1 1 1 −S (A) /W = Ero4 + Cra4 + A(V 2 − v 2 ) 4 4 2 1 2 2 + Fro ra + Dro2 V + K ra2 V 2 + v dro2 + kra2 + aV (41)

and

− P φˆ 1/2 (x 2 + y 2 )(x y − x y) 2

yi = y (A) /M = −(1/W ) ∂ Soi(A) /∂ y(a)

with

− R(x y − x y)2 ˆ 1/2

xi = x (A) /M = −(1/W ) ∂ Soi(A) /∂ xa

(39)

ro2 = xo2 + yo2 ;

ra2 = xa2 + ya2

V = xo xa + yo ya ;

v = xo ya − xa yo

(42)

xi = xa Cra2 + 2K V + 2kv + (F − A)ro2 + xo K ra2 + 2AV + av + Dro2 − yo kra2 + aV + dro2 (43) 2 yi = ya Cra + 2K V + 2kv + (F − A)ro2 + xo kra2 + aV + dro2

Each coefficient A, C, . . . , d, k represents a different type of geometric aberration. Although all lenses suffer from every aberration, with the exception of the anisotropic aberrations described by k, a, and d, which are peculiar to magnetic lenses, the various aberrations are of very unequal importance when lenses are used for different purposes. In microscope objectives, for example, the incident electrons are scattered within the specimen and emerge at relatively steep angles to the optic axis (several milliradians or tens of milliradians). Here, it is the spherical (or aperture) aberration C that dominates, and since this aberration does not vanish on the optic axis, being independent of ro , it has an extremely important effect on image quality. Of the geometric aberrations, it is this spherical aberration that determines the resolving power of the electron microscope. In the subsequent lenses of such instruments, the image is progressively enlarged until the final magnification, which may reach 100,000× or 1,000,000×, is attained. Since angular magnification is inversely proportional to transverse magnification, the

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angular spread of the beam in these projector lenses will be tiny, whereas the off-axis distance becomes large. Here, therefore, the distortions D and d are dominant. A characteristic aberration figure is associated with each aberration. This figure is the pattern in the image plane formed by rays from some object point that cross the aperture plane around a circle. For the spherical aberration, this figure is itself a circle, irrespective of the object position, and the effect of this aberration is therefore to blur the image uniformly, each Gaussian image point being replaced by a disk of radius MCra3 . The next most important aberration for objective lenses is the coma, characterized by K and k, which generates the comet-shaped streak from which it takes its name. The coefficients A and F describe Seidel astigmatism and field curvature, respectively; the astigmatism replaces stigmatic imagery by line imagery, two line foci being formed on either side of the Gaussian image plane, while the field curvature causes the image to be formed not on a plane but on a curved image surface. The distortions are more graphically understood by considering their effect on a square grid in the object plane. Such a grid is swollen or shrunk by the isotropic distortion D and warped by the anisotropic distortion d; the latter has been evocatively styled as a pocket handkerchief distortion. Figure 8 illustrates these various aberrations. Each aberration has a large literature, and we confine this account to the spherical aberration, an eternal preoccupation of microscope lens designers. In practice, it is more convenient to define this in terms of angle at the specimen, and recalling that x(z) = xo s(z) + xa t(z), we see that xo = xo s (z o ) + xa t (z o ) Hence, C xi = C xa xa2 + ya2 = 3 xo xo2 + yo2 + · · · to

(44)

and we therefore write Cs = c/to3 so that xi = Cs xo xo2 + yo2 ; yi = Cs yo xo2 + yo2 (45) It is this coefficient Cs that is conventionally quoted and tabulated. A very important and disappointing property of Cs is that it is intrinsically positive: The formula for it can be cast into positive-definite form, which means that we cannot hope to design a round lens free of this aberration by skillful choice of geometry and excitation. This result is known as Scherzer’s theorem. An interesting attempt to upset the theorem was made by Glaser, who tried setting the integrand that occurs in the formula for Cs , and that can be written as the sum of several squared terms, equal to zero and solving the resulting differential equation for the field (in the magnetic case). Alas, the field distribution that emerged was not suitable for image formation, thus confirming the truth of the theorem, but it has been found useful in β-ray spectroscopy. The full implications of the theorem were established by Werner Tretner, who estab-

FIGURE 8 Aberration patterns: (a) spherical aberration; (b) coma; (c–e) distortions.

lished the lower limit for Cs as a function of the practical constraints imposed by electrical breakdown, magnetic saturation, and geometry. Like the cardinal elements, the aberrations of objective lenses require a slightly different treatment from those of condenser lenses and projector lenses. The reason is easily understood: In magnetic objective lenses (and probeforming lenses), the specimen (or target) is commonly immersed deep inside the field and only the field region downstream contributes to the image formation. The

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spherical aberration is likewise generated only by this part of the field, and the expression for Cs as an integral from object plane to image plane reflects this. In other lenses, however, the object is in fact an intermediate image, formed by the next lens upstream, and the whole lens field contributes to the image formation and hence to the aberrations. It is then the coupling between incident and emergent asymptotes that is of interest, and the aberrations are characterized by asymptotic aberration coefficients. These exhibit an interesting property: They can be expressed as polynomials in reciprocal magnification m (m = 1/M), with the coefficients in these polynomials being determined by the lens geometry and excitation and independent of magnification (and hence of object position). This dependence can be written    4 C m  2 K   3  m m     k    2    A  = Q m a  = q  m  (46)      F   1   m  d D 1 in which  x  0  Q= 0  0 0

Q and q have the patterns  x x x x   x x x x x    0 x x x ; q = 0  0 x x x 0 0 0 x x

x x 0

 x  x  , (47) x

where an x indicates that the matrix element is a nonzero quantity determined by the lens geometry and excitation. Turning now to chromatic aberrations, we have m (P) =

∂m (2) ∂m (2) φ + B ∂φ ∂B

(48) 2. Lie Methods

and a straightforward calculation yields γ φo B0 (c) x = −(Cc xo + CD xo − Cθ yo ) −2 B0 φˆ o γ φo B0 y (c) = −(Cc yo + CD yo + Cθ xo ) −2 B0 φˆ o (49) for magnetic lenses or x (c) = −(Cc xo + CD xo )

φo φˆ o

chromatic aberration Cc , which causes a blur in the image that is independent of the position of the object point, like that due to Cs . The coefficient Cc also shares with Cs the property of being intrinsically positive. The coefficients CD and Cθ affect projector lenses, but although they are pure distortions, they may well cause blurring since the term in φo and Bo represents a spread, as in the case of the initial electron energy, or an oscillation typically at main frequency, coming from the power supplies. Although a general theory can be established for the parasitic aberrations, this is much less useful than the theory of the geometric and chromatic aberrations; because the parasitic aberrations are those caused by accidental, unsystematic errors—imperfect roundness of the openings in a round lens, for example, or inhomogeneity of the magnetic material of the yoke of a magnetic lens, or imperfect alignment of the polepieces or electrodes. We therefore merely point out that one of the most important parasitic aberrations is an axial astigmatism due to the weak quadrupole field component associated with ellipticity of the openings. So large is this aberration, even in carefully machined lenses, that microscopes are equipped with a variable weak quadrupole, known as a stigmator, to cancel this unwelcome effect. We will not give details of the aberrations of quadrupoles and prisms here. Quadrupoles have more independent aberrations than round lenses, as their lower symmetry leads us to expect, but these aberrations can be grouped into the same families: aperture aberrations, comas, field curvatures, astigmatisms, and distortions. Since the optic axis is straight, they are third-order aberrations, like those of round lenses, in the sense that the degree of the dependence on xo , xo , yo , and yo is three. The primary aberrations of prisms, on the other hand, are of second order, with the axis now being curved.

(50)

with a similar expression for y (c) for electrostatic lenses. In objective lenses, the dominant aberration is the (axial)

An alternative way of using Hamiltonian mechanics to study the motion of charged particles has been developed, by Alex Dragt and colleagues especially, in which the properties of Lie algebra are exploited. This has come to be known as Lie optics. It has two attractions, one very important for particle optics at high energies (accelerator optics): first, interrelations between aberration coefficients are easy to establish, and second, high-order perturbations can be studied systematically with the aid of computer algebra and, in particular, of the differential algebra developed for the purpose by Martin Berz. At lower energies, the Lie methods provide a useful check of results obtained by the traditional procedures, but at higher energies they give valuable information that would be difficult to obtain in any other way.

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C. Instrumental Optics: Components 1. Guns The range of types of particle sources is very wide, from the simple triode gun with a hairpin-shaped filament relying on thermionic emission to the plasma sources furnishing high-current ion beams. We confine this account to the thermionic and field-emission guns that are used in electron-optical instruments to furnish modest electron currents: thermionic guns with tungsten or lanthanum hexaboride emitters, in which the electron emission is caused by heating the filament, and field-emission guns, in which a very high electric field is applied to a sharply pointed tip (which may also be heated). The current provided by the gun is not the only parameter of interest and is indeed often not the most crucial. For microscope applications, a knowledge of brightness B is much more important; this quantity is a measure of the quality of the beam. Its exact definition requires considerable care, but for our present purposes it is sufficient to say that it is a measure of the current density per unit solid angle in the beam. For a given current, the brightness will be high for a small area of emission and if the emission is confined to a narrow solid angle. In scanning devices, the writing speed and the brightness are interrelated, and the resulting limitation is so severe that the scanning transmission electron microscope (STEM) came into being only with the development of high-brightness field-emission guns. Apart from a change of scale with φˆ 2 /φˆ 1 in accelerating structures, the brightness is a conserved quantity in electronoptical systems (provided that the appropriate definition of brightness is employed). The simplest and still the most widely used electron gun is the triode gun, consisting of a heated filament or cathode, an anode held at a high positive potential relative to the cathode, and, between the two, a control electrode known as the wehnelt. The latter is held at a small negative potential relative to the cathode and serves to define the area of the cathode from which electrons are emitted. The electrons converge to a waist, known as the crossover, which is frequently within the gun itself (Fig. 9). If jc is the current density at the center of this crossover and αs is the angular spread (defined in Fig. 9), then B = jc παs2

(51)

It can be shown that B cannot exceed the Langmuir limit Bmax = jeφ/π kT , in which j is the current density at the filament, φ is the accelerating voltage, k is Boltzmann’s constant (1.4 × 10−23 J/K), and T is the filament temperature. The various properties of the gun vary considerably with the size and position of the wehnelt and anode and the potentials applied to them; the general behavior has

FIGURE 9 Electron gun and formation of the crossover.

been satisfactorily explained in terms of a rather simple model by Rolf Lauer. The crossover is a region in which the current density is high, and frequently high enough for interactions between the beam electrons to be appreciable. A consequence of this is a redistribution of the energy of the particles and, in particular, an increase in the energy spread by a few electron volts. This effect, detected by Hans Boersch in 1954 and named after him, can be understood by estimating the mean interaction using statistical techniques. Another family of thermionic guns has rare-earth boride cathodes, LaB6 in particular. These guns were introduced in an attempt to obtain higher brightness than a traditional thermionic gun could provide, and they are indeed brighter sources; they are technologically somewhat more complex, however. They require a slightly better vacuum than tungsten triode guns, and in the first designs the LaB6 rod was heated indirectly by placing a small heating coil around it; subsequently, however, directly heated designs were developed, which made these guns more attractive for commercial purposes. Even LaB6 guns are not bright enough for the needs of the high-resolution STEM, in which a probe only a few tenths of a nanometer in diameter is raster-scanned over a thin specimen and the transmitted beam is used to form an image (or images). Here, a field-emission gun is indispensable. Such guns consist of a fine tip and two (or more) electrodes, the first of which creates a very high electric field at the tip, while the second accelerates the electrons thus extracted to the desired accelerating voltage. Such guns operate satisfactorily only if the vacuum is very good indeed; the pressure in a field-emission gun must be some five or six orders of magnitude higher than that in a thermionic triode gun. The resulting brightness is

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FIGURE 10 Electrostatic einzel lens design: (A) lens casing; (B and C) insulators.

appreciably higher, but the current is not always sufficient when only a modest magnification is required. We repeat that the guns described above form only one end of the spectrum of particle sources. Others have large flat cathodes. Many are required to produce high currents and current densities, in which case we speak of spacecharge flow; these are the Pierce guns and PIGs (Pierce ion guns). 2. Electrostatic Lenses Round electrostatic lenses take the form of a series of plates in which a round opening has been pierced or circular cylinders all centered on a common axis (Fig. 10). The potentials applied may be all different or, more often, form a simple pattern. The most useful distinction in practice separates lenses that create no overall acceleration of the beam (although, of course, the particles are accelerated and decelerated within the lens field) and those that do produce an overall acceleration or deceleration. In the first case, the usual configuration is the einzel lens, in which the outer two of the three electrodes are held at anode potential (or at the potential of the last electrode of any lens upstream if this is not at anode potential) and the central electrode is held at a different potential. Such lenses were once used in electrostatic microscopes and are still routinely employed when the insensitivity of electrostatic systems to voltage fluctuations that affect all the potentials equally is exploited. Extensive sets of curves and tables describing the properties of such lenses are available. Accelerating lenses with only a few electrodes have also been thoroughly studied; a configuration that is of interest today is the multielectrode accelerator structure. These accelerators are not intended to furnish very high particle energies, for which very different types of accelerator are employed, but rather to accelerate electrons to energies beyond the limit of the simple triode structure, which cannot be operated above ∼150 kV. For microscope and microprobe instruments with accelerating voltages in the range of a few hundred kilovolts up to a few megavolts, therefore, an accelerating structure must be inserted

between the gun and the first condenser lens. This structure is essentially a multielectrode electrostatic lens with the desired accelerating voltage between its terminal electrodes. This point of view is particularly useful when a field-emission gun is employed because of an inconvenient aspect of the optics of such guns: The position and size of the crossover vary with the current emitted. In a thermionic gun, the current is governed essentially by the temperature of the filament and can hence be varied by changing the heating current. In field-emission guns, however, the current is determined by the field at the tip and is hence varied by changing the potential applied to the first electrode, which in turn affects the focusing field inside the gun. When such a gun is followed by an accelerator, it is not easy to achieve a satisfactory match for all emission currents and final accelerating voltages unless both gun and accelerator are treated as optical elements. Miniature lenses and guns and arrays of these are being fabricated, largely to satisfy the needs of nanolithography. A spectacular achievement is the construction of a scanning electron microscope that fits into the hand, no bigger than a pen. The optical principles are the same as for any other lens. 3. Magnetic Lenses There are several kinds of magnetic lenses, but the vast majority have the form of an electromagnet pierced by a circular canal along which the electrons pass. Figure 11 shows such a lens schematically, and Fig. 12 illustrates a more realistic design in some detail. The magnetic flux

FIGURE 11 Typical field distribution in a magnetic lens.

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FIGURE 12 Modern magnetic objective lens design. (Courtesy of Philips, Eindhoven.)

is provided by a coil, which usually carries a low current through a large number of turns; water cooling prevents overheating. The magnetic flux is channeled through an iron yoke and escapes only at the gap, where the yoke is terminated with polepieces of high permeability. This arrangement is chosen because the lens properties will be most favorable if the axial magnetic field is in the form of a high, narrow bell shape (Fig. 11) and the use of a high-permeability alloy at the polepieces enables one to create a strong axial field without saturating the yoke. Considerable care is needed in designing the exact shape of these polepieces, but for a satisfactory choice, the properties of the lens are essentially determined by the gap S, the bore D (or the front and back bores if these are not the same), and the excitation parameter J ; the latter 1/2 is defined by J = NI/φˆ o , where NI is the product of the number of turns of the coil and the current carried by it and φˆ o is the relativistic accelerating voltage; S and D are typically of the order of millimeters and J is a few amperes per (volts)1/2 . The quantity NI can be related to the axial field strength with the aid of Amp`ere’s circuital theorem (Fig. 13); we see that ∞ B(z) dz = µ0 NI so that NI ∝ B0 −∞

the maximum field in the gap, the constant of proportion-

FIGURE 13 Use of Ampere’s ` circuital theorem to relate lens excitation to axial field strength.

ality being determined by the area under the normalized flux distribution B(z)/B0 . Although accurate values of the optical properties of magnetic lenses can be obtained only by numerical methods, in which the field distribution is first calculated by one of the various techniques available—finite differences, finite elements, and boundary elements in particular—their variation can be studied with the aid of field models. The most useful (though not the most accurate) of these is Glaser’s bell-shaped model, which has the merits of simplicity, reasonable accuracy, and, above all, the possibility of expressing all the optical quantities such as focal length, focal distance, the spherical and chromatic aberration coefficients Cs and Cc , and indeed all the third-order aberration coefficients, in closed form, in terms of circular functions. In this model, B(z) is represented by B(z) = B0 (1 + z 2 /a 2 ) (52) and writing w2 = 1 + k 2 , k 2 = η2 B02 a 2 /4φˆ 0 , z = a cot ψ the paraxial equation has the general solution x(ψ) = (A cos ψ + B sin ψ)/ sin ψ

(53)

The focal length and focal distance can be written down immediately, and the integrals that give Cs and Cc can be evaluated explicitly. This model explains very satisfactorily the way in which these quantities vary with the excitation and with the geometric parameter a. The traditional design of Fig. 12 has many minor variations in which the bore diameter is varied and the yoke shape altered, but the optical behavior is not greatly affected. The design adopted is usually a compromise between the optical performance desired and the technological needs of the user. In high-performance systems, the specimen is usually inside the field region and may be inserted either down the upper bore (top entry) or laterally through the gap (side entry). The specimen-holder mechanism requires a certain volume, especially if it is of one of the sophisticated models that permit in situ experiments: specimen heating, to study phase changes in alloys, for example, or specimen cooling to liquid nitrogen or liquid helium temperature, or straining; specimen rotation and tilt are routine requirements of the metallurgist. All this requires space in the gap region, which is further encumbered by a cooling device to protect the specimen from contamination, the stigmator, and the objective aperture drive. The desired optical properties must be achieved subject to the demands on space of all these devices, as far as this is possible. As Ugo Valdr`e has said, the interior of an electron microscope objective should be regarded as a microlaboratory. Magnetic lenses commonly operate at room temperature, but there is some advantage in going to very

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FIGURE 14 Superconducting lens system: (1) objective (shielding lens); (2) intermediate with iron circuit; (3) specimen holder; and (4) corrector device.

low temperature and running in the superconducting regime. Several designs have been explored since Andr`e Laberrigue, Humberto Fern´andez-Mor´an, and Hans Boersch introduced the first superconducting lenses, but only one has survived, the superconducting shielding lens introduced by Isolde Dietrich and colleagues at Siemens (Fig. 14). Here, the entire lens is at a very low temperature, the axial field being produced by a superconducting coil and concentrated into the narrow gap region by superconducting tubes. Owing to the Meissner–Ochsenfeld effect, the magnetic field cannot penetrate the metal of these superconducting tubes and is hence concentrated in the gap. The field is likewise prevented from escaping from the gap by a superconducting shield. Such lenses have been incorporated into a number of microscopes and are particularly useful for studying material that must be examined at extremely low temperatures; organic specimens that are irretrievably damaged by the electron beam at higher temperatures are a striking example. Despite their very different technology, these superconducting lenses have essentially the same optical properties as their warmer counterparts. This is not true of the various magnetic lenses that are grouped under the heading of unconventional designs; these were introduced mainly by Tom Mulvey, although the earliest, the minilens, was devised by Jan Le Poole. The common feature of these lenses, which are extremely varied in appearance, is that the space occupied by the lens is very different in volume or shape from that required by a traditional lens. A substantial reduction in the volume can be achieved by increasing the current density in the coil; in the minilens (Fig. 15), the value may be ∼80 mm2 , whereas in a conventional lens, 2 A/mm2 is a typical figure. Such lenses are employed as auxiliary lenses in zones already occupied by other elements, such as bulky traditional lenses. After the

initial success of these minilenses, a family of miniature lenses came into being, with which it would be possible to reduce the dimensions of the huge, heavy lenses used for very high voltage microscopes (in the megavolt range). Once the conventional design had been questioned, it was natural to inquire whether there was any advantage to be gained by abandoning its symmetric shape. This led to the invention of the pancake lens, flat like a phonograph record, and various single-polepiece or “snorkel” lenses (Fig. 16). These are attractive in situations where the electrons are at the end of their trajectory, and the single-polepiece design of Fig. 16 can be used with a target in front of it or a gun beyond it. Owing to their very flat shape, such lenses, with a bore, can be used to render microscope projector systems free of certain distortions, which are otherwise very difficult to eliminate. This does not exhaust all the types of magnetic lens. For many years, permanent-magnet lenses were investigated in the hope that a simple and inexpensive microscope could be constructed with them. An addition to the family of traditional lenses is the unsymmetric triplepolepiece lens, which offers the same advantages as the single-polepiece designs in the projector system. Magnetic lens studies have also been revivified by the needs of electron beam lithography. 4. Aberration Correction The quest for high resolution has been a persistent preoccupation of microscope designers since these instruments came into being. Scherzer’s theorem (1936) was therefore a very unwelcome result, showing as it did that the principal resolution-limiting aberration could never vanish in

FIGURE 15 Minilens.

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FIGURE 17 Correction of spherical aberration in a scanning transmission electron microscope. (Left) Schematic diagram of the quadrupole–octopole corrector and typical trajectories. (Right) Incorporation of the corrector in the column of a Vacuum Generators STEM. [From Krivanek, O. L., et al. (1997). Institute of Physics Conference Series 153, 35. Copyright IOP Publishing.]

FIGURE 16 Some unconventional magnetic lenses.

round lenses. It was Scherzer again who pointed out (1947) the various ways of circumventing his earlier result by introducing aberration correctors of various kinds. The proof of the theorem required rotational symmetry, static fields, the absence of space charge, and the continuity of certain properties of the electrostatic potential. By relaxing any one of these requirements, aberration correction is in principle possible, but only two approaches have achieved any measure of success. The most promising type of corrector was long believed to be that obtained by departing from rotational symmetry, and it was with such devices that correction was at last successfully achieved in the late 1990s. Such correctors fall into two classes. In the first, quadrupole lenses are employed. These introduce new aperture aberrations, but by adding octopole fields, the combined aberration of the round lens and the quadrupoles can be cancelled. At least four quadrupoles and three octopoles are required.

A corrector based on this principle has been incorporated into a scanning transmission electron microscope by O. Krivanek at the University of Cambridge (Fig. 17). In the second class of corrector, the nonrotationally symmetric elements are sextupoles. A suitable combination of two sextupoles has a spherical aberration similar to that of a round lens but of opposite sign, and the undesirable second-order aberrations cancel out (Fig. 18). The technical difficulties of introducing such a corrector in a high-resolution transmission electron microscope have been overcome by M. Haider (Fig. 19). Quadrupoles and octopoles had seemed the most likely type of corrector to succeed because the disturbance to the existing instrument, already capable of an unaided resolution of a few angstroms, was slight. The family of correctors that employ space charge or charged foils placed across the beam perturb the microscope rather more. Efforts continue to improve lenses by inserting one or more

FIGURE 18 Correction of spherical aberration in a transmission electron microscope. Arrangemnent of round lenses and sextupoles (hexapoles) that forms a semiaplanatic objective lens. The distances are chosen to eliminate radial coma. [From Haider, M., et al. (1995). Optik 99, 167. Copyright Wissenschaftliche Verlagsgesellschaft.]

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FIGURE 20 Foil lens and polepieces of an objective lens to be corrected. [From Hanai, T., et al. (1998). J. Electron Microsc. 47, 185. Copyright Japanese Society of Electron Microscopy.]

FIGURE 19 (a) The corrector of Fig. 18 incorporated in a transmission electron microscope. (b) The phase contrast transfer function of the corrected microscope. Dashed line: no correction. Full line: corrector switched on, energy width (a measure of the temporal coherence) 0.7 eV. Dotted line: energy width 0.2 eV. Chromatic aberration remains a problem, and the full benefit of the corrector is obtained only if the energy width is very narrow. [From Haider, M., et al. (1998). J. Electron Microsc. 47, 395. Copyright Japanese Society of Electron Microscopy.]

foils in the path of the electrons, with a certain measure of success, but doubts still persist about this method. Even if a reduction in total Cs is achieved, the foil must have a finite thickness and will inevitably scatter the electrons traversing it. How is this scattering to be separated from that due to the specimen? Figure 20 shows the design employed in an ongoing Japanese project. An even more radical solution involves replacing the static objective lens by one or more microwave cavities. In Scherzer’s original proposal, the incident electron beam was broken into short pulses and the electrons far from the axis would hence arrive at the lens slightly later than those traveling near the axis. By arranging that the axial electrons encounter the maximum field so that the peripheral electrons experience a weaker field, Scherzer argued, the effect of Cs could be eliminated since, in static lenses, the peripheral electrons are too strongly focused. Unfortunately, when we insert realistic figures into the corresponding equations, we find that the necessary frequency is in the gigahertz range, with the result that the electrons spend a substantial part of a cycle, or more than a cycle, within the microwave field. Although this means that the simple explanation is inadequate, it does not invalidate the principle, and experiment and theory both show that microwave cavity lenses can have positive or negative spherical aberration coefficients. The principal obstacles to their use are the need to produce very short pulses containing sufficient current and, above all, the fact that the beam emerging from such cavity lenses has a rather large energy

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electrostatic designs can be combined in many ways, of which we can mention only a small selection. A very ingenious arrangement, which combines magnetic deflection with an electrostatic mirror, is the Castaing–Henry analyzer (Figs. 23a–23c) which has the constructional convenience that the incident and emergent optic axes are in line; its optical properties are such that an energy-filtered image or an energy spectrum from a selected area can be obtained. A natural extension of this is the magnetic filter (Fig. 23d), in which the mirror is suppressed; if the particle energy is not too high, use of the electrostatic analog of this can be envisaged (Fig. 23e). It is possible to eliminate many of the aberrations of such filters by arranging the system not only symmetrically about the mid-plane (x − x in Fig. 23d), but also antisymmetrically about the planes midway between the mid-plane and the optic axis. A vast number of prism combinations have been explored by Veniamin Kel’man and colleagues in Alma-Ata in the quest for high-performance mass and electron spectrometers. Energy analysis is a subject in itself, and we can do no more than mention various other kinds of energy or FIGURE 21 Microwave cavity lens between the polepieces of a magnetic lens. (Courtesy of L. C. Oldfield.)

spread, which makes further magnification a problem. An example is shown in Fig. 21. Finally, we mention the possibility of a posteriori correction in which we accept the deleterious effect of Cs on the recorded micrograph but attempt to reduce or eliminate it by subsequent digital or analog processing of the image. A knowledge of the wave theory of electron image formation is needed to understand this idea and we therefore defer discussion of it to Section III.B. 5. Prisms, Mirrors, and Energy Analyzers Magnetic and electrostatic prisms and systems built up from these are used mainly for their dispersive properties in particle optics. We have not yet encountered electron mirrors, but we mention them here because a mirror action is associated with some prisms; if electrons encounter a potential barrier that is high enough to halt them, they will be reflected and a paraxial optical formalism can be developed to describe such mirror optics. This is less straightforward than for lenses, since the ray gradient is far from small at the turning point, which means that one of the usual paraxial assumptions that off-axis distance and ray gradient are everywhere small is no longer justified. The simplest magnetic prisms, as we have seen, are sector fields created by magnets of the C-type or pictureframe arrangement (Fig. 22) with circular poles or sector poles with a sector or rectangular yoke. These or analogous

FIGURE 22 (a) C-Type and (b) picture-frame magnets URE typically having (c) sector-shaped yoke and poles.

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FIGURE 23 Analyzers: (a–c) Castaing–Henry analyzer; (d) filter; and (e) electrostatic analog of the filter.

685

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lens” (SOL) have been developed, in which the target lies within the field region.

III. WAVE OPTICS A. Wave Propagation The starting point here is not the Newton–Lorentz equations but Schr¨odinger’s equation; we shall use the nonrelativistic form, which can be trivially extended to include relativistic effects for magnetic lenses. Spin is thus neglected, which is entirely justifiable in the vast majority of practical situations. The full Schr¨odinger equation takes the form − FIGURE 24 Mollenstedt ¨ analyzer.

momentum analyzers. The Wien filter consists of crossed electrostatic and magnetic fields, through which particles of a particular energy will pass undeflected, whereas all others will be deviated from their path. The early β-ray spectrometers exploited the fact that the chromatic aberration of a lens causes particles of different energies to be focused in different planes. The M¨ollenstedt analyzer is based on the fact that rays in an electrostatic lens far from the axis are rapidly separated if their energies are different (Fig. 24). The Ichinokawa analyzer is the magnetic analog of this and is used at higher accelerating voltages where electrostatic lenses are no longer practicable. In retardingfield analyzers, a potential barrier is placed in the path of the electrons and the current recorded as the barrier is progressively lowered.

h2 2 eh ∇ + A · grad m0 im 0 e2 2 ∂ + −e + =0 A − ih 2m 0 ∂t

and writing (x, y, z, t) = ψ(x, y, z)e−iωt

(55)

we obtain −

h2 2 eh ∇ ψ+ A · grad ψ 2m 0 im 0 e2 2 + −e + A ψ = Eψ 2m 0

(56)

with E = hω

(57)

where h = h/2π and h is Planck’s constant. The freespace solution corresponds to

6. Combined Deflection and Focusing Devices In the quest for microminiaturization, electron beam lithography has acquired considerable importance. It proves to be advantageous to include focusing and deflecting fields within the same volume, and the optical properties of such combined devices have hence been thoroughly studied, particularly, their aberrations. It is important to keep the adverse effect of these aberrations small, especially because the beam must be deflected far from the original optical axis. An ingenious way of achieving this, proposed by Hajime, Ohiwa, is to arrange that the optic axis effectively shifts parallel to itself as the deflecting field is applied; for this, appropriate additional deflection, round and multipole fields must be superimposed and the result may be regarded as a “moving objective lens” (MOL) or “variable-axis lens” (VAL). Perfected immersion versions of these and of the “swinging objective

(54)

p = h/λ or 1/2

λ = h/(2em 0 φ0 )1/2 ≈ 12.5/φ0

(58)

where p is the momentum. As in the case of geometric optics, we consider the paraxial approximation, which for the Schr¨odinger equation takes the form 2 ∂ ψ 1 ∂ 2ψ −h 2 + em 0 (φ + η2 B 2 )(x 2 + y 2 )ψ + 2 2 ∂x ∂y 2 ∂ψ =0 ∂z and we seek a wavelike solution: −i hp ψ − 2i hp

ψ(x, y, z) = a(z) exp[i S(x, y, z)/ h].

(59)

(60)

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After some calculation, we obtain the required equation describing the propagation of the wave function through electrostatic and magnetic fields: 3/2 p0 i pg (z) 2 2 ψ(x, y, z) = (x exp + y ) 2πi hh(z) p 1/2 2 hg(z)

∞ ×

ψ(xo , yo , z o ) exp

−∞

× (x − xo g)2 + (y − yo g) d xo dyo (61) This extremely important equation is the basis for all that follows. In it, g(z) and h(z) now denote the solutions of the geometric paraxial equations satisfying the boundary conditions g(z o ) = h (z o ) = 1, g (z o ) = h(z o ) = 0. Reorganizing the various terms, Eq. (61) can be written

× exp

∞ ψ(xo , yo , z o ) −∞

iπ g(z) xo2 + yo2 λh(z)

− 2(xo x + yo y)

+ r h (z)(x + y ) 2

2

d xo dyo

(62)

with λ = h/ po and r = p/ po = (φ/φo )1/2 . Let us consider the plane z = z d in which g(z) vanishes, g(z d ) = 0. For the magnetic case (r = 1), we find Ed ψ(xd , yd , z d ) = ψ(xo , yo , z o ) iλh(z o ) 2i × exp − (xo xd + yo yd ) d xo dyo λh(z d ) (63) with E d = exp[iπ h (z d )(xd2 + yd2 )/λh(z d )], so that, scale factors apart, the wave function in this plane is the Fourier transform of the same function in the object plane. We now consider the relation between the wave function in the object plane and in the image plane z = z i conjugate to this, in which h(z) vanishes: h(z i ) = 0. It is convenient to calculate this in two stages, first relating ψ(xi , yi , z i ) to the wave function in the exit pupil plane of the lens, ψ(xa , ya , z a ) and then calculating the latter with the aid of Eq. (62). Introducing the paraxial solutions G(z), H (z) such that G(z a ) = H (z a ) = 1;

1 ψ(xa , ya , z a ) ψ(xi , yi , z i ) = iλH (z i ) iπ × exp G(z i ) xa2 + ya2 λH (z) − 2(xa xi + ya yi )

i po 2 hg(z)h(z) 2

1 ψ(x, y, z) = iλr h(z)

we have

G (z a ) = H (z a ) = 0

+ H (z i )

xi2

+

yi2

d xa dya

(64)

Mψ(xi , yi , z i )E i = ψ(xo , yo , z o )K (xi , yi ; xo , yo )E o d xo dyo

(65)

Using Eq. (62), we find

where M is the magnification, M = g(z i ), and iπ ga h i − gi h a 2 E i = exp xi + yi2 λM ha iπga 2 E o = exp xo + yo2 λh a

(66)

These quadratic factors are of little practical consequence; they measure the curvature of the wave surface arriving at the specimen and at the image. If the diffraction pattern plane coincides with the exit pupil, then E o = 1. We write h(z a ) = f since this quantity is in practice close to the focal length, so that for the case z d = z a , iπgi 2 2 E i = exp − (67) x + yi λM i The most important quantity in Eq. (65) is the function K (xi , yi ; xo , yo ), which is given by 1 K (x, y; xo , yo ) = 2 2 A(xa , ya ) λ f 2πi x × exp − xo − xa λf M y ya d xa dya (68) + yo − M or introducing the spatial frequency components ξ = xa /λ f ; we find K (x, y; xo , yo ) =

η = ya /λ f

(69)

A(λ f ξ, λ f η)

x × exp −2πi xo − ξ M y + yo − η dξ dη (70) M

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In the paraxial approximation, the aperture function A is simply a mathematical device defining the area of integration in the aperture plane: A = 1 inside the pupil and A = 0 outside the pupil. If we wish to include the effect of geometric aberrations, however, we can represent them as a phase shift of the electron wave function at the exit pupil. Thus, if the lens suffers from spherical aberration, we write A(xa , ya ) = a(xa , ya ) exp[−iγ (xa , ya )] (71) in which 2π γ = λ

2 xa + ya2 2 1 xa2 + ya2 1 Cs − 4 f2 2 f2

πλ (72) Cs λ2 (ξ 2 + η2 )2 − 2(ξ 2 + η2 ) 2 the last term in allowing for any defocus, that is, any small difference between the object plane and the plane conjugate to the image plane. All the third-order geometric aberrations can be included in the phase shift γ , but we consider only Cs and the defocus . This limitation is justified by the fact that Cs is the dominant aberration of objective lenses and proves to be extremely convenient because Eq. (65) relating the image and object wave functions then has the form of a convolution, which it loses if other aberrations are retained (although coma can be accommodated rather uncomfortably). It is now the amplitude function a(xa , ya ) that represents the physical pupil, being equal to unity inside the opening and zero elsewhere. In the light of all this, we rewrite Eq. (65) as 1 yi xi E i ψ(xi , yi , z i ) = K − xo , − yo E o M M M =

×ψo (xo , yo , z o ) d xo dyo

(73)

Defining the Fourier transforms of ψo , ψi , and K as follows, ψ˜ o (ξ, η) =

E o ψo

× exp[−2πi(ξ xo + ηyo )] d xo dyo ψ˜ o (ξ, η) = E i ψi (M xi , M yi ) × exp[−2πi(ξ xi + ηyi )] d xi dyi 1 = 2 E i ψi (xi , yi ) M (ξ xi + ηyi ) × exp −2πi d xi dyi M K˜ (ξ, η) = K (x, y) × exp[−2πi(ξ x + ηy)] d x d y

(74)

in which small departures from the conventional definitions have been introduced to assimilate inconvenient factors, Eq. (65) becomes 1 ˜ (75) K (ξ, η)ψ˜ o (ξ, η) ψ˜ i (ξ, η) = M This relation is central to the comprehension of electron-optical image-forming instruments, for it tells us that the formation of an image may be regarded as a filtering operation. If K˜ were equal to unity, the image wave function would be identical with the object wave function, appropriately magnified; but in reality K˜ is not unity and different spatial frequencies of the wave leaving the specimen, ψ(xo , yo , z o ), are transferred to the image with different weights. Some may be suppressed, some attenuated, some may have their sign reversed, and some, fortunately, pass through the filter unaffected. The notion of spatial frequency is the spatial analog of the temporal frequency, and we associate high spatial frequencies with fine detail and low frequencies with coarse detail; the exact interpretation is in terms of the fourier transform, as we have seen. We shall use Eqs. (73) and (75) to study image formation in two types of optical instruments, the transmission electron microscope (TEM) and its scanning transmission counterpart, the STEM. This is the subject of the next section. B. Instrumental Optics: Microscopes The conventional electron microscope (TEM) consists of a source, condenser lenses to illuminate a limited area of the specimen, an objective to provide the first stage of magnification beyond the specimen, and projector lenses, which magnify the first intermediate image or, in diffraction conditions, the pattern formed in the plane denoted by z = z d in the preceding section. In the STEM, the role of the condenser lenses is to demagnify the crossover so that a very small electron probe is formed on the specimen. Scanning coils move this probe over the surface of the latter in a regular raster, and detectors downstream measure the current transmitted. There are inevitably several detectors, because transmission microscope specimens are essentially transparent to electrons, and thus there is no diminution of the total current but there is a redistribution of the directions of motion present in the beam. Electronoptical specimens deflect electrons but do not absorb them. In the language of light optics, they are phase specimens, and the electron microscope possesses means of converting invisible phase variations ot amplitude variations that the eye can see. We now examine image formation in the TEM in more detail. We first assume, and it is a very reasonable first approximation, that the specimen is illuminated by a parallel

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uniform beam of electrons or, in other words, that the wave incident on the specimen is a constant. We represent the effect of the specimen on this wave by a multiplicative specimen transparency function S(xo , yo ), which is a satisfactory model for the very thin specimens employed for highresolution work and for many other specimens. This specimen transparency is a complex function, and we write S(xo , yo ) = [1 − s(xo , yo )] exp[iϕ(xo , yo )] (76a) = 1 − s + iϕ

(76b)

for small values of s and ϕ. The real term s requires some explanation, for our earlier remarks suggest that s must vanish if no electrons are halted by the specimen. we retain the term in s for two reasons. First, some electrons may be scattered inelastically in the specimen, in which case they must be regarded as lost in this simple monochromatic and hence monoenergetic version of the theory. Second, all but linear terms have been neglected in the approximate expression (76b) and, if necessary, the next-higher-order term (− 12 ϕ 2 ) can be represented by s. The wave leaving the specimen is now proportional to S normalizing, so that the constant of proportionality is unity; after we substitute ψ(xo , yo , z o ) = 1 − s + iϕ

(77)

into Eq. (75). Again denoting Fourier transforms by the tilde, we have 1 ˜ ψ˜ i (ξ, η) = K (ξ, η)[δ(ξ, η) − s˜ (ξ, η) + i ϕ(ξ, ˜ η)] M 1 = a exp(−iγ )(δ − s˜ + i ϕ) ˜ (78) M and hence ψi (M xi , M yi ) =

1 a exp(−iγ )(δ − s˜ + i ϕ) ˜ M × exp[2πi(ξ xi + ηyi )] dξ dη (79)

The current density at the image, which is what we see on the fluorescent screen of the microscope and record on film, is proportional to ψi ψi∗ . We find that if both ϕ and s are small, M

2

ψi ψi∗

∞ ≈ 1−2

a s˜ cos γ

−∞

× exp[2πi(ξ x + ηy)] dξ dη ∞ +2

a ϕ˜ sin γ

−∞

× exp[2πi(ξ x + ηy)] dξ dη

(80)

FIGURE 25 Function sin γ at Scherzer defocus = (Cs λ)1/2 .

and writing j = M 2 ψi ψi∗ and C = j − 1, we see that C˜ = −2a s˜ cos γ + 2a ϕ˜ sin γ

(81)

This justifies our earlier qualitative description of image formation as a filter process. Here we see that the two families of spatial frequencies characterizing the specimen, ϕ˜ and s˜ , are distorted before they reach the image by the linear filters cos γ and sin γ . The latter is by far the more important. A typical example is shown in Fig. 25. The distribution 2 sin γ can be observed directly by examining an amorphous phase specimen, a very thin carbon film, for example. The spatial frequency spectrum of such a specimen is fairly uniform over a wide range of frequencies so that C˜ ∞ sin γ . A typical spectrum is shown in Fig. 26, in which the radial intensity distribution is proportional to sin2 γ . Such spectra can be used to estimate the defocus and the coefficient Cs very accurately. The foregoing discussion is idealized in two respects, both serious in practice. First, the illuminating beam has been assumed to be perfectly monochromatic, whereas in reality there will be a spread of wavelengths of several parts per million; in addition, the wave incident on the specimen has been regarded as a uniform plane wave, which is equivalent to saying that it originated in an ideal ponint source. Real sources, of course, have a finite size, and the single plane wave should therefore be replaced by a spectrum of plane waves incident at a range of small angles to the specimen. The step from point source and monochromatic beam to finite source size and finite wavelength spread is equivalent to replacing perfectly coherent illumination by partially coherent radiation, with the wavelength spread corresponding to temporal partial coherence and the finite source size corresponding to spatial partial coherence. (We cannot discuss the legitimacy of separating these effects here, but simply state that this is almost always permissible.) Each can be represented by an envelope function, which multiplies the coherent transfer functions sin γ and cos γ . This is easily seen for the temporal spatial coherence. Let us associate a probability distribution H ( f ), H ( f ) d f = 1, with the current density at each point in the image, the argument f being some

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FIGURE 26 Spatial frequency spectrum (right) of an amorphous phase specimen (left).

convenient measure of the energy variation in the beam incident on the specimen. Hence, d j/j = H ( f ) d f . From Eq. (80), we find j = 1 − a s˜ Ts exp[2πi(ξ x + ηy)] d ξ d η +

a ϕT ˜ ϕ exp[2πi(ξ x + ηy)] d ξ d η

where

microscope. Beyond the first zero of the function, information is no longer transferred faithfully, but in the first zone the transfer is reasonably correct until the curve begins to dip toward zero for certain privileged values of the defocus, = (Cs λ)1/2 , (3Cs λ)1/2 , and (5Cs λ)1/2 ; for the first of these values, known as the Scherzer defocus,

(82)

Ts = 2

cos γ (ξ, η, f )H ( f ) d f

Tϕ = 2

(83) sin γ (ξ, η, f )H ( f ) d f

and if f is a measure of the defocus variation associated with the energy spread, we may set equal to o + f , giving Ts = 2 cos γ H ( f ) cos[πλ f (ξ 2 + η2 )] d f (84) Tϕ = 2 sin γ H ( f ) cos[πλ f (ξ 2 + η2 )] d f if H ( f ) is even, and a slightly longer expression when it is not. The familiar sin γ and cos γ are thus clearly seen to be modulated by an envelope function, which is essentially the Fourier transform of H ( f ). A similar result can be obtained for the effect of spatial partial coherence, but the demonstration is longer. Some typical envelope functions are shown in Fig. 27. An important feature of the function sin γ is that it gives us a convenient measure of the resolution of the

FIGURE 27 Envelope functions characterizing (a) spatial and (b) temporal partial coherence.

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the zero occurs at the spatial frequency (Cs λ3 )−1/4 ; the reciprocal of this multiplied by one of various factors has long been regarded as the resolution limit of the electron microscope, but transfer function theory enables us to understand the content of the image in the vicinity of the limit in much greater detail. The arrival of commercial electron microscopes equipped with spherical aberration correctors is having a profound influence on the practical exploitation of transfer theory. Hitherto, the effect of spherical aberration dictated the mode of operation of the TEM when the highest resolution was required. When the coefficient of spherical aberration has been rendered very small by correction, this defect is no longer the limiting factor and other modes of operation become of interest. We now turn to the STEM. Here a bright source, typically a field-emission gun, is focused onto the specimen; the small probe is scanned over the surface and, well beyond the specimen, a far-field diffraction pattern of each elementary object area is formed. This pattern is sampled by a structured detector, which in the simplest case consists of a plate with a hole in the center, behind which is another plate or, more commonly, an energy analyzer. The signals from the various detectors are displayed on cathode-ray tubes, locked in synchronism with the scanning coils of the microscope. The reason for this combination of annular detector and central detector is to be found in the laws describing electron scattering. The electrons incident on a thin specimen may pass through unaffected; or they may be deflected with virtually no transfer of energy to the specimen, in which case they are said to be elastically scattered; or they may be deflected and lose energy, in which case they are inelastically scattered. The important point is that, on average, inelastically scattered electrons are deflected through smaller angles than those scattered elastically, with the result that the annular detector receives mostly elastically scattered particles, whereas the central detector collects those that have suffered inelastic collisions. The latter therefore have a range of energies, which can be separated by means of an energy analyzer, and we could, for example, form an image with the electrons corresponding to the most probable energy loss for some particular chemical element of interest. Another imaging mode exploits electrons that have been Rutherford scattered through rather large angles. These modes of STEM image formation and others that we shall meet below can be explained in terms of a transfer function theory analogous to that derived for the TEM. This is not surprising, for many of the properties of the STEM can be understood by regarding it as an inverted TEM, the TEM gun corresponding to the small central

detector of the STEM and the large recording area of the TEM to the source in the STEM, spread out over a large zone if we project back the scanning probe. We will not pursue this analogy here, but most texts on the STEM explore it in some detail. Consider now a probe centered on a point xo = ξ in the specimen plane of the STEM. We shall use a vector notation here, so that xo = (xo , yo ), and similarly for other coordinates. The wave emerging from the specimen will be given by ψ(xo ; ξ) = S(xo )K (ξ − xo )

(85)

in which S(xo ) is again the specimen transparency and K describes the incident wave and, in particular, the effect of the pupil size, defocus, and aberrations of the probeforming lens, the last member of the condenser system. Far below the specimen, in the detector plane (subscript d) the wave function is given by ψd (xd , ξ) = S(xo )K (ξ − xo ) × exp(−2πixd · xo /λR) dxo

(86)

in which R is a measure of the effective distance between the specimen and the detector. The shape of the detector (and its response if this is not uniform) can most easily be expressed by introducing a detector function D(xd ), equal to zero outside the detector and equal to its response, usually uniform, over its surface. The detector records incident current, and the signal generated is therefore proportional to jd (ξ) = |ψd (xd ; ξ)|2 D(xd ) dxd =

S(xo )S ∗ (xo )K (ξ − xo )K ∗ (ξ − xo )

× exp[−2πixd · (xo − x o )/λR] ×D(xd ) dxo dxo dxd

(87)

or introducing the Fourier transform of the detector response, jd (ξ) = S(xo )S ∗ xo )K (ξ − xo )K ∗ (ξ − xo ) ˜ ×D

xo − xo λR

dxo dxo

(88)

We shall use the formula below to analyze the signals collected by the simpler detectors, but first we derive the STEM analog of the filter Eq. (81). For this we introduce

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the Fourier transforms of S and K into the expression for ψd (xd , ξ). Setting u = xd /λR, we obtain ψd (λRu; ξ) = S(xo )K (ξ − xo ) exp(−2πiu · xo ) dxo ) =

˜ K˜ (q) S(p) × exp[−2πixo · (u − p + q)]

× exp[(−2πiq · ξ) dp dq dxo ˜ K˜ (q)δ(u − p + q) = S(p) × exp(2πiq · ξ) dp dq ˜ K˜ (p − u) exp[2πiξ · (p − u)] dp = S(p) (89) After some calculation, we obtain an expression for jd (ξ) = j(xd ; ξ)D(xd ) dxd and hence for its Fourier transform ˜j d (p) = ˜ d ; p)D(xd ) dxd j(x (90) Explicitly, j˜d (p) =

| K˜ (xd /λR)|2 D(xd )δ(p) − s˜(p)

qs (xd /λR; p D(xd ) dxd

+ i ϕ(p) ˜

qϕ (xd /λR; p)D(xd ) dxd

(91)

for weakly scattering objects, s 1, ϕ 1. The spatial frequency spectrum of the bright-field image signal is thus related to s and ϕ by a filter relation very similar to that obtained for the TEM. We now return to Eqs. (87) and (88) to analyze the annular and central detector configuration. For a small axial detector, we see immediately that 2 ˜ o )K (ξ − xo ) dxo jd (ξ) ∝ S(x (92) which is very similar to the image observed in a TEM. For an annular detector, we divide S(xo ) into an unscattered and a scattered part, S(xo ) = 1 + σs (xo ). The signal consists of two main contributions, one of the form [σs (xo +σs∗ (xo ))] |K (ξ − xo )|2 dxo , and the other |σs (xo )|2 |K (ξ − xo )|2 dxo . The latter term usually dominates. We have seen that the current distribution in the detector plane at any instant is the far-field diffraction pattern

of the object element currently illuminated. The fact that we have direct access to this wealth of information about the specimen is one of the remarkable and attractive features of the STEM, rendering possible imaging modes that present insuperable difficulties in the TEM. The simple detectors so far described hardly exploit this wealth of information at all, since only two total currents are measured, one falling on the central region, the other on the annular detector. A slightly more complicated geometry permits us to extract directly information about the phase variation ϕ(xo ) of the specimen transparency S(xo ). Here the detector is divided into four quadrants, and by forming appropriate linear combinations of the four signals thus generated, the gradient of the phase variation can be displayed immediately. This technique has been used to study the magnetic fields across domain boundaries in magnetic materials. Other detector geometries have been proposed, and it is of interest that it is not necessary to equip the microscope with a host of different detectors, provided that the instrument has been interfaced to a computer. It is one of the conveniences of all scanning systems that the signal that serves to generate the image is produced sequentially and can therefore be dispatched directly to computer memory for subsequent or on-line processing if required. By forming the far-field diffraction pattern not on a single large detector but on a honeycomb of very small detectors and reading the currents detected by each cell into framestore memory, complete information about each elementary object area can be recorded. Framestore memory can be programmed to perform simple arithmetic operations, and the framestore can thus be instructed to multiply the incoming intensity data by 1 or 0 in such a way as to mimic any desired detector geometry. The signals from connected regions of the detector—quadrants, for example—are then added, and the total signal on each part is then stored, after which the operation is repeated for the next object element under the probe. Alternatively, the image of each elementary object area can be exploited to extract information about the phase and amplitude of the electron wave emerging from the specimen. A STEM imaging mode that is capable of furnishing very high resolution images has largely superseded the modes described above. Electrons scattered through relatively wide angles (Rutherford scattering) and collected by an annular detector with appropriate dimensions form an “incoherent” image of the specimen structure, but with phase information converted into amplitude variations in the image. Atomic columns can be made visible by this technique, which is rapidly gaining importance. The effect of partial coherence in the STEM can be analyzed by a reasoning similar to that followed for the TEM; we will not reproduce this here.

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C. Image Processing 1. Interference and Holography The resolution of electron lenses is, as we have seen, limited by the spherical aberration of the objective lens, and many types of correctors have been devised in the hope of overcoming this limit. It was realized by Dennis Gabor in the late 1940s, however, that although image detail beyond the limit cannot be discerned by eye, the information is still there if only we could retrieve it. The method he proposed for doing this was holography, but it was many years before his idea could be successfully put into practice; this had to await the invention of the laser and the development of high-performance electron microscopes. With the detailed understanding of electron image formation, the intimate connection between electron holography, electron interference, and transfer theory has become much clearer, largely thanks to Karl-Joseph Hanszen and colleagues in Braunschweig. The electron analogs of the principal holographic modes have been thoroughly explored with the aid of the M¨ollenstedt biprism. In the hands of Akira Tonomura in Tokyo and Hannes Lichte in T¨ubingen, electron holography has become a tool of practical importance. The simplest type of hologram is the Fraunhofer in-line hologram, which is none other than a defocused electron image. Successful reconstruction requires a very coherent source (a field-emission gun) and, if the reconstruction is performed light-optically rather than digitally, glass lenses with immense spherical aberration. Such holograms should permit high-contrast detection of small weak objects. The next degree of complexity is the single-sideband hologram, which is a defocused micrograph obtained with

693 half of the diffraction pattern plane obscured. From the two complementary holograms obtained by obscuring each half in turn, separate phase and amplitude reconstruction is, in principle, possible. Unfortunately, this procedure is extremely difficult to put into practice, because charge accumulates along the edge of the plane that cuts off half the aperture and severely distorts the wave fronts in its vicinity; compensation is possible, but the usefulness of the technique is much reduced. In view of these comments, it is not surprising that offaxis holography, in which a true reference wave interferes with the image wave in the recording plane, has completely supplanted these earlier arrangements. In the in-line methods, the reference wave is, of course, to be identified with the unscattered part of the main beam. Figure 28 shows an arrangement suitable for obtaining the hologram; the reference wave and image wave are recombined by the electrostatic counterpart of a biprism. In the reconstruction step, a reference wave must again be suitably combined with the wave field generated by the hologram, and the most suitable arrangement has been found to be that of the Mach– Zehnder interferometer. Many spectacular results have been obtained in this way, largely thanks to the various interference techniques developed by the school of A. Tonomura and the Bolognese group. Here, the reconstructed image is made to interfere with a plane wave. The two may be exactly aligned and yield an interference pattern representing the magnetic field in the specimen, for example; often, however, it is preferable to arrange that they are slightly inclined with respect to one another since phase “valleys” can then be distinguished from “hills.” In another arrangement, the twin images are made to interfere, thereby amplifying the corresponding phase shifts twofold (or more, if higher order diffracted beams are employed).

FIGURE 28 (Left) Ray diagram showing how an electron hologram is formed. (Right) Cross-section of an electron microscope equipped for holography. [From Tonomura, A. (1999). “Electron Holography,” Springer-Verlag, Berlin/New York.]

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FIGURE 29 Arrangement of lenses and mirrors suitable for interference microscopy. [From Tonomura A. (1999). “Electron Holography,” Springer-Verlag, Berlin/New York.]

Electron holography has a great many ramifications, which we cannot describe here, but we repeat that many of the problems that arise in the reconstruction step vanish if the hologram is available in digital form and can hence be processed in a computer. We now examine the related techniques, although not specifically in connection with holography. 2. Digital Processing If we can sample and measure the gray levels of the electron image accurately and reliably, we can employ the computer to process the resulting matrix of image graylevel measurements in many ways. The simplest techniques, usually known as image enhancement, help to adapt the image to the visual response or to highlight features of particular interest. Many of these are routinely available on commercial scanning microscopes, and we will say no more about them here. The class of methods that allow image restoration to be achieved offer solutions of more difficult problems. Restoration filters, for example, reduce the adverse effect of the transfer functions of Eq. (81). Here, we record two or more images with different values of the defocus and hence with different forms of the transfer function and seek the weighted linear combinations of these images, or rather of their spatial frequency spectra, that yield the best estimates (in the least-squares sense) of ϕ˜ and s˜ . By using a focal series of such images, we can both cancel, or at least substantially reduce, the effect of the transfer functions sin γ and cos γ and fill in the information missing from each individual picture around the zeros of these functions. Another problem of considerable interest, in other fields as well as in electron optics, concerns the phase of the object wave for strongly scattering objects. We have seen that the specimens studied in transmission microscopy

Charged -Particle Optics

are essentially transparent: The image is formed not by absorption but by scattering. The information about the specimen is therefore in some sense coded in the angular distribution of the electron trajectories emerging from the specimen. In an ideal system, this angular distribution would be preserved, apart from magnification effects, at the image and no contrast would be seen. Fortunately, however, the microscope is imperfect; contrast is generated by the loss of electrons scattered through large angles and intercepted by the diaphragm or “objective aperture” and by the crude analog of a phase plate provided by the combination of spherical aberration and defocus. It is the fact that the latter affects the angular distribution within the beam and converts it to a positional dependence with a fidelity that is measured by the transfer function sin γ that is important. The resulting contrast can be related simply to the specimen transparency only if the phase and amplitude variations are small, however, and this is true of only a tiny class of specimens. For many of the remainder, the problem remains. It can be expressed graphically by saying that we know from our intensity record where the electrons arrive (amplitude) but not their directions of motion at the point of arrival (phase). Several ways of obtaining this missing information have been proposed, many inspired by the earliest suggestion, the Gerchberg– Saxton algorithm. Here, the image and diffraction pattern of exactly the same area are recorded, and the fact that the corresponding wave functions are related by a Fourier transform is used to find the phase iteratively. First, the known amplitudes in the image, say, are given arbitrary phases and the Fourier transform is calculated; the amplitudes thus found are then replaced by the known diffraction pattern amplitudes and the process is repeated. After several iterations, the unknown phases should be recovered. This procedure encounters many practical difficulties and some theoretical ones as well, since the effect of noise is difficult to incorporate. This and several related algorithms have now been thoroughly studied and their reliability is well understood. In these iterative procedures, two signals generated by the object are required (image and diffraction pattern or two images at different defocus values in particular). If a STEM is used, this multiplicity of information is available in a single record if the intensity distribution associated with every object pixel is recorded and not reduced to one or a few summed values. A sequence of Fourier transforms and mask operations that generate the phase and amplitude of the electron wave has been devised by John Rodenburg. A very different group of methods has grown up around the problem of three-dimensional reconstruction. The twodimensional projected image that we see in the microscope often gives very little idea of the complex spatial relationships of the true structure, and techniques have therefore

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been developed for reconstructing the latter. They consist essentially in combining the information provided by several different views of the specimen, supplemented if possible by prior knowledge of an intrinsic symmetry of the structure. The fact that several views are required reminds us that not all specimens can withstand the electron onslaught that such multiple exposure represents. Indeed, there are interesting specimens that cannot be directly observed at all, because they are destroyed by the electron dose that would be needed to form a discernible image. Very low dose imaging must therefore be employed, and this has led to the development of an additional class of image restoration methods. Here, the aim is first to detect the structures, invisible to the unaided eye, and superimpose low-dose images of identical structures in such a way that the signal increases more rapidly than the noise and so gradually emerges from the surrounding fog. Threedimensional reconstruction may then be the next step. The problem here, therefore, is first to find the structures, then to align them in position and orientation with the precision needed to achieve the desired resolution. Some statistical check must be applied to be sure that all the structures found are indeed the same and not members of distinct groups that bear a resemblance to one other but are not identical. Finally, individual members of the same group are superposed. Each step demands a different treatment. The individual structures are first found by elaborate cross-correlation calculations. Cross-correlation likewise enables us to align them with high precision. Multivariate analysis is then used to classify them into groups or to prove that they do, after all, belong to the same group and, a very important point, to assign probabilities to their membership of a particular group.

IV. CONCLUDING REMARKS Charged-particle optics has never remained stationary with the times, but the greatest upheaval has certainly been that caused by the widespread availability of large, fast computers. Before, the analysis of electron lenses relied heavily on rather simple field or potential models, and much ingenuity was devoted to finding models that were at once physically realistic and mathematically tractable. Apart from sets of measurements, guns were almost virgin territory. The analysis of in-lens deflectors would have been unthinkable but fortunately was not indispensable since even the word microminiaturization has not yet been coined. Today, it is possible to predict with great accuracy the behavior of almost any system; it is even possible to obtain aberration coefficients, not by evaluating the corresponding integrals, themselves obtained as a result of exceedingly long and tedious algebra, but by solving the

exact ray equations and fitting the results to the known aberration pattern. This is particularly valuable when parasitic aberrations, for which aberration integrals are not much help, are being studied. Moreover, the aberration integrals can themselves now be established not by long hours of laborious calculation, but by means of one of the computer algebra languages. A knowledge of the fundamentals of the subject, presented here, will always be necessary for students of the subject, but modern numerical methods now allow them to go as deeply as they wish into the properties of the most complex systems.

SEE ALSO THE FOLLOWING ARTICLES ACCELERATOR PHYSICS AND ENGINEERING • HOLOGRAPHY • QUANTUM OPTICS • SCANNING ELECTRON MICROSCOPY • SCATTERING AND RECOILING SPECTROSCOPY • SIGNAL PROCESSING, DIGITAL • WAVE PHENOMENA

BIBLIOGRAPHY Carey, D. C. (1987). “The Optics of Charged Particle, Beams,” Harwood Academic, London. Chapman, J. N., and Craven, A. J., eds. (1984). “Quantitative Electron Microscopy,” SUSSP, Edinburgh. Dragt, A. J., and Forest, E. (1986). Adv. Electron. Electron. Phys. 67, 65–120. Feinerman, A. D. and Crewe, D. A. (1998). “Miniature electron optics.” Adv. Imaging Electron Phys. 102, 187–234. Frank, J. (1996). “Three-Dimensional Electron Microscopy of Macromolecular Assemblies,” Academic Press, San Diego. Glaser, W. (1952). “Grundlagen der Elektronenoptik,” Springer-Verlag, Vienna. Glaser, W. (1956). Elektronen- und Ionenoptik, Handb. Phys. 33, 123– 395. Grivet, P. (1972). “Electron Optics,” 2nd Ed. Pergamon, Oxford. Hawkes, P. W. (1970). Adv. Electron. Electron Phys., Suppl. 7. Academic Press, New York. Hawkes, P. W., ed. (1973). “Image Processing and Computer-Aided Design in Electron Optics,” Academic Press, New York. Hawkes, P. W., ed. (1980). “Computer Processing of Electron Microscope Images,” Springer-Verlag, Berlin and New York. Hawkes, P. W., ed. (1982). “Magnetic Electron Lenses,” Springer-Verlag, Berlin and New York. Hawkes, P. W., and Kasper, E. (1989, 1994). “Principles of Electron Optics,” Academic Press, San Diego. Hawkes, P. W., ed. (1994). “Selected Papers on Electron Optics,” SPIE Milestones Series, Vol. 94. Humphries, S. (1986). “Principles of Charged Particle Acceleration.” (1990). “Charged Particle Beams,” Wiley-Interscience, New York and Chichester. Lawson, J. D. (1988). “The Physics of Charged-Particle Beams,” Oxford Univ. Press, Oxford. Lencov´a, B. (1997). Electrostatic Lenses. In “Handbook of Charged Particle Optics” (J. Orloff, ed.), pp. 177–221, CRC Press, Boca Raton, FL.

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696 Livingood, J. J. (1969). “The Optics of Dipole Magnets,” Academic Press, New York. Orloff, J., ed. (1997). “Handbook of Charged Particle Optics,” CRC Press, Boca Raton, FL. Reimer, L. (1997). “Transmission Electron Microscopy,” SpringerVerlag, Berlin and New York. Reimer, L. (1998). “Scanning Electron Microscopy,” Springer-Verlag, Berlin and New York. Saxton, W. O. (1978). Adv. Electron. Electron Phys., Suppl. 10. Academic Press, New York.

Charged -Particle Optics Septier, A., ed. (1967). “Focusing of Charged Particles,” Vols. 1 and 2, Academic Press, New York. Septier, A., ed. (1980–1983). Adv. Electron. Electron Phys., Suppl. 13A–C. Academic Press, New York. Tonomura, A. (1999). “Electron Holography,” Springer-Verlag, Berlin. Tsuno, K. (1997). Magnetic Lenses for Electron Microscopy. In “Handbook of Charged Particle Optics” (J. Orloff, ed.), pp. 143–175, CRC Press, Boca Raton, FL. Wollnik, H. (1987). “Optics of Charged Particles,” Academic Press, Orlando.

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I. II. III. IV. V. VI.

One-Dimensional Considerations Stress Strain Hooke’s Law and Its Limits Strain Energy Equilibrium and the Formulation of Boundary Value Problems VII. Examples

GLOSSARY Anisotropy A medium is said to be anisotropic if the value of a measured, physical field quantity depends on the orientation (or direction) of measurement. Eigenvalue and eigenvector Consider the matrix equation AX = λX, where A is an n × n square matrix, and X is an n-dimensional column vector. In this case, the scalar λ is an eigenvalue, and X is the associated eigenvector. Isotropy A medium is said to be isotropic if the value of a measured, physical field quantity is independent of orientation. ELASTICITY THEORY is the (mathematical) study of the behavior of those solids that have the property of recovering their size and shape when the forces that cause the deformation are removed. To some extent, almost all solids display this property. In this article, most of the discussion will be limited to the special case of linearly elastic solids, where deformation is proportional to ap-

plied forces. This topic is usually referred to as classical elasticity theory. This branch of mathematical physics was formulated during the nineteenth century and, since its inception, has been developed and refined to form the background and foundation for disciplines such as structural mechanics; stress analysis; strength of materials; plates and shells; solid mechanics; and wave propagation and vibrations in solids. These topics are fundamental to solving present-day problems in many branches of modern engineering and applied science. They are used by structural (civil) engineers, aerospace engineers, mechanical engineers, geophysicists, geologists, and bioengineers, to name a few. The deformation, vibrations, and structural integrity of modern high-rise buildings, airplanes, and highspeed rotating machinery are predicted by applying the modern theory of elasticity.

I. ONE-DIMENSIONAL CONSIDERATIONS If we consider a suitably prepared rod of mild steel, with (original) length L and cross-sectional area A, subjected 801

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Within the context of the international system of units (Syst`eme International, or SI), the unit of stress is the pascal (Pa). One pascal is equal to one newton per square meter (N m−2 ). The unit of strain is meter per meter, and thus strain is a dimensionless quantity. We note that 1 N m−2 = 1 Pa = 1.4504 × 10−4 psi and 1 psi = 6894.76 Pa. Typical values of the Young’s (elastic) modulus E and yield stress in tension Y for some ductile materials are shown in Table II in Section IV. The tension test of a rod and naive definitions of stress and strain are associated with one-dimensional considerations. Elasticity theory is concerned with the generalization of these concepts to the general, three-dimensional case.

II. STRESS

FIGURE 1 (a) Tension rod. (b) Stress–strain curve (ductile material).

to a longitudinal, tensile force of magnitude F, then the rod will experience an elongation of magnitude L, as shown in Fig. la. So that we can compare the behavior of rods of differing cross section in a meaningful manner, it is convenient to define the (uniform) axial stress in the rod by σ = F/A and the (uniform) axial strain by ε = L/L. We note that the unit of stress is force per unit of (original) area and the unit of strain is change in length divided by original length. If, in a typical tensile test, we plot stress σ versus strain ε, we obtain the curve shown in Fig. 1b. In the case of mild steel, and many other ductile materials, this curve has a straight line portion that extends from 0 < σ < σp , where σp is the proportional limit. The slope of this line is σ/ε = E, where E is known as Young’s modulus (Thomas Young, 1773–1829). When σp < σ , the stress–strain curve is no longer linear, as shown in Fig. 1b. When the rod is extended beyond σ = σp (the proportional limit), it suffers a permanent set (deformation) upon removal of the load F. At σ = Y (the yield point), the strain will increase considerably for relatively small increases in stress (Fig. 1b). For the majority of structural applications, it is desirable to remain in the linearly elastic, shape-recoverable range of stress and strain (0 ≤ σ ≤ σp ). The mathematical material model that is based on this assumption is said to display linear material characteristics. For example, an airplane wing will deflect in flight because of air loads and maneuvers, but when the loads are removed, the wing reverts to its original shape. If this were not the case, the wing’s lifting capability would not be reliably predictable, and, of course, this would not be desirable. In addition, if the load is doubled, the deflection will also double.

Elastic solids are capable of transmitting forces, and the concept of stress in a solid is a sophistication and generalization of the concept of force. We consider a material point P in the interior of an elastic solid and pass an oriented plane II through P with unit normal vector n (see Fig. 2). Consider the portion of the solid which is shaded. Then on a (small) area A surrounding the point P, there will act a net force of magnitude F, and the stress vector at P is defined by the limiting process T(n) = lim

A→0

F . A

(1)

It is to be noted that the magnitude as well as the direction of the stress vector T depends upon the orientation of n. If we resolve the stress vector along the (arbitrarily chosen) (x, y, z) = (x1 , x2 , x3 ) axes, then we can write

FIGURE 2 Stress vector and components.

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T1 = τ11 e1 + τ12 e2 + τ13 e3 T2 = τ21 e1 + τ22 e2 + τ23 e3

(2)

T3 = τ31 e1 + τ32 e2 + τ33 e3 , where Ti = T(ei ) for i = 1, 2, 3; that is, the Ti are stress vectors acting upon the three coordinate planes and ei are unit vectors associated with the coordinate axes (x, y, z) = (x1 , x2 , x3 ). We note that here and in subsequent developments, we use the convenient and common notation τ12 ≡ τx y , T1 ≡ Tx , T2 ≡ T y , etc. In other words, the subscripts 1, 2, 3 take the place of x, y, z. We can also write T1 = τ11 n 1 + τ12 n 2 + τ13 n 3 T2 = τ21 n 1 + τ22 n 2 + τ23 n 3

(3)

T3 = τ31 n 1 + τ32 n 2 + τ33 n 3 , where n = e1 n 1 + e2 n 2 + e3 n 3 and T(n) = e1 T1 + e2 T2 + e3 T3 = T 1 n 1 + T 2 n 2 + T3 n 3 .

FIGURE 3 Stress tensor components.

(4)

This last expression is known as the lemma of Cauchy (A. L. Cauchy, 1789–1857). The stress tensor components   τ11 τ12 τ13   [τi j ] = τ21 τ22 τ23  (5) τ31 τ32 τ33 can be visualized with reference to Fig. 3, with all stresses shown acting in the positive sense. We note that τi j is the stress component acting on the face with normal ei , in the direction of the vector e j . With reference to Fig. 2, it can also be shown that relative to the plane II, the normal component N and the shear component S are given by Tn ≡ N = T · n =

3

Ti n i

i=1

=

3 3

n i n j τi j

At every interior point of a stressed solid, there exist at least three mutually perpendicular directions for which all shearing stresses τi j , i = j, vanish. This preferred axis system is called the principal axis system. It can be found by solving the algebraic eigenvalue–eigenvector problem characterized by       τ11 − σ τ12 τ13   0   n1    τ22 − σ τ23  n 2 = 0 , (7)  τ21     τ31 τ32 τ33 − σ  n 3   0  where n, n 2 , and n 3 are the direction cosines of the principal axis system such that n 21 + n 22 + n 23 = 1; and σ1 , σ2 , and σ3 are the (scalar) principal stress components. The necessary and sufficient condition for the existence of a solution for Eq. (7) is obtained by setting the coefficient determinant equal to zero. The result is σ 3 + I1 σ 2 + I2 σ − I3 = 0,

(6a)

where the quantities

i=1 j=1

I1 = τ11 + τ22 + τ33 , τ τ τ I2 = 11 12 + 22 τ21 τ22 τ32

and Ts ≡ S = T · s =

3

Ti si

i=1

=

3 3

n i s j τi j ,

(8)

and (6b)

i=1 j=1

where n · s = 0 and n, s are unit vectors normal and parallel to the plane II, respectively.

τ11 I3 = τ21 τ31

τ12 τ22 τ32

τ13 τ23 τ33

τ23 τ33 + τ33 τ13

τ31 , τ11

(9a) (9b)

(9c)

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are known as the first, second, and third stress invariants, respectively. For example, we consider these stress tensor components at a point P of a solid, relative to the x, y, z axes:   3 1 1   [τi j ] = 1 0 2 . (10) 1 2 0 Thus, I1 = 3,

I2 = −6,

I3 = −8

and σ 3 − 3σ 2 − 6σ + 8 = (σ − 4)(σ − 1)(σ + 2) = 0. Consequently, the principal stresses at P are σ1 = 4, σ2 = 1, and σ3 = −2. With the aid of Eq. (7), it can be shown that the principal directions at P are given by the mutually perpendicular unit vectors n(1) n(2) n(3)

2 1 1 = e1 √ + e 2 √ + e 3 √ 6 6 6 1 1 1 = e1 − √ + e 2 √ + e3 √ 3 3 3 1 1 + e3 √ . = e1 (0) + e2 − √ 2 2

(11)

When the Cartesian axes are rotated in a rigid manner from x1 , x2 , x3 , to x1 , x2 , x3 , as shown in Fig. 4, the components of the stress tensor transform according to the rule τ p q =

3 3

a p i aq j τi j ,

(12)

i=1 j=1

where a p i = cos(x p , x1 ) = cos(e p , ei ) are the nine direction cosines that orient the primed coordinate system relative to the unprimed system. For example, consider the rotation of axes characterized by the table of direction cosines  [ai j ] =

2 3 1 3 2 3

− 23

− 13

2 3 1 3

2 3



 − 23  .

FIGURE 4 Principal axes.

when referred to x , y , z axes, according to the law of transformation [Eq. (12)]. The extreme shear stress at a point is given by τmax = 12 (σ1 − σ3 ) and this value is τmax = 12 (4 + 2) = 3 for the stress tensor [Eq. (10)]. It should be noted that the principal stresses are ordered, that is, σ1 ≥ σ2 ≥ σ3 , and that σ1 (σ3 ) is the largest (smallest) normal stress for all possible planes through the point P. If we now establish a coordinate system coincident with principal axes then in “principal stress space,” the normal stress N and the shear stress S on a plane characterized by the outer unit normal vector n are, respectively, N = n 21 σ1 + n 22 σ2 + n 23 σ3 and S 2 = n 21 n 22 (σ1 − σ2 )2 + n 22 n 23 (σ2 − σ3 )2 + n 23 n 21 (σ − σ1 )2 , (15b) where σ1 , σ2 , and σ3 are principal stresses. We now visualize eight planes, the normal to each of which makes equal angles with respect to principal axes. The shear stress acting upon these planes is known as the octahedral shear stress τ0 , and its magnitude is

(13)

The stress components τi j in Eq. (10) relative to the x, y, z axes will become   0.889 0.778 0.222   [τ p q ] = 0.778 −1.444 1.444 (14) 0.222 1.444 3.556

(15a)

τ0 =

1 3

(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2

1/2

≥ 0. (16)

It can be shown that the octahedral shear stress is related to the average of the square of all possible shear stresses at the point, and the relation is 3 (τ )2 5 0

= S 2 .

(17)

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It can also be shown that 9τ02

or

=

2I12

− 6I2 ,

where I1 and I2 are the first and second stress invariants, respectively [see Eqs. (9a) and (9b)]. We also note the bound 3 τ0 2 1≤ ≤√ (19) 2 τmax 3 and the associated implication that 32 τ0 ∼ = 1.08 τmax with a maximum error of about 7%. Returning to the stress tensor [Eq. (10)], we have τmax = 3 and 9τ02 = 2I12 − 6I2 = (2)(9) + (6)(6) = 54, or τ0 = and

√

√ 1≤

6 = 2.4495,

3/2τ0 = τmax

√

2 3/2(2.4495) ≤√ , 3 3

1 = 1 ≤ 1.1547.

(18)

III. STRAIN In our discussion of the concept of stress, we noted that stress characterizes the action of a “force at a point” in a solid. In a similar manner, we shall show that the concept of strain can be used to quantify the notion of “deformation at a point” in a solid. We consider a (small) quadrilateral element in the unstrained solid with dimensions d x, dy, and dz. The sides of the element are taken to be parallel to the coordinate axes. After deformation, the volume element has the shape of a rectangular parallelepiped with edges of length (d x + du), (dy + d v), (dz + d w). With reference to Fig. 5, the material point P in the undeformed configuration is carried into the point P in the deformed configuration. A projection of the element sides onto the x–y plane, before and after deformation, is shown in Fig. 5. We note that all changes in length and angles are small, and they

FIGURE 5 Strain.

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have been exaggerated for purposes of clarity. We now define extensional strain εx x = ε11 as change in length per unit length, and therefore for the edge PA (in Fig. 5), we have [d x + (∂u/∂ x) d x] − d x ∂u εx x = = dx ∂x and [d x + (∂v/∂ y) dy] − dy ∂v ε yy = = , dy ∂y and a projection onto the y–z plane will result in ∂w [dz + (∂w/∂z) dz] − dz εzz = = . dz ∂z The shear strain is defined as one-half of the decrease of the originally right angle APB. Thus, with reference to Fig. 5, we have 2εx y = 2ε yx = =

(∂u/∂ y) dy (∂v/∂ x) d x + d x + (∂u/∂ x) d x dy + (∂v/∂ y) dy

∂v/∂ x ∂u/∂ y ∂v ∂u + = + 1 + (∂u/∂ x) 1 + (∂v/∂ y) ∂x ∂y 1 ∂v/∂ y

3 3

εi j n i n j

(21a)

i=1 j=1

and the shear strain relative to the vectors n and s is S=

3 3 1 εi j n i s j . 2 i=1 j=1

(21b)

Equation (21a) expresses the extensional strain and Eq. (21b) expresses the shearing strain for an arbitrarily chosen element; therefore, we can infer that the nine (six independent) quantities εi j (i = 1, 2, 3; j = 1, 2, 3) provide a complete characterization of strain associated with a material point in the solid. It can be shown that the nine quantities εi j constitute the components of a tensor of order two in a three-dimensional space, and the appropriate law of transformation under a rotation of coordinate axes is 3 3

a p i aq j εi j ;

(22)

i=1 j=1

(small rotations).

In a similar manner, using projections onto the planes y–z and z–x, we can show that 2ε yz

N=

ε p q =

because it is assumed that 1 ∂u/∂ x;

point P in a solid (see Fig. 2), then it can be shown that the extensional strain N in the direction n is given by the formula

∂v ∂w = + ∂z ∂y

and ∂w ∂u + . ∂x ∂z Consequently, the complete (linearized) strain-displacement relations are given by   εx x εx y εx z   ε yx ε yy ε yz  εzx εzy εzz   ∂u ∂v ∂w 1 ∂u 1 ∂u   + +  2 ∂y ∂x 2 ∂z ∂x   ∂x    ∂u ∂v 1 ∂v ∂w   1 ∂v = . + +  2 ∂x ∂y ∂y 2 ∂z ∂y      1 ∂w ∂v ∂w  1 ∂w ∂u  + + 2 ∂x ∂z 2 ∂y ∂z ∂z 2εzx =

(20) Equation (20) characterizes the deformation of the solid at a point. If we define the mutually perpendicular unit vectors n and s with reference to a plane II through a

p = 1, 2, 3;

q = 1, 2, 3,

where the a p i are direction cosines as in Eq. (12). As in the case of stress, there will be at least one set of mutually perpendicular axes for which the shearing strains vanish. These axes are principal axes of strain. They are found in a manner that is entirely analogous to the determination of principal stresses and axes. (See Section II.) It should be noted that a single-valued, continuous displacement field for a simply connected region is guaranteed provided that the six equations of compatibility of A. J. C. Barr´e de Saint-Venant (1779–1886) are satisfied: ∂ 2 ε yy ∂ 2 εx y ∂ 2 εx x , (23a) + = 2 ∂ y2 ∂x2 ∂ x∂ y ∂e yx ∂εx y ∂ 2 εx x ∂ ∂εx z = − + + , ∂ y∂z ∂x ∂x ∂y ∂z (23b) and there are two additional equations for each of Eqs. (23a) and (23b), which are readily obtained by cyclic permutation of x, y, z.

IV. HOOKE’S LAW AND ITS LIMITS The most general linear relationship between stress tensor and strain tensor components at a point in a solid is given by

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τi j =

3 3

TABLE II Some Material Properties for Ductile Materialsa

Ci jkl εkl ;

i = 1, 2, 3;

Yield point stress, σ Y (tension, Pa)

Young’s modulus, E (Pa)

Strain at yield point, εY (tension)

Aluminum alloy (2024 T 4) Brass

290 × 106

7.30 × 1010

0.00397

103 × 106

10.3 × 1010

0.00100

Bronze

138 × 106

10.3 × 1010

0.00134

Magnesium alloy

138 × 106

Steel (low carbon, structural Steel (high carbon)

248 × 106

4.50 × 1010 20.7 × 1010

0.00307 0.00120

414 × 106

20.7 × 1010

0.00200

k=1 l=1

j = 1, 2, 3,

(24)

where the 34 = 81 constants Ci jkl are the elastic constants of the solid. If a strain energy density function exists (see Section V), and in view of the fact that the stress and strain tensor components are symmetric, the elastic constants must satisfy the relations Ci jkl = Ci jlk ,

Ci jkl = C jikl ,

Ci jkl = Ckli j , (25)

and therefore the number of independent elastic constants is reduced to 12 (62 − 6) + 6 = 21 for the general anisotropic elastic solid. If, in addition, the elastic properties of the solid are independent of orientation, the number of independent elastic constants can be reduced to two. In this case of an isotropic elastic solid, the relation between stress and strain is given by Eεx x = τx x − ν(τ yy + τzz )

Material

a Adapted from Reismann, H., and Pawlik, P. S. (1980). “Elasticity: Theory and Applications,” Wiley (Interscience), New York.

materials. In the case of a ductile material with a welldefined yield point (see Fig. 1b), there are at least two failure theories that yield useful results. A. The Hencky-Mises Yield Criterion

Eε yy = τ yy − ν(τzz + τx x ) Eεzz = τzz − ν(τx x + τ yy ) (26)

2Gεx y = τx y 2Gε yz = τ yz 2Gεzx = τzx ,

where G = E /2(1 + ν) is the shear modulus, E is Young’s modulus (see Section I), and ν is Pohisson’s ratio (S. D. Poisson, 1781–1840). Equation (26) is known as Hooke’s law (Robert Hooke, 1635–1693) for a linearly elastic, isotropic solid. A listing of typical values of the elastic constants is provided in Table I. Many failure theories for solids have been proposed, and they are usually associated with specific classes of

This theory predicts failure (yielding) at a point of the solid when 9τ02 ≥ 2Y 2 , where τ0 is the octahedral shear stress [see Eq. (16)] and Y is the yield stress in tension (see Fig. 1b). In this case, the ratio of yield stress in tension √Y to the yield stress in pure shear τ has the value Y /τ = 3. B. The Tresca Yield Criterion This theory postulates that yielding occurs when the extreme shear stress τmax at a point attains the value τmax ≥ Y /2. We note that for this theory the ratio of yield stress in tension to the yield stress in pure shear is equal to Y /τ = 2. A listing of the values of Y for some commonly used materials is given in Table II.

V. STRAIN ENERGY TABLE I Typical Values of Elastic Constantsa Material

v

E (Pa)b

G (Pa)b

Aluminum Concrete

0.34 0.20

6.89 × 1010 0.76 × 1010

2.57 × 1010 1.15 × 1010

Copper

0.34

8.96 × 1010

3.34 × 1010

Glass Nylon Rubber

0.25 0.40 0.499

6.89 × 1010 2.83 × 1010 1.96 × 106

2.76 × 1010 1.01 × 1010 0.654 × 106

Steel

0.29

20.7 × 1010

8.02 × 1010

a Adapted from Reismann, H., and Pawlik, P. S. (1980). “Elasticity: Theory and Applications,” Wiley (Interscience), New York. b Note that 1 Pa = 1 N m−2 = 1.4504 × 10−4 lb in.−2 .

We now consider an interior material point P in a stressed, elastic solid. We can construct a Cartesian coordinate system x, y, z with origin at P, which is coincident with principal axes at P. The point P is enclosed by a small, rectangular parallelepiped with sides of length d x, dy, and dz. The areas of the sides of the parallelepiped are dA z = d x d y, dA x = dy dz, dA y = dz d x, and the volume is d V = d x d y dz. The potential (or strain) energy stored in the linearly elastic solid is equal to the work of the external forces. Consequently, neglecting heat generation, if W is the strain energy per unit volume (strain energy density), we have

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W d V = 12 (τx x A x )(d xεx x ) + 12 (τ yy A y )(dyε yy )

VI. EQUILIBRIUM AND THE FORMULATION OF BOUNDARY VALUE PROBLEMS

+ 12 (τzz A z )(dzεzz ) = 12 (τx x εx x + τ yy ε yy + τzz εzz ) d V, and therefore the strain energy density referred to principal axes is W =

1 (τ ε 2 xx xx

+ τ yy ε yy + τzz εzz ).

In the general case of arbitrary (in general, nonprincipal) axes, this expression assumes the form W = 12 (τx x εx x + τx y εx y + τx z εx z ) + 12 (τ yx ε yx + τ yy ε yy + τ yz ε yz ) + 12 (τzx εzx + τzy εzy + τzz εzz ),

External agencies usually deform a solid by two distinct types of loadings: (a) surface tractions and (b) body forces. Surface tractions act by virtue of the application of normal and shearing stresses to the surface of the solid, while body forces act upon the interior, distributed mass of the solid. For example, a box resting on a table is subjected to (normal) surface traction forces at the interface between tabletop and box bottom, whereas gravity causes forces to be exerted upon the contents of the box. Consider a solid body B bounded by the surface S in a state of static equilibrium. Then at every internal point of B, these partial differential equations must be satisfied: ∂τx y ∂τx x ∂τx z + + + Fx = 0 ∂x ∂y ∂z

or, in abbreviated notation, W =

3 3 1 τi j εi j . 2 i=1 j=1

∂τ yx ∂τ yy ∂τ yz + + + Fy = 0 ∂x ∂y ∂z

(27)

In view of the relations in Eqs. (24) and (27), the expression for strain energy density can be written in the form W =

3 3 3 3 1 Ci jkl εi j εkl . 2 i=1 j=1 k=1 l=1

(28a)

In the case of an isotropic elastic material [see Eq. (26)], this equation reduces to 3 3 1 W = λ(ε11 + ε22 + ε33 )2 + 2G εi j εi j , 2 i=1 j=1 (28b) where λ = E/(1 + v)(1 − 2v). Thus, with reference to Eq. (28), we note that the strain energy density is a quadratic function of the strain tensor components, and W vanishes when the strain field vanishes. Equation (28) serves as a potential (generating) function for the generation of the stress field, that is, τi j =

3 3 ∂ W (εi j ) = Ci jkl εkl ; ∂εi j k=1 l=1

i = 1, 2, 3; j = 1, 2, 3 (29)

[see Eq. (24)]. The concept of strain energy serves as the starting point for many useful and important investigations in elasticity theory and its applications. For details, the reader is referred to the extensive literature, a small selection of which can be found in the Bibliography.

(30)

∂τzy ∂τzx ∂τzz + + + Fz = 0, ∂x ∂y ∂z where τx y = τ yx , τ yz = τzy , τzx = τx z , and F = Fx ex + Fy e y + Fz ez is the body force vector per unit volume. The admissible boundary conditions associated with Eq. (30) may be stated in the form: T ≡ (T1 , T2 , T3 )

on

S1

and u ≡ (u, v, w)

on

S2 ,

(31)

where T is the surface traction vector [see Eq. (4)], u is the displacement vector, and S = S1 + S2 denotes the bounding surface of the solid. The solution of a problem in (three-dimensional) elasticity theory requires the determination of  the displacement vector field u  the stress tensor field τi j in B. (32)  and the strain tensor field εi j This solution is required to satisfy the equations of equilibrium [Eq. (30)], the equations of compatibility [Eq. (23)], the strain-displacement relations [Eq. (20)], and the stress–strain relations [Eq. (26) or (24)], as well as the boundary conditions [Eq. (31)]. This is a formidable task, even for relatively simple geometries and boundary conditions, and the exact or approximate solution requires extensive use of advanced analytical as well as numerical mathematical methods in most cases.

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where δ is the z displacement of the top of the cylinder. With the aid of Eq. (20), we obtain the strain field

VII. EXAMPLES A. Example A

εx x = ε yy = 0;

We consider an elastic cylinder of length L with an arbitrary cross section. The cylinder is composed of a linearly elastic, isotropic material with Young’s modulus E and Poisson’s ratio ν. The cylinder is inserted into a perfectly fitting cavity in a rigid medium, as shown in Fig. 6, and subjected to a uniformly distributed normal stress τzz = T on the free surface at z = L. We assume that the bottom of the cylinder remains in smooth contact with the rigid medium, and that the lateral surfaces between the cylinder and the rigid medium are smooth, thus capable of transmitting normal surface tractions only. Moreover, normal displacements over the lateral surfaces are prevented. Thus, we have the displacement field u = v = 0,

εi j ≡ 0,

εzz = δ/L; i = j.

(33)

In view of Eqs. (26) and (33), we have τx x − ν(τ yy + τzz ) = 0, τ yy − ν(τx x + τzz ) = 0, and τzz − ν(τ yy + τx x ) = E(δ/L), and therefore, ν τzz 1−ν δ (1 − ν) τzz = E = T, L (1 − 2ν)(1 + ν) τx x = τ yy =

w = (δ/L)z,

τi j = 0

(34)

for i = j.

In the case of a copper cylinder, we have (see Table I) ν = 0.34, E = 8.96 × 1010 Pa; and for an axial strain εzz = δ/L = 0.0005, we readily obtain τx x = τ yy = 35.53 × 106 Pa and τzz = 68.9 × 106 Pa. Thus, when we compress the copper cylinder with a stress τzz = T = −68.9 × 106 Pa, there will be induced a lateral compressive stress τx x = τ yy = −35.53 × 106 Pa. We note that the strain field [Eq. (33)] satisfies the equations of compatibility [Eq. (23)] and the stress field [Eq. (34)] satisfies the equations of equilibrium [Eq. (30)] provided the body force vector field F vanishes (or is negligible). B. Example B We consider the case of plane, elastic pure bending (or flexure) of a beam by end couples as shown in Fig. 7. In the reference state, the z axis and the beam longitudinal axis coincide. The cross section of the beam (normal to the z axis) is constant and symmetrical with respect to the y axis. Its area is denoted by the symbol A, and the centroid of A is at (0, 0, z). The beam is acted upon by end moments Mx = M such that Mx = τzz y d A = M A

and FIGURE 6 Transversely constrained cylinder.

My =

τzz x d A = 0. A

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FIGURE 7 Pure bending of a beam.

The present situation suggests the stress field    0 0 0 τx x τx y τx z   0 0 0  , (35) τ yx τ yy τ yz  =   My  τzx τzy τzz 0 0 I 2 where I = A y d A, on account of physical reasoning and (elementary) Euler-Bernoulli beam theory. Upon substitution of Eq. (35) into Eq. (26), and in view of Eq. (20), we obtain ∂u ν ν M εx x = − τzz = − y= E E I ∂x ∂v ν ν M ε yy = − τzz = − y= (36) E E I ∂y M ∂w τzz = y= , E EI ∂z and all shearing strains vanish. We now integrate the partial differential equations in (36), subject to the following boundary conditions: At (x, y, z) = (0, 0, 0) we require u = v = w = 0 and εzz =

∂u ∂y ∂u = = = 0. ∂z ∂z ∂y Thus, the beam displacement field is given by Mν xy EI M 2 v=− (37) [z + ν(y 2 − x 2 )] 2E I M w= yz. EI We note that the strain field (36) satisfies the equation of compatibility (23) and the stress field (35) satisfies the equations of equilibrium (30) provided the body force vector field F vanishes (or is negligible). With reference to Fig. 7, in the reference configuration, the top surface of the beam is characterized by the plane u=−

y = b. Subsequent to deformation, the top surface of the beam is characterized by v Mb2 M 2 (z − νx 2 ) − , 2EI 2EI and for (x, y, z) = (0, b, 0) we have v=−

(38)

ν Mb2 . 2EI We now write Eq. (38) in the form v(0, b, 0) = −

ν Mb2 M 2 =− (z − νx 2 ), (39) 2EI 2EI and we note that V denotes the deflection of the (originally) plane top surface of the beam. The contour lines V = constant of this saddle surface are shown in Fig. 8a. We note that the contour lines consist of two families of hyperbolas, each having two branches. The asymptotes are straight √ lines characterized by V = 0, so that tan α = z/x = ν. An experimental technique called holographic interferometry is uniquely suited to measure sufficiently small deformations of a beam loaded as shown in Fig. 7. In Fig. 8b we show a double-exposure hologram of the deformed top surface of a beam loaded as shown in Fig. 7. This hologram was obtained by the application of a two (light) beam technique, utilizing Kodak Holographic 12002 plates. The laser was a 10-mW He-Ne laser, 632.8 nm, with beam ratio 4:1. The fringe lines in Fig. 8b correspond to the contour lines of Fig. 8a. The close correspondence between theory and experiment is readily observed. We also note that this technique results in the nondestructive, experimental determination of Poisson’s ratio ν of the beam. V =v+

C. Example C We wish to find the displacement, stress field, and strain field in a spherical shell of thickness (b − a) > 0 subjected to uniform, internal fluid (or gas) pressure p. The shell is

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We now divide this equation by (θ)2 and r then take the limit as r → 0 and τrr /r → dτrr /dr . The result of these manipulations is the stress equation of equilibrium

(a)

dτrr 2 + (τrr − τθ θ ) = 0. dr r

(40)

In view of the definition of strain in Section III, the straindisplacement relations for the present problem are (dr + du) − dr du = dr dr u 2π (r + u) − 2πr = , = 2πr r

εrr = (b)

εθ θ

(41)

where the letter u denotes radial displacement. For our present purpose, we now write Hooke’s law (26) in the following form: τrr = (λ + 2G)εrr + 2λεθθ (42)

τθ θ = 2(λ + G)εθ θ + λεrr , where λ= FIGURE 8 (a) Contour lines; V, constant. (b) Double-exposure hologram of deformed plate surface. (Holographic work was performed by P. Malyak in the laboratory of D. P. Malone, Department of Electrical Engineering, State University of New York at Buffalo.) [This hologram is taken from Reismann, H., and Pawlik, P. S. (1980). “Elasticity: Theory and Applications,” Wiley (Interscience), New York.]

bounded by concentric spherical surfaces with outer radius r = b and inner radius r = a, and we designate the center of the shell by O. In view of the resulting point-symmetric displacement field, there will be no shear stresses acting upon planes passing through O and upon spherical surfaces a ≤ r ≤ b. Consequently, at each point of the shell interior, the principal stresses are radial tension (or compression) τrr and circumferential tension (or compression) τθθ , the latter having equal magnitude in all circumferential directions. To obtain the pertinent equation of equilibrium, we consider a volume element (free body) bounded by two pairs of radial planes passing through O, each pair subtending a (small) angle θ , and two spherical surfaces with radii r and r + r . Invoking the condition of (radial) static equilibrium, we obtain (τrr + τrr )[(r + r )θ ]2 − τrr (r θ)2 r = 2τθθ r + r (θ )2 . 2

Eν 2Gν = . (1 + ν)(1 − 2ν) (1 − 2ν)

If we substitute Eq. (41) into Eq. (42) and then substitute the resulting equations into Eq. (40), we obtain the displacement equation of equilibrium d 2u 2 2 du − 2 u = 0. + 2 dr r dr r

(43)

The spherical shell has a free boundary at r = b and is stressed by internal gas (or liquid) pressure acting upon the spherical surface r = a. Consequently, the boundary conditions are τrr (a) = − p,

(44)

where p ≥ 0 and τrr (b) = 0. The solution of the differential equation (43) subject to the boundary conditions (44) is u=

pa 3r pa 3 b3 , + 3K (b3 − a 3 ) 4G(b3 − a 3 )r 2

a ≤ r ≤ b,

(45) where K = E/[3(1 − 2ν)] = (3λ + 2G)/3 is the modulus of volume expansion, or bulk modulus. Upon substitution of Eq. (45) into Eq. (41), we obtain the strain field εrr = εθ θ

pa 3 b3 pa 3 − 3K (b3 − a 3 ) 2G(b3 − a 3 )r 3

pa 3 pa 3 b3 = , + 3K (b3 − a 3 ) 4G(b3 − a 3 )r 3

(46)

and upon substitution of Eq. (46) into Eq. (42), we obtain the stress field

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τrr

τθθ

3 pa 3 b = σ 2 = σ3 ≤ 0 = 3 1− 3 (b − a ) r pa 3 1 b 3 = σ1 ≥ 0. = 3 1+ (b − a 3 ) 2 r

(47)

The criterion due to Tresca (see Section IV) predicts failure when τmax = Y/2. With the aid of Eq. (50), this results again in Eq. (53), and we conclude that for the present example, the failure criteria of Hencky-Mises and Tresca predict the same pressure at incipient failure of the shell given by the formula (53).

We also note the following relations: τrr + 2τθθ =

3 pa 3 , (b3 − a 3 )

τrr + 2τθθ = 3K . εrr + 2εθθ

εrr + 2εθθ =

pa 3 , K (b3 − a 3 ) (48)

With reference to Eq. (16), the octahedral shear stress is 1/2 1 (σ1 − σ )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 3 √ 3 2 pa 3 b = , (49) 3 3 2 (b − a ) r

τ0 =

and the maximum shear stress (as a function of r ) is 3 1 3 pa 3 b τmax = (σ1 − σ3 ) = , (50) 3 3 2 4 (b − a ) r and √ we note that for the present case we have τ0 /τmax = (2 2)/3 ∼ = 0.9428 and [see Eq. (19)] 3 τ0 2 1< =√ . (51) 2 τmax 3 We now apply the failure criterion due to Hencky-Mises √ (see Section IV): Yielding will occur when 3τ0 = 2Y , where Y denotes the yield stress in simple tension of the shell material. Upon application of this criterion and with the aid of Eq. (49), we obtain 2 (b3 − a 3 ) r 3 p= Y, (52) 3 a3 b and the smallest value of p results when r = a. Thus we conclude that the Hencky-Mises failure criterion predicts yielding on the surface r = a when 3 2 a p= Y. (53) 1− 3 b

SEE ALSO THE FOLLOWING ARTICLES ELASTICITY, RUBBERLIKE • FRACTURE AND FATIGUE • MECHANICS, CLASSICAL • MECHANICS OF STRUCTURES • NUMERICAL ANALYSIS • STRUCTURAL ANALYSIS, AEROSPACE

BIBLIOGRAPHY Boresi, A. P., and Chong, K. P. (1987). “Elasticity in Engineering Mechanics,” Elsevier, Amsterdam. Brekhovskikh, L., and Goncharov, V. (1985). “Mechanics of Continua and Wave Dynamics,” Springer-Verlag, Berlin and New York. Filonenko-Borodich, M. (1963). “Theory of Elasticity,” Peace Publishers, Moscow. Fung, Y. C. “Foundations of Solid Mechanics,” Prentice-Hall, Englewood Cliffs, NJ. Green, A. E., and Zerna, W. (1968). “Theoretical Elasticity,” 2nd ed., Oxford Univ. Press, London and New York. Landau, L. D., and Lifshitz, F. M. (1970). “Theory of Elasticity” (Vol. 7 of Course of Theoretical Physics), 2nd ed., Pergamon, Oxford. Leipholz, H. (1974). “Theory of Elasticity,” Noordhoff-International Publications, Leyden, The Netherlands. Lur’e, A. I. (1964). “Three-Dimensional Problems of the Theory of Elasticity,” Wiley (Interscience), New York. Novozhilov, V. V. (1961). “Theory of Elasticity,” Office of Technical Services, U.S. Department of Commerce, Washington, D.C. Parkus, H. (1968). “Thermoelasticity,” Ginn (Blaisdell), Boston. Parton, V. Z., and Perlin, P. I. (1984). “Mathematical Methods of the Theory of Elasticity,” Vols. I and II, Mir Moscow. Reismann, H., and Pawlik, P. S. (1974). “Elastokinetics,” West, St. Paul, Minn. Reismann, H., and Pawlik, P. S. (1980). “Elasticity: Theory and Applications,” Wiley (Interscience), New York. Solomon, L. (1968). “Elasticit´e Lin´eaire,” Masson, Paris. Southwell, R. V. (1969). “An Introduction to the Theory of Elasticity,” Dover, New York. Timoshenko, S. P., and Goodier, J. M. (1970). “Theory of Elasticity,” 3rd ed., McGraw-Hill, New York.

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Electromagnetic Compatibility J. F. Dawson A. C. Marvin C. A. Marshman University of York

I. II. III. IV. V.

Sources of Electromagnetic Interference Effects of Interference Interference Coupling Paths and Their Control Design for Electromagnetic Compatibility Electromagnetic Compatibility Regulations and Standards VI. Measurement and Instrumentation

GLOSSARY Antenna factor The factor by which the received voltage at a specified load is multiplied to determine the received field at the antenna. Common-mode current/voltage The component of current/voltage which exists equally and in the same direction on a pair of conductors or multiconductor bundle, i.e., the return is via a common ground connection (cf. differential mode). Crosstalk Unintentional transfer of energy from one circuit to another by inductive or capacitive coupling or by means of a common impedance (e.g., in a common return conductor). Differential mode current/voltage The component of current/voltage which exists equally and in opposite directions on a pair of conductors (cf. common mode). Shielding effectiveness The ratio of electric or magnetic field strength without a shield to that with the shield present (larger numbers mean better shielding).

Skin depth The depth of the layer in which radiofrequency current flows on the surface of a conductor. Skin effect The confinement, at high frequencies, of current to a thin layer close to the surface of a conductor. Source The source of electromagnetic interference. Victim A circuit or system affected by electromagnetic interference.

ELECTROMAGNETIC COMPATIBILITY (EMC) is the ability of electrical and electronic systems to coexist with each other without causing or suffering from malfunction due to electromagnetic interference (EMI) from each other or from natural causes. As we rely more and more upon electronic systems for the day-to-day operation of our factories, houses, and transport systems, the need to achieve electromagnetic compatibility has increased in importance. This has resulted in the design, analysis, and measurement techniques discussed in this article.

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The limits of electromagnetic (EM) emissions from equipment and immunity to EMI that an equipment must tolerate in an operating environment are determined by standards organizations, in particular, the International Electrotechnical Commission (IEC) and its CISPR committee (Comit´e International Special Perturbations Radioelectrique). The guidelines laid down in the standards may be enforced through regulations.

I. SOURCES OF ELECTROMAGNETIC INTERFERENCE A. Natural Sources 1. Electrostatic Discharge When differing materials are in sliding contact one material may lose electrons to the other—this is the triboelectric effect. This results in a buildup of electrical charge. The electric field due to the charge can cause electrical breakdown of the air (or other insulating material) surrounding the source of the charge, resulting in an electrostatic discharge (ESD). The rate of charge transfer depends on the materials in contact. Electrostatic discharge can be reduced by using materials which are closely matched in the triboelectric series or by using materials with a low conductivity which allow the charge to leak away before it accumulates sufficiently to discharge due to a breakdown of insulation. A common cause of electrostatic discharge is the use of synthetic clothing and furniture. Electric charge is induced on the human body due to friction between clothing or shoes and furniture or floor coverings; the body capacitance (a few hundred picofarads) can charge to voltages as high as 15 kV. When the body comes in close proximity to electronic equipment a spark between the body and metal on the equipment may occur. This can result in a large current flow with a very fast rise time (2–3 solar masses, a black hole. The typical signal from such an explosion is broadband and peaked at around 1 kHz. Detection of such a signal has been the goal of detector development over the last three decades. However, we still know little about the efficiency with which this process produces gravitational waves. For example, an exactly spherical collapse will not produce any gravitational radiation at all. The key issue is the kinetic energy of the nonspherical motions since the gravitational wave amplitude is proportional to this [Eq. (30)]. After 30 years of theoretical and numerical attempts to simulate gravitational collapse, there is still no great progress in understanding the efficiency of this process in producing gravitational waves. For a conservative estimate of the energy in nonspherical motions during the collapse, relation (31) leads to events of an amplitude detectable in our galaxy, even by bar detectors. The next generation of laser interferometers would be able to detect such signals from the Virgo cluster at a rate of a few events per month. The main source of nonsphericity during the collapse is the angular momentum. During the contraction phase, the angular momentum is conserved and the star spins up to rotational periods of the order of 1 msec. In this case, a number of consequent processes with large luminosity might take place in this newly born neutron star. A number of instabilities, such as the so-called bar mode instability and the r-mode instability, may occur which radiate copious amounts of gravitational radiation immediately after

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84 the initial burst. Gravitational wave signals from these rotationally induced stellar instabilities are detectable from sources in our galaxy and are marginally detectable if the event takes place in the nearby cluster of about 2500 galaxies, the Virgo cluster, 15 Mpc away from the earth. Additionally, there will be weaker but extremely useful signals due to subsequent oscillations of the neutron star; f, p, and w modes are some of the main patterns of oscillations (normal modes) of the neutron star that observers might search for. These modes have been studied in detail, and once detected in the signal, they would provide a sensitive probe of the neutron star structure and its supranuclear equation of state. Detectors with high sensitivity in the kilohertz band will be needed in order to fully develop this so-called gravitational wave asteroseismology. If the collapsing central core is unable to drive off its surrounding envelope, then the collapse continues and finally a black hole forms. In this case the instabilities and oscillations discussed above are absent and the newly formed black hole radiates away within a few milliseconds any deviations from axisymmetry and ends up as a rotating or Kerr black hole. The characteristic oscillations of black holes (normal modes) are well studied, and this unique ringing down of a black hole could be used as a direct probe of their existence. The frequency of the signal is inversely proportional to the black hole mass. For example, it was stated earlier that a 100-solar-mass black hole will oscillate at a frequency of ∼100 Hz (an ideal source for LIGO), while a supermassive one with mass 107 solar masses, which might be excited by an infalling star, will ring down at a frequency of 10−3 Hz (an ideal source for LISA). The analysis of such a signal should reveal directly the two parameters that characterize any (uncharged) black hole, namely its mass and angular momentum. B. Radiation from Spinning Neutron Stars A perfectly axisymmetric rotating body does not emit any gravitational radiation. Neutron stars are axisymmetric configurations, but small deviations cannot be ruled out. Irregularities in the crust (perhaps imprinted at the time of crust formation), strains that have built up as the stars have spun down, off-axis magnetic fields, and/or accretion could distort the axisymmetry. A bump that might be created at the surface of a neutron star spinning with frequency f will produce gravitational waves at a frequency of 2 f and such a neutron star will be a weak but continuous and almost monochromatic source of gravitational waves. The radiated energy comes at the expense of the rotational energy of the star, which leads to a spindown of the star. If gravitational wave emission contributes considerably to the observed spindown of pulsars, then we can estimate the amount of the emitted energy. The corresponding am-

Gravitational Wave Physics

plitude of gravitational waves from nearby pulsars (a few kpc away) is of the order of h ∼ 10−25 −10−26 , which is extremely small. If we accumulate data for sufficiently long time, e.g., 1 month, then the effective amplitude, which increases as the square root of the number of cycles, could easily go up to the order of h c ∼ 10−22 . We must admit that we are extremely ignorant of the degree of asymmetry in rotating neutron stars, and these estimates are probably very optimistic. On the other hand, if we do not observe gravitational radiation from a given pulsar we can place a constraint on the degree of nonaxisymmetry of the star. C. Radiation from Binary Systems Binary systems are the best sources of gravitational waves because they emit copious amounts of gravitational radiation, and for every system we know exactly the amplitude and frequency of the gravitational waves in terms of the masses of the two bodies and their separation (see Section II.E). If a binary system emits detectable gravitational radiation in the bandwidth of our detectors, we can easily identify the parameters of the system. According to the formulas of Section II.E, the observed frequency change 5/3 will be f˙ ∼ f 11/3 Mchirp and the corresponding amplitude 5/3 5/3 2/3 will be h ∼ Mchirp f /r = f˙/ f 3r , where Mchirp = µM 2/3 is a combination of the total and reduced mass of the system called the chirp mass. Since both frequency f and its rate of change f˙ are measurable quantities, we can immediately compute the chirp mass (from the first relation), thus obtaining a measure of the masses involved. The second relation provides a direct estimate of the distance of the source. These relations have been derived using the Newtonian theory to describe the orbit of the system and the quadrupole formula for the emission of gravitational waves. Post-Newtonian theory—inclusion of the most important relativistic corrections in the description of the orbit—can provide more accurate estimates of the individual masses of the components of the binary system. When analyzing the data of periodic signals, the effective amplitude is not the amplitude of the signal alone, but √ h c = n · h, where n is the number of cycles of the signal within the frequency range where the detector is sensitive. A system consisting of two typical neutron stars will be detectable by LIGO when the frequency of the gravitational waves is ∼10 Hz until the final coalescence around 1000 Hz. This process will last for about 15 min and the total number of observed cycles will be of the order of 104 , which leads to an enhancement of the detectability by a factor of 100. Binary neutron star systems and binary black hole systems with masses of the order of 50 solar masses are the primary sources for LIGO. Given the anticipated

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sensitivity of LIGO, binary black hole systems are the most promising sources and could be detected as far as 200 Mpc away. The event rate with the present estimated sensitivity of LIGO is probably a few events per year, but future improvement of detector sensitivity (the LIGO II phase) could lead to the detection of at least one event per month. Supermassive black hole systems of a few million solar masses are the primary sources for LISA. These binary systems are rare, but due to the huge amount of energy released, they should be detectable from as far as the boundaries of the observable universe.

D. Cosmological Gravitational Waves One of the strongest pieces of evidence in favor of the Big Bang scenario is the 2.7 K cosmic microwave background radiation. This thermal radiation first bathed the universe around 1 million years after the Big Bang. By contrast, the gravitational radiation background anticipated by theorists was produced at Planck times, i.e., at 10−43 sec or earlier after the Big Bang. Such gravitational waves have traveled almost unimpeded through the universe since they were generated. The observation of cosmological gravitational waves will be one of the most important contributions of gravitational wave astronomy. These primordial gravitational waves will be, in a sense, another source of noise

for our detectors and so they will have to be much stronger than any other internal detector noise in order to be detected. Otherwise, confidence in detecting such primordial gravitational waves could be gained by using a system of two detectors and cross-correlating their outputs. The two LIGO detectors are well placed for such a correlation.

SEE ALSO THE FOLLOWING ARTICLES COSMOLOGY • GLOBAL GRAVITY MODELING • GRAVITATIONAL WAVE ASTRONOMY • NEUTRON STARS • PULSARS • RELATIVITY, GENERAL • SUPERNOVAE

BIBLIOGRAPHY Blair, D. G. (1991). “The Detection of Gravitational Waves,” Cambridge University Press, Cambridge. Marck, J.-A., and Lasota, J.-P. (eds.). (1997). “Relativistic Gravitation and Gravitational Radiation,” Cambridge University Press, Cambridge. Saulson, P. R. (1994). “Fundamentals of Interferometric Gravitational Wave Detectors,” World Scientific, Singapore. Thorne, K. S. (1987). “Gravitational radiation.” In “300 Years of Gravitation” (Hawking, S. W., and Israel, W., eds.), Cambridge University Press, Cambridge.

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Heat Transfer George Alanson Greene Brookhaven National Laboratory

I. II. III. IV. V.

Conduction Heat Transfer Heat Transfer by Convection Thermal Radiation Heat Transfer Boiling Heat Transfer Physical and Transport Properties

GLOSSARY Boiling The phenomenon of heat transfer from a surface to a liquid with vaporization. Conduction Heat transfer within a solid or a motionless fluid by transmission of mechanical vibrations and free electrons. Convection Heat transfer within a flowing fluid by translation of macroscopic fluid volumes from hot regions to colder regions. Heat transfer coefficient An engineering approximation defined by Newton’s law of cooling which relates the heat flux to the overall temperature difference in a system. Thermal radiation The transport of thermal energy from a surface by nonionizing electromagnetic waves. Temperature A scalar quantity which defines the internal energy of matter.

HEAT TRANSFER plays an essential role in everything that we do. Our bodies are exposed to a changing environment yet to live we must remain at 98.6◦ F. The dynamics of our planet’s atmosphere and oceans are driven by

the seasonal variations in heat flux from our Sun. These dynamics, in turn, dictate whether it will rain or snow, whether there will be hurricanes or tornadoes, drought or floods, if crops will grow or die. We burn fuels to heat our homes, our power plants create steam to turn turbines to make electricity, and we reverse the process to cool our dwellings with air conditioning for comfort. There are several modes for the transport of heat which we experience daily. Among these are conduction of heat in solids by molecular vibrations, convection of heat in fluids by the motion of fluid elements from hot to cold regions, thermal radiation in which heat is transferred from surface to surface by electromagnetic radiation, and boiling heat transfer in which heat is transferred from a surface by causing a liquid-to-vapor phase change in an adjacent fluid.

I. CONDUCTION HEAT TRANSFER Heat transfer in opaque solids occurs exclusively by the process of conduction. If a solid (or a stationary liquid or gas) is transparent or translucent, heat can be transferred in a solid by both conduction and radiation, and in fluids, heat can be transferred by conduction, radiation, and

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280 convection. In general, materials in which heat is transferred by conduction only are solid bodies; the addition of convection and radiation to these systems enters the solutions through conditions imposed at the boundaries. A. Physics of Conduction and Thermal Conductivity The mechanisms of heat conduction depend to a great extent on the structure of the solid. For metals and other electrically conducting materials, heat is conducted through a solid by atomic and molecular vibrations about their equilibrium positions and by the mobility of free conductionband electrons through the solid, the same electrons which conduct electricity. There is, in fact, a rigorous relationship between the thermal and the electrical conductivities in metals, known as the Franz–Wiedemann law. In nonmetallic or dielectric materials, lattice vibrations induced by atomic vibrations, otherwise known as phonons, are the principle mechanism of heat conduction. A phonon can be considered a quantum of thermal energy in a thermoelastic wave of fixed frequency passing through a solid, much like a photon is a quantum of energy of a fixed frequency in electromagnetic radiation theory, hence the origin of the name. It is the absence of free conduction-band electrons that make dielectrics poor heat and electrical conductors, relying only on phonons or lattice vibrations to transfer heat energy through a solid. This is intuitively less efficient than conduction in a metal or conductor as will be discussed. It is also clear that in dielectrics which rely on phonon transport for heat transfer, anything which reduces the phonon transport in a material will correspondingly reduce its heat transfer efficiency. An example is the effect of dislocations or impurities in crystals and alloying in metals. The roots of our understanding of thermal conductivity by phonon transport come from kinetic theory and particle physics. Phonon transport in a dielectric is analogous to the thermal conductivity of a gas which depends on collisions between gas molecules to transfer heat. Considering the phonons as particles in the spirit of the dual particle-wave nature of electromagnetic theory, the thermal conductivity of a dielectric solid can be shown to be represented by the relationship given as k p = ρcv vλ/3, where ρ is the phonon density, cv is the heat capacity at constant volume, v is the average phonon velocity, and λ is the mean free path of the phonon. For heat conduction by phonon transport, the phonon velocity is on the order of the sound speed in the solid and the mean free path is on the order of the interatomic spacing. Although the phonon density increases with increasing temperature, the thermal conductivity may remain unchanged or even decrease as the temperature increases if

Heat Transfer

the effect of the vibrations is to diminish the mean free path by an equivalent factor or more. If a dielectric is raised to a very high temperature, heat conduction is increased by thermal excitation of bound electrons which causes them to take on the characteristics of free electrons as in metals, hence the increased thermal conductivity as we find in metals. In extreme cases, this can be accompanied by electron or x-ray emission from the solid. In metals, conduction by phonons is enhanced by conduction by electrons, as just described for hightemperature dielectrics. The derivation from quantum mechanics is parallel to that for phonon transport, except that c is the electron heat capacity, v is the Fermi velocity of the free electrons, and λ is the electronic mean free path of the valence electrons. Due to its complexity, the details of the derivation of the electron contribution to the thermal conductivity will not be presented. However, it is easy to show that in a metal, the total thermal conductivity is the sum of the phonon contribution and the electron contribution, k = k p + ke . In pure metals, the electron contribution to the thermal conductivity may be 30 times greater than the phonon contribution at room temperature. For a more rigorous derivation of the thermal conductivity, the reader is directed to the literature on solid-state physics and quantum mechanics. B. Fundamental Law of Heat Conduction The second law of thermodynamics requires that heat is transferred from one body to another body only if the two bodies are at different temperatures, and that the heat flows from the body at the highest temperature to the body at the lowest temperature. In essence, this is a statement that a thermal gradient must exist in the solid and that heat flows down the thermal gradient. In addition, the first law of thermodynamics requires that the thermal energy is conserved in the absence of heat sources or sinks in the body. It follows from this that a body has a temperature distribution which is a function of space and time, T = T (x, y, z, t), and that the thermal field within the solid is constructed by the superposition of an infinite number of isothermal surfaces which never intersect, lest some point of intersection in space be simultaneously at two or more temperatures which is impossible. Consider a semi-infinite solid whose boundaries are parallel and isothermal at different temperatures. Eventually the temperature distribution within the body will become invariant with time and the heat flow from surface one to surface two becomes q1−2 = −k A(T1 − T2 )/d, where q1−2 is the heat flux, A is the area normal to the heat flux, T1 and T2 are the temperatures of the two isothermal bounding surfaces and d is the separation between the surfaces.

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In the limit that the coordinate normal to the isothermal plane approaches zero, this equation then becomes ∂T ∂n

(1a)

∂T , ∂n

(1b)

qn = −k A or qn = −k

which is known as Fourier’s heat conduction equation. The heat flow per unit area per unit time across a surface is called the heat flux q and has units of W/m2 . The heat flux is a vector and can be calculated for any point in a solid if the temperature field and thermal conductivity are known. C. Differential Heat Conduction Equation The differential heat conduction equations derive from the application of Fourier’s law of heat conduction, and the basic character of these equations is dependent upon shape and varies as a function of the coordinate system chosen to represent the solid. If Fourier’s equation is applied to a simple, isotropic solid in Cartesian coordinates and if the thermal conductivity is assumed to be constant, the equation for the transient conservation of thermal energy due to conduction of heat in a solid with a heat source (or heat sink) can be derived as follows,

∂ T ∂ T ∂ T q 1 ∂T + + 2 + = , ∂x2 ∂ y2 ∂z k α ∂t 2

2

2

(2)

where q is the volumetric heat source and α is the thermal diffusivity, α = k/ρc. If the heat source is equal to zero, this reduces to the Fourier equation. If the temperature in the solid is invariant with respect to time, this becomes the Poisson equation. Furthermore, if the temperature is timeinvariant and the heat source is zero, this becomes the Laplace equation. Other forms of the thermal energy conservation equations in a solid can be derived for other coordinate systems and the eventual solutions depend on the initial and boundary conditions which are imposed. It is the solutions to these equations as given previously that is modeled in commercially available computer analysis packages for heat transfer solutions in solids. The reader is referred to VanSant for a thorough listing of analytical solutions to the heat conduction equations in many coordinate systems and subject to numerous initial and boundary conditions in order to experience the elegance of analytical solutions to problems in heat transfer physics and to appreciate the relationship between heat transfer and mathematics. A curiosity of the parabolic differential form of the heat conduction equation just presented is that it implies that

the velocity of propagation of a thermal wave in a solid is infinite. This is a consequence of the fact that the solution predicts that the effects of a thermal disturbance in a solid are felt immediately at a distance infinitely removed from the disturbance itself. This is in spite of the definition of the thermal conductivity which is based upon finite speed of propagation of free electrons or phonons in matter. In practical applications, this outcome is inconsequential because the effect at infinity is generally small. However, there are circumstances in which this peculiarity in the equations may actually become significant and lead to erroneous results, for instance, in heat transfer problems at very low temperatures or very short time scales, in which cases the finite speed of propagation of heat becomes important. Two examples of such circumstances which can be encountered in practice are cryogenic heat transfer near absolute zero and rapid energy transfer in materials due to subatomic particles which travel at the speed of light. It has been suggested that the form of the differential equations for conduction heat transfer should be the damped-wave or the hyperbolic heat conduction equation, often called the telegraph equation, which includes the finite speed of propagation of heat, C, as shown below without derivation. 1 ∂T ∂2T ∂2T ∂2T 1 ∂2T + + + . = C 2 ∂t 2 α ∂t ∂x2 ∂ y2 ∂z 2

(3)

For most practical problems in heat conduction, the solutions to the parabolic and hyperbolic heat conduction equations are essentially identical; however, the cautions offered in Eq. (3) should be evaluated in circumstances where the finite propagation speed could become important, especially when using commercial equation solvers which will undoubtedly not model the hyperbolic effect just described. Rendering the damped-wave equation dimensionless will reveal to the analyst when the wave propagation term and the diffusion term on the LHS of the hyperbolic heat conduction equation are of the same order of magnitude and both must be included in the solution, for instance, when t ∼ (α/C 2 ). A continued discussion of conduction heat transfer in solids would require the solutions to many special heat transfer cases for which the parabolic heat conduction equation can be easily integrated. Examples of these can be found in every heat transfer text book, and they will not be solved here. However, two examples will be discussed which illustrate powerful techniques for solving contemporary heat transfer problems. These are the lumped heat capacity approximation for transient heat conduction in solids with convection, and the numerical decomposition of the differential heat conduction equation for finite difference computer analysis. By necessity, these discussions will be brief but illustrative.

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D. Lumped Heat Capacity Approximation in Transient Conduction Some heat transfer systems, usually involving a “small” body or a body which is thin in the direction of heat transfer, can be analyzed under the assumption that their temperature is uniform spatially, only a function of time. This is called the lumped heat capacity assumption. The system can be analyzed as a function of time only, greatly simplifying the analysis. This situation can be illustrated by considering a small, spherical object at an initial temperature T0 which is suddenly submerged in a fluid at temperature T∞ which imposes a heat transfer coefficient at the surface of the sphere h with the units W/m2 K. If the sphere has density ρ , specific heat c p , surface area A, and volume V , the transient energy conservation equation can be written as follows: dT (t) hA =− (T (t) − T∞ ), dt ρc p V

(4)

FIGURE 1 Cartesian coordinate grid for finite difference analysis.

and the solution for the time-dependent dimensionless temperature of the sphere θ becomes as follows, θ = exp(−τ ) = exp(−Bi · Fo), where Bi is the Biot number (Bi = h /k) which is the ratio of the internal heat transfer resistance to the external heat transfer resistance, and Fo is the Fourier number (Fo = αt/ 2 ), the dimensionless time. In order to simplify the solution for the transiently cooled sphere, a condition was imposed that the spatial variations of the temperature in the sphere were small. This condition is satisfied if the resistance to heat transfer inside the object is small compared to the external resistance to heat transfer from the sphere to the fluid. Mathematically, this is stated that the Biot number ≤0.1, a factor of an order of magnitude. If this condition is satisfied, heat transfer solutions can be greatly simplified.

An analytical solution to such a simple heat transfer problem could be a formidable task; however, analysis by decomposing the energy equation into a form suitable for finite difference numerical analysis will greatly simplify the task. In other words, an energy balance is performed on each shaded control volume (one for every node), allowing for heat transfer across each face of the control volume from surrounding nodes or, in the case of the convective boundary, the surrounding fluid. Assuming for convenience that x = y, the temperature is not a function of time and the thermal conductivity is a constant, the finite difference equations can be easily derived, and they are presented here for the three nodes in the example in Fig. 1.

E. Finite Difference Representation of Steady-State Heat Conduction In practice, it is frequently not possible to achieve analytical solutions to heat transfer problems in spite of simplifications and approximations. It is often necessary to resort to numerical solutions because of complexities involving geometry and shape, variable physical and transport properties, and complex and variable initial and boundary conditions. Figure 1 shows a rectilinear two-dimensional solid, divided into a grid of equally spaced nodes; it will be assumed that a steady-state temperature field exits. Three nodes are depicted in Fig. 1: (a) an interior node which is surrounded by other nodes in the solid, (b) a node on the insulated boundary of the solid, and (c) a node on the convective boundary of the solid.

Interior node:

0 = T2 + T3 + T4 + T5 − 4T1

Insulated boundary:

0 = T2 + T4 + 2T3 − 4T1

(5)

Convective boundary: 1 0 = (2T3 + T2 + T4 ) + 2 h · x − + 2 T1 . k

h · x T∞ k

Note the appearance in the convective boundary node equation of the term (h x/k), the finite difference form of the Biot number which was introduced in the preceding section. Such equations can be written for all the nodes and assembled in a manner convenient for iterative solution. Simple examples such as those shown here will rapidly converge; more complex problems will require more complex algorithms and stringent convergence criteria.

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II. HEAT TRANSFER BY CONVECTION In the preceding section, we discussed the mechanisms of conduction heat transfer as the sole agent which transports heat energy within a solid. Convection was only considered insofar as it entered the problem through the boundary conditions. For fluids, however, this is true only under the conditions that the fluid is motionless, a condition almost never realized in practice. In general, fluids are in motion either by pumping or by buoyancy, and the heat transfer in fluids in motion is enhanced over conduction because the moving fluid particles carry heat with them as internal energy; the transport of heat through a fluid by the motion of macroscopic fluid particles is called convection. We will now consider methods of modeling convective heat transfer and the concept of the heat transfer coefficient which is the fundamental variable in convection. The analysis of convective heat transfer is more complex than conduction in solids because the motion of the fluid must be considered simultaneously with the energy transfer process. The general approach assumes that the fluid is a continuum instead of the more basic and complex approach assuming individual particles. Although fundamental issues such as the thermodynamic state and the transport properties of the fluid cannot be solved theoretically by the continuum approach, the solutions to the fluid mechanics and heat transfer are made more tractable; parallel studies at the molecular level can resolve the thermodynamic and transport issues. In practice, the thermodynamic and transport properties, although available from theoretical studies on a molecular level, are generally input to the study of heat transfer empirically.

A. Internal and External Convective Flows There are two general classes of problems in convective heat transfer: internal convection in channels and pipes in which the flow patterns become fully developed and spatially invariant after traversing an initial entrance length and the heat flux is uniform along the downstream surfaces, and external convection over surfaces which produces a shear or boundary layer which continues to grow in the direction of the flow and which never becomes fully developed or spatially invariant. Both internal duct flows and external boundary layer flows can be either laminar or turbulent, depending upon the magnitude of a dimensionless parameter of the fluid mechanics known as the Reynolds number. For internal flows, the flow is laminar if the Reynolds number is less than 2 × 103 ; for external flows, the rule of thumb is that the flow is laminar if the Reynolds number is less than 5 × 105 . Schematic representations of both an internal flow case and an external boundary layer are shown in Figs. 2a,b, respectively. B. Fluid Mechanics and the Reynolds Number Steady flow in a channel (internal flow) and external flow over a boundary are governed by a balance of forces in the fluid in which inertial forces and pressure forces are balanced by viscous forces on the fluid. This leads to the familiar concept of a constant pressure drop in a water pipe which provides the force to overcome friction along the pipe walls and thus provides the desired flow rate of water out the other end. For Newtonian fluids, viscous or shear forces in the fluid are described by a relationship between the stress (force/unit area) between the fluid layers which

FIGURE 2 Schematic of internal flow and external flow boundary layers.

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results in a shear of the velocity field in the fluid as follows, τ = µ · ∂u/∂ y, where τ is the shear stress in the fluid, ∂u/∂ y is the rate of strain of the fluid, and µ, the constant of proportionality, is a fluid transport property known as the dynamic viscosity. This is the Newtonian stress–strain relationship and it forms the basis for the fundamental equations of fluid mechanics. The force balance in the direction of flow which provides for a state of equilibrium on a fluid element in the flow can be written as a balance of differential pressure forces normal to the fluid element by tangential shear forces on the fluid element as shown in the following: ∂P D ∂τ − + = Fx = (ρu). (6) ∂x ∂y Dt Substituting the Newtonian stress–strain relationship into this force balance, we find that ∂P ∂u ∂ 2u ∂u − +µ 2 =ρ u +v (7a) ∂x ∂y ∂x ∂y and in dimensionless form, this becomes 2 ∗ ∗ ∗ ∂ P∗ µ ∂ u ∗ ∂u ∗ ∂u − ∗ + = u + v , ∂x ρUL ∂ y ∗2 ∂x∗ ∂ y∗

(7b)

where the quantity (ρU L/µ) is called the Reynolds number of the flow, and it represents the ratio of inertial forces to viscous forces in the fluid. The Reynolds number, sometimes written as Re = U L/ν, where ν is the kinematic viscosity, ν = µ/ρ, is the similarity parameter of fluid mechanics which provides the convenience of similarity solutions to general classes of fluid mechanics problems (i.e., pressure drop in laminar or turbulent pipe flow can be scaled by the Reynolds number, regardless of the velocity, diameter, or viscosity) and is the parameter which predicts when laminar conditions transition to turbulence. The Reynolds number plays a fundamental role in predicting convection and convective heat transfer. C. The Convective Thermal Energy Equation and the Nusselt Number In order to solve for heat transfer in convective flows, an energy balance is constructed on an elemental fluid element. In Cartesian coordinates, this usually involves the balance of convection of heat into and out of the elemental volume in the direction of the flow (x-direction) and conduction of heat into and out of the elemental volume transverse to the flow (y-direction). Taylor series expansions of the convective heat flux in the x-direction and the conduction heat flux in the y-direction permit the representation of the heat balance on the differential fluid volume in differential form. The statement of thermal energy conservation on the differential unit volume of fluid

d x · dy can be written in the form of the laminar boundary layer equation as ∂T ∂T ∂2T (8) +v =k 2. ρc p u ∂x ∂y ∂y Furthermore, if the velocity, temperature, and coordinates are nondimensionalized by the characteristic scales of the problem, such as the maximum velocity U , the overall length L, and the overall temperature difference Tw − T∞ , the dimensionless form of the laminar boundary layer equation becomes u∗

∂θ ∂θ 1 ∂ 2θ + v∗ ∗ = , ∗ ∂x ∂y Re · Pr ∂ y ∗2

(9)

subject to the appropriate boundary conditions. Note the appearance in Eq. (9) of the familiar Reynolds number, Re = U L/ν, and the appearance of another dimensionless parameter, the Prandtl number, Pr = µ·c/k, which includes the physical and transport properties of fluid mechanics and heat transfer. In simple terms, the Prandtl number represents the ratio of the thickness of the hydrodynamic boundary layer to the thickness of the thermal boundary layer. If the Prandtl number equals unity, both boundary layers grow at the same rate. For all problems of convective heat transfer in fluids, the dominant dimensionless scaling or modeling parameters are the Reynolds number and the Prandtl number. Solutions to the thermal energy equations of convective heat transfer are complex and can only be described here in general terms. To express the overall effect of convection on heat transfer, we call upon Newton’s law of cooling given by ∂T q = h A(Tw − T∞ ) = −k A , (10a) ∂y w where h is the heat transfer coefficient which has units of (W/m2 K). The heat transfer coefficient can also be written as −k(∂ T /∂ y)w h= . (10b) (Tw − T∞ ) The dimensional solutions to the laminar boundary layer equations (and the thermal convective energy equations in general) involve solutions as shown previously for the heat transfer coefficient. The solution for the heat transfer coefficient may be nondimensionalized by multiplying by the characteristic length scale of the problem and dividing by the thermal conductivity. In this manner, the dimensionless heat transfer coefficient is introduced, and is called the Nusselt number, h·x = f (Re, Pr), (11) k which is the general form of most solutions in forced-flow convective heat transfer. Nu =

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In closing the discussion on convective heat transfer, it would be useful to present two examples of the dimensionless heat transfer coefficient, the solution to the thermal energy equation, for the two cases presented earlier in Fig. 2. The first example is the laminar flow external boundary layer which was depicted in Fig. 2b. For the case of laminar boundary convective heat transfer over a horizontal, flat surface, the dimensionless heat transfer coefficient, which is the solution to the thermal energy equation becomes as follows: h(x) · x (12) = 0.332 Pr1/3 Re(x)1/2 . k The second example is fully-developed flow in a smooth circular pipe which was depicted in Fig. 2a. For the case of fully developed turbulent flow in a smooth circular pipe, the solution of the convective thermal energy equation becomes as follows: h·d N ud = = 0.023 Re(d)0.8 Prn , (13) k where n = 0.4 for heating and n = 0.3 for cooling. Equation (13) is called the Dittus–Boelter equation for turbulent heat transfer in a pipe. The derivations given in this chapter were, by necessity, simplifications of more rigorous derivations which may be found in the bibliography. N u(x) =

III. THERMAL RADIATION HEAT TRANSFER In the preceding sections, we examined two fundamental modes of heat transfer, conduction and convection, and have shown how they are developed from a fundamental theoretical approach. We now turn our attention to the third fundamental mode of heat transfer, thermal radiation. A. Physical Mechanisms of Thermal Radiation Thermal radiation is the form of electromagnetic radiation that is emitted by a body as a result of its temperature. There are many types of electromagnetic radiation, some is ionizing and some is nonionizing. Electromagnetic radiation generally becomes more ionizing with increasing frequency, for instance x-rays and γ -rays. At lower frequencies, electromagnetic radiation becomes less ionizing, for instance, visible, thermal, and radio wave radiation. However, this is not a hard and fast rule. The spectrum of thermal radiation includes the portion of the frequency band of the electromagnetic spectrum which includes infrared, visible, and ultraviolet radiation. Regardless of the type of electromagnetic radiation being considered, all electromagnetic radiation is propagated at the speed of light, c = 3 × 1010 cm/s, and this speed is equal to the

product of the wavelength and frequency of the radiation, c = λν, where λ is the wavelength and is ν the frequency. The portion of the electromagnetic spectrum which is considered thermal covers the range of wavelength from 0.1 to 100 µm; in comparison, the visible light portion of the thermal spectrum in which humans can see is very narrow, covering only from 0.35 to 0.75 µm. If we were insects, we might see in the infrared range of the spectrum. If we did, warm bodies would look like multicolored objects but the glass windows in our homes would be opaque because infrared is reflected by glass just like visible light is reflected by mirrors. Since the windows in our homes are transparent to visible light, they let in solar radiation which is emitted in the visible spectrum; since they are opaque to infrared radiation, the surfaces in your house which radiate in the infrared do not radiate out to space at night; your house loses heat by conduction through the walls and convection from the outside surfaces. The emission or propagation of thermal energy takes place as discrete photons, each having a quantum of energy E given by E = hν, where h is Plank’s constant (h = 6.625 × 10−34 J · s). An analogy is sometimes used to characterize the propagation of thermal radiation as particles such as the molecules of a gas, each having mass, momentum, and energy, a so-called photon “gas.” In this fashion, we have that the energy of the photons in the “photon gas” is E = hν = mc2 , the photon mass is m = hν/c2 , and the photon momentum is p = hν/c. It follows from statistical thermodynamics that the radiation energy density per unit volume per unit wavelength can be derived as uλ =

3π hcλ−5 , exp(hc/λkT )−1

(14)

where k is Boltzmann’s constant (k = 1.38 × 10−23 J/molecule · K). If the energy density of the radiating gas is integrated over all wavelengths, the total energy emitted is proportional to the absolute temperature of the emitting surface to the fourth power as Eb = σ T 4,

(15)

where E b is the energy radiated by an ideal radiator (or black body) and σ is the Stefan–Boltzmann constant (σ = 5.67 × 10−8 W/m2 K4 ). E b is called the emissive power of a black body. The term “black body” should be taken with caution for although most surfaces which look black to the eye are ideal radiators, other surfaces such as ice and some white paints are also black at long wavelengths. Equation (15) is known as the Stefan–Boltzmann law of radiation heat transfer for an ideal thermal radiator. We have now developed three fundamental laws of classical heat transfer:

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r Fourier’s law of conduction heat transfer r Newton’s law of convective cooling r Stefan–Boltzmann law of thermal radiation

B. Radiation Properties The properties of thermal radiation are not dissimilar to our experience with visible light. When thermal radiation is incident upon a surface, part is reflected, part is absorbed, and part is transmitted. The fraction reflected is called the reflectivity ρ, the fraction absorbed is called the absorptivity α, and the fraction transmitted is called the transmissivity τ . These three variables satisfy the identity that α +ρ + τ = 1. Since most solid bodies do not transmit thermal radiation, τ = 0 and the identity reduces to α + ρ = 1. There are two types of surfaces when it comes to the reflection of thermal radiation from a surface: specular and diffuse. If the angle of incidence of incoming radiation is equal to the angle of reflected radiation, the reflected radiation is called specular. If the reflected radiation is distributed uniformly in all directions regardless of the angle of incidence, the reflected radiation is called diffuse. In general, polished smooth surfaces are more specular and rough surfaces are more diffuse. The emissive power of a surface E is defined as the energy emitted from the surface per unit area per unit time. If you consider a body of surface area A inside a black enclosure and in thermal equilibrium with the enclosure, an energy balance on the enclosed surface states that E · A = qi · A · α; in other words, the energy emitted from the body is equal to the fraction of the incident energy absorbed from the black enclosure. If the surface inside the black enclosure is itself a black body, the statement of thermal equilibrium then becomes as follows: E b · A = qi · A · (1), where α = 1 for the enclosed black body. Dividing these two statements of thermal equilibrium, we get, E/E b = α; in other words, the ratio of the emissive power of a body to the emissive power of a black body at the same temperature is equal to the absorptivity of the surface, α. If this ratio holds such that the absorptivity α is equal to the emissivity ∈ for all wavelengths, we have Kirchhoff’s law, ∈ = α, for a grey body or for grey body radiation. In other words, the surface is a grey body such that the monochromatic emissivity of the surface ∈λ is a constant for all wavelengths. In practice, the emissivities of various surfaces can vary by a great deal as a function of wavelength, temperature, and surface conditions. A graphical example of the variations in the total hemispherical emissivity of Inconel 718 as a function of surface condition and temperature as reported by the author is shown in Fig. 3. However,

FIGURE 3 Total hemispherical emissivity of Inconel 718: (a) shiny, (b) oxidized in air for 15 min at 815◦ C, (c) sandblasted and oxidized in air for 15 min at 815◦ C.

the convenience of the assumptions of a grey body, one whose monochromatic emissivity ∈λ is independent of wavelength, and Kirchhoff’s law, that ∈ = α, make many practical problems more tractable to solution. Plank has developed a formula for the monochromatic emissive power of a black body from quantum mechanics as shown in the following: E bλ =

C1 λ−5 , exp(C2 /λT )−1

(16)

where C1 = 3.743 × 108 W · µm4 /m2 and C2 = 1.439 × 104 µm · K, and λ is the wavelength in micrometers. Plank’s distribution function given in Eq. (16) predicts that the maximum of the monochromatic black body emissive power shifts to shorter wavelengths as the absolute temperature increases, and that the peak in E bλ increases as the wavelength decreases. An illustrative example of the trends of Plank’s distribution function is that while a very hot object such as a white-hot ingot of steel radiates in the visible spectrum, λ ∼ 1 µm, as it cools it will radiate in increasingly longer wavelengths until it is in the infrared spectrum, λ ∼ 100 µm. A relationship between the temperature and the peak wavelength of Plank’s black body emissive power distribution function known as Wein’s displacement law is given here: λmax · T = 2897 . 6 µm · K.

(17)

This relationship determines the peak wavelength of the emissive power distribution for a black body at any temperature T . If the body is grey with an average emissivity ∈, the value of E bλ is simply multiplied by ∈ to get E λ , but as a first approximation, the peak of Plank’s distribution function remains unchanged.

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C. Radiation Shape Factors Surfaces that radiate thermal energy radiate to each other, and it is necessary to know how much heat leaving Surface 1 gets to Surface 2 and vice versa, in order to determine the surface heat flux. The function that determines the amount of heat leaving Surface 1 that is incident on Surface 2 is called the radiation shape factor, F1−2 . Consider two black surfaces A1 and A2 at two different temperatures T1 and T2 . The energy leaving A1 arriving at A2 is E b1 A1 F1−2 and the energy leaving A2 arriving at A1 is E b2 A2 F2−1 . Since the surfaces are black and all incident energy is absorbed (∈1 = ∈2 = 1), the net radiative energy exchange between the two surfaces is q1−2 = E b1 A1 F1−2 − E b2 A2 F2−1 . Setting both surfaces to the same temperature forces q1−2 to zero and E b1 = E b2 , therefore A1 F1−2 = A2 F2−1 . This relationship is known as the reciprocity relationship for radiation shape factors and can be written in general as Am Fm,n = An Fn,m . This relationship is geometrical and applies for grey diffuse surfaces as well as black surfaces. Since our surfaces were black, we can substitute the black body emissive power, E bi = σ Ti4 , to get the result q1−2 = A1 F1−2 σ T14 − T24 . (18) In general, the solution for shape factors involves geometrical calculus. However, many shape factors have been tabulated in books which simplify the analyses significantly. There are relations between shape factors which are useful for constructing complex shape factors from an assembly of more simple shape factors. Considerable time could be spent in this discussion; however, these will not be discussed here and the reader is directed to the references for more details. D. Heat Exchange Between Nonblack Bodies We have just derived a useful equation for the heat flux between two black, diffuse surfaces q1−2 = A1 F1−2 (E b1 − E b2 ). In analogy to Ohm’s law, this can be rewritten in the form of a resistance to heat transfer as q1−2 = (E b1 − E b2 )/Rspatial where Rspatial = 1/(A1 F1−2 ). It is implied in this formulation that since both bodies are black and thus perfect emitters, they have no surface resistance to radiation, only a geometrical spatial resistance. If both surfaces were grey, they would have ∈ = 1, and there would be associated with each surface a resistance due to the emissivity of each surface, a thermodynamic resistance in addition to the spatial resistance just shown. Let us examine this problem in more general terms. The problem of determining the radiation heat transfer between black surfaces becomes one of determining the geometric shape factor. The problem becomes more complex when consid-

ering nonblack bodies because not all energy incident on a surface is absorbed, some is reflected back and some is reflected out of the system entirely. In order to solve the general problem of radiation heat transfer between grey, diffuse, isothermal surfaces, we must define two new concepts, the radiosity J and irradiation G. The radiosity J is defined as the total radiation which leaves a surface per unit area per unit time, and the irradiation G is defined as the total energy incident on a surface per unit area per unit time. Both are assumed uniform over a surface for convenience. Assuming that τ = 0 and ρ = (1 − ∈), the equation for the radiosity J is as follows: J = εE b + ρG = εE b + (1 − ε)G.

(19)

Since the net energy leaving the surface is the difference between the radiosity and the irradiation, we find, q (20) = J − G = εE b + (1 − ε)G − G, A and solving for G from Eq. (19) and substituting in Eq. (20), we get the following solution for the surface heat flux: εA Eb − J q= (E b − J ) = . (21) (1 − ε) (1 − ε)/ε A In another analogy to Ohm’s law, the LHS of Eq. (21) can be considered a current, the RHS-top a potential difference, and the RHS-bottom a surface resistance to radiative heat transfer. We now consider the exchange of radiant energy between two surfaces A1 and A2 . The energy leaving A1 which reaches A2 is J1 A1 F1−2 , and the energy leaving A2 which reaches A1 is J2 A2 F2−1 . Therefore, the net energy transfer from A1 to A2 is q1−2 = J1 A1 F1−2 − J2 A2 F2−1 , and using the reciprocity relation for shape factors we find, q1−2 = (J1 − J2 )A1 F1−2 =

(J1 − J2 ) , (1/A1 F1−2 )

(22)

where (1/A1 F1−2 ) is the spatial resistance to radiative heat transfer between A1 and A2 . A resistance network may now be constructed for two isothermal, grey, diffuse surfaces in radiative exchange with each other by dividing the overall potential difference by the sum of the three resistances as follows: q1−2 =

=

E b1 − E b2 (1 − ε1 )/ε1 A1 + 1/A1 F1−2 + (1 − ε2 )/ε2 A2 σ T14 − T24 . (1 − ε1 )/ε1 A1 + 1/A1 F1−2 + (1 − ε2 )/ε2 A2 (23)

This approach can be readily extended to include more than two surfaces exchanging radiant energy but the

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equations quickly become unwieldy so no example will be presented here. One example of a problem which may be easily solved with this network method and which frequently arises in practice, such as in the design of experiments, is the problem of two grey, diffuse, and isothermal infinite parallel surfaces. In this problem, A1 = A2 and F1−2 = 1 since all the radiation leaving one surface reaches the other surface. Substituting for F1−2 in Eq. (23) and dividing by A = A1 = A2 , we find that the net heat flow per unit area becomes σ T14 − T24 q1−2 = . (24) A 1/ε1 + 1/ε2 − 1 A second example which serves to illustrate this technique is the problem of two long concentric cylinders, with A1 being the inner cylinder and A2 the outer cylinder. Once again, applying Eq. (23) noting that F1−2 = 1, we find that, σ T14 − T24 q1−2 = . (25) A1 1/ε1 + (A1 / A2 )(1/ε2 − 1) In the limit that ( A1 /A2 ) → 0, for instance, for a small convex object inside a very large enclosure, this reduces to the simple solution shown in Eq. (26). q1−2 = σ ε1 T14 − T24 . A1

(26)

These are only two simple examples of the power of the radiation network approach to solving radiative heat transfer problems with many mutually irradiating surfaces. The study of thermal radiative heat transfer goes on to consider radiative exchange between a gas and a heat transfer surface, complex radiation networks in absorbing and transmitting media, solar radiation and radiation within planetary atmospheres, and complex considerations of combined conduction–convection–radiation heat transfer problems. The reader is encouraged to investigate these and other topics in radiative heat transfer further.

IV. BOILING HEAT TRANSFER The phenomenon of heat transfer from a surface to a liquid with a phase change to the vapor phase by the formation of bubbles is called boiling heat transfer. When a pool of liquid at its saturation temperature is heated by an adjacent surface which is at a temperature just above the liquid saturation temperature, heat transfer may proceed without phase change by single-phase buoyancy or natural convection.

FIGURE 4 Pool boiling curve for water.

A. Onset of Pool Boiling As the surface temperature is increased, bubbles appear on the heater surface signaling the onset of nucleation and incipient pool boiling. The rate of heat transfer by pool boiling as this is called is usually represented graphically by presenting the surface heat flux, qw , versus surface superheat, Tw − Tsat . This is referred to as the boiling curve. The process of boiling heat transfer is quite nonlinear, the result of the appearance of a number of regimes of boiling which depend fundamentally upon different heat transfer processes. The components of the pool boiling curve have been well established and are shown graphically in Fig. 4. The first regime of the boiling curve is the natural convection regime, essentially a regime just preceding boiling, in which the heat transfer is by single-phase flow without vapor generation. In this regime, buoyancy of the hot liquid adjacent to the surface of the heater forces liquid to rise in the cooler liquid pool followed by fresh cold liquid passing over the heater to repeat the process. B. Nucleate Boiling A further increase in the surface superheat or the surface heat flux will drive the system to the onset of nucleate boiling (ONB), the point on the boiling curve at which bubbles first appear on the heater surface. The rate of vapor bubble growth, the area density of nucleation sites which become active, the bubble frequency and bubble departure diameter manifest themselves as dominant parameters controlling the heat flux as the pool enters the nucleate boiling regime, all of which are increasing functions of the surface superheat. Without further discussion, it is mentioned that the rate of heat transfer in the nucleate boiling regime is extremely sensitive to various properties and conditions including system pressure, liquid agitation, and subcooling; surface finish, age, and

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coatings; dissolved noncondensible gases in the liquid; size and orientation of the heater; and nonwetting and treated surfaces. Heat fluxes in the nucleate pool boiling regime increase very rapidly with small increases in the surface superheat. The literature contains numerous efforts by various people to develop generalized correlations for nucleate pool boiling applicable to a wide range of liquids and generalized to include many of the properties and conditions previously listed. As a minimum, any successful correlation must include provisions which reflect the character or conditions of the heater surface as well as the properties of the boiling fluids themselves, requirements which have presented formidable obstacles to the development of any universally applicable correlation. One of the earliest attempts at such a correlation was developed by Rohsenow (1952), as seen in the following equation, for its historical significance and continued applicability: 0.5 0.33 c (Tw − Tsat ) q σ = Cs f Prs . i g µ i g g(ρ − ρg ) (27) where Cs f ∼ 0.013, s = 1 for water and s = 1.7 for all other fluids. Examination of this correlation reveals a theme underlying all of heat transfer and that is the essential requirement for accurate values of the physical and transport properties of the fluids of interest. C. Critical Heat Flux As the surface superheat in nucleate pool boiling continues to increase, the resulting increase in the boiling heat flux is accompanied by an increase in active nucleation sites on the surface of the heater, thus resulting in an increasing vapor production rate per unit area. The boiling heat flux will continue to increase up to a point at which the liquid can no longer remove any more heat from the surface due to vapor blanketing of the surface, restriction of liquid flow to the surface, and flooding effects which push liquid droplets away from the surface. There is no general agreement as to which of these mechanisms is responsible for the boiling crisis which ensues, and indeed each may be controlling under different geometric conditions. Regardless, soon the pool boiling curve reaches a peak heat flux which is called the critical heat flux (CHF). The critical heat flux in pool boiling is predominantly a hydrodynamic phenomenon, in which insufficient liquid is able to reach the heater surface due to the rate at which vapor is leaving the surface. As such, it is an unstable condition in pool boiling which should be avoided in engineered systems through design. There are two routes by which CHF can be reached. The first is by controlling the

temperature of the heater surface, in which case the system will simply return to nucleate boiling if the superheat is reduced or enter into transition boiling if the superheat exceeds CHF. D. Film Boiling It is more likely, however, that in most engineering systems the actual independent variable would be the heat flux, not the surface temperature. In this case, any increase in the surface heat flux above the CHF limit would induce a huge temperature excursion in the surface as it became vaporblanketed and heated-up adiabatically. This temperature excursion would continue until the imposed heat load was able to be transferred to the boiling liquid by thermal radiation from the surface almost exclusively. As a result of the vapor blanketing of the heater preventing liquid– solid contact, this boiling regime is called film boiling, and the occurrence of the thermal excursion from CHF into film boiling is known as burnout. This term comes from the fact that the resulting surface temperatures are, in general, so high that the surface and thus the equipment is damaged. Film boiling as a heat transfer process does not enjoy wide commercial application because such high temperatures are generally undesirable. In film boiling, a continuous vapor film blankets the heater surface which prevents direct contact of liquid with the surface. Vapor is generated at the interface between the vapor film and the overlying liquid pool by conduction through the vapor film and thermal radiation across the vapor film from the hot surface. It is of interest to note that in film boiling, the boiling heat flux is insensitive to the surface conditions unlike nucleate boiling, in which surface conditions or surface finish may play a dominant role. In transition boiling, the unstable regime between nucleate and film boiling, surface conditions do influence the data providing evidence that there is some liquid–surface contact in transition boiling which is not manifested in the film boiling regime. However, due to this decoupling of the boiling process from the heater surface conditions, film boiling is more tractable to analysis. The classical analysis of film boiling from a horizontal surface was performed by Berenson (1961). Many others have since contributed to the understanding of film boiling, notably by extending his work to very high superheats as well as to include liquid subcooling effects. The original model derived by Berenson for the film boiling heat transfer coefficient is reproduced in the following: h = 0.425

k g3 ρg (ρ − ρg )g(i g + 0.4c p,g Tsat µ Tsat (σ/g(ρ − ρg )1/2

0.25 . (28)

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For extension to higher temperatures at which thermal radiation becomes significant, a simple correction is made to the calculated film boiling heat flux by adding a radiative heat transfer contribution.

V. PHYSICAL AND TRANSPORT PROPERTIES Accurate and reliable thermophysical property data play a significant role in all heat transfer applications. Whether designing a laboratory experiment, analyzing a theoretical problem, or constructing a large-scale heat transfer facility, it is crucial to the success of the project that the physical properties that go into the solution are accurate, lest the project be a failure with adverse financial consequences as well as environmental and safety implications. In the solutions of heat transfer problems, numerous physical and transport properties enter into consideration, all of which are functions of system parameters such as temperature and pressure. These properties also vary significantly from material to material when intuition suggests otherwise, such as between alloys of a similar base metal. Physical and transport properties of matter are surprisingly difficult to measure accurately, although the literature abounds with measurements which are presented with great precision and which frequently disagree with other measurements of the same property by other investigators by a wide margin. Although this can sometimes be the result of variations in the materials or the system

parameters, all too often it is the result of flawed experimental techniques. Measurements of physical properties should be left to specialists whenever possible. It should come as no surprise, therefore, that the dominant sources of uncertainties or errors in analytical and experimental heat transfer frequently come from uncertainties or errors in the thermophysical properties themselves. This concluding section presents four tables of measured physical properties for selected materials under various conditions to illustrate the variability which can be encountered between materials and, in one case, the variability of a single material property as a function of temperature alone. Table I presents the most frequently used physical and transport properties of selected pure metals and several common alloys at 300 K. Listed are commonly quoted values for the density, specific heat, thermal conductivity, and thermal diffusivity for 18 metals and alloys. Heat transfer applications frequently require these properties at ambient temperature due to their use as structural materials. For properties at other temperatures, the reader is referred to Touloukian’s 13-volume series on the thermophysical properties of matter. It can be seen in Table I that some of the properties vary quite significantly from metal to metal. A judicious choice of metal or alloy for a particular application usually involves optimization of not only the thermophysical properties of that metal but also the mechanical properties and corrosion resistance. Table II presents the most frequently used physical and transport properties for selected gases at 300 K. Once again, 300 K represents a temperature routinely

TABLE I Physical Properties of Pure Metals and Selected Alloys at 300 K Material Aluminum Bismuth Copper Gold Iron 304 SS 316 SS Lead Nickel Inconel 600 Inconel 625 Inconel 718 Platinum Silver Tin Titanium Tungsten Zirconium

Tmelt (K) 933 545 1358 1336 1810 1670 ∼1670 601 1728 1700 — 1609 2045 1235 505 1953 3660 2125

ρ (kg/m3 )

c p (J/kg · K)

2702 9780 8933 19300 7870 7900 8238 11340 8900 8415 8442 8193 21450 10500 7310 4500 19300 6570

903 122 385 129 447 477 468 129 444 444 410 436 133 235 227 522 132 278

k (W/m · K) 237 7.9 401 317 80 14.9 13.4 35 91 14.9 9.8 11.2 72 429 67 22 174 23

α · 106 (m2 /s) 97.1 6.6 117 127 23 3.95 3.5 24 23 4.0 2.8 3.1 25 174 40 9.3 68 12.4

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ρ (kg/m3 )

cp (kJ/kg · K)

µ · 105 (kg/m · s)

ν · 106 (m2 /s)

k (W/m · K)

α · 106 (m2 /s)

Pr

Air Hydrogen Oxygen Nitrogen CO2

1.18 0.082 1.30 1.14 1.80

1.01 14.3 0.92 1.04 0.87

1.98 0.90 2.06 1.78 1.50

16.8 109.5 15.8 15.6 8.3

0.026 0.182 0.027 0.026 0.017

0.22 1.55 0.22 0.22 0.11

0.708 0.706 0.709 0.713 0.770

encountered in practical applications. The table considers five common gases and lists seven properties of general interest and frequent use. The reader is cautioned against the use of these properties at temperatures other than 300 K. All these properties with the exception of the Prandtl number are strong functions of temperature and significant errors can result if they are extrapolated to other conditions. The reader is once again directed to Touloukian for detailed property data. Applications of heat transfer at very low temperatures such as at liquid nitrogen (76 K) and liquid helium (4 K) temperatures present unique challenges to the experimentalist and require knowledge of the cryogenic properties of matter. Table III presents a summary of the temperaturedependent specific heat of six common cryogenic materials over the temperature range from 2 to 40 K to illustrate the extreme sensitivity of this property in particular (and most cryogenic properties in general) to even slight variations in temperature. Clearly, experiments, analyses, or designs which do not use precise, accurate, and reliable data for the physical properties of materials at cryogenic temperatures will suffer from large uncertainties. In addition, precise temperature control is a necessity at these temperatures. For research applications, these uncertainties could easily render experimental results and research conclusions TABLE III Specific Heat of Selected Materials at Cryogenic Temperatures Specific heat (J/kg · K) T (K) 2 4 6 8 10 15 20 30 40

Al

Cu

α-Iron

Ti

0.05 0.26 0.50 0.88 1.4 4.0 8.9 31.5 77.5

0.0066 0.0217 0.0545 0.114 0.205 0.663 1.76 6.53 14.2

0.183 0.382 0.615 0.900 1.24 2.49 4.50 12.4 29.0

0.146 0.317 0.540 0.840 1.26 3.30 7.00 24.5 57.1

Ice 0.12 0.98 3.3 7.8 15 54 114 229 340

Quartz — — — — 0.7 4.0 11.3 22.1 65.3

invalid. The reader is cautioned to seek out the most reliable data for thermophysical properties when operating under cryogenic conditions. Finally, there are occasions in heat transfer when it is advantageous to utilize liquid metals as a heat transfer medium. Table IV lists 12 low melting point metals commonly encountered in practice and lists their melting temperatures and boiling temperatures for comparison. The choice of a suitable liquid metal for a particular application does not provide for the flexibility which engineers and scientists have come to expect from other fluids at ordinary temperatures. Often the choice of an appropriate liquid metal for a particular application depends on the phase change temperatures as shown in Table IV. When and if no suitable liquid metal is found among the pure metals, alloys can be used instead. These alloys or mixtures usually have physical properties and melting/boiling temperatures which are significantly different from their constituent elements. No data for liquid metal alloys are given here; indeed, the data for liquid metal alloys are meager. It is crucial to the success of any heat transfer experiment, analysis, or facility that careful and judicious choices are made in the selection of the materials to be TABLE IV Melting and Boiling Temperatures of Some Common Liquid Metals Metal

Tmelt (K)

T boil (K)

Lithium Sodium Phosphorus Potassium Gallium Rubidium Indium Tin Cesium Mercury Lead Bismuth

452 371 317 337 303 312 430 505 302 234 601 544

1590 1151 553 1035 2573 969 2373 2548 1033 630 2023 1833

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292 used and that the data on their thermophysical properties are accurate and precise. It is difficult and expensive to determine these properties on an application by application basis, and property data which have not been measured by specialists may suffer large uncertainties and errors. The results of experiments and analyses can only be as accurate and reliable as their data and frequently that accuracy and reliability are limited by the thermophysical properties which were used.

SEE ALSO THE FOLLOWING ARTICLES CRYOGENICS • DIELECTRIC GASES • ELECTROMAGNETICS • FUELS • HEAT EXCHANGERS • HEAT FLOW • THERMAL ANALYSIS • THERMODYNAMICS • THERMOMETRY

Heat Transfer

BIBLIOGRAPHY Berenson, P. J. (1961). “Film boiling heat transfer from a horizontal surface,” J. Heat Transfer 83, 351–358. Eckert, E. R. G., and Drake, R. M. (1972). “Analysis of Heat and Mass Transfer,” McGraw-Hill, New York. Hartnett, J. P., Irvine, T. F., Jr., Cho, Y. I., and Greene, G. A. (1964– present). “Advances in Heat Transfer,” Academic Press, Boston. Rohsenow, W. M., Hartnett, J. P., and Ganic, E. N. (1985). “Handbook of Heat Transfer Fundamentals,” McGraw-Hill, New York. Rohsenow, W. M. (1952). “A method of correlating heat transfer data for surface boiling of liquids,” Trans ASME 74, 969. Sparrow, E. M., and Cess, R. D. (1970). “Radiation Heat Transfer,” Brooks/Cole Publishing Company, Belmont, CA. Touloukian, Y. S., et al. (1970). “Thermophysical Properties of Matter,” TPRC Series (v. 1–13), Plenum Press, New York. VanSant, J. R. (1980). “Conduction Heat Transfer Solutions,” Lawrence Livermore National Laboratory, UCRL-52863.

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Liquids, Structure and Dynamics Thomas Dorfmuller ¨ University of Bielefeld

I. II. III. IV. V. VI. VII. VIII. IX.

Introduction Structure of Liquids Dynamic Properties of Liquids Molecular Interactions and Complex Liquids Compartmented Liquids Glass-Forming Liquids Gels Dynamics in Complex Liquids Conclusions

GLOSSARY Amphiphiles Molecules consisting of a hydrophobic and a hydrophilic moiety. Complex liquids Liquids that consist of molecules whose anisotropic shape, specific interactions, and intramolecular conformations determine their properties to a significant degree. Dynamic spectroscopy Spectroscopic technique that uses the shape of spectral lines to obtain information about the dynamics of molecules. Gels Polymer–liquid mixtures displaying no steady-state flow. Gels are cross-linked solutions. Glass Solidlike amorphous state of matter. Liquid crystals Liquids consisting of highly anisotropic molecules whose orientations are strongly correlated. Micelles Aggregates formed in an aqueous solution of a detergent. Microemulsions Ternary solution of water, oil, and a detergent that forms dropletlike aggregates.

Pair correlation function Function of distance r that describes the probability of finding a molecule at a place located at a distance r from the origin, given that another molecule is located at the origin. See Fig. 1. Phase diagram Diagram having the coordinates pressure and temperature, in which the solid, liquid, and gaseous phases occupy different regions. Plastic crystals Crystalline phases displaying a liquidlike rotational mobility of the molecules. Simple liquids Liquids consisting of spherical or nearly spherical molecules interacting with central forces and without conformational degrees of freedom. Time correlation function Statistical quantity used to describe the temporal evolution of a random process. See Eq. (13). THE LIQUID STATE is a condensed state of matter that is roughly characterized by high molecular mobility and

779

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780 a low degree of order compared with solids. Liquids can be distinguished from gases by their high density. Liquid structure is studied by various scattering methods and liquid dynamics by a large number of spectroscopic techniques. The liquid phase of matter is of paramount importance in physics, chemistry, biology, and engineering sciences. Especially, liquid or liquidlike systems such as micelles, microemulsions, gels, and membranes play important roles in biology and in industrial applications. Although much detailed knowledge of liquids has been accumulated, the study of many fundamental issues in liquid-state physics and chemistry is actively pursued in many fields of science.

I. INTRODUCTION The liquid state of matter cannot be easily defined in an unambiguous and consistent way. It is often defined in terms of the phase-diagram (i.e., with respect to the solid and gaseous state). However, the distinction between the liquid phase and the gas phase is not sharp in the critical region of the phase-diagram, and the distinction of a liquid from a solid is also unclear for substances that have a tendency to supercool and are able to form glasses at low temperatures. Furthermore, many fluid substances are known that display specific structures, such as liquid crystals and micelles, where some of the criteria usually attributed to liquids do not apply. One could give a wide, but still to some extent ambiguous, definition of liquids by saying that a liquid is a disordered condensed phase. This would then include glasses, which due to the low mobility of the constituent molecules are usually regarded as amorphous solid systems. Another case where the limits of the liquid state are ill-defined is that of disordered clusters of molecules and of two-dimensional disordered arrangements on surfaces. The question of whether these phases should be considered liquids is a matter of the context in which they are studied. Liquids can be classified according to the properties of the molecules that constitute them. We thus distinguish between atomic and molecular liquids, among nonpolar, polar, and ionic liquids, and between liquids whose molecules do or do not display hydrogen bonding. Since interparticle interactions play a central role in determining the properties of liquids, we can broadly classify simple and complex liquids according to the way in which the molecules or atoms of the liquid interact. The simplest liquids are those consisting of atoms of noble gases. Thus, liquid argon, being considered as the prototype of a simple liquid, has been the object of many studies because of the absence of any complicating features in the

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intermolecular interaction. What distinguishes, for example, liquid argon from most other liquids is the spherical shape of its atoms leading to central interaction forces, the dispersive character of the interparticle forces, and the absence of internal degrees of freedom. Other liquids that can be considered simple should consist of atoms or molecules with shapes not deviating much from a sphere; they should not display noncentral, angledependent, or specific saturable interparticle forces like, for example, those that lead to the formation of hydrogen bonds, and, finally, the internal degrees of freedom, especially configurational degrees of freedom, should not much influence the properties of the liquid. According to these criteria, the majority of liquids are complex, the concept of a simple liquid being the result of an extrapolation of the properties of a relatively small number of liquids whose molecules comply to some extent to the abovementioned requirements of simple liquids. The concept of a simple liquid has been very fruitful in contributing to the development of the concepts that are necessary to describe the essentials of the liquid state. However, since most interesting and important liquids must be reckoned among the complex liquids, the study of these is extremely important.

II. STRUCTURE OF LIQUIDS The ordered structure of crystalline solids is a consequence of interparticle interactions leading to a dependence of the free energy of a system of interacting particles on their arrangement in space. The stability of a particular crystalline structure at thermodynamic equilibrium results from the minimum of the free energy achieved in this state. Because such interactions are negligible in gases at low pressure, we observe chaos instead of order in gaseous systems. The case of liquids is intermediate between the two extremes of perfect order in ideal crystalline solids and complete lack of order in ideal gases. More precisely, the molecules of the liquids interact, and as a consequence they tend to arrange themselves in a constrained structure. On the other hand, the thermal energy of a liquid is high enough so that the molecules rearrange themselves rapidly and continuously. As a result, order in most liquids is not constant in time, nor does it extend over distances larger than a few molecular diameters. If the shapes of the molecules and the attractive forces between molecules are anisotropic, then we may observe orientational order in a liquid, with the molecules with aligning themselves so that the orientation of neighboring molecules is not completely random. The opposite case is observed in plastic crystals, which are positionally ordered systems in which the molecules are, however, relatively free to rotate; so we do not have orientational order as in classical crystals.

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Both translationally and rotationally ordered structures can be described by appropriate functions of spatial variables such as interparticle distance and relative orientation. Translational order is described by the radial pair correlation function g(r ) and orientational order by the static orientational correlation function. The function g(r ) is the probability of finding a molecule at a point A at a distance located between r and r + dr from another point B, given that another molecule is at B. The static orientational correlation is a number reflecting the probability that a molecule located at a distance r from another molecule is oriented in such a way that the two molecules form an angle between ϑ and ϑ + dϑ. For this definition to apply, the molecules must have a symmetry allowing the definition of a physically identifiable orientation axis. This is the case, for example, with linear and cylindrically symmetric molecules. Thus, the size of this quantity gives us a clue as to whether molecules located at a distance r from each other tend to align in parallel, antiparallel, or perpendicular orientations and characterizes the average angle between such molecules. The importance of both quantities stems from the fact that they can be measured by the diffraction of electromagnetic radiation and by scattering of slow neutrons, both having a wave-length of approximately the average interparticle distance. Figure 1 displays a characteristic shape of g(r ), illustrating the exclusion of neighbors at small distances from the reference molecule. The first, more pronounced, peak at r1 corresponds to the shell of the nearest neighbors. The second at r2 and the further peaks come from more distant and hence more diffusely distributed shells. The oscillations characteristic of the radial distribution decay after a small number of maxima and minima showing that no long-range order is present at distances

FIGURE 1 A typical radial pair correlation function of a simple liquid. Note the maxima at r 1 and r 2 , which illustrate the increased probability of finding a molecule at these distances from the central molecule. Note also the value 0 at small distances and the limiting value of 1 at large distances. The first is a consequence of the repulsive interaction of molecules, and the second illustrates the randomization of the mean particle density at large distances.

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FIGURE 2 The pair potential energy between two molecules A and B. The two molecules, represented by the spheres, are shown at the equilibrium distance r AB if the potential is a Lennard– Jones potential. In terms of the hard core repulsive potential (vertical dashed line) this is the contact position with the centerto-center distance equal to r AB . The Lennard–Jones potential: 12 6 V(r ) = 4EAB [(r 0 /r ) − (r 0 /r ) ] results from the superposition −12 of the r repulsive branch (upper part of the solid curve) and the r −6 attractive branch (lower half of the solid curve).

as large as a few molecular diameters. For large values of the distance r the product ρg(r ) approaches the value of the average number density ρ = N /V of the equilibrium distribution, where N is the number of molecules contained in the volume V . In contrast to this, a system with long-range order would display nondecaying oscillations of g(r ) over significant distances. Although the general form of the intermolecular forces is known, it is very difficult to derive directly from this approximate knowledge the exact shape of the radial distribution function. As a useful approximation, however, the repulsive branch of the potential has been approximated by a hard core potential and the attractive branch expressed by simple inverse power of the intermolecular center-to-center distance. This is illustrated in Fig. 2. With such an approximation it was found that the main features of the radial pair distribution function can be explained qualitatively even if we completely neglect attraction. It thus appears that most of the liquid structure is the result of the steep repulsive intermolecular pair potential.

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Steep in this context means a potential energy function that can be expressed as an inverse power of the intermolecular distance with an exponent that is significantly larger than the value n = 6 found in dispersion forces. Usually, steep potentials are approximated by an exponent n = 12 or larger because this is the case for the repulsive part of the Lennard–Jones potential illustrated in Fig. 2. In most cases the repulsive branch of the potentials was shown to be much steeper than the attractive branch, although some complex liquids, such as associating liquids, do have steep attractive branches that are essential in determining their structure. In this latter case the potential has the shape of a narrow, steep-walled well leading to relatively stable dimers or higher aggregates. In a system of particles interacting through central pair forces, we can derive a simple equation between the excess internal energy U per molecule due to intermolecular interactions, the pair potential V (r ) and the radial distribution function g(r ): ∞ U = N 2πρ g(r )V (r )r 2 dr. (1) 0

We can also derive an expression for the equation of state for this system in terms of the radial distribution function and the gradient of V (r ): 1 d V (r ) PV = NkT 1 − g(r )r dr . (2) 6kT V dr In this equation k is the Boltzmann constant and P the pressure. The compressibility equation, Eq. (3), connects the isothermal compressibility defined as βT = (∂ V /∂ P)T V −1 with the radial pair distribution function: kTβT = 1 + ρ [g(r ) − 1] dr. (3) V

The liquid structure that is inherent in g(r ) can also be described in terms of the structure factor S(k), which is a quantity used to describe neutron and X-ray scattering experiments. The elastic scattering of, for example, neutrons ˚ is determined by the having a typical wavelength of 1 A local arrangement of the scattering atoms. The structure factor is connected to the radial distribution function by means of the equation S(k) = 1 + ρ exp(−ikr)[g(r ) − 1] dr. (4) V

This equation shows that the structure factor can be expressed in terms of the Fourier transform in space of the radial distribution function. For isotropic liquids the radial pair distribution function is a function of the modulus r and the structure factor of the modulus of the vector k. If the liquid is anisotropic, we must use instead the full vectors r and k.

Actually, S(k) describes the liquid structure in k-space, which is the reciprocal of ordinary r -space. The role of this function in describing the structure of the liquid is determined by the character of the incident radiation which is characterized by its wavevector ki and that of the scattered radiation by ks . The conservation of momentum of the system, liquid + incident radiation + scattered radiation, leads to a scattering intensity for a given angle of observation that depends only on S(k), where the vector k is defined by k = ks − ki .

(5)

S(k) curves can be calculated by model theories that can then be tested against the experimental curves obtained from neutron scattering experiments. In the cases of linear (e.g., nitrogen and carbon disulfide) and tetrahedal molecules (e.g., yellow phosphorus and carbon trichloride), the diffraction methods have shown that the former indeed tend to align in the liquid, whereas the latter in some cases form interlocked structures. The structural information obtained by diffraction methods is important but far from complete, and the confirmation by model calculations is essential. The calculation of accurate structure factors from diffraction experiments is often hampered by correction problems and problems of interpretation. The use of isotopically substituted molecules has proved essential in obtaining the necessary data to calculate the more detailed atom–atom pair distribution functions of molecular liquids. With increasing complexity of the molecules, the problems increase too. In the case of some more complex molecules, however, such as acetonitrile, chloroform, methylene chloride, and methanol, the diffraction methods have given structure factors that compare favorably with theoretical data. Especially, one can confirm the formation of hydrogen-bonded chainlike structures, which are expected from the physical properties of these substances and from some dynamic data. The above relations can be extended to describe more complex polyatomic molecular liquids if appropriate parameters for the description of the molecular coordinates (i.e., either the relative position of the center of mass r and the angles describing the orientation of the molecule or the set of parameters specifying the position of all the atoms in the molecule) are introduced. The radial pair distribution function then becomes a function of all these coordinates and is generally much too complex to calculate or even to visualize. A very useful simplification of the description and hence the calculation of liquid structure using site correlation functions was introduced with the so-called RISM theory. This theory incorporates the chemical structure of the molecules into the model by approximating them to objects consisting of hard fused spheres modeling their

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chemical structures. The usefulness of this view in describing the pair correlation function stems from the fact that it is mainly the shape of the molecule that is critical in determining the structure of the liquid. If, on the other hand, the role of the long-range attractive part of the intermolecular potential is important (e.g., in the calculation of thermodynamic properties or in the case of strongly structured liquids such as water or some other polar liquids), different methods must be used, such as perturbation theories. These are based upon the assumption that we can split the intermolecular interactions into a simple well-defined part and into another that is considered a small perturbation, but which confers to the more complex system its specific properties, which we want to calculate. The main problem lies in the question whether it is possible to find an adequate reference-state leading to a known structure so that the interactions of the more complex liquid can be obtained by adding a perturbation term to the reference potential. The hard-sphere potential was often chosen as a reference, but its usefulness for polyatomic liquids has been seriously questioned. The strengths and limitations of theoretical models that are used to obtain a quantitative description of liquid structure are often assessed by comparing the results with X-ray and neutron diffraction data and with the results of computer simulation calculations. The latter provide us with a method of calculating numerically the properties of model liquids consisting of molecules with a well-defined intermolecular potential. In molecular dynamics computer simulations, the classical trajectories of an ensemble of molecules are calculated by solving the equation of motion. The static properties (i.e., pair distribution functions, equations of state, and internal energy) are calculated as averages over a sufficiently large number of equilibrium configurations created from the trajectories. The value of computer simulation lies in the possibility of calculating separately the effects of different features of real molecules: shape, the potential energy parameters, the dipole moment, and several others. This proved to be very valuable for the understanding of the important factors affecting the structure of liquids. On the other hand, our incomplete knowledge of the intermolecular potential of real molecules prevents the computer simulations from giving an exact replica of the real liquid. Furthermore, due to the necessary restrictions in computer capacity, the ensembles that can be reasonably handled consist of 1000 molecules or less. The averages over such ensembles are considered to represent, to a sufficient degree of accuracy, statistical averages in a bulk liquid consisting of some 1020 molecules. This entails problems of a statistical nature that have been only partially solved.

It is fair to say that our knowledge of the structure of liquids has advanced in the last two decades, the essential mechanisms determining liquid structure being understood in principle. What is still lacking is an accurate knowledge of intermolecular potentials derived either from experiments in the liquid state or by ab initio quantum-mechanical calculations. Furthermore, we must be aware that most of our models are approximations and that we still are unable to predict whether a given approximation is adequate to give good thermodynamic, structural, or dynamical data. We also do not know why most of the currently used models give good results for data of one of the above-mentioned classes but poor results for another class.

III. DYNAMIC PROPERTIES OF LIQUIDS The description of the equilibrium state of a liquid by means of the radial pair distribution function or the structure factor can be extended to include time-dependent properties of liquids. This can be done by the use of the Van Hoove correlation function G(r, t) which has been introduced as a tool for the description of quasi-elastic neutron-scattering results. This function is both space- and time-dependent. In analogy to the definition of the static radial distribution function by means of Eq. (1), the Van Hoove correlation function is defined as a time-dependent density–density correlation function: G(r, t) =

ρ(r, t) · ρ(0, 0) , ρ(0, 0) 2

(6)

where G(r, t) is the probability of finding a particle i in a region dr around a point r at time t, given that there was a particle j at the origin at time t = 0. To separate the motion of particles in a laboratory-fixed frame of reference from the relative motion of the particles, it is convenient to separate G(r, t) into a self and a distinct part: G(r, t) = G s (r, t) + G d (r, t).

(7)

Figure 3 illustrates the behavior of G s (r, t) and G d (r, t) on three time scales. The time scales are considered with respect to the so-called structural relaxation time τ , which is defined as the average time required to change the local configuration of the liquid. In Fig. 3, the following cases are distinguished: 1. The time scale is short with respect to the structural relaxation time. 2. The time scale is similar to the structural relaxation time.

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FIGURE 3 The shapes of (a) the self term and (b) the distinct term of the Van Hoove space-time correlation function. At times that are short with respect to the structural relaxation time, Gs (r, t) is sharply peaked since the reference molecule did not have time to change its position significantly, whereas Gd (r, t) is zero due to the repulsion of the reference molecule at the origin, which does not allow another molecule to occupy the same position. With increasing time, as t becomes similar to τ , Gs (r, t) broadens because the probability of finding the reference molecule away from the origin increases. On the other hand, at small distances from the origin, Gd (r, t) increases as the probability of finding another molecule at the origin is no longer zero. Finally, at long times (t τ ), the probability of finding a given molecule at a distance r is small and independent of the distance from the origin, and the probability of finding some molecule at r is 1.

3. The time scale is long with respect to the structural relaxation time. One sees that at short times G s is sharply peaked about r = 0, whereas G d displays the characteristic oscillations similar to the time-in-dependent radial pair distribution function in Fig. 1. At long times, both functions vary little in space and approach the steady-state value as the local distribution is nearly averaged out at times long as compared to τ . The described generalization leading to the space- and time-dependent Van Hoove correlation function is readily extended to the structure factor which then becomes frequency dependent. The use of a frequency dependent dynamic structure factor S(k, ω) stems from the study of the spectra of scattered slow neutrons. The symbol ω is for the angular frequency. Thermal neutrons are well suited to the study of the dynamics of liquids because their energy is comparable to kT , and the wavelength associated with the neutrons is comparable to intermolecular distances at liquid densities. The measurable derivative d 2 σ/dO dϑ of the differential cross section dσ/dO is directly related to S(k, ω) by the equation d 2σ 2 k1 S(k, ω), (8) =b dO dϑ k0 where σ is the total cross section and b the scattering length typical for the scattering atom (of the order of magnitude of the nuclear radius). If the molecule is heteronuclear, this

Liquids, Structure and Dynamics

relation has the form of a sum over all j atomic species with scattering lengths b j , O represents the solid angle under which the scattered radiation is detected and k1 and k0 the moduli of the wavevectors of the neutrons before and after the scattering event, respectively. The dynamic structure factor can be separated into a self and a distinct part Ss (k, ω) and Sd (k, ω) corresponding to the self and distinct parts of the Van Hoove correlation function. This separation acknowledges the fact that the molecular motion detected by neutron scattering involves both single-particle and collective motions. We can use two extreme models to describe the situation. A. The Perfect Gas Model The assumption of a free motion of the molecules √ with a mass M and the most probable velocity v0 = 2kT /M leads to the following expression for the Van Hoove correlation function: −3 −r 2 G s (r, t) = π −2 v0 τ exp (9a) (v0 τ )2 G d (r, t) = σ.

(9b)

From this expression we can obtain the following expression for the dynamic structure factor for this model: −1 −ω2 S(k, ω) = kv0 π 1/2 exp . (kv0 )2

(10)

In liquids, this limit of a free (i.e., collisionless) motion is realized in the limit r → 0 and t → 0 corresponding to k → ∞ and ω → ∞. Such behavior is approximated in scattering experiments with wavelengths significantly shorter than the average interparticle spacing of a few Angstroms (thermal neutrons) and probing times shorter that the average time between successive collisions. B. Single-Particle Motion in the Hydrodynamic Limit The opposite extreme to this limit is obtained at long times and large distances corresponding to k → 0 and ω → 0. In this range the molecular interactions and not their masses determine the motion that is now monitored by the scattering of long-wavelength radiation (e.g., light scattering). This latter range of liquid dynamics corresponds to the hydrodynamic limit. In this limit the liquid can be treated as a continuum to which the hydrodynamic equations apply, the molecular details being formally introduced as extensions of the classical Navier–Stokes equations (e.g., by using a frequency-dependent viscosity to take care of molecular relaxation processes). In the hydrodynamic limit (i.e., when r and t are sufficiently large), the singleparticle correlation function obeys a diffusion equation

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similar to Fick’s differential diffusion equation. Under appropriate boundary conditions, we obtain the following integrated form:

relation function over the time from t = 0 to t = ∞. If the process described by the above correlation function is diffusive, then the correlation function is exponential:

−r 2 . (11) 4Dt This is also Gaussian, differing, however, in the time dependence from the case of the free motion in Eq. (10). The corresponding dynamic structure factor is given by

−t . (14) τ The time constant τ describing the decay of the correlation is termed a relaxation time. The corresponding spectrum I (ω) has a Lorentzian shape given by

G s (r, t) = (4π Dt)−3/2 exp

(1/π )Dk 2 . (12) ω2 + Dk 2 This expression represents the spectrum of the scattered intensity at a fixed value of the wavevector (i.e., at a fixed angle of observation). The time correlation function formalism has been shown to be adequate for representing liquid dynamics in a convenient way. Thus, some experimental methods such as photon correlation spectroscopy directly give time correlation functions; others such as infrared and Raman bandshape analysis operate in the frequency domain, and the obtained spectra can be Fourier transformed to give time correlation functions. Figure 4 visualizes the relation between the two domains. This procedure is based on the fluctuation-dissipation theorem of statistical mechanics which connects random thermal fluctuations in a medium to the power spectrum characterizing the frequency spectrum of the process. A time correlation function of a dynamical molecular variable A(t) (e.g., a dipole moment) is defined by Ss (k, ω) =

C(t) = A(0)A(t) .

(13)

The correlation time τ of a process described by the above correlation function is defined as the integral of the cor-

FIGURE 4 Relation between (a) the power spectrum (frequency domain) of a relaxation process and (b) the correlation function (time domain) of a dynamical variable describing this process. In dynamic spectroscopy, the half width at half height of the spectral line is measured, and the correlation function is obtained by Fourier transforming the spectral profile. The relaxation time can be directly calculated from the linewidth by the relation indicated in the figure if the spectral profile is a Lorentzian. In photon correlation spectroscopy, which operates in the time domain, a correlation function is directly measured.

C(t) = C(0) exp

I (ω) =

1 . 1 + (ω − ω0 )2 τ 2

(15)

The Lorentzian bandshape as indicated in the figure is characterized by the relaxation time that thus can be extracted from the half width at half height of the spectral band by 1 . (16) 2π The spectral band function I (ω) and the corresponding correlation function C(t), as illustrated in Fig. 4, are a Fourier transform pair (i.e., they can be uniquely transformed into each other). The macroscopic transport coefficients such as the mass diffusion coefficient, the thermal conductivity coefficient, and the macroscopic shear viscosity have been related to the time integral of pertinent correlation functions. Thus, the mass diffusion coefficient D is given by 1 ∞ D= v(0)v(t) dt. (17) 3 0 τ=

In this equation v(t) is the molecular velocity at time t. We have presently at our disposition several sources of information about the molecular dynamics of liquids. Among them the most important experimental techniques are Rayleigh and Raman light scattering; infrared and far infrared spectroscopy; NMR spectroscopy; fluorescence anisotropy methods, either stationary or time dependent; and time-dependent spectroscopy from the nanosecond to the femtosecond time scale. On the other hand, one of the most important sources of dynamical information on liquids is the computer simulation by means of molecular dynamics. The method aiming at extracting dynamical information from the shape of spectra is termed dynamical spectroscopy. The dynamical information contained in spectral band-shapes is in most cases complex, since the spectra reflect rotational, translational, and vibrational broadening mechanisms that cannot be uniquely sorted out. Each spectroscopic method has its strengths and its inherent restrictions, which is why most of the progress has been obtained by the simultaneous application of several methods on the same liquid. One of the most serious restrictions is that some methods (e.g., Rayleigh scattering) probe collective motions that are very difficult to relate

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786 exactly to single molecule motions. On the other hand, collective motions, essential to the understanding of liquid dynamics, are of great interest in themselves. On the other extreme, NMR data probe essentially single molecule motions, however, in contrast to optical spectroscopic methods, they do not give all the information contained in full time correlation functions but only correlation times. In all the dynamical methods, we must be aware of instrumental restrictions with regard to the accessible time scale. The extension of the time scale over as many decades as possible is vital to the understanding of the underlying molecular mechanism and is one of the main experimental goals in this area. For this reason it is important to obtain reliable data with different methods on time scales complementing each other and which can be properly adjusted to each other. This is for example the case with the simultaneous study of the depolarized light scattering and flourescence anisotropy of the same label molecule dissolved in a liquid which can be used to monitor molecular rotations on a time scale extending from a few picoseconds to approximately 1 µs (i.e., over six decades). In the last years the use of ultrashort laser pulses has become an increasingly important tool for the study of liquid dynamics. By this method it has become possible to study directly in the time-domain processes taking place in the picosecond and subpicosecond time range. These include orientational processes in liquids, the rates of charge transfer processes, the rate of recombination of ions to molecules in a liquid cage, and a number of solvent-dependent photophysical processes. Thus, vibration–rotation coupling and the rates of vibrational energy and phase relaxation were studied by picosecond spectroscopy, and the corresponding rates could be determined in some cases. Although the data obtained in the time domain and those in the frequency domain are rigorously linked, the former sometimes allow us to circumvent serious instrumental complications. In combination with spectroscopic lineshape analysis, real-time techniques have improved significantly our understanding of liquid-state dynamics. Intermolecular dynamics is manifested in so-called interaction-induced spectra. This phenomenon, which leads to the occurrence of forbidden spectral lines appearing in high-density gases and in liquids has been studied extensively in the last years and has been shown to be helpful in obtaining information mainly about the shorttime dynamics of liquids. The main mechanism by which these spectra are produced is the induction of a timedependent dipole on a molecule by electric fields of other molecules in its immediate neighborhood and the interaction of this dipole with the electric field of the light.The intermolecular inducing fields may be coulomb-, dipole-, or higher-multipole fields, as the case may be. Furthermore,

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high-energy collisions produce distortions of the colliding molecules, thus inducing transient dipoles that also contribute to interaction-induced spectra. It appears that the wealth of information concealed in interaction-induced spectra is presently the main problem encountered in the analysis of such data.

IV. MOLECULAR INTERACTIONS AND COMPLEX LIQUIDS The structures as well as the dynamics of liquids in equilibrium are determined by interactions of the molecules. Knowledge of these interactions is essential in obtaining a theoretically founded description of the physics of liquids. However, it is still very difficult to carry out quantum mechanical ab initio calculations of the intermolecular potential, although the basic understanding of the interactions between molecules is available. Thus, such calculations have been fruitful only for a small number of molecules consisting of a relatively small number of atoms. The alternative to theoretical calculations is to determine intermolecular potentials by accurate gas-phase experiments that probe essentially two-particle interactions. However, this has also proved at least ambiguous since it is generally not possible to use unmodified gas-phase potentials for liquids. On the other hand, the reverse method of determining intermolecular potentials from liquid-state data is also unyielding because the method is modeldependent and, additionally, because most measurable quantities present themselves as integrals over ensembles of molecules from which the integrand cannot be uniquely determined. For the present we must use empirical potentials that represent averages over those effects that cannot yet be explicitly taken into account. A major problem that is still unsolved is the calculation of many-particle interactions in an ensemble of interacting molecules. This is critical for the description of a liquid since at liquid densities, due to the small distances between interacting molecules, we are, in principle, not allowed to express the total interaction in the liquid as a sum of pairwise additive interactions, neglecting the many-body character of the problem. Some experimental results have been interpreted by assuming that many-particle interactions are important; however, such interpretations are still far from being quantitative.

V. COMPARTMENTED LIQUIDS The nature of molecular interactions is very important in determining the structure and the properties of complex liquids. Many molecules consist of two different parts, the

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one interacting strongly with water (the hydrophile) and the other not (the hydrophobe). In aqueous solutions we observe compact or lamellar liquid aggregated structures, depending on the nature of the solute, the temperature, and the concentration. The micelles thus formed are assumed to have a rather compact hydrophobic core surrounded by a hydrophilic shell. The hydrophiles are generally ionic or polar groups, whereas the hydrophobe is often an aliphatic chain. The driving force of aggregation in water of such molecules, called amphiphiles or detergents, is the minimization of the total free energy resulting from contributions from the water–water, water–hydrophile, and hydrophile–hydrophile interactions. The phase diagram of amphiphile–water mixtures displays several distinct phases with quite different properties, depending on the temperature and the concentration of its constituents. The structure of the resulting aggregates is a function of the nature of the hydrophile and of the length of the hydrophobic chain (see Fig. 5). Also the structure of the liquid within the micellar core seems to be in some cases different from that in bulk liquids with the effect that solubilized species may display a specific behavior as regards reactivity, acidity, mobility, etc. Another example of compartmentation of practical importance is the case of microemulsions, which are formed when water, oil, and a detergent are mixed in appropriate proportions. Such systems are used to solubilize otherwise unsoluble substances and to promote chemical reactions by capturing them in their interior and thus increasing the local concentration of the reactants. Catalytic reactions in micelles and microemulsions play an increasing role in chemistry. The occurrence of localized compartmented liquidlike phases is a very important phenomenon and plays a major role in biological systems where both fluidity and compartmentation are essential. The internal fluidity of

FIGURE 5 The structure of a micelle in water.

compartmented liquid phases is studied intensively by spectroscopic methods such as ESR spectroscopy and fluorescence anisotropy decay of convenient dissolved or chemically bound labels. Even in the absence of distinct phases, molecules that consist of groups with different affinities to the solvent give rise to more localized and less randomized structures that affect the physical and chemical properties of liquid mixtures. Hydrogen-bonding molecules can belong to this category. Charge transfer interactions may affect in a similar fashion the local structure of a liquid mixture.

VI. GLASS-FORMING LIQUIDS Amorphous substances with a solidlike rigidity play an important technological role, and their study is one of the major fields in materials science. Such substances are generally obtained when a liquid is cooled below its melting point while preventing crystallization. Glass-forming liquids (i.e., those that can be obtained in the glassy state) must have special properties connected with the symmetry, configuration, and flexibility of the molecules or their ability to form intermolecular bonds. Polymeric liquids are among the best studied glass-forming systems. The properties of liquids at different temperatures can basically be understood in terms of the kinetic energy and the intermolecular potential of the molecules. In some cases, however, during the process of cooling, the viscosity of a liquid increases by several orders of magnitude in a rather narrow temperature range. The explanation usually given for this extreme slowing down of most of the molecular dynamics is that in such cases the thermal energy kT becomes similar to or smaller than the intermolecular potential energy required by the molecules to accommodate in the respective equilibrium configuration. Thus, the source of the high viscosity is the freezing of intramolecular configurations, while at sufficiently high temperatures the molecules are able to move past each other, allowing local stresses to be relieved at a much faster rate. In the case of nonrigid molecules such as most polymers this process is supported by the adaptation of the molecule to the constraints produced by the environment and external forces. In the high-viscosity state, on the other hand intramolecular barriers may prevent the molecule from undergoing configurational changes, this process leading to an increasing rigidity of the molecule itself. As the relaxation of the undercooled liquid to thermodynamic equilibrium becomes slower than the cooling rate, instead of crystal formation we observe the formation of a rigid amorphous glass. The temperature Tg at which this occurs is known as the glass point. Several authors define the glass point as the temperature at which

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terms of a slow α-process, a faster β-process, and other processes indicated by the Greek letters γ , δ, and so on. Especially the α-process is crucial in determining the mechanical properties of glass-forming systems, and attempts are made to synthesize molecules with a given temperature dependence of the α-relaxation process. This would allow us to obtain materials with definite useful mechanical properties in a given temperature range. The interpretation of these processes at a molecular level, however, is a still unsolved problem. Many attempts to rationalize the data in a semiphenomenological manner are based on the concept of the free volume available to the molecular motion and hence to the relaxation of nonequilibrium configurations. All these phenomena, which have been extensively studied in polymer melts, are also observed in several glass-forming low-molecular-weight liquids (e.g., o-terphenyl, decalin, salol, and polyalcohols). An important observation in supercooled liquids is the dependence of the physical properties of these substances on the thermal history of the sample. The question whether glass formation is the effect of kinetic constraints only, or whether other factors play a role, is still open. The existence of metallic glasses and the observation of a glassy phase in computer simulations of molecules as simple as argon may be important clues to this question.

VII. GELS When a low-molecular-weight liquid is dissolved in a high-molecular-weight system (the stationary component), which often is a cross-linked polymer, under certain conditions we observe the formation of a gel. This is a macroscopically homogeneous liquid with high internal mobility but no macroscopic steady-state flow. Gel formation requires the presence of more or less stable cross-links to prevent viscous flow as well as a fluid component that must be a good solvent for the stationary component. Figure 7 displays schematically the structure of a gel.

VIII. DYNAMICS IN COMPLEX LIQUIDS

FIGURE 6 Plot of the elasticity modulus versus temperature, showing the transition from the glassy to the rubbery to the liquid state.

One of the dynamical problems studied theoretically and experimentally very extensively is the rotational motion of molecules in liquids. The molecular rotation in gases is solely determined by the kinetic energy and the moment(s) of inertia of the molecule. Under the action of frequent random collisions with other molecules, the rotation of a particular molecule in a liquid is continuously perturbed, and this is reflected in an exponential time dependence of the correlation function of the orientation of the molecular axes. Generally, at short times of the order of 0.1 ps and less, the motion is determined by the molecular

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pared with the corresponding macroscopic data (e.g., the macroscopic shear viscosity). The translational motion of very large molecules in liquids, such as diluted polymers or supramolecular aggregates like micelles and microemulsions, has been studied by light-scattering methods to obtain information about the molecular weight and size of the diffusing entity, its polydispersity, the interactions with other species (e.g., ions) or, at higher concentrations, interactions between the diffusing molecules themselves, and, finally, the internal flexibility and the rate of configurational changes.

IX. CONCLUSIONS FIGURE 7 The structure of a gel. The points represent the molecules of the liquid. The cross-linked polymer is represented by the lines.

moment of inertia and the temperature, whereas at longer times, the motion is determined by angular momentum exchange due to the frequent collisions with other molecules. This can be described by a friction exerted on the rotating molecule by its neighbors. It was shown by Debye that under certain simplifying assumptions this so-called rotational diffusion can be described by a relaxation time τOR which is connected to the macroscopic viscosity η of the liquid: τOR =

ηV . kT

In this equation V is a characteristic volume, the hydrodynamic volume of the molecule. If the shape and size of the rotating molecule are known, this relation can be used to probe the local viscosity in liquid systems (e.g., in micelles and membranes). Such a local viscosity can be different from the macroscopic viscosity and is accessible only through measurements done on label molecules. Since molecular labels are convenient indicators of the local microdynamics of the liquid in their neighborhood, they can also be used to test theoretical models of liquid-state dynamics. The experimental methods currently used are NMR relaxation, Raman linewidth measurements, dynamic light-scattering spectroscopy, fluorescence anisotropy, and dielectric relaxation. The theory of rotating molecules in a liquid medium interacting in an uncorrelated random fashion with the surroundings has been described by models amenable to analytical calculations in the case of simple liquids. The quantities that enter the calculation are molecular moments of inertia, molecular masses, and intermolecular forces. In the case of more complex liquids, the assumption of a diffusive motion in a continuum is made, and the parameters of the model are hydrodynamic quantities that can be com-

The liquid state includes a large number of phenomenologically very different systems such as simple liquids, micelles, microemulsions, polymer melts, liquid crystals, and gels. All these systems have in common (1) a rather high molecular mobility which may, however, be restricted in different ways depending on the system, and (2) molecular disorder which may also be restricted in different ways. The study of the liquid state involves most of the modern physical methods, and the theory of its molecular aspect requires elaborate statistical mechanical methods. The study of the liquid state is progressing at a rapid rate although several basic problems still remain unanswered.

SEE ALSO THE FOLLOWING ARTICLES FLUID DYNAMICS • GLASS • HYDROGEN BOND • LIQUID CRYSTALS (PHYSICS) • MICELLES • MOLECULAR HYDRODYNAMICS • PERMITTIVITY OF LIQUIDS • POTENTIAL ENERGY SURFACES • RHEOLOGY OF POLYMERIC LIQUIDS • X-RAY SMALL-ANGLE SCATTERING

BIBLIOGRAPHY Barnes, A. J., Orville-Thomas, W. J., and Yarwood, J., eds. (1983). “Molecular Dynamics and Interactions,” D. Reidel, Dordrecht. Berne, B. J., and Pecora, R. (1976). “Dynamic Light Scattering,” Wiley, New York. Birnbaum, G. (1985). “Phenomena Induced by Intermolecular Interactions,” Plenum, New York. Enderby, J. E., and Barnes, A. C. (1990). Reports on Progress in Physics 53(1 & 2), 85–180. Hansen, J. P., and McDonald, I. R. (1976). “Theory of Simple Liquids,” Academic Press, New York. Rothschild, W. G. (1984). “Dynamics of Molecular Liquids,” Wiley, New York. Rowlinson, J. S. (1982). “Liquids and Liquid Mixtures,” Butterworth, London. Wang, C. H. (1985). “Spectroscopy of Condensed Media,” Academic Press, Orlando.

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I. II. III. IV. V. VI. VII. VIII.

Kinematics Newton’s Laws of Motion Applications Work and Energy Momentum Rigid Body Motion Central Forces Alternate Forms

GLOSSARY Conservation Certain quantities—e.g., energy and momentum—remain the same for a system before, during, and after some interaction (often a collision). Such quantities are said to be conserved. Dynamics Explanation of the cause of motion. This involves forces acting on massive bodies and the motion that ensues. Energy Ability to do work; stored-up work. Kinematics Description of motion. Momentum Mass multiplied by the velocity. Momentum is a vector. Statics Study of forces acting on bodies at rest. Work Distance a body moves multiplied by the component of force in the direction of the motion. Work is a scalar.

CLASSICAL MECHANICS is the study of ordinary, massive objects, for example, the study of objects roughly

the size of a bread box traveling at roughly sixty miles an hour. It is to be distinguished from quantum mechanics, which deals with particles or systems of particles that are extremely small, and it should also be distinguished from relativity, which deals with extremely high velocities. Classical mechanics can be divided into statics, kinematics, and dynamics. Statics is the study of forces on a body at rest. Kinematics develops equations that merely describe the motion without question to its cause. Dynamics seeks to explain the cause of the motion.

I. KINEMATICS Motion of an object must be described in terms of its position relative to some reference frame. If the motion occurs in one dimension (as along a straight highway or railroad track) the position will usually be written as x. If the motion occurs in two or three dimensions (as an airplane circling an airport or a spacecraft on its way to Jupiter) the position will be written as r. The position is a

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vector. For the one-dimensional case, the vector nature of the position shows up as the sign of x. For example, the position may be considered positive to the right; then it will be negative to the left. Position is commonly measured in meters. Of course, position may also be measured in kilometers, centimeters, feet, or miles as the need arises. Velocity is the time rate of change of the position of an object. It can be written as v = x/t or

position are wanted. For constant acceleration, a, in one dimension the velocity and position at some time t can be found from v = v0 + at x = x0 + v0 t + 12 at 2 , where x0 is the initial position at t = 0 and v0 is the initial velocity at t = 0. Often it is useful to determine the velocity at some position, rather than at some time. These two equations can be solved to provide v 2 = v02 + 2a(x − x0 ).

v = d x/dt for the one-dimensional case or, for the three-dimensional case, as v = r/t

B. Nonconstant Acceleration Acceleration is connected to position through a secondorder differential equation

or

a=

v = dr/dt, where x or r is the position of the object of interest. Velocity describes how fast the object is moving and in which direction. That means that velocity is a vector. Speed is the magnitude ( just the “how fast” without the direction) of velocity. Speed is a scalar. Both are commonly measured in m/s. Acceleration is the time rate of change of the velocity of an object. It can be written as a = v/t

or d 2r dt 2 so the solution of x(t) or r(t) from a or a may, indeed, be rather difficult. If the acceleration is known as a function of time, a(t), then it may be integrated directly to yield t v(t) = v0 + a(t) dt a=

and

or

a = v/t or

x(t) = x0 +

a = dv/dt for the one-dimensional case or, for the three-dimensional case, as

d2x dt 2

t

v(t) dt.

A few other, special cases exist, which may be solved directly. In general, though, ideas from dynamics, such as energy conservation or momentum conservation, are usually necessary in solving for or understanding the motion of an object when its acceleration is not constant.

a = dv/dt. Acceleration is commonly measured in meters per second per second (m/s2 ). An acceleration of 10 m/s2 means that the velocity increases by 10 m/s every second. Other systems of units could be used. For example, automotive engineers may find it useful to express a car’s acceleration in miles/h/s. An acceleration of 4.3 miles/h/s means that a car’s velocity increases by 4.3 miles/h every second. A. Constant Acceleration If the position is known as a function of time, then the velocity and acceleration are quite easy to determine by applying their definitions. However, it is more usually the case that the acceleration is known and the velocity and

II. NEWTON’S LAWS OF MOTION A. Inertia Once a car is moving a braking system is necessary to bring it back to a stop. Or a book lying on a desk requires a push or a shove from the outside to start it moving. Both of these situations are examples of inertia. Because of inertia, an object tends to continue to do what it is presently doing. This seems to have been first understood by Galileo and was first clearly stated by Sir Isaac Newton in the first of his three laws of motion.“In the absence of forces from the outside, a body at rest will remain at rest and a body in motion will continue in motion along the same straight line with the same velocity.”

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Friction is a force, which is nearly always present and sometimes masks this idea of inertia. If a book is given a shove across a table it may stop before reaching the edge. The law of inertia, Newton’s first law of motion, is still valid. But there is a force from the outside, the force of friction. An ordinary car will not coast forever; it will eventually come to rest. But there are forces from the outside which cause this (namely friction due to the air and friction between tires and roadway.) The idea of inertia is important because it asserts that motion continues because of the motion present. There need not be an active, continuing agent present at all times as the motion continues.

C. Action–Reaction

B. Force (F = ma)

III. APPLICATIONS

While inertia is important, motion is far more interesting when there is a force present. If there are several forces acting on a body, it is the net force—the vector sum of all the forces—that is important. Newton’s second law of motion states that “in the presence of a net or unbalanced force a body will experience an acceleration. That acceleration is inversely proportional to the mass of the body and is directly proportional to the force and in the direction of the force.” This can be written as a = F/m although it is more commonly written as F = ma. There is little exaggeration to say that almost all of classical mechanics derives directly from Newton’s second law. Velocity and acceleration are easily and often confused. Most people are more familiar with velocity or speed. However, it is the acceleration that is of most use in determining and describing motion and its cause. A force is a push or a pull. Force is anything that causes an acceleration. Newton’s second law can be used as the definition of a force. Newton’s second law also provides an operational definition of the mass of an object. It is a measure of how much “stuff” there is in an object. It is a measure of how difficult it is to accelerate an object. By definition, a particular block of platinum–iridium alloy has been designated to have a mass of exactly 1 kg. If the same force is applied to this (or an identical) block and to another block and the other block is found to have an acceleration exactly one-half that of the standard block, then the other block’s mass is 2 kg. The mass of an object is always the same. It is independent of altitude or position. As we shall see, there is an important distinction between mass and weight. In the metric system (or SI units), mass is measured in kilograms, force in newtons, and acceleration in m/s2 . A force of one newton could cause a 1 kg mass to accelerate at 1 m/s2 .

Newton’s third law of motion states that “if object 1 exerts a force on object 2 then object 2 also exerts a force back on object 1. The two forces are identical in magnitude and opposite in direction.” An example of this is the force you exert down on a chair when you sit on it and the force the chair exerts up on you. When an airplane propeller pushes back on the air, the air pushes forward on the propeller. As the sun pulls on earth, earth also pulls back on the sun. It is impossible to exert a force on an object without an additional force being exerted by that object. Notice that the forces in question are always exerted on different objects.

A. Straight-Line Motion Any constant force produces a constant acceleration so the kinematic equations for constant acceleration are immediately useable. 1. Free Fall Freely falling objects near the earth’s surface are found to have a constant acceleration of 9.8 m/s2 (or 32 ft/s2 ) downward if air resistance can be neglected. This acceleration is usually labeled “g”; that is, g = 9.8 m/s2 = 32 ft/s2 . To produce the same acceleration, the forces on two different bodies must be proportional to the masses. That means that the force of gravity must be proportional to the mass of a body. This force of gravity is called weight W and W = mg. The kinematics equations that describe an object in free fall, then, are simply v = v0 − gt x = x0 + v0 t − 12 gt 2 , where the acceleration a has just been replaced with −g (the minus sign merely indicates downward). 2. Simple Harmonic Motion A spring exerts a linear restoring force. As a spring is stretched or compressed, the force it exerts is proportional to how far it has been stretched or compressed from its unstretched, uncompressed, equilibrium position. And the force is directed to move the stretched or compressed spring back to that equilibrium position. This force can be described by the equation F = −kx,

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where x is the displacement from equilibrium (positive for stretch and negative for compression), k is a spring constant that describes the strength of the spring, and F is the force. If an object of mass m is attached to such a spring the motion that it undergoes is known as simple harmonic motion. That motion can be described by x(t) = A sin(ωt + ϕ). A is called the amplitude of the motion and is the maximum displacement from the equilibrium position. The motion is symmetric; the object will move as far on one side of the equilibrium position as on the other side. ϕ is a phase angle determined by the initial conditions (x0 , v0 ). ω is the angular frequency in rad/s. It is related to the more usual frequency f of cycles per second by

from and lands back on the same level surface. The range R is given by R = v02 sin 2θ g, where v0 is the initial speed and θ is the angle above the horizontal at which the projectile is thrown. Note that the range is the same for complimentary angles; that is, θ and 90◦ − θ give the same range. Maximum range is found for θ = 45◦ .

IV. WORK AND ENERGY A. Work

For such a mass on a spring the angular frequency is equal to ω = k/m.

Work done by a constant force F is defined as the distance D an object moves multiplied by the component of force in that direction. Pushing on a wall may tire your body but no work has been done according to this definition. If a yo-yo swings in a circle, the string continually exerts a force perpendicular to the direction of motion and no work is done. Work is a scalar quantity. The units of work are newton-meters or joules.

The period T is the amount of time required for a single cycle; therefore, the period and frequency are related by

B. Kinetic Energy

f = 2πω.

f = 1/T. A small object of mass m suspended by a cord of length l and allowed to swing back and forth is called a simple pendulum. For small amplitudes the motion of such a simple pendulum is also simple harmonic motion. For the simple pendulum the angular frequency is given by ω = g/l. Note that for both examples of simple harmonic motion, the frequency is independent of the amplitude. B. Three-Dimensional Motion Newton’s second law makes it easy to extend the ideas of straight-line motion to projectile motion, the motion followed by a body thrown and released near the earth’s surface. Observe this motion from far, far away in the plane of the motion and it looks like the object has simply been thrown upward. The force of gravity acts to accelerate it downward. The vertical part of the motion appears to be simply free fall and that is just motion with constant acceleration. Observe this motion from far above and it looks like the object is moving at constant velocity. There is no horizontal force. The horizontal part of the motion appears to be simply constant velocity. The path a projectile takes is a parabola. The range is the horizontal distance an object will go if it is thrown

The amount of work done on a body is equal to an increase in the quantity 12 mv 2 . That is, W = 12 mvf2 − 12 mv02 , where vf is the final speed after the work has been done and v0 was the original speed before. Because the object is moving, it has the ability to do work on something else—it could exert a force on another object over some distance. This ability to do work is called energy. Energy is a scalar. The quantity 12 mv 2 is called the kinetic energy; it is energy due to motion. The kinetic energy associated with the random motion of molecules due to heat is called thermal energy. C. Potential Energy Doing work on an object may change its position or condition. Lifting an object requires doing work against gravity. Because of its higher position, the object can then do work on something else as it falls; thus, it has gravitational potential energy. If an object of mass m is lifted from an initial height y0 to a final height y its potential energy is changed by an amount PE = mg(y − y0 ). Stretching or compressing a spring requires work to be done. That work done is stored up in the spring; the spring can be released and can do work on something

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else. This elastic potential energy of a spring stretched or compressed a distance x from its equilibrium position is PE = 12 kx 2 . D. Conservation of Energy Work and energy are useful because the work done on a system by forces from outside the system is equal to the change in the total energy of the system. The total energy of a system is the sum of the potential, kinetic, and thermal energies. If the work from external forces is zero then the total energy of the system remains constant—the total energy is conserved. A simple pendulum is an example of a system for which the external forces do no work. The force exerted by the supporting string on the mass of a pendulum is always perpendicular to the direction of motion so no work is done. Therefore, the energy must be conserved. If a pendulum is lifted some distance and released, it begins with some amount of gravitational potential energy. As it swings that potential energy decreases but its speed increases, which means the kinetic energy increases. The sum of the kinetic and potential energies remains constant. A roller coaster offers another example of a system that alternately changes potential energy (height) into kinetic energy (speed) and vice versa. For both a roller coaster and a pendulum, friction will eventually cause the system to stop. Friction can be considered an external force or we can look at the thermal energy associated with the slight increase in temperature of the wheels and rails as a roller coaster runs.

V. MOMENTUM

ficult to measure or predict. But conservation of momentum means that the vector sum of the momenta of the two objects before the collision will be the same as the vector sum of the momenta of the two objects after the collision. By itself, this is not sufficient to completely solve for the velocities of the two objects after the collision (assuming the conditions before the collision are given). But the final velocities can be found in two very useful extremes. If the kinetic energy is also conserved, that is, no energy is lost to heat or deforming the objects, the collision is termed elastic. The additional information provided is enough to solve for the final motion. If the two objects stick together the collision is termed inelastic and the maximum amount of kinetic energy is lost. Note that momentum is always conserved whether the collision is totally elastic, totally inelastic, or anywhere in between. B. Rocket Propulsion A car’s motion can be understood by looking at the wheels as they push on the pavement and understanding that the pavement pushes back on the wheels. But how, then, does a rocket move and accelerate in space? There is nothing else around for it to push on that can push back on it. A rocket burns fuel that is exhausted from the rocket’s engine at high velocity. As momentum is carried in one direction by the fuel, an equal amount of momentum is carried in the opposite direction by the rocket. If you stand in a child’s wagon and throw bricks in one direction you will be moved in the other direction. As momentum is carried in one direction by the bricks, an equal amount of momentum is carried in the opposite direction by you and the wagon. The idea is the same as that used in explaining rocket propulsion. If gravity can be neglected, a rocket’s final velocity is given by

Momentum, usually designated by p, is defined by multiplying the mass m of an object by its velocity v, p = mv. It is similar to kinetic energy in that momentum increases with increasing speed. But it is different in that momentum is a vector quantity. Like energy, momentum is useful because it is conserved. In the absence of external forces, the total momentum of a system of particles remains constant. Even though the internal forces between the particles may be very complicated, the vector sum of all the momenta of all the particles remains constant. Conservation of momentum is related to Newton’s third law of motion (action and reaction). A. Collisions When two objects collide—as two billiard balls hitting or two cars crashing into each other—the forces are very dif-

v = v0 + u ln (m 0 /m), where v0 is its initial velocity, u is the exhaust velocity of the burned gases, m 0 is the initial mass of the rocket, and m is the final mass of the rocket.

VI. RIGID BODY MOTION A. Center of Mass For a system of particles of mass m i each located at position ri , the mass-weighted average position of the particles is called the center of mass and is defined by R= m i ri mi ,

i

i

where i means to sum over all values of i. The total mass of the system of particles is M = i m i .

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For a rigid body the summation over individual masses is replaced by an integral over the volume of the body. The center of mass is then defined by 1 R= ρr d V, M V where M is the total mass of the body, given by M= ρ d V.

inertia is I = MR 2 . For a solid cylinder, I = 12 MR 2 . Force is replaced by a “rotational force” that depends upon the force and its placement from the axis of rotation; this is called a torque T . While a small force applied at the doorknob side opens a door easily, a large force will be required if it is applied back near the hinge; the rotational effect in the two cases is the same. Torque is given by T = rF sin θ,

V

ρ is the mass density (mass per unit volume), r is just the location vector, and V is the volume of the body. As with all vector equations, this may be easier to understand in component form. The three coordinates (X, Y, Z ) of the center of mass are 1 X = ρx d V, M V 1 Y = ρy d V, M V 1 Z = ρz d V. M V The center of mass is a “uniquely interesting point” for even though the motion of individual particles or rotations of the body may be frustratingly complicated, the motion of the center of mass will be that of a single point particle with mass M.

where r is the distance from the axis of rotation, F is the force, and θ is the angle between the two. Just as a distance x labels the position of a mass on a straight track, an angle θ (measured in radians) labels the angular position of a rotating object. Angular velocity ω describes its speed of rotation in rad/s and angular acceleration α describes the rate of change of angular velocity in rad/s2 . The rotational equivalent of F = ma is T = I α. The angular momentum for rotation about a fixed axis is L = I ω, which closely parallels P = Mv for the linear case. 2. Rotation in General In general, however, rotation can be more complicated than straight-line motion. Angular momentum remains a conserved quantity. But in general angular momentum is given by L = {I}ω,

B. Angular Momentum Just as linear momentum was useful in understanding and predicting translational motion because of its conservation, so another conserved quantity (called the angular momentum) will be useful in discussing rotational motions. The angular momentum L of a small particle relative to some origin is given by L = r × p, where r is the location of the particle from the origin, p is its momentum, and × indicates the vector cross product. For a system of particles, the total angular momentum is the vector sum of the individual angular momenta. For an extended body, the total angular momentum requires evaluating an integral over the volume of the body. 1. Rotation about a Fixed Axis For rotation about a fixed axis, there is a strong correlation with straight-line motion. The mass is replaced by a “rotational mass” that depends upon the geometry of the mass (how far it is located from the axis of rotation.) This “rotational mass” is called the moment of inertia I . For a hollow cylinder of mass M and radius R, the moment of

where {I} is now a tensor. This brings about the interesting case in which the angular momentum L and the angular velocity ω may not necessarily be parallel to each other. This can be seen by tossing a book or tennis racket in the air spinning about each of three mutually perpendicular axes. For the longest and shortest axes, L and ω will be in the same direction; for the medium length axis they will not be in the same direction.

VII. CENTRAL FORCES A. Definitions A central force is one whose direction is always along a radius; that is, either toward or away from a point that can be used as an origin (or force center), and whose magnitude depends solely upon the distance from that origin, r . A central force can always be written as F = F(r )ˆr, where rˆ is a unit vector in the radial direction. Central forces are important because many real situations involve central forces. The gravitational force between two masses

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and the electrostatic force between two charges are both central forces. Motion due to a central force will always be confined to a plane.

B. Gravity Gravity is the force of attraction between two massive bodies. First described by Sir Isaac Newton, the force of gravity between two bodies with masses m 1 and m 2 separated by a distance r is given by FG = (Gm 1 m 2 ) r 2 , where G is a universal constant (G = 6.672 × 10−11 N m2 /kg2 ). This expression is valid for calculating the force earth exerts on an apple near its surface or the force earth exerts on our moon or the force our sun exerts on Jupiter. 1. Kepler’s Laws of Planetary Motion Before Newton discovered this law of universal gravitation, Johannes Kepler found, based upon careful observational data, that the motion of the planets in our solar system could be explained by three laws: 1. Planets move in orbits that are ellipses with the sun at one focus (elliptical orbits). 2. Areas swept out by the radius vector from the sun to a planet in equal times are equal (equal areas in equal times). 3. The square of a planet’s period is proportional to the cube of the semimajor axis of its orbit (T 2 ∝ r 3 ). It was a great triumph of Newton’s law of universal gravitation that it could explain and predict Kepler’s laws of planetary motion. Kepler’s second law is true for any central force; it is the result of conservation of angular momentum. The other two laws depend upon gravity being an inverse square force.

C. Harmonic Oscillator A mass suspended between three sets of identical, mutually perpendicular springs forms an isotropic, threedimensional simple harmonic oscillator. The springs provide a restoring force of the form F = −kr so this three-dimensional harmonic oscillator experiences a central force. Examples of such a system are atoms in certain crystals, where the interatomic bonds act as the springs in this simple case. If the springs (or interatomic bonds for a crystal) are not all identical, then the force due to a displacement in one direction will be different than that for another direction. The harmonic oscillator is anisotropic and can no longer be described as a central force.

VIII. ALTERNATE FORMS Newton’s second law of motion, F = ma, can be used to solve for the motion in many situations. But the same information can be written in different forms and used in situations where direct solution of F = ma is very difficult or perhaps impossible.

A. Lagrange’s Equations Lagrange’s equations of motion can be written as d ∂L ∂L = , dt ∂ q˙ k ∂qk where qk is a “generalized coordinate” and L is called the Lagrangian function. The Lagrangian function is the difference between the kinetic energy and the potential energy; L = KE − PE. The dot means a time derivative; q˙ k = dqk /dt. B. Hamilton’s Equation

2. Orbits Planets travel in elliptical orbits about the sun. Satellites travel in elliptical orbits about their planet. If the speed of a satellite is suddenly increased the shape of the elliptical orbit elongates. If a satellite has enough velocity to escape and never return to the planet the path it travels is a parabola or a hyperbola. Escape velocity is the minimum velocity that will allow a satellite to travel away from its planet and never return. If a satellite leaves earth’s surface with a velocity of about 40,000 km/h (25,000 miles/h) it will escape from earth and never return.

Hamilton’s equations of motion can be written as q˙ k = ∂ H/∂ pk and p˙ k = −∂ H/∂qk , where, again, qk is a “generalized coordinate,” pk is a “generalized momentum,” and H is called the Hamiltonian function. For many situations, the Hamiltonian H is the total energy of the system.

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C. Poisson Brackets and Quantum Mechanics Hamilton’s equations can be rewritten in terms of Poisson brackets as

OF STRUCTURES • NONLINEAR DYNAMICS MECHANICS • RELATIVITY, GENERAL •

• QUANTUM STATISTICAL

MECHANICS • VIBRATION, MECHANICAL

q˙ k = [qk , H ]

BIBLIOGRAPHY

and ˙ = [ pk , H ], pk where the Poisson brackets are defined by ∂A ∂B ∂B ∂A [A, B] = . − ∂qk ∂ pk ∂qk ∂ pk k This formulation is especially interesting because it allows for an easy and direct transfer of ideas from classical mechanics to quantum mechanics.

SEE ALSO THE FOLLOWING ARTICLES CELESTIAL MECHANICS • CRITICAL DATA IN PHYSICS CHEMISTRY • ELECTROMAGNETICS • MECHANICS

AND

Arya, A. P. (1997). “Introduction to Classical Mechanics,” Prentice Hall, New York. Kwatny, H. G., and Blankenship, G. L. (2000). “Nonlinear Control and Analytical Mechanics: Computational Approach,” Birkhauser, Boston. Brumberg, V. A. (1995). “Analytical Techniques of Celestial Mechanics,” Springer-Verlag, Berlin. Chow, T. L. (1995). “Classical Mechanics,” Wiley, New York. Doghri, I. (2000). “Mechanics of Deformable Solids: Linear and Nonlinear, Analytical and Computational Aspects,” Springer-Verlag, Berlin. Hand, L. N., and Finch, J. D. (1998). “Analytical Mechanics,” Cambridge University Press, Cambridge, UK. Jos´e, J. V., and Saletan, E. J. (1998). “Classical Dynamics: A Contemporary Approach,” Cambridge University Press, Cambridge, UK. Torok, J. S. (1999). “Analytical Mechanics: With an Introduction to Dynamical Systems,” Wiley, New York.

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Molecular Hydrodynamics Sidney Yip

Jean Pierre Boon

Massachusetts Institute of Technology

Universit´e Libre de Bwxelles

I. II. III. IV. V. VI. VII.

Motivation Density Correlation Function Linearized Hydrodynamics Generalized Hydrodynamics Kinetic Theory Mode Coupling Theory Lattice Gas Hydrodynamics

GLOSSARY Diffusion Dissipation of thermal fluctuations by essentially random (or stochastic) motions of atoms (as in thermal or concentration diffusion). Generalized hydrodynamics Theoretical description of fluctuations in fluids based on the extension of the equations of linearized hydrodynamics to finite frequencies and wavelengths. (k, ω) space Region of wavenumber and frequency where thermal fluctuations are being studied. Lattice gas automata Class of cellular automata designed to model fluid systems using discrete space and time implementation. Memory function Space–time dependent kernel appearing in the equation of motion for time correlation function, which contains the effects of static and dynamical interactions. Mode coupling A theory in which the interatomic interactions are expressed in terms of products of two or more modes of thermal fluctuations, such as the densities of particle number, current, and energy.

Propagation Cooperative motion of atoms characterized by a peak at finite frequency in the frequency spectrum of density fluctuations (as in pressure wave propagation). Thermal fluctuations Spontaneous localized fluctuations in the particle number, momentum, and energy densities of atoms in a fluid at thermal equilibrium. Time correlation function Function that expresses the correlation of dynamical variables evaluated at two different time (and space) points. Uncorrelated binary collisions Sequence of two-body collisions in which the collisions are taken to be independent even though pairs of atoms can recollide one or more times.

MOLECULAR HYDRODYNAMICS is the theoretical description of spontaneous localized fluctuations in space and time of the particle number density, the current density, and the energy density in a fluid at thermal equilibrium. Its domain of applicability ranges from low frequencies and long wavelengths, where the linearized equations of

141

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142 hydrodynamics are applicable, to frequencies and wavelengths comparable to interatomic collision frequencies and mean free paths.

I. MOTIVATION When a fluid is disturbed locally from equilibrium, it will relax by allowing the perturbation to dissipate throughout the system. At the macroscopic level this response involves the processes of mass diffusion, viscous flow, and thermal conduction, which are the mechanisms by which the transport of mass, momentum, and energy can take place. In the absence of an external disturbance, we can still speak of the dynamical behavior of a fluid in these terms. The reason is that the same processes also govern the dissipation of spontaneous fluctuations that are always present on the microscopic level in a fluid at finite temperature. So the fluid can be considered as a “reservoir of thermal excitations” extending over a broad range of wavelengths and frequencies from the hydrodynamic scale down to the range of the intermolecular potential. Thus, the study of thermal fluctuations is fundamental to the understanding of the molecular basis of fluid dynamics. The conventional theory of fluid dynamics invariably begins with the equations of hydrodynamics. The basic assumption of hydrodynamics is that changes in the fluid take place sufficiently slowly in space and time that the system can be considered to be in a state of local thermodynamic equilibrium. Under this condition we have a closed set of equations describing the space–time variations of the conserved variables, namely, the mass, momentum, and energy densities. These equations become explicit, when the thermodynamic derivatives and the transport coefficients occurring in them are known; however, such constants are not determined within the hydrodynamic theory, and therefore must be provided by either measurement or more fundamental calculations. The equations of hydrodynamics have an extremely wide range of scientific and technological applications. They are valid for disturbances of arbitrary magnitude provided the space and time variations are slow on the molecular scales with lengths measured in collision mean free path l and times in inverse collision frequency ωc−1 . In terms of the wavelength, 2π/k, and frequency ω, of the fluctuations, the hydrodynamic description is valid only in the region of low (k, ω), where kl 1 and ω ωc . When the condition of slow variations is not fully satisfied, we expect the fluid behavior to show molecular or nonhydrodynamic effects. Unless the fluctuations are far removed from the hydrodynamic region of (k, ω), the discrepancies often appear only in a subtle and gradual manner. This suggests that extensions or generalizations

Molecular Hydrodynamics

of the hydrodynamic description may be useful and may be accomplished by retaining the basic structure of the equations, while replacing the thermodynamic derivatives and transport coefficients by functions that directly reflect the molecular structure of the fluid and the effects of individual intermolecular collisions. The result is then a theory that is valid even on the scales of collision mean free path and mean free time, a theory that may be called molecular hydrodynamics. In essence, molecular hydrodynamics is a description that considers both the macroscopic behavior of mass, momentum, and energy transport, and the microscopic properties of local structure and intermolecular collisions. There are several reasons why a study of the extension of hydrodynamics is important. First, we obtain a better understanding of the validity of hydrodynamics. Second, an appreciation of how the details of molecular structure and collisional dynamics can affect the behavior of the conserved variables is essential to the study of transport phenomena on the molecular level. Finally, it is one of the basic aims of nonequilibrium statistical mechanics to develop a unified theory of liquids that treats not only the processes in the hydrodynamic region of (k, ω), but also the molecular behaviors that manifest at higher values of wavenumber and frequency.

II. DENSITY CORRELATION FUNCTION The fundamental quantities in the study of thermal fluctuations in fluids are space and time-dependent correlation functions. These functions are the natural quantities for theoretical analyses as well as laboratory measurements. They are well defined for a wide variety of physical systems, and they possess both macroscopic properties and interpretations at the microscopic level. For the fluid system of interest we imagine an assembly of N identical particles (molecules), each of mass m, contained in a volume . The molecules have no internal degrees of freedom, and they are assumed to interact through a two-body, additive, central potential u(r ). The fluid is in thermal equilibrium, at a state far from any phase transition. Also, there are no external fields imposed, so the system is invariant to spatial translation, rotation, and inversion. A time correlation function is the thermodynamic average of a product of two dynamical variables, each expressing the instantaneous deviation of a fluid property from its equilibrium value. The dynamical variables that we wish to consider are the number density, N 1 n(r, t) = √ δ(r − Ri (t)) N i=1

(1)

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where Ri (t) denotes the position of particle i at time t, and the current density, N 1 j(r, t) = √ vi (t)δ(r − Ri (t)) N i=1

(2)

where vi (t) is the velocity of particle i at time t. The thermodynamic average of a dynamical variable A(r, t) is defined as A(r, t) = d 3 R1 . . . d 3 R N d 3 P1 . . . × d 3 PN f eq (R N , P N )A(r, t)

(3)

where f eq is an equilibrium distribution of particle positions R N = (R1 , . . . , RN ), and momenta P N = (P1 . . . P N ). Typically we adopt the canonical ensemble in evaluating Eq. (3), f eq (R N , P N ) = Q −1 N exp(−βU )

N

f 0 (Pi )

(4)

i=1

with β = (k B T )−1 , T being the fluid temperature and k B the Boltzmann’s constant. U (R N ) is the potential energy, f 0 (P) is the normalized Maxwell–Boltzmann distribution f 0 (P) = (β/2π m)3/2 exp(−βP 2 /2m) and Q N is the configurational integral Q N = d 3R1 . . . d 3 R N exp(−βU )

(5)

(6)

where U is the potential energy of the system. Applying Eq. (3) √ to Eqs. (1) and (2) gives the average values n(r, t) = N /V , and j(r, t) = 0. Notice that in general a dynamical variable depends on the particle positions R N and momenta P N , and also on the position r and time t, where the property is being considered. On the other hand, the average values are independent of r and t because the system is uniform and in equilibrium. Given the dynamical variable n(r, t) we define the timedependent density correlation function as G(|r − r |, t − t ) = V δn(r , t )δn(r, t) N 1 = δ(r − Ri (t )) n i, j × δ(r − R j (t)) − n

(7)

where δn(r, t) = n(r, t) − n(r, t), and n = N /V is the average number density of the fluid at equilibrium. Despite its rather simple appearance this function contains all the

FIGURE 1 The time correlation function circle with its interconnected segments of theory, experiment, and atomistic simulation.

structural and dynamical information concerning density fluctuations. Note that G depends on the separation |r − r | because of rotational invariance and it is a function of t − t because of time translational invariance. Without loss of generality we can take r = 0 and t = 0. The density correlation function is the leading member of a group of time correlation functions that have received attention in recent studies of nonequilibrium statistical mechanics. These functions have become the standard language for experimentalists and theorists alike, because they can be measured directly and they are welldefined quantities for which microscopic calculations can be formulated. Moreover, time correlation functions are accessible by atomistic simulations. Figure 1 shows the complementary nature of theoretical, experimental, and simulation studies of time correlation functions. In this article we are primarily concerned with the theoretical developments, which, however, rely on simulation data and scattering measurements for guidance and validation. It is instructive to note the simple physical interpretation of G(r, t), which we can deduce from its definition. Consider a laboratory coordinate system placed in the fluid such that at time t = 0 a particle is at the origin. At a later time t, place an element of volume d 3r at the position r. Then G(r, t)d 3r is the average (or expected) number of particles in the element of volume at r at time t, given that a particle was located at the origin initially. The initial value of G is G(r, 0) = δ(r) + ng(r )

(8)

where g(r ) is the equilibrium pair distribution function n 2 g(r ) = δ(r − Ri )δ(R j ) (9) i,j i = j

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resulting equations as an initial value problem with initial values,

III. LINEARIZED HYDRODYNAMICS The linearized hydrodynamic equations for a fluid with no internal degrees of freedom consist of the continuity equation, which expresses mass or particle number conservation, ∂ρ1 (r, t) + ρ0 ∇ · v(r, t) = 0 ∂t

(10)

the Navier–Stokes equation, which expresses momentum or current conservation, ρ0

∂v(r, t) c02 + ∇ρ1 (r, t) ∂t γ +

c02 αρ0 ∇T1 (r, t) − η∇∇ · v(r, t) = 0 γ

(11)

and the energy transport equation, which expresses kinetic energy conservation, ρ0 Cv

∂ T1 (r, t) Cv (γ − 1) ∂ρ1 (r, t) − − λ∇ 2 T1 (r, t) = 0 ∂t α ∂t (12)

In these equations, ρ = ρ0 + ρ1 is the local number density, T = T0 + T1 is the local temperature, and v is the velocity, with subscripts 0 and 1 denoting the equilibrium value and instantaneous deviation, respectively. The ratio of specific heats at constant pressure and constant volume C p /Cv is γ . The combination of shear and bulk viscosities 4 η + ηB is denoted by η, and c0 , α, λ are, respectively, 3 s the adiabatic sound speed, the thermal expansion coefficient, and the thermal conductivity. Equations (10)–(12) are linearized in the sense that ρ1 , T1 , and v are assumed to be small, and therefore, only terms to first order in these quantities need be kept. This assumption makes the description valid only for small-amplitude disturbances such as thermal fluctuations. The parameters of the hydrodynamic description are thermodynamic coefficients α, γ , and c0 , the thermal expansion coefficient, the ratio of specific heats at constant pressure and volume, and the adiabatic sound speed, and transport coefficients, ηs , ηB , and λ, the shear and bulk viscosities with η = 43 ηs + ηB , and the thermal conductivity. Once these are specified, the equations can be used to calculate explicitly the spatial and temporal distributions of the particle number, current, and energy densities for a given set of boundary and initial conditions. We will be interested in the decay of a density pulse created by thermal fluctuations in a uniform, infinite fluid medium. For this problem it will be most convenient to discuss the solutions in wavevector space by taking the Fourier transform in configuration space and solving the

ρ1 (r, t = 0) = δ(r) + n[g(r ) − 1] v(r, t = 0) = 0

(13)

T1 (r, t = 0) = 0 Equation (13) states that at time t = 0 a density pulse occurs in the fluid in the form of a particle localized at the origin of the coordinate system plus a distribution of particles according to n[g(r ) − 1]; also, there are no current or temperature perturbations. The meaning of ρ1 (r, t) as the density response to this initial condition is the spatial distribution of this density pulse as time evolves. Notice that any particle in the fluid can contribute to ρ1 (r, t) for t > 0, not just the particle originally located at the origin. After taking the Fourier transform of Eqs. (10)–(12), we can solve for n(k, t) = d 3r eik·r ρ1 (r, t) (14) The calculation is best carried out by taking the Laplace transform in time, for example, ∞ n(k, s) = dt e−st n(k, t) (15) 0

thus obtaining a system of coupled algebraic equations for the Laplace–Fourier transformed densities n(k, s), v(k, s), and T1 (k, s). The system of equations is homogeneous, and for nontrivial solutions the transform variable s has to satisfy a cubic equation. Since the hydrodynamics description is applicable only when spatial and temporal variations of the densities occur smoothly, it is appropriate to look for roots of the cubic equation to lowest orders in the wavenumbers. To order two, s± = ±ic0 k − k 2 s3 = −λk 2 /ρ0 c p

(16)

where = [η + λ(Cv−1 − C −1 p )]/2ρ0 . As a result of the density pulse, both pressure and temperature fluctuations are induced. The pair of complex roots s± describes the propagation of pressure fluctuations as damped sound waves, with speed c0 and attenuation . The root s3 describes the diffusion of temperature fluctuations with attenuation λ/ρ0 C p . Using Eq. (16) we can invert the Laplace transformed solution for n(k, s) and compute the correlation function. The result is F(k, t) = n(k, t)n(−k) = d 3r eik·r G(r , t)

(17)

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C p − Cv λk 2 F(k, t) = S(k) exp − t Cp ρ0 C p

Cv 2 + exp(−k t) cos c0 kt Cp

(18)

where S(k) = n(k)n(−k) = F(k, t = 0)

(19)

is known as the static structure factor of the fluid. In the long wavelength limit, it is a thermodynamic quantity: S(k → 0) → nkB T xT , where xT is the isothermal compressibility. Equation (18) shows that there are two components in the time decay of density fluctuations, an exponential decay associated with heat diffusion, and a damped oscillatory decay associated with pressure (sound) propagation. The dynamics of density fluctuations can be studied directly by scattering beams of thermal neutrons or laser light from the fluid and measuring the frequency spectrum of the scattered radiation. In such experiments, the frequency spectrum of the density fluctuation is measured, ∞ S(k, ω) = dt e−iωt F(k, t) (20) −∞

In contrast to S(k), which is what we obtain from a neutron or X-ray diffraction experiment, S(k, ω) is called the dynamic structure factor because it gives information about both structure and dynamics of the fluid. Since we can probe the fluid structure at different wavenumbers, the frequency behavior of S(k, ω) can vary considerably from the hydrodynamic regime of long wavelengths (kl 1), to the regime of free particle flow (kl 1). The frequency spectrum of density fluctuations in the hydrodynamic regime is characterized by three welldefined spectral lines, corresponding to the three modes in F(k, t) or the three roots to the dispersion equation as given in Eq. (16). From (18) and (20), one obtains C p − Cv λk 2 /ρ0 C p S(k, ω) = S(k)

C p ω2 + λk 2 /ρ0 C p 2 k 2 Cv + C p (ω + c0 k)2 + (k 2 )2

k 2 (21) + (ω − c0 k)2 + (k 2 )2 The spectrum is composed of a central peak with maximum at ω = 0 and whose full width at half maximum is 2λk 2 /ρ0 C p . This peak is called the Rayleigh line; its intensity is given by S(k)[1 − 1/γ ]. There are also two

equally displaced side peaks with maxima at ω± = ±c0 k and whose full width at half maximum is 2k 2 ; these are called the Brillouin doublet and their integrated intensity is given by S(k)/γ . The intensity ratio of the Rayleigh component to the Brillouin components is γ − 1, a quantity known as the Landau–Placzek ratio. Note that a more accurate solution contains cross terms involving heat diffusion and pressure propagation and gives rise to an asymmetry in the Brillouin components. There are other time correlation functions of interest, such as the transverse current correlation, 1 T Jt (k, t) = v j (t)vkT (0)eik·[R j (t)−Rk (0)] (22) N j,k where v Tj (t) is the transverse component (direction perpendicular to k) of the velocity of the jth particle at time t. From the Navier–Stokes equation (11) we find [∂ Jt (k, t)/∂t] = −νk 2 Jt (k, t)

(23)

where ν = η/ρ0 . The corresponding frequency spectrum is a Lorentzian function. Jt (k, ω) = 2v02 νk 2 [ω2 + (νk 2 )2 ] (24) with J (k, t = 0) = v02 = (βm)−1 . We see that at long wavelengths transverse current fluctuations in a fluid dissipate by simple diffusion at a rate given by νk 2 .

IV. GENERALIZED HYDRODYNAMICS The hydrodynamic description of fluctuations in fluids is expected to become inappropriate at finite values of (k, ω) when kl 1 and ω ωc , where l is the collision mean free path and ωc the mean collision frequency (see Section I). Nevertheless, we can extend the hydrodynamic description by allowing the thermodynamic coefficients in Eqs. (12) and (13) to become wavenumber dependent and the transport coefficients to become k- and ω-dependent. This method of extension is called generalized hydrodynamics. The basic idea of generalized hydrodynamics can be simply presented by considering the case of the transverse current fluctuations. One of the fundamental differences between simple liquids and solids is that the former cannot support a shear stress, which is another way of saying that they have zero shear modulus. On the other hand, it is also known that at sufficiently short wavelengths or high frequencies shear waves can propagate through a simple liquid because then the system behaves like a viscoelastic medium. We have seen that according to hydrodynamics the frequency spectrum of the transverse current correlation function, (24), describes a diffusion process at all

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frequencies. The absence of a propagating mode in (24) is an example of the inability of linearized hydrodynamics to treat viscoelastic behavior at finite (k, ω). In the approach of generalized hydrodynamics we extend (23) by postulating the equation t ∂ dt K t (k, t − t )Jt (k, t ) (25) Jt (k, t) = −k 2 ∂t 0

the Maxwell relaxation time in viscoelastic theories. Furthermore, we expect τ (k) to be a decreasing function of k on the grounds that fluctuations at shorter wavelengths generally dissipate more rapidly. The simple interpolation expression

The kernel K t (k, t) is called a memory function; it is itself a time correlation function like Jt (k, t). The role of K t is to enable Jt to take on a short-time behavior that is distinctly different from its behavior at long times. It is reasonable that a quantity such as K t (k, t) should be present in the extension of hydrodynamics. With the introduction of a suitable K t (k, t), we expect that (25) will give shear wave propagation at finite (k, ω), while in the limit of small (k, ω) we recover Eq. (23). On a phenomenological basis, without specifying completely K t (k, t) by a systematic derivation, we can require this function to satisfy requirements that incorporate certain properties of the function that we can readily derive. The two properties of K t (k, t) most relevant to the present discussion are

would be consistent with this expectation and entails no further parameters. There exist more elaborate models for τ (k) as well as for K t (k, t), but the model Eq. (30) with (31) has the virtue of simplicity. Then Eq. (25) gives

K t (k, t = 0) = (nm)−1 G ∞ (k) and

lim

k→0 0

∞

dt K t (k, t) = ν

(26)

(27)

where G ∞ (k) is the high-frequency shear modulus. Both G ∞ and ν are actually properties of Jt (k, ω), ∞ (kv0 )2 1 G ∞ (k) = dω ω2 Jt (k, ω) (28) nm 2π −∞ 2 ω 2 2v0 ν = lim lim Jt (k, ω) (29) ω→0 k→0 k Moreover, Eq. (28) can be reduced to a kinetic contribution (kv02 )2 and an integral over g(r ) and potential function derivative that can be evaluated by quadrature. Equations (26) and (27) may be regarded as constraints or “boundary conditions” on K t (k, t), but by themselves they do not determine the memory function. Empirical forms have been proposed for K t (k, t) with adjustable parameters determined by imposing Eqs. (26) and (27). As an example, we consider the exponential or single relaxation time model, K t (k, t) = [G ∞ (k)/nm] exp[−t/τ (k)]

(30)

where we are still free to specify the wavenumberdependent relaxation time τ (k). Notice that Eq. (26) has already been incorporated. Applying Eq. (27) we obtain τ (k = 0) = nmν/G ∞ (0), a quantity sometimes called

1 τt2 (k)

Jt (k, ω) =

=

1 τt2 (0)

+ (kv0 )2

(31)

2v02 k 2 K t (k, 0) τt (k)

2 1 × ω − k K t (k, 0) − 2 2τt (k)

−1 1 2 2 τt (k) + k K t (k, 0) − 2 4τt (k) 2

2

(32) The effect of the memory function now may be seen in the spectral behavior of Jt (k, ω). Whenever k 2 K t (k, 0) >

1 2τt2 (k)

(33)

there will exist a finite frequency, where the denominator in Eq. (32) is a minimum, and Jt (k, ω) will show a resonant peak. The resonant structure indicates a propagating mode associated with shear waves. Notice that Eq. (33) cannot hold at sufficiently small k; thus, in the long wavelength limit Eq. (32) can only describe diffusion, in agreement with Eq. (24). Figure 2 shows the data of molecular dynamics simulation; we see clear evidence of the onset of shear waves as k increases. Generalized hydrodynamic descriptions for other time correlation functions also can be developed by using memory function equations such as Eq. (25). We will briefly summarize the results for density and longitudinal current fluctuations. The continuity equation, Eq. (6), is an exact expression, unlike the Navier–Stokes or the energy transport equation. One of its implications is a rigorous relation between the density correlation function, F(k, t), and the longitudinal current correlation function Jl (k, t). The latter is defined in a similar way as Eq. (22), with the transverse component v Tj replaced by the longitudinal component (direction parallel to k). In terms of the dynamic structure factor S(k, ω), the relation is Jl (k, ω) = (ω/k)2 S(k, ω)

(34)

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the propagation frequency would be c0 k and the damping constant governed by the sound attenuation coefficient [cf. Eq. (21)] instead of ν as in Eq. (35). The inadequacy of the hydrodynamic description Eq. (35) at finite (k, ω) values is more subtle than is the case of Jt (k, ω). We find that Eq. (35) gives an overestimate of the damping of fluctuations, and it does not describe any of the effects associated with the intermolecular structure as manifested through the static structure factor S(k). The extension of Eq. (35) can proceed if we write t ∂ Jl (k, t) dt K l (k, t − t )Jl (k, t ) (36) =− ∂t 0 with K l (k, t) =

(kv0 )2 + k 2 φl (k, t) S(k)

(37)

The form of K l (k, t) is motivated by the coupling of Eqs. (10) and (11), and the generalization of the isothermal compressibility x T , nk B T x T → S(k). Combining Eqs. [(35) and (36)] gives Jl ( k , ω) 2v02 (ωk)2 φl (k, ω) 2 (kv0 )2 2 2 ω − + ωk φl (k, ω) + [ωk 2 φl (k, ω)]2 S(k) (38)

=

where φl and φl are the real and imaginary parts of ∞ φl (k, s) = dt e−st φl (k, t) (39) 0

FIGURE 2 Normalized transverse current correlation function of liquid argon at various wavenumbers, molecular dynamics simulation data (circles), and exponential memory function model with τt (k) given by Eq. (41) (solid curves) and by a more elaborate expression (dashed curves).

Since this holds in general, we will focus our attention on Jl (k, ω). For purposes of illustration we assume that temperature fluctuations can be ignored. This means that we can set T1 = 0 in Eq. (11) and obtain Jl (k, ω) = 2v02

(ωk)2 [ω2 − (cT k)2 ]2 + [ωνk 2 ]2

(35)

with cT = c0 /ν being the isothermal sound speed. We see that in the hydrodynamic description the longitudinal current fluctuations, in contrast to the transverse current fluctuations, propagate at a frequency essentially given by ω ∼ cT k 2 . If temperature fluctuations were not neglected,

with s = iω, and they describe the dissipative and reactive responses, respectively. It is evident from a comparison of Eq. (38) with Eq. (35) that in addition to the generalization of the isothermal compressibility, the longitudinal viscosity has become a complex k- and w-dependent quantity. Through φl (k, t) we can again introduce physical models and use various properties to determine the k dependence. One way to characterize the breakdown of hydrodynamics in the case of Jl (k, ω) is to follow the frequency of the propagating mode as k increases. Notice first that by virtue of Eq. (34) Jl (k, ω) always shows a peak at a nonzero frequency. At small k this peak is associated with sound propagation. If we define ωm (k) c(k) = (40) k where ωm (k) is the peak position, then c(k) in the long wavelength limit is the adiabatic sound speed. This being the case, it is reasonable to regard Eq. (40) as the speed at which collective modes propagate in the fluid at any wavenumber. In terms of c(k) we have a well-defined

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V. KINETIC THEORY

FIGURE 3 Variation of propagation velocities with wavenumber in liquid argon. Generalized hydrodynamics results are given as the solid curve denoted by c (k ) and by the dashed curve, neutron scattering measurements are denoted by the closed circles and slash marks, and computer simulation data are denoted by open circles. The quantities c 0 (k ) and c ∞ (k ) are defined in the text.

quantity for discussing the variation of propagation speed at finite k. Notice that we do not refer to the propagating fluctuations at finite k as sound waves, because the latter are excitations that manifest clearly in S(k, ω). There exist computer simulation results and neutron inelastic scattering data on simple liquids from which c(k) can be determined. Figure 3 shows a comparison of these results with a generalized hydrodynamics calculation. Also shown are the adiabatic sound speed c0 (k) and the high-frequency sound speed c∞ (k), c0 (k) = v0 [γ /S(k)]1/2 1/2 1 4 c∞ (k) = G ∞ (k) + K ∞ (k) nm 3

(41) (42)

where K ∞ is the high-frequency bulk modulus. It is seen in Fig. 3 that c0 (k) and c∞ (k) provide lower and upper bounds on c(k). The fact that c(k) deviates from both may be attributed to dynamical effects, which cannot be described through static properties such as in Eqs. (41) and (42). Relative to the adiabatic sound speed c0 (k → 0) we see in c(k) first an enhancement as k increases up to about 1 A−1 , then a sharp decrease at larger k. The former behavior, a positive dispersion, is due to shear relaxation, whereas the latter, a strong negative dispersion, is due to structural correlation effects represented by S(k). From this discussion we may conclude that an expression such as Eq. (38), with rather simple physical models for φl (k, t), provides a semiquantitatively correct description of density and current fluctuations at finite (k, ω).

In the theory of particle and radiation transport in fluids there exists a well established connection between the continuum approach as represented by the hydrodynamics equations and the molecular approach as represented by kinetic equations in phase space, an example of which is the Boltzmann equation in gas dynamics. Through this connection we can obtain expressions for calculating the input parameters in the continuum equations, such as the transport coefficients in Eqs. (11) and (12). We can also solve the kinetic equations directly to analyze thermal fluctuations at finite (k, ω), and in this way take into account, explicitly, the effects of spatial correlations and detailed dynamics of molecular collisions. In contrast to generalized hydrodynamics, the kinetic theory method allows us to derive, rather than postulate, the space–time memory functions like K (k, t). The essence of the kinetic theory description is that particle motions are followed in both configuration and momentum space. Analogous to Section II we begin with the phase space density A(rpt) =

N

δ(r − Ri (t))δ(p − Pi (t))

(43)

i=1

and the time-dependent phase-space density correlation function [cf. Eq. (7)]

C(r − r , pp , t) = δ A(rpt)δ A(r p 0)

(44)

with A = n f 0 ( p). The fundamental quantity in the analysis is now C(r, pp , t), from which the time correlation functions of Section II can be obtained by appropriate integration over the momentum variables. For example, G(r, t) = d 3 pd 3 p C(r, pp , t) (45) Various methods have been proposed to derive the equation governing C(r, pp , t). All the results can be put into the generic form k·p z− C(kpp z) − d 3 p φ(kpp z)C(kp p z) m = −iC0 (kpp ) where C(kpp z) =

(46)

∞

d3r

dt ei(k·r−zt) C(r, pp , t)

(47)

with the initial condition C0 (kpp ) = d 3 r eik·r C(r, pp , t = 0) = n f 0 ( p)δ(p − p ) + n 2 f 0 ( p) f 0 ( p )h(k) (48)

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and nh(k) = S(k) − 1. In Eq. (46) the function φ(kpp z) is the phase-space memory function, which plays the same role as the memory function K (k, t) in Eq. (25) or Eq. (36). It contains all the effects of molecular interactions. If φ were identically zero, then Eq. (46) would describe a noninteracting system in which the particles move in straight line trajectories at constant velocities. We can also think of φ as the collision kernel in a transport equation. There are a number of formal properties of φ pertaining to symmetries, conservation laws, and asymptotic behavior, which one can analyze. Also, explicit calculations have been made under different conditions, such as low density, weak coupling, or relaxation time models. In general, it is useful to separate φ into an instantaneous, or static, part and a time-varying, or collisional, part, φ(kpp z) = φ (s) (kp) + φ (c) (kpp z) k·p n f 0 ( p)C(k) m

FIGURE 4 Frequency spectrum of dynamic structure factor in xenon gas at 349.6 K and 1.03 atm; light scattering data for 6328 A˚ incident light and scattering angle of 169.4◦ are shown as closed circles while the full curve denotes results obtained using the linearized Boltzmann equation for hard spheres. Calculated spectrum has been convolved with the resolution function shown by the dashed curve.

(49)

where φ (s) (kp) = −

20:58

(50)

The quantity C(k) = [S(k) − 1]/nS(k) is known as the direct correlation function. Physically φ (s) represents the effects of mean field interactions with nC(k) as the effective potential of the fluid system. The calculation of φ (c) is a difficult problem because we have to deal with the details of collision dynamics. It can be shown that in the limit of low densities, low frequencies, and small wavenumbers, φ (c) reduces to the collision kernel in the linearized Boltzmann equation. This connection is significant because the Boltzmann equation is the fundamental equation in the study of transport coefficients and of the response of a gas to external perturbations. The basic assumption underlying the Boltzmann equation is that intermolecular interactions can be treated as a sequence of uncorrelated binary collisions. This assumption renders the equation much more tractable, but it also limits the validity of the equation to low-density gases. Figure 4 shows the frequency spectrum of density fluctuations in xenon gas at 349.6 K and 1.03 atm calculated according to the procedure: 1 S(k, ω) = Re d 3 pd 3 p C(kpp z)z=iω (51) π where C is determined from Eq. (46) with φ (c) given by the binary collision kernel for hard sphere interactions. At such a low density it is valid to ignore φ (s) and the second term in Eq. (48). Also shown in Fig. 4 are the experimental data from light scattering spectroscopy. The good agreement is evidence that the linearized Boltzmann equation provides an accurate description of thermal fluctuations in low-density

gases in the kinetic regime where kl ∼ 1. The agreement is less satisfactory when the data are compared with the results of hydrodynamics; in this case the calculated spectrum shows essentially no structure. This again indicates that at finite (k, ω) the hydrodynamic theory overestimates the damping of density fluctuations. Generally speaking, kinetic theory calculations have been quantitatively useful in the analysis of light scattering experiments on gases and gas mixtures. For moderately dense systems, typically fluids at around the critical density, the Boltzmann equation needs to be modified to take into account the local structure of the fluid. In the case of hard spheres, the modified equation generally adopted is the generalized Enskog equation, which involves g(σ ), the pair distribution function at contact (with σ the hard sphere diameter); the collision term differs from the collision integral in the linearized Boltzmann equation for hard spheres only in the presence of two phase factors, which represent the nonlocal spatial effects in collisions between molecules of finite size. Figure 5 shows the frequency spectra of density fluctuations obtained from simulation and kinetic theory at rather long wavelengths in hard sphere fluids and at three densities, corresponding roughly to half the critical density, 1.7 times critical density, and liquid density at the triple point. The √ kσ values are such that using the expression l −1 = 2π nσ 2 g(σ ) for the collision mean free path, we find that for the three cases (a)–(c) a molecule on the average would have suffered about 1, 5, and 20 collisions, respectively, in traversing a distance equal to the hard sphere diameter. On this basis we might expect the spectra in (b) and (c) to be dominated by hydrodynamic behavior, while that in (a) should show significant deviations.

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should become important at high densities. Just like the onset of shear wave propagation, this characteristic feature is part of the viscoelastic behavior expected of dense fluids. In order to describe such effects in the present context, it is now recognized that correlated collisions will have to be included in the kinetic equation. Aside from density and thermal fluctuations, it is also known that the transport coefficients derived from the Enskog equation are in error up to a factor of 2 at the liquid density when compared to computer simulation data hard spheres. Moreover, simulation studies have revealed a nonexponential, long-time decay of the velocity autocorrelation function that cannot be explained by the Enskog theory. Any attempt to treat correlated collision effects necessarily leads to nonlinear kinetic equations. For practical calculations it appears that only the correlated binary collisions, called ring collisions, are tractable. To incorporate these dynamical processes in the kinetic theory, we can develop a formalism wherein φ (c) is given as the sum φ (c) = φE + φR , where φE is the memory function

FIGURE 5 Frequency spectra of dynamic structure factor S(k, ω) in hard sphere fluids at three densities; simulation data are shown as open circles while the solid curves denote results obtained using the generalized Enskog equation. Dimensionless frequency ω* is defined as ωτE /kσ , with the Enskog collision time τE−1 = 4 √ π nv0 σ 2 g(σ ). Only inputs to the calculations are S(k ) and g (σ ), which can be obtained from the simulation data. In (b) the effects of ignoring entirely the static part of the memory function and of using the conventional Enskog equation are also shown. For (a) nσ 3 = 0.1414, kσ = 0.412, S(k ) = 0.563; g (σ ) = 1.22; (b) nσ 3 = 0.471, kσ = 0.616; S(k ) = 0.149, g (σ ) = 4.98; and (c) nσ 3 = 0.884, kσ = 0.759, S(k ) = 0.0271, g (σ ) = 2.06.

The theoretical curves in Fig. 5 are kinetic model solutions to the generalized Enskog equation. They are seen to describe quantitatively the computer simulation data. We could have expected good agreement in the lowest density case, which is nevertheless two orders of magnitude higher in density than a gas under standard conditions. That the theory is still accurate at condition (b) is already somewhat unexpected. So it is rather surprising that a kinetic theory that treats the interactions as only uncorrelated binary collisions is applicable at liquid density, as shown in (c). Indeed, at three times the present value of kσ a characteristic discrepancy appears in the high-density case, as shown in Fig. 6. The failure of the generalized Enskog equation to account properly for the simulation results at low frequencies can be traced to the presence of a slower decaying component in the data for F(k, t). It seems reasonable to associate this with the relaxation of clusters of particles, which

FIGURE 6 The density correlation function and normalized transverse current correlation in a hard sphere fluid at a density of nσ 3 = 0.884. The k value is 2.28 σ −1 . Computer simulation data are given by the circles, while calculations using the generalized Enskog equation or the mode coupling theory are denoted by the dashed and solid curves, respectively.

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for the generalized Enskog equation, and φR describes the ring collision contribution. In essence φR can be expressed schematically as φR = VCCV, where V is an effective interaction, which involves the actual intermolecular potential and the equilibrium distribution function of the fluid, and C is the phase space correlation function. The important point to note is that the memory function now depends quadratically on C, thereby making Eq. (46) a nonlinear kinetic equation. The appearance of nonlinearity, or feedback effects, is not so surprising when we recognize that in a dense medium the motions of a molecule will have considerable effects on its surroundings, which in turn will react and influence its subsequent motions. The inclusion of correlated collisions is a significant development in the study of transport called renormalized kinetic theory. The presence of ring collisions unavoidably makes the analysis of time correlation functions considerably more difficult. Nevertheless, it can be shown analytically that we obtain a number of nontrivial collective properties characteristic of a dense fluid, such as a power law decay of the velocity autocorrelation function, and nonanalytic density expansions of sound dispersion and transport coefficients.

VI. MODE COUPLING THEORY There exists another method of analyzing time correlation functions, which has common features with both generalized hydrodynamics and renormalized kinetic theory. In this approach we formulate an approximate expression for the space–time memory function that is itself nonlinear in the time correlation functions. The method is called mode coupling because the correlation functions describe the hydrodynamic modes, the conserved variables of density, momentum, and energy, in the small (k, ω) limit, and they are brought together to represent higher order correlations, which are important in a strongly coupled system such as a liquid. The mode coupling approach has been particularly successful in describing the dynamics of dense fluids; it is the only tractable microscopic theory of dense simple fluids. To describe the mode coupling formalism, we consider the density correlation function or its Laplace transform ∞ S(k, z) ≡ i dt ei zt F(k, t) ≡ [F(k, t)] (52) 0

and similarly for Jl (k, z), the longitudinal current correlation function. Using the continuity equation we find 2 S(k) k S(k, z) = − Jl (k, z) (53) + z z

which is just another form of Eq. (34). Now we write an equation for Jl (k, z) [cf. Eq. (36)] in the form −1 20 (k) 2 Jl (k, z) = −v0 z − + D(k, z) (54) z where 20 (k) = (kv0 )2 /S(k) and D(k, z) is the memory function. Combining Eqs. (53) and (54) gives −1 20 (k) S(k, z) = −S(k) z − (55) z + D(k, z) which is an exact equation. The basic assumption underlying the mode coupling theory is the approximate expression derived for D(k, z). In essence one obtains 20 (k) D(k, z) + d 3 k V (k, k )[F(k , t) v ×F(|k − k |, t)]

(56)

where v is a characteristic collision frequency usually taken from the Enskog theory, and V (k, k ) is an effective interaction. Equation (56) is an example of a two-mode coupling approximation involving two density modes F(k, t). Depending on the problem, we can have other products containing modes from the group {F(k, t), Fs (k, t), Jt (k, t), Jl (k, t), and the energy fluctuation}, and for each mode coupling term there will be an appropriate vertex interaction V (k, k ). The calculation of S(k, z) is fully specified by combining Eqs. (55) and (56). By expressing the memory function back in terms of the correlation function, we obtain a selfconsistent description capable of treating feedback effects. These are the effects that become important at high densities, and that we have tried to treat in the kinetic theory approach through the ring collisions. Mode coupling calculations were first applied to analyze the transverse and longitudinal current correlation functions in liquid argon and liquid rubidium. The theory was found to give a satisfactory account of the computer simulation results on shear wave propagation in Jt and the dispersion behavior of ωm (k) in Jl . The theory was then reformulated for the case of hard spheres and extensive numerical results were obtained and compared in detail with simulation data. It was shown that the viscoelastic behavior discussed previously, which could not be explained by the generalized Enskog equation, is now well described. The improvement due to mode coupling can be seen in Fig. 6. Another problem where the capability of mode coupling analysis to treat dense medium effects can be demonstrated is the Lorentz model. This is the study of the diffusion of a tagged particle in a random medium of stationary scatterers and of its localization when the scatterer density n exceeds a critical value n c . The system can be

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more difficult so that at a certain density the local structure will no longer relax on the time scale of observation. This condition of structural arrest is a fundamental characteristic of solidification, and it is appropriate to ask if mode coupling theory can describe such a highly cooperative process. Indeed, a certain self-consistent approximation in mode-coupling theory will lead to a model that exhibits a freezing transition. The signature of the transition is that the system becomes nonergodic at a critical value of the density or temperature. To demonstrate that the mode-coupling formalism can describe a transition from ergodic to nonergodic behavior, we consider a schematic model for the normalized dynamic structure factor ϕ(z) ≡ S(k,z)/S(k), and ignore the wavenumber dependence in the problem. In analogy with Eq. (55) we write FIGURE 7 Density variation of the diffusion coefficient in the two-dimensional Lorentz model where the hard disks can overlap: mode coupling theory (solid curve) and computer simulation data. D 0 is the diffusion coefficient given by the Enskog theory.

characterized by the diffusion coefficient of the tagged particle D, which plays the role of an order parameter. The model then exhibits two distinct phases, a “diffusion” phase D = 0, when n < n c , and for n > n c a “localization” phase with D = 0. Figure 7 shows the density variation of D in the case of the two-dimensional Lorentz model with hard disk scatterers that can overlap. The mode coupling theory gives satisfactory results if the density is scaled according to n c . As for the prediction of n c , the theory gives n ∗c = nσ d = 0.64 and 0.72 for d = 2 and d = 3, respectively, while molecular dynamics simulations give 0.37 and 0.72. Here the tagged particle and the stationary scatterers are both hard spheres of diameter σ , and d is the dimensionality of the system. The fact that the theory does not give an accurate value for n c in two dimensions indicates that the statistical distribution of system configurations in which the particle becomes trapped requires a more complicated treatment than the simplest mode coupling approximation. On the other hand, the density variation of the velocity autocorrelation function observed by simulation, particularly its nonexponential decay at long times for n < n c , can be calculated very satisfactorily. In view of the successful attempts at describing the dense, hard sphere fluids and the Lorentz model, we might wonder what mode coupling theory will give at densities beyond the normal liquid density, typically taken to be the triple point density of a van der Waals liquid, n ∗ = nσ 3 = 0.884. On intuition alone we expect that as the atoms in the fluid are pushed more closely against each other, structural rearrangement becomes more and

−ϕ(z)−1 = z + K (z) −20 K (z)−1

= z + M(z)

(57) (58)

with M(z) playing the role of D(k, z). We will consider two different approximations to the memory function M(z), M(z) M0 (z) + m(z)

(59)

and M(z) M0 (z) +

m(z) 1 − (z) m(z)

(60)

with M0 (t) = ω¯ δ(t) m(t) =

4λ20

(61) 2

F (t)

(62)

(t) = λ F(t) J (t)

(63)

Equations (59) and (62) constitute the original modecoupling approximation, henceforth denoted as the LBGS model, in which only the coupling of density fluctuation modes, with F(t) defined by Eq. (18), is considered. Equations (60) and (63) constitute an extension in which the coupling to longitudinal current modes, with J (t) given by Eq. (34), is also considered. We will refer to this as the extended mode-coupling approximation. In both models M0 (z) = i ω¯ is the Enskog-theory contribution to the memory function, where ω¯ is an effective collision frequency. The coupling coefficients λ and λ will be treated as density- and temperature-dependent constants. Comparing the two models, we see that the difference lies in the presence of (z) in Eq. (60). It may seem remarkable that an apparently simple approximation of coupling two density modes can provide a dynamical model of freezing. To see how this comes about, notice that the quantity of interest in the analysis is the relaxation behavior of the time-dependent

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density correlation function G(r, t), or its Fourier transform F(t) = F(k, t). Under normal conditions one expects F(t → ∞) = 0 because all thermal fluctuations in an equilibrium system should die out if one waits long enough. When freezing occurs, this condition no longer holds as some correlations now can persist for all times. The condition that F(t) stays finite as t → ∞ means the system has become nonergodic. To see that Eq. (59) can give such a transition, we look for a solution to the closed set of equations (57), (58), and (59) of the form ϕ(z) = − f /z + (1 − f )ϕv (z)

(64)

where the first term is that component of F(t) which does not vanish at long times, F(t → ∞) = f , and ϕv (z) is a well-behaved function that is not singular at small z. Since ϕv (z) is not pertinent to our discussion, we do not need to show it explicitly. Inserting this result into (59) yields M z = −4λ20 f /z + Mv (z)

(65)

with Mv (z) representing all the terms that are nonsingular at small z; we obtain ϕ(z) = −

20:58

4λ f 2 1 1 − (z) 1 + 4λ f 2 z 1 + 4λ f 2

(66)

with (z) also well behaved. For Eqs. (64) and (66) to be compatible, we must require f =

4λ f 2 1 + 4λ f 2

(67)

This is a simple quadratic equation for f , with solution f = 1/2 + (1/2)(1 − 1/λ)1/2 . Therefore, we see that in order for the postulated form of the density correlation function to be acceptable solution to Eqs. (57), (58), and (59), f must be real, or λ > 1. The implication of this analysis is that in the LBGS model the ergodic phase is defined by the region λ < 1, where the nondecaying component f must vanish, and a nonergodic phase exists for λ > 1. The onset of nonergodicity signifies the freezing in of some of the structural degrees of freedom in the fluid; therefore, it may be regarded as a transition from a liquid to a glass. The origin of this transition is purely dynamical since it arises from a nonlinear feedback mechanism introduced through m(z). The freezing or localization of the particles shows up as a simple pole in the low-frequency behavior of ϕ(z), a consequence of the fact that M(z) ≈ 1/z at low frequencies. The LBGS model is the first mode-coupling approximation providing a dynamical description of an ergodic to nonergodic transition. The transition also has been derived using a nonlinear fluctuating hydrodynamics formulation. Analysis of the LBGS model shows that the diffusion coefficient D has a power-law density dependence, D ∼ (n c − n)α , with exponent α 1.76, and cor-

respondingly the reciprocal of the shear viscosity coefficient η behaves in the same way. There exist experimental and molecular dynamics simulation data that provide evidence supporting the density and temperature variation of transport coefficients predicted by the model. Specifically, diffusivity data for the supercooled liquid methy-c yclohexane and for hard-sphere and Lennard–Jones fluids obtained by simulation are found to have density dependence that can be fitted to the predicted power law. The fact that the mode-coupling approximation is able to give a reasonable description of transport properties in liquids at high densities and low temperatures beyond the triple point is considered rather remarkable. The LBGS model also has been found to provide the theoretical basis for interpreting recent neutron and light scattering measurements on dynamical relaxations in dense fluids. These experiments show that the temporal relaxation of the density correlation function F(k, t) is nonexponential, F(k, t) exp[−t/τ )β ], with β distinctly less than unity. This behavior of scaling, in the sense of F being a function of t/τ , where τ is a temperature-dependent relaxation time, and of stretching, in the sense of β < 1, is also given by Eq. (59) provided a term λ F(t) is added to Eq. (62). Thus, the ability of the mode-coupling approximation to describe the dynamical features of relaxation in dense fluids has considerable current experimental support. The successes of the approximation Eq. (59) notwithstanding, it does have an important physical shortcoming, namely, it does not treat the hopping motions of atoms when they are trapped in positions of local potential minima. These motions are expected to be dominant at sufficiently low temperature of supercooling; their presence means that the system should remain in the ergodic phase, albeit the relaxation times can become exceedingly long. For this reason, the predicted transition of the LBGS model is called the ideal glass transition; in reality one does not expect such a transition to be observed. The extended mode-coupling model, Eq. (60), in fact provides a cutoff for the ideal glass transition by virtue of the presence of (z). One can see this quite simply from the small-z behavior of Eqs. (57), (58), and (60). With nonzero, ϕ(z) no longer has a singular component varying like 1/z, so F(t) will always vanish at sufficiently long times. Even though the two approximations, Eqs. (59) and (60), give different predictions for the transition, one has to resort to numerical results in order to see the differences between the two models in their descriptions of F(k, t) in the time region accessible to computer simulation and neutron and light scattering measurements. In Fig. 8 we show the intermediate scattering function F(k, t) of a fluid calculated by simulation using a truncated Lennard–Jones interaction at various fluid densities

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FIGURE 8 Relaxation of density correlation function F(t) at wavenumber k = 2 A−1 obtained by molecular dynamics simulation at various reduced densities n∗ and reduced temperature T ∗ = 0.6. Time unit τ is defined as (mσ 2 /ε)1/2 .

(n ∗ = nσ 3 ). One sees that as n ∗ increases, the relaxation of F(k, t) becomes increasingly slow. Compared to these simulation results, the corresponding mode-coupling calculations, using a model equivalent to Eq. (59), show the same qualitative behavior of slowing down of relaxation; however, the mode-coupling model predicts a freezing effect that is too strong. This discrepancy is not seen in a model equivalent to Eq. (60). Thus, there is numerical evidence that the cutoff mechanism of the transition, represented by , is rather significant. To what extent can the dynamical features of supercooled liquids be described by mode-coupling models such as Eqs. (59) and (60)? Although these approximations seem to give semiquantitative results when compared to the available experimental and simulation results, it is also recognized that hopping motions should be incorporated in order that the theory be able to give a realistic account of the liquid-to-glass transition.

VII. LATTICE GAS HYDRODYNAMICS Fluctuations extend continuously from the molecular level to the hydrodynamic scale, but we have seen that there are experimental and theoretical limitations to the ranges where they can be probed and computed. Indeed, no theory provides a fully explicit analytical description of space– time dynamics establishing the bridge between kinetic theory and hydrodynamic theory, and scattering techniques have limited ranges of wavelengths over which fluctuation correlations can be probed. With numerical computational methods one can realize molecular dynamics simulations that in principle, could cover the whole desired range, but in practice there are computation time and memory requirement limitations. Lattice gas automata (LGA) are discrete models constructed as an extremely simplified version of a many-

Molecular Hydrodynamics

particle system where pointlike particles residing on a regular lattice move from node to node and undergo collisions when their trajectories meet at the same node. The remarkable fact is that, if the collisions occur according to some simple logical rules and if the lattice has the proper symmetry, this automaton shows global behavior very similar to that of real fluids. Furthermore, the lattice gas automaton exhibits two important features: (i) It usually resides on large lattices, and so possesses a large number of degrees of freedom; and (ii) its microscopic Boolean nature, combined with the (generally) stochastic rules that govern its microscopic dynamics, results in intrinsic fluctuations. Therefore, the lattice gas can be considered as a “reservoir of thermal excitations” in much the same way as an actual fluid, and so can be used as a “virtual laboratory” for the analysis of fluctuations, starting from a microscopic description. A lattice gas automaton consists of a set of particles moving on a regular d-dimensional lattice L at discrete time steps, t = nt, with n an integer. The lattice is composed of V nodes labeled by the d-dimensional position vectors r ∈ L. Associated to each node there are b channels (labeled by indices i, j, . . . , running from 1 to b). At a given time, t, channels are either empty (the occupation variable n i (r, t) = 0) or occupied by one particle [n i (r, t) = 1]. If channel i at node r is occupied, then there is a particle at the specified node r, with a velocity ci . The set of allowed velocities is such that the condition r + ci t ∈ L is fulfilled. It may be required that the b set {ci }i=1 be invariant under a certain group of symmetry operations in order to ensure that the transformation properties of the tensorial objects that appear in the dynamical equations are the same as those in a continuum [such as the Navier–Stokes equation (11)]. The “exclusion principle” requirement that the maximum occupation be of one particle per channel allows for a representation of the automaton configuration in terms of a set of bits {n i (r, t); i = 1, . . . , b; r ∈ L}. The evolution rules are thus simply logical operations over sets of bits, which can be implemented in an exact manner in a computer. The time evolution of the automaton takes place in two stages: propagation and collision. We reserve the notation b n(r, t) ≡ {n i (r, t)}i=1 for the precollisional configuration b of node r at time t, and n∗ (r, t) ≡ {n i∗ (r, t)}i=1 for the configuration after collision. In the propagation step, particles are moved according to their velocity n i (r + ci t, t + t) = n i∗ (r, t)

(68)

The (local) collision step is implemented by redistributing the particles occupying a given node r among the channels associated to that node, according to a given

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prescription, which can be stochastic. The collision step can be represented symbolically by n i∗ (r, t) = σi ξn(r,t)→σ (69) σ

where ξn(r,t)→σ is a random variable equal to 1 if, starting b from configuration n(r, t), configuration σ ≡ {σi }i=1 is the outcome of the collision, and 0 otherwise. The physics of the problem is reflected in the choice of transition matrix ξs→σ . Taking an average over the random variable (assuming homogeneity of the stochastic process in both space and time), and using Eq. (68), we obtain n i (r + ci , t + 1) = σi ξ s→σ δ[n(r, t), s] (70) σ,s

where automaton units (t = 1) are used. These microdynamic equations constitute the basis for the theoretical description of correlations in lattice gas automata. Starting from Eq. (70), by performing an ensemble average over an arbitrary distribution of initial occupation numbers (denoted by angular brackets), one derives a hierarchy of coupled equations for the n-particle distribution functions, analogous to the BBGKY hierarchy in continuous systems. The first two equations in this hierarchy are f i (r + ci , t + 1) = σi ξ s→σ δ(n(r, t), s), (71) σ,s f i(2) j (r + ci , r + c j , t + 1) = (1 − δ(r, r )) σi σ jξ s→σ ξ s →σ σ,s,σ ,s

× δ(n(r, t), s)δ(n(r , t), s ) + δ(r, r )

σi σ j ξ s→σ δ(n(r, t), s)

(72)

s,σ

where f i (r, t) = n i (r, t) f i(2) j (r, r , t)

= n i (r, t)n j (r , t)

(73) (74)

are the one- and two-particle distribution functions, respectively. The fluctuations of the channel occupation number are δn i (r, t) = n i (r, t) − f i (r, t), and the corresponding pair correlation function reads G i j (r, r , t) = δn i (r, t)δn j (r , t)

(75)

Using a cluster expansion and neglecting three-point correlations, the hierarchy of equations can be approximately truncated to yield the generalized Boltzmann equation for

the single particle distribution function f i (r, t), f i (r + ci , t + 1) − f i (r, t) = i(1,0) (r, t) (1,2) + i,kl (r, t)G kl (r, r,t)

(76)

k α, and then falls at the rapid rate of (1/ f )(1/ f 3 ) = (1/ f )4 above f = β, where the joint effects of E a and E b are active. If α and β are not so very different, there is a transition region of intermediate slope between the level behavior at low frequencies and the 1/ f 4 high-frequency fall-off. Figure 5 shows the results of this behavior, as found in the practical world of guitar playing. Here the bridge force is shown as a function of frequency for a string that is plucked at a fairly normal point 18 of the way along it by a plectrum, whose width 1 of the gives a local string curvature extending over 16

FIGURE 5 Overall spectrum envelope of the guitar bridge driving force. The trend is from constancy at low frequencies to a highfrequency rolloff proportional to 1/ f 3 .

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248 “open” (maximum) string length. For reasons that will become clearer as we go along, a line is drawn on this graph to suggest that the high-frequency behavior of the spectrum is well approximated by a 1/ f 3 falloff rate, with a breakpoint located at about five times the open string first-mode frequency. The spectral notches are shown explicitly only for the E a ( f ) aspect of the behavior. As remarked earlier, when a player is performing music on a guitar, he tends to pluck the strings at a roughly constant distance from the bridge. The bridge driving-force spectrum for all the notes played on any one string will then share a single spectral envelope of the form shown in Fig. 5, including the notches. While the spectrum envelopes for the adjacent strings (tuned to different pitches) are alike in form, the notches and breakpoints are displaced bodily to higher or lower frequencies by amounts depending on the tunings of these other strings. Taken as a group, however, the force exerted on the bridge by all the strings has an overall envelope that is frequency independent at low frequencies and varies roughly as 1/ f 3 at high frequencies. The transition region between those two behaviors is blurred somewhat, due to the differences between breakpoint frequencies belonging to the individual strings. While the gross envelope for the bridge driving-force spectrum is in fact made up of six distinct parts (one for each string), the mechanism for conversion of this excitation into the room-average sound is very much the same for all strings. It is all mediated by the same set of body resonances, and these can produce fluctuations in the radiated sound above and below the essentially frequency-independent trend line of the overall radiation process for sounds emitted by a platelike object. Thus the observed room-average spectrum is found to have fluctuations above and below an overall envelope whose shape is very similar to that of the curve in Fig. 5. It is easy to estimate the expected number of fluctuations over any frequency span of interest. This is equal to the number of body modes found in this span, augmented by the corresponding number of notches in the drive-force spectrum. It is appropriate here to inject an additional piece of information about the human auditory processor. A music listener is (in a laboratory situation) readily able to detect the strengthening or weakening of one or more sinusoidal components of a harmonic collection. However, in the processing of music or speech he does not “pay attention” very much to the presence of holes or notches in the spectra of the sounds that he processes as a means for recognizing them or assessing their tone color. More precisely, failure to provide notches will be noticed and criticized in an attempted sound synthesis, but (except for certain special cases) their mere presence and their

Musical Acoustics

mean spacing are of more significance than their exact positions. B. The Harpsichord and Piano The harpsichord and piano are acoustically similar to the guitar in that they have a set of vibrating strings that only gradually communicate their energy to a two-dimensional platelike structure (the soundboard), which in turn passes on the vibration in the form of audible sound to the room. Once again, for musico-neurological reasons, it is important that the primary string mode oscillations take place at frequencies that are in whole-number relation to one another. The major difference that distinguishes these instruments from the guitar is the fact that they lack frets, so that each string is used at only a single vibrating length; also, the place and manner of plucking or striking is chosen by the instrument’s maker rather than by its player. The strings of a harpsichord are excited by a set of plectra ( jacks) operated from a keyboard, so that in many respects the dynamical behavior of a harpsichord is identical with that of the guitar. As a result, the main features of the spectral envelope of harpsichord sounds in a room are the same as those of a guitar. The curve shown in Fig. 5 and the accompanying discussion apply equally well to the harpsichord, the chief difference being in a slightly different distribution of spectral notches associated with the excitation mechanism, and a greatly decreased mean spacing ( 20 Hz) of the radiation irregularities that are associated with the plate modes of the soundboard. The (as yet undiscussed) air resonances of the cavity under the harpsichord’s soundboard play a very much smaller role in determining the overall tone than is the case for the guitar. The mechanical structure of the piano is quite analogous to that of the harpsichord, but the use of hammers rather than plectra to excite its strings causes several modifications to the overall envelope function. While it is commonly believed that the striking point should have an effect on the envelope similar to that given by E a ( f ) for the plucked string, in fact the corresponding function is much less dependent on frequency, with only small dips appearing at the frequencies of the “notches” of E a ( f ). However, the width and softness of the hammer join with the string’s stiffness to give rise to an envelope function E a ( f ) exactly as given for the plucked strings. There is, however, one more dynamical influence on the spectral envelope of a struck string: When the hammer strikes the string, it bounces off again after a time that is jointly determined by the hammer mass and elasticity, the string tension, the position of the striking point along the string, and the length of the string. The details of this dependency of the hamer contact time on these parameters are complicated, and it will suffice for us to present only its main spectral

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consequences. It gives rise to an envelope function E c ( f ) that is satisfactorily represented by a simplified formula as cos(π f /4 f c ) Ec ( f ) = . (8) 1 + ( f / f c )2 Here f c is very nearly equal to the reciprocal of the hammer contact time. The nature of this function is shown in the right-hand part of Fig. 4. The envelope function has a familiar form, being essentially constant at low frequencies, having a number of deep notches, and ultimately falling away as 1/ f 2 at high frequencies with a breakpoint γ for the main trend at f = f c . The analog to Eq. (6) for the piano is then F( f ) = K · E b ( f ) · E c ( f ).

FIGURE 6 Measured room-average spectrum envelope of piano tones. Above about 800 Hz the components weaken as 1/ f 3 .

(9)

At very high frequencies ( f α, β, and γ ), the driveforce spectrum falls away as (1/ f 3 )(1/ f 2 ) = (1/ f 5 ). As before, the behavior at somewhat lower frequencies can have an apparent fall rate represented by some intermediate exponent that depends on the maker’s choice of β and γ . In the piano, the distance of the hammer’s striking point from the string end varies smoothly (sometimes in two or more segments) from about 10 or 12% of the string length at the bass end of the scale to about 8% at the treble end. Similarly, the mass, width, and softness of the hammers fall progressively in going up the scale from the bass end. Taken together, these four varying parameters of piano design provide the maker with his chief means for achieving what he calls a “good” tone for the instrument simultaneously with uniformity of loudness and of keyboard “feel.” The hammer mass has a direct influence on the feel of the keys. It also plays a major role in determining the loudness of the note via its effect on the kinetic energy that it converts into vibrational energy of the string. The hammer mass (as well as its softness to some extent) joins with the striking point and string parameters to control the contact time during a hammer blow. At C2 near the bottom of the scale, the design is such that the hammer 1 mass is about 30 of the total string mass, and the hammer’s contact time is around 4 ms (γ 250 Hz); at the midscale C4 , the string and hammer masses are about equal, and the contact time is about 1.5-ms; at C7 , near the top of scale, it has fallen to 1 ms (γ 1000 Hz). In a related manner, the string stiffness joins with hammer softness to determine the string-curvature envelope E b ( f ) and its corresponding breakpoint frequency β. Figure 6 shows the remarkable uniformity in the trend line for the measured room-average spectra (defined in Section I) of notes taken from the musically dominant midrange portion of a grand piano’s scale. These notes, running scale-wise from G3 up to G5 (having repetition

rates 192 to 768 Hz), will be recognized as lying in the region in which the auditory processor is particularly quick, precise, and confident. As a result, any regularities shown by the spectra of these notes have strong implications about the manner in which the ear deals with such notes. In the figure, the dots located along the zero-decibel line represent the normalized amplitudes of the first-mode frequency components of all 14 played notes. The remaining dots then give the relative amplitudes of the remaining higher harmonic components (expressed in decibels relative to the “fundamental” components). About half of the notes shown here were played and measured several times, over a period of five years, using a variety of analysis techniques. The variability due to all causes (irregularity of striking the keys, statistical fluctuations of the room measurement, wear of the piano, and differences due to altered analysis technique) may be shown to be about ±2 dB for the position of any one dot on the curve. For this reason it is possible to attribute the observed scattering of the points about their basic trend line almost wholly to the excitatory spectrum notches and to the radiation effects of soundboard resonances in the body of the piano. The magnitude of these fluctuations is consistent with estimates based on the resonance properties of a sound board. Figure 6 illustrates a spectral property that is shared by nearly all of the familiar midrange musical instruments. Here, as in the guitar and harpsichord, we find fluctuations about an essentially constant low-frequency average trend, plus a rolloff with a (1/ f 3 ) dependency at high frequencies. Dividing the two spectral regions, there is also a well-defined break point that lies close to 780 Hz for the piano. To recapitulate, the proportioning of the piano’s string length and strike point and its hammer’s breadth, mass, and softness cause the critical frequency parameters f b and f c to vary widely for strings over the midrange playing scale. Nevertheless, the maker has arranged to distribute them in

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250 such a way as to preserve the absolute spectral envelope over a wide range of playing notes. Since it is clearly not an accident that the piano and harpsichord have developed with proportions of the type implied previously, we are led to inquire as to what are the perceptual constraints imposed on the design by the needs of the listner’s musical ear. The answers to this inquiry (implied in large measure by the auditory properties outlined in Section I) will be made more explicit in the remaining course of this article. C. Radiation Behavior of Platelike Vibrators A number of significant musical properties of the guitar, harpsichord, and piano have been elucidated by an examination of the ways in which their platelike body or soundboard structures communicate the vibrations imposed on them by the string to the surrounding air in the room (violins of course also communicate this way). It has already been asserted that the trend of radiating ability of such structures driven by oscillating forces is essentially independent of frequency except at low frequencies. Because a number of musical complexities are associated with this apparently simple trend, it is worthwhile to devote some space to a brief outline of the radiation physics that is involved. To begin with, consider a thin plate of limitless extent, driven at some point by a sinusoidal driving force of fixed magnitude F0 and variable frequency f . Analysis shows that the vibrational velocity produced at the driving point of such a plate is proportional to the magnitude of the driving force, but independent of its frequency. For definiteness, let the plate be of spruce about 3-mm thick (as is the case for the guitar and violin top plate, or the soundboard of a harpsichord). Also, we temporarily limit the driving frequency to values that lie below about 3000 Hz. Despite the fact that the entire surface of the plate is set into vibration by the excitory force, only a small patch near the driving point is actually able to emit sound into the air! The radius rrad of this radiatively effective patch is about 16 cm at 100 Hz, and it varies inversely as the square root of the frequency. Thus, the area of the active patch varies as 1/ f . Since the radiation ability of a small vibrating piston is proportional to its area, velocity amplitude, and vibrational frequency, the sound emitted by the board not only comes from a tightly localized spot at the point of excitation, but also the amount radiated is entirely independent of frequency. When the size of the plate is restricted by any kind of boundary (free, hinged, or clamped), additional radiation becomes possible from a striplike region extending a distance rrad (defined previously) inward from these boundaries. Any sort of rigid blocking applied at some point, or hole cut in the plate, also gives rise to a radiatively

Musical Acoustics

active region of width rrad around the discountinuity, and the system retains its essentially frequency-independent radiating behavior. The fact that the system is now of finite extent means that it has a large number of vibrational modes (whose mean spacing is set mainly by the thickness and total plate area of the structure). The system’s net radiated power then fluctuates symmetrically above and below the large-plate trend line in a manner controlled by the size, width, and damping of the modal response peaks and dips. Curiously enough, the general level of radiation is hardly influenced by the plate damping produced by its own internal friction. Above a certain coincidence frequency f coinc (the previously mentioned 3000 Hz for a spruce plate 3-mm thick), the entire vibrating plate abruptly becomes able to radiate into air. For a limitless plate, the radiating power becomes enormous just above f coinc , and it then drops off to a new frequency-independent value that is considerably greater than that found below f coinc . However, for a finite-sized system broken up into many parts (as in the musical structures), there is no readily detectable alteration in the overall radiating ability as the drive frequency traverses f coinc , although many details of the directional distribution of sound are drastically changed. The coincidence frequency is inversely proportional to the plate thickness, and the radius rrad of the radiatively active regions is proportional to the square root of the thickness; this means that for a piano (whose plate thickness is about triple that of a harpsichord), f coinc falls to about 1000 Hz and rrad is about 27 cm at 100 Hz. The practical implications of the radiation properties briefly discussed here are numerous. To begin with, it should be clear that the boxlike (and therefore irregular) structure of the guitar and violin joins with the sound holes and miscellaneous internal bracing to greatly increase the sound output from what would otherwise be very softvoiced instruments (as are the lutes and viols of simpler construction that were developed earlier). By the beginning of the seventeenth century, the harpsichord soundboard had already accquired numerous heavy struts along with structural discontinuities provided by the bridges needed to serve two complete sets of strings (the so-called 4- and 8-ft arrays) and a hitch-pin rail between the bridges to bear the tension of the 4-ft strings. These strong and heavy discontinuities play an important role in providing the free, clear sound that is characteristic of a really fine harpsichord. The instruments by the French builder, Pascal-Joseph Taskin (1723–1793), which are noted for the fullness of their tone, are provided with an unusually rigid set of bracings. It is important to notice that despite the completely counterintuitive nature of the lumpy and discontinuous structures that favor sound production, the best makers nevertheless discovered

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and adhered to designs whose underlying acoustical virtues have only been elucidated in the late twentieth century. A quick look at the modern piano shows a similar adaptation of the vibrating structure to its radiating task, although we might today ask (by analogy) whether or not a few properly placed extra braces might improve the sound somewhat. Some of the difficulty often faced by sound engineers when making recordings of the piano is readily understood in terms of the ever-shifting patchwork of sources that are active. The tendency of many engineers to place their microphones close to the piano means that rapidly changing, and often perceptually conflicting, signals are registered in the two recording channels. The ear is so accustomed to assembling the sounds from all over the soundboard, via the mediation of a room, that anything else can confuse it. As a matter of fact, the ear is so insistent on receiving piano sounds from a random patchwork of shifting sources that successful electronic syntheses of piano music can be done using the simplest of waveform sources as long as signals are randomly distributed to an array of six to ten small loudspeakers placed on a flat board! D. Piano Onset and Decay Phenomena When a hammer strikes a piano string it is subjected to an impulsive blow that contains many frequency components. This blow, which is transmitted to the bridge in the form of a continuously distributed drive-force spectrum whose shape is exactly the same as the spectrum envelope of the discrete string vibration drive forces, is heard in the room as a distinct thump. Those parts of each measured room-average spectrum that lie between the strongly represented string-vibration components clearly show the envelope of the thump part of the net sound. As a matter of fact, the similarity between the shapes of the thumpenvelopes of various notes, and of each one of these to the overall envelope displayed in Fig. 6, serves as a good confirmation of our picture of the piano sound-generation process. Despite the fact that the piano sound is produced by an impulsive excitation taking place in only a very few milliseconds, the buildup of radiated sound in its neighborhood takes place over a period of time that is 10 to 50 times longer. We will begin our search for an explanation for this slow buildup by outlining the vibrational energy budget of a piano tone. Setting aside temporarily the energy associated with the initial thump, we can recognize that each string mode is abruptly supplied with its share of energy at the time of the hammer blow. Over the succeeding seconds, some of this energy is dissipated unproductively as heat within the body of the wire and

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251 at its anchorages. There is also a flow of vibrational energy into the soundboard, to set up its vibrations. Overall, the board vibrations build up under the stimulus of the string until the board’s own energy loss rate to internal friction and to radiation into the room (and to some extent back into the string) are equal to the input rate from the string. Globally speaking then, we would expect the radiated sound to rise in amplitude for a while (as the soundboard comes into equilibrium with the string vibrations) and then to decay gradually as the string gives up its energy. The initial part of a curve showing this behavior calculated for a typical pianolike string and soundboard is shown by the solid line in Fig. 7. This sort of calculation is able to give good account of the main feature of the onset times (35 to 50 ms). However, the measured behavior for a piano shows considerably more complexity, for the following reasons: 1. The initial thump is instantly transmitted to the soundboard, and the resulting wave travels across it and suffers numerous reflections. 2. During the initial epoch, while both the thump and the main tonal components are spreading across the soundboard, and making their first few reflections, all frequency components are able to radiate fairly efficiently. As a result, components in the sound output have a significant representation at very early times. This behavior is schematized by the irregular line in Fig. 7. 3. The cross-influences of the three piano strings and bridge belonging to a typical piano note make the strings’ own decay quite irregular. This irregularity is then reflected in the longterm decay of the tone. All this is an example of the statistical fluctuation behavior that was outlined for rooms in Section I of this article.

FIGURE 7 Smooth beaded curve: Trend of initial buildup of soundboard vibrational energy after excitation of a string. Irregular curve: Schematic representation of the actual buildup.

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252

III. THE SINGING VOICE The instruments discussed so far have belonged to a class of musical sound generators in which the primary source of acoustical energy (the vibrating string) is abruptly set in motion and allowed to die away over the next few seconds. There is another, much wider class of musical instruments (including the human voice, the woodwinds and brasses, and the violin family) in which the oscillations of the primary vibrator are able to sustain themselves over relatively long periods of time, drawing their energy from a nonoscillatory source such as the musician’s wind supply or the steady motion of his bow arm. The tone production system of the singing voice provides an excellent introduction to this class of continuoustone instruments for two reasons. First, discussion is simplified by the fact that the primary exciter (the singer’s larynx) maintains its own oscillations in a manner that is quasi-independent of the vocal tract air passages that it excites. Second, a further expository simplification comes about because the frequency of its oscillations is controlled by a set of muscles that are distinct from those that determine the shape of the upper airway. Fundamentally, the larynx acts as a self-operating flow control valve that admits puffs of compressed air into the vocal tract in a regular sequence whose repetition rate f 0 determines the pitch of the note being sung. Since this flow signal is of a strictly periodic nature, the frequencies of its constituent sinusoids are exact integer multiples n f 0 of the laryngeal oscillation rate, and they therefore produce a sound of the type that is very well suited to the auditory processes of musical listening. Aside from its ability to generate continuous rather than decaying sounds, the singer’s voice-production mechanism, with its signal path from primary sound source (larynx) to concert hall via the vocal passages, is quite analogous in its physical behavior to that which leads the sound from vibrating string to the room by way of a soundboard or guitar body. Because the behavior of the sound transmission path through the vocal tract is far more important for present purposes than is the spectral description of the excitatory pulse train, we temporarily limit ourselves to the simple remark that the source component (of frequency n f 0 ) is related to the lowest component (of frequency f 0 ) by a factor (1/n 2 )A(n), where A(n) has a relatively constant trend line plus a few irregularly spaced notches or quasi-notches. The perceptual significance of these notches is relatively small, as in the case of the stringed instrument spectra. In short, the spectrum envelope of any voiced sound in the room includes a factor 1/ f 2 due the source spectrum as a major contributor to its overall shape.

Musical Acoustics

The vocal tract air column extending from the laryngeal source to the singer’s open mouth may be analyzed as a nonuniform but essentially one-dimensional waveguide, whose detailed shape can be modified by actions of the throat, jaw, tongue, and lip muscles. One end of this duct is bounded by the high acoustical impedance presented by the larynx, and the other by the low impedance of the singer’s open mouth aperture. Acoustical theory shows that such a bounded, one-dimensional medium has its natural frequencies spaced in a roughly uniform manner. Furthermore, the 15-cm length of this region implies that this mean spacing be about 1000 Hz, so that we expect no more than three or four such resonance frequencies in the region below 4000 Hz that contains the musically significant part of the voice spectrum. The signal transmission path from larynx to the listening room has a transfer function Tr ( f ) that is the product of three factors. One is a term T1 ( f ) falling smoothly as 1/ f 1/2 associated with acoustical energy losses that take place at the walls of the vocal tract. Another factor, T2 ( f ), has to do with the efficacy of sound emission from the mouth aperture into the room. This rises smoothly with a magnitude proportional to the signal frequency f . The third factor, T3 ( f ), fluctuates above and below a constant trend line, and it depends on the shape given to the vocal tract passage by its controlling muscles. The peaks in T3 ( f ) lie at frequencies that correspond to the normal-mode frequencies of the vocal tract if it is imagined to be closed off at the larynx and open at the mouth. The dips in the transmission function lie, on the other hand, at the modal frequencies of the vocal tract considered as an air column that is open at both ends. Both the peaks and dips have widths of about 50 Hz in the frequency range below 1500 Hz, rising to about 200 Hz at 4000 Hz. These peaks and dips tend to rise or fall above and below the trend line by about ±10 dB (i.e., factors of about 3±1 ). The nature of the overall vocal tract transfer function Tlr ( f ) between larynx and the room is summarized as Tlr ( f ) = (1/ f )1/2 × ( f ) × (peaks and dips) = ( f )1/2 × (peaks and dips).

(10)

Because the vocal source spectrum has a relatively featureless 1/ f 2 behavior, it is convenient to display graphically the product (1/ f 2 )Tlr ( f ), representing the sound normally measurable via the room-averaging procedure. Figure 8 presents such curves computed for three configurations of the vocal tract. The pattern of peaks and dips in the Tlr ( f ) function is of major perceptual significance: Each vowel or other voice sound is associated with a particular vocal tract configuration, and so a particular Tlr ( f ). Speech then consists of a

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253 cance in speech. First of all, the source spectrum shape, which is approximated by An = A1 /n 2 for the “mildest” and most speechlike tone color, may be modified to the form 1 + (1/γ )2 An = A1 . (11) 1 + (n/γ )2

FIGURE 8 Schematic representation of spectrum envelope curves for three sung vowels.

rapidly changing set of spectral envelope patterns, which are recognized by the listener in a manner that depends almost not at all on the nature of the laryngeal source spectrum. Thus, a singer who produce the vowel aah at a pitch of A2 ( f 0 = 110 Hz) supplies his listeners with a generated sound whose harmonic components can be evaluated from the curve for aah (see Fig. 8) at the discrete frequencies 110, 220, 330, . . . Hz, as indicated by the small circles on the curve. Similarly, a singer producing the vowel ooh at G4 ( f 0 = 392 Hz) emits a sound whose spectrum has component amplitudes that are related in the manner indicated by the small x’s. The vowel pattern recognition abilities of the human listener are highly developed. Whispered speech is perfectly comprehensible, even though the source signal consists of a densely distributed random collection of sinusoids (white noise) rather than the discrete and harmonic collection of voiced speech. Furthermore, a radio announcer is completely intelligible whether the receiver tone controls are set to “treble boost, bass cut” (nearly equivalent to multiplying Tlr ( f ) by f ) or to “treble cut, bass boost” (which is nearly the same as multiplying the spectrum by 1/ f ). The recognition process requires, as a matter of fact, only the existence of properly located peaks relative to the local trend of the spectrum, while the positions and depths of the dips are essentially irrelevant, to the point where many electronic speech synthesizers omit them entirely. Thus, all that is really necessary is to specify the frequencies of the lowest three or four transmission peaks for each sound. These are denoted by F1 , F2 , F3 , and F4 and are called the format frequencies (and are about 17% higher for women than for men). So far, no clear distinction needs to be made between speech and song beyond the need in music for precisely defined pitch (and thence values of f 0 ). The musician has, however, three special resources that have little signifi-

The components for which n is less than γ have then essentially the same amplitude as A1 , while the 1/n 2 falloff is postponed to the higher, h δ components. In the extreme case, γ can be as large as 3. The second resource of the singer is the use of what is known as format tuning. Consider a soprano who is asked to sing the vowel aah at the pitch D5 at the top of the treble staff. This means that her sound consists of sinusoids having frequencies close to n f 0 = 587, 1175, 1762, . . . Hz. Her normally spoken “aah” would have a second-formant frequency F2 , close to 1260 Hz, but she may choose to alter her vocal tract shape (and thus modify the vowel sound) somewhat, in order to place F2 exactly on top of the 1175-Hz second harmonic of the sung note. This sort of tuning is effective only for notes sung at a pitch high enough that a formant frequency can be adjusted to one of the first three voice harmonics, which assures that the rapid 1/n 3/2 fall-off in amplitude has not significantly reduced the prominence of the tuned component in the net sound. The most obvious benefit to be gained from formant tuning is almost trivial: The net loudness of the note is increased to an extent that can be useful if the singer is struggling for audibility against an overpowering accompanist. Subtler, and more significant musically, is a sort of glow and fullness that is imparted to the tone of a formanttuned note. The perceptual reasons for this are not entirely clear, but the fame of many fine sopranos is enhanced by their skillful use of the technique. The third spectral modification that is available to singers (especially for tenors, and for the highest notes of other males) is a rather curious one: A systematic modification of the vocal tract region near the larynx and/or a manner of vowel production that makes the second and third formant frequencies almost coincident can give rise to an extremely strong transmission peak in the neighborhood of 3000 Hz. This peak is referred to as the singer’s formant, regardless of its mode of production. The presence of such a formant considerably increases the net sound output power of the voice, a fact that joins with certain features of the ear’s perception mechanism to produce a large increase in the loudness of all tones sung in this way. It also produces what is usually referred to as tonal brilliance and penetrating character. When used flexibly and tastefully by true artists all three of these vocal resources greatly enhance the beauty and

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254 expressiveness of the musical line. For them to have such effects, they must be used subtly, with close attention to the meaning of the music and the words. As a group, lesser singers do not make use of formant tuning except to increase their loudness. Among this same group of lessthan-satisfactory performers, the other two forms of vocal production are used incessantly, in part to call attention to themselves by mere loudness, and in part as evidence that they have what is called a “trained voice.” It is a curious fact that if any other musical instrument (or a loudspeaker) had a strong and invariable peak around 3000 Hz in its spectral envelope, it would be subject to instant and bitter criticism.

IV. THE WIND INSTRUMENTS The first and most important feature of the family of wind instruments (from the point of view of physics and perception psychology) is the fact that its tones are selfsustaining. The duration of these tones is limited only by the desire of the player and the sufficiency of his air supply. As hinted already in connection with the oscillations in the singer’s larynx, self-sustaining oscillators of necessity give rise to sounds made up of exactly harmonic components. The second distinguishing feature of the wind instruments is that the air column whose natural frequencies control the frequency and wave shape of the primary oscillation is also the device that transmits the resulting sounds to the listening room. It is no longer possible (as with the stringed instruments and with the voice) to describe a vibration source that is essentially independent of the transmission mechanisms that convert its output into the sounds that we hear. A. The Structure of a Wind Instrument Figure 9 will serve to introduce the essential features of a musical wind instrument as it is seen by a physicist.

FIGURE 9 Basic structure of a wind instrument: Air supply, flow controller, air column, and dynamical coupling between the latter two.

Musical Acoustics

To begin with, the player is responsible for providing a supply of compressed air to the instrument’s reed system. This reed system functions as a flow controller that admits puffs of air into an adjustable air column belonging to the instrument itself. The system oscillates because the flow controller is actuated by an acoustical signal generated within the upper end of the air column; this signal is in fact the air column’s response to the excitory flow injected via the reed. The structural features that serve to distinguish between the two major families of wind instruments may be summarized as follows: 1. A woodwind is recognized by the fact that the length of its air column is adjusted by means of a sequence of toneholes that are opened or closed in various combinations to determine the desired notes. The oboe, clarinet, saxophone, and flute are all members of this family. 2. A brass instrument is distinguished by the fact that its air column continues uninterrupted from mouthpiece to bell, the necessary length adjustments being provided either via segments of additional tubing that are added into the bore by means of valves as in the trumpet or by means of a sliding extension of the sort found on the trombone. The sound production process in all wind instruments involves the action of an air flow controller under the influence of acoustical disturbances produced within the air column. This provides another description of the various kinds of wind instrument in terms of the flow controllers that are found on the different instruments. 1. The cane reed is found on the clarinet, oboe, bassoon, and saxophone. It is not (for present purposes) necessary to further distinguish between single and double reeds; they both share the dynamical property that the valve action is such as to decrease the air flow through them when the pressure is increased within the player’s mouth. 2. The lip reed normally used on brass instruments and the cornetto is the second major type of flow controller. Here the valve action is such that the transmitted flow is increased by an increment of the pressure in the player’s mouth. 3. Flutes, recorders, and most organ pipes are kept in oscillation through the action of a third type of controller that may aptly be described as an air reed. Here we find an air jet whose path is deflected into and out of the air column through the action of the velocity of the air as it oscillates up and down the length of the governing air column. It should be emphasized that while the nature of the flow controller itself is very important, it does not usefully

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distinguish the instrumental wind families from each other in the essential features of their oscillatory behavior. B. The Oscillation Process: Time-Domain Version There are two main ways of describing the oscillation processes of a self-sustained instrument. The time-domain description deals with the temporal growth and evolution of a small initial impulse that leaves the flow controller and then is reflected back and forth between the two terminations of the air column—the tone holes and/or bell at its lower-end termination and the flow-controlling reed system at its upper end. When the initial impulse is reflected from the lower termination, it suffers a change of form and an enfeeblement of its vigor (as does each of its successors). The change in form arises because of the acoustical complexity of the termination, and the loss in amplitude occurs because some of the incident wave energy has been lost in the journey and some transmitted into the outside air. At the reed end, another very different form of terminating complexity produces a change in the reflected shape of each returning impulse that travels up to it from the bottom end of the instrument. The size of this regenerated pulse also increases because it receives energy supplied by the incoming compressed air from the player’s lungs. As the tonal start-up process evolves toward the condition of steady oscillation, the wave shape stabilizes into one in which each reflection at the lower end of the air column is modified in such a way as to “undo” the modification that takes place at the reed. C. The Frequency-Domain Description of Wind Instrument Oscillation The time-domain description of the oscillation process is readily susceptible to mathematical analysis and permits detailed calculation of the sound spectra produced by a given reed and air column. However, it is ill-adapted to the task of showing general relations between the mechanical structure of an instrument and its playing behavior, nor will it guide its maker in adjusting it for improved tone and response. Fortunately, a second way of picturing the oscillatory system—the frequency-domain version—can readily deal with such questions and is well suited for our present descriptive purposes. In the frequency-domain analysis, we start by relating the proportions of an instrument to the natural frequencies and dampings of the various vibrational modes of the controlling air column. For present purposes, it suffices to describe the flow controller merely by reiterating that the increment of flow produced by an

increment of control signal is not in proportion to it. In particular, a sinusoidal control signal of frequency f = P gives rise to a pulsating flow that may be analyzed into constituent sinusoids having a harmonic set of frequencies P, 2P, 3P, . . . . Additional components appear when the excitatory signal is itself the superposition of several sinusoids. If these have the frequencies P, Q, R, S, . . . , the resulting flow signal will contain an elaborate collection of components having frequencies that can be described by f = |α P ± β Q ± γ R ± δS|.

(12)

Here α, β, γ , and δ are integers that can take on any values between zero and an upper limit N , which can be as high as 4 or 5. Clearly, hundreds of these frequencies can be present in the flow. (Because of their cross-bred ancestry, they are known as heterodyne frequencies.) It is also clear from their very number and their computational origins that they are distributed over a frequency range extending from zero to more than N times the highest of the stimulus frequencies, and that the amplitude of each of these new components is determined by a combination of the amplitudes of all the original components. This means that the energy associated with each flow component is determined jointly by all members of the controlling set of sinusoids. It is this cross-coupling of stimuli and responses having widely different frequencies that underlies the dynamical behavior of all self-sustained musical instruments, and so governs their musical properties. Consider the behavior of a reed coupled to an air column designed in such a way as to have only a single resonant mode of oscillation. When blown softly, the system will oscillate at a frequency f 0 that is very nearly equal to the modal frequency. Because the strengths of the higher numbered heterodyne components always fall toward zero under conditions of weak excitation, almost the entire efficacy of the flow controller is focused on supplying excitation to the air column at the frequency f 0 of its own maximum response, and the system can oscillate efficiently. However, the system is perfectly stable, because any tendency of the system to “run away” leads to the production of heterodyne components that dissipate energy. These components do not replenish themselves because the air column does not respond strongly to them and so does not “instruct” the reed to reproduce them. If we try to play loudly by blowing harder on such a single-mode instrument, the sinusoid at f 0 hardly changes in strength, but some hissing noise appears and the reed either chokes up entirely (on the cane reed instruments) or blows wide open (on the brasses), with a complete cessation of tone. Similar behavior is observed for a multimode air column if the mode frequencies are randomly placed. Usually the system starts as though only the strongest resonance was present, and the choking-up of the oscillation is more

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abrupt because of the enormously increased number of unproductive heterodyne components that are produced when the blowing pressure is increased. The discussion so far has shown the conditions under which a reed-plus-air-column system can not play; it is time now to describe the requirements for a system that will produce sounds other than weak sinusoids. Suppose that the air column has a shape such that its natural frequencies themselves form a (very nearly) harmonic set, in the manner f n = n f 1 + εn ,

(13)

where the discrepancy εn is a measure of the inharmonicity. The heterodyne frequencies will now form small clumps closely grouped around the exactly harmonic frequencies n f n . As a group, the modes thus appear able then to cooperate with the reed and so to regenerate the flowstimulus energy (distributed in narrow clumps of evergrowing complexity). What actually happens is that the modes quickly lock together to produce a strictly harmonic oscillation with a repetition rate f 0 such that the overall energy production of the system is maximized. Such a mode-locked regime of oscillation turns out to be increasingly quick-starting and stable in all respects if the aircolumn mode frequencies are increasingly well aligned into harmonic relationship. It will also run over a wide range of blowing pressures (and so produce a musically useful range of loudness). D. Musically Useful Air-Column Shapes We have just learned that for a self-sustained multimode oscillation to exist at all, the air-column shape must be such that the natural frequencies of its modes are in very nearly exact harmonic relationships. There are very few possible basic shapes that can meet this criterion. For instruments of the reed woodwind type there are two, for the brass instruments there are two, and for the air-reed (flute) family there is only one. It can be shown that because the canereed and lip-reed instruments have pressure-operated flow controllers, the relevant air column’s natural frequencies are those calculated or measured under the condition that its blowing end be closed off by means of an air-tight plug, while the downstream end is left in open communication with the outside air via the tone holes and bell. On the other hand, for the velocity-operated air reed of the flute family, it is necessary to consider the air-column modal frequencies for the condition when both ends are open. The clarinet family is the sole representative of the cylindrical bore (first) type of possible reed woodwind while the oboe, bassoon, and saxophone belong to the basically conical second group. The trumpet, trombone, and French horn are representatives of the outward-flaring

hyperbolic-shaped air columns suitable for brass instruments, while the fl¨ugelhorn and certain baritone horns are familiar examples of the conical second group. The flutes, on the other hand, are all based on a straight-sided tube, which can have positive, negative, or zero taper. That is, they can either expand or contract conically in going downstream from the blowing end, or can be untapering (i.e., cylindrical). Because of the acoustical complexities of the tone holes at the lower ends of all the woodwinds and of the mouthpiece and reed structures at the upper ends of both brasses and woodwinds, the actual air-column shapes of the various instruments differ in many small ways from their prototypical bases. In all cases, however the differences are such as to align the modal frequencies of the complete air column in the required harmonic relationship. E. Sound Spectra in the Mouthpiece/Reed Cavity It would require a lengthy discussion to explain the ways in which the mouthpiece/reed cavity sound pressure spectrum (which “instructs” the flow-controlling reed) takes its form, but it is not difficult to describe its general nature for the various kinds of wind instrument. It is clear that a lot of the f 0 fundamental component will be present in the mouthpiece spectrum: not only is it directly generated via the lowest frequency air-column resonance but also by difference-heterodyne action between every adjacent pair of the higher harmonics. In similar fashion, there will be a fair amount of heterodyne contribution to the second harmonic component arising from (at the very least) the nonlinear interaction of every alternate pair of harmonics. Analogous contributions are likewise made to the higher tonal components by ever more complex combinations of pairs of peaks in the resonance curve. Details aside, the foregoing considerations are themselves able to imply a tendency for the successive harmonics of the generated tone to be progressively weaker. A closer look brings in information about the manner in which the properties of the air column and reed act to determine the spectrum. For a cane-reed woodwind (such as a clarinet, saxophone, or oboe) or for a lip-reed brass instrument it is possible to show that as long as the playing level is low enough that the reed does not pound completely closed during any part of its swing, the behavior of the pressure amplitudes pn of the various harmonics is well caricatured by n Z n F( f n ) + Mn p1 pn = . (14) p0 [F( f n ) − Z n ] + Dn Here Z n is the height of the flow-induced resonance response curve (input impedance) at the frequency f n of the

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nth harmonic component of the played tone; the factor ( pi / p0 )n gives the influence of the playing level on the overall spectrum; p1 is the amplitude of the fundamental component of the tone; and p0 is a reference pressure defined such that ( p1 / p0 ) = 1 when the reed just closes at one extreme of its cyclic swing. We shall postpone an explanation of the musical recognizability of p0 or its relation to the ordinary loudness specifications that run from pianissimo to fortissimo in the player’s natural vocabulary and will note only that an important measure of the strength of any component is the magnitude of its associated air-column resonance peak. The functions Mn and Dn are slowly changing, and they describe the already-mentioned nonlinear processes of energy exchange between spectral components that assure amplitude stability and well-defined waveforms at all playing levels. The fuction F( f n ) describes the relation between the reed’s primary flow-controlling ability and the frequency of some signal component that may be acting upon it: F( f n ) = ±K r [1 − ( f n / f r )2 ].

(15)

Here K r is a constant, f n is the frequency of the signal component, and f r is the (player-controllable) natural frequency of the reed taken by itself. The plus sign in the defining equation for F applies to the cane reeds (which are pushed open by an increase in mouthpiece pressure), while the minus sign applies to the lip reeds belonging to the ordinary brass instruments (which are pushed closed by an increase in mouthpiece pressure). Since F must be positive if oscillation is to be supported, the lowest few harmonics of the tone of cane reed instruments must have f n ≤ f r , whereas for the brasses f n ≥ f r for all the components. Figure 10 illustrates the state of affairs for the woodwind family of instruments. The measured input impedance curve Z ( f ) is shown for an English horn fingered to give

FIGURE 10 Measured resonance curve for a typical woodwind air column (English horn), along with a curve showing the nature of the primary flow-control function F( f ).

FIGURE 11 Measured resonance curve for a brass instrument air column (trumpet). It is shown along with a typical brass instrument flow-control function F( f ).

the air column for playing its own (written) note C4 . Also shown is a typical flow-control curve F( f ) with the reed frequency set at 1650 Hz. Notice (for future reference) that the air column almost completely lacks resonance peaks above what is known as its cutoff frequency f c , which lies near 1200 Hz. Figure 11 shows in a similar fashion the impedance curve and F( f ) for a trumpet in the case in which the player has set f r 350 Hz in preparation for sounding his written note G4 . This note has its major energy production associated with the cooperation of resonance peaks 3, 6, and 9 (which are in accurately harmonic relationship on a good trumpet). Once again we call attention to the absence of air-column resonances above a cutoff frequency, this time lying near 1500 Hz. The pressure spectrum in the trumpet’s mouthpiece is difficult to guess by eye because it depends on the product of the heights of the Z n peaks and the rising F( f ) curve; however, it is clear that the components having frequencies above the 1500-Hz cutoff are very weak. There are thus two reasons (acting for both woodwinds and brasses) why the spectrum should have a strong first harmonic and progressively weaker second, third, and fourth components, with rapidly disappearing components above that. In addition to the falloff associated with the weakening heterodyne contribution, we have at high frequencies a progressive reduction in the resonance peak heights, and above f c all energy production ceases. So far, resonance curves have been presented for only one of the many air-column configurations that are possible via the fingerings available to a player. It goes almost without saying that when a player desires to sound a lower note, he lengthens the air column of a woodwind by closing a tonehole, or by adding an extra length of tubing to the bore of a brass instrument by means of a valve piston or slide. In this way the frequencies of all the resonance peaks are shifted downward by a factor of 1/1.05946 for every semitone lowering of the desired pitch. An example

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FIGURE 12 Clarinet resonance curves measured for the air columns used in playing the notes C4 to G4 .

of this behavior is presented in Fig. 12, where the resonance curves are presented for the written notes lying between C4 and G4 of a clarinet. The leftmost peak labeled by the numeral 1 is the first-mode resonance for the air column used to produce C4 ; the leftmost peak marked with a 2 similarly indicates the second-mode peak belonging to the same column arrangement, and so on for the higher-numbered peaks. In an exactly parallel way, the rightmost numerals 1, 2, 3, . . . indicate the corresponding resonance peaks for the note G4 . There are two noteworthy features in this set of impedance curves. The first is shared by all wind instrument resonance curves: The cutoff frequency (above which there are no peaks) remains the same for all fingerings. The second feature is characteristic of the clarinet family alone: While peaks 1, 2, 3, . . . are in the strict whole-number relationship demanded by the cooperative nature of wind instrument tone production, the modal frequencies f 1 , f 2 , f 3 , . . . lie in a 1, 3, 5, . . . sequence, with dips in the resonance curves appearing at the positions of the even multiples of the mode-1 frequency. An immediate consequence of this fact is that despite the restorative powers of the alternate-component heterodynes, the even-numbered members of the generated mouth-piece pressure spectrum are weaker than the odd-numbered ones. Because of the fixed cutoff frequency shared by all the air columns used to play notes on a given instrument, and because all the notes share the same mouthpiece and reed structure, it is possible to construct spectrum envelope formulas for the notes of the various classes of instrument. These formulas have a mathematical structure very much like those presented earlier in connection with the bridge-driving forces exerted by the strings of a guitar, harpsichord, or piano. The basic physics that determines them is of course entirely different here, since wind instrument oscillations are active, nonlinear, and self-sustaining. It is fairly obvious from the mathematical nature of the ( pi / p0 )n factor that if the player sounds his notes progres-

Musical Acoustics

sively more softly (without changing the tension of his lips or the setting of the reed), the higher-n components fall away very much more quickly than the lower members of the sequence. In decibel language, we can say that for every decibel that the p1 component is weakened, the level of pn falls by n dB. At the softest possible level, then, we expect the tone within the mouthpiece to have degenerated into a single sinusoid of frequency f 1 . In actual practice, the oboe is essentially unplayable at levels for which ( pi / p0 ) < 1, while the bassoon is almost never played thus. The saxophone can be so used, but normally it too is used in the domain where pi / p0 > 1. Only the clarinet is played at the levels discussed so far, and for it, the customary forte instruction gives a tone for which pi / p0 is little or no larger than unity. This raises the immediate question of what happens to the spectral envelope when an instrument is played at the higher dynamic levels. The answer varies with the instrument, and while it is fairly well known, it would take us too far afield to discuss it here. The solid curve in Fig. 13 presents the general shape of the internal (mouthpiece) spectral envelope for the nonclarinet woodwinds. The corresponding internal spectral envelope for the brasses is shown in the same figure by a closely dotted curve, while the behavior of the odd and even components of the clarinet’s spectrum is shown by the pair of dashed lines. All these curves are calculated on the assumption that the factor ( p1 / p0 )n belongs to the instrument’s normal mezzoforte playing level. The interplay between the direct energy processes at the air-column resonance peaks and the heterodyne transfer of energy between components is made vivid by the following observations. The heights of the various resonance peaks in Fig. 11 show clearly that there can be little direct

FIGURE 13 Internal (mouthpiece or reed-cavity) spectrum envelopes. Nonclarinet woodwinds, solid curve; brasses, dotted curve; clarinet, dashed curves, one for the odd-numbered components of the played note, and one for the evens.

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production of energy by the first two or three components of the played tone C4 (led cooperatively by peaks 2, 4, 6, . . .). For brass instruments, on the other hand, the dotted spectrum envelope curve given in Fig. 13 shows, that the actual strength of the generated fundamental component in the mouthpiece is the strongest of all, while components 2, 3, and 4 are progressively weaker. Clearly, a great deal of heterodyne action is needed to transfer the majority of the high-frequency generated power into the low-frequency part of the spectrum. We shall meet similar behavior in the tone production processes of the violin. Consider next what happens when the trumpet player sounds what he calls the F2 pedal note, whose repetition rate is exactly one-third that of C4 . The harmonic components 3, 6, 9, . . . of the pedal tone lie at air-column peaks 2, 4, 6, . . . and so can act as direct producers of acoustical energy. Meanwhile, the tonal components (1, 2), (4, 5), (7, 8), . . . are away from any resonance peaks, and so exist only because of the heterodyne conversion process. The shape of the measured mouthpiece spectrum envelope for components 3, 6, 9 . . . for F2 is almost identical with that belonging to C4 . The remaining (heterodyned) components have a very similar envelope, but one that is many decibels weaker. F. Transformation of the Mouthpiece Spectrum Envelope into the Room-Average Envelope Attention has already been called to the fact that regardless of the air-column configuration, each musical wind instrument has a cutoff frequency f c above which it lacks resonance response peaks. The same air-column physics that produces a falling away of the resonance peak heights immediately below f c (and so a reduced production of the corresponding mouthpiece pressure components) also plays a significant role in the transformation of the mouthpiece sound into the one enjoyed by listeners in the concert hall. It is a property of the sequence of open tone holes at the lower end of a woodwind that, at low frequencies, sound is almost exclusively radiated from only the first of the holes, while the lower holes come into active play one by one as the signal frequency is raised. Above the cutoff frequency f c , all of the holes are fully active as radiators, and the sound emission not only becomes nearly independent of frequency, but also essentially complete. While the acoustical laws of sound transmission and radiation from a brass instrument bell are quite different from those governing the woodwind tone holes, here too we find very weak emission of low-frequency sounds. The bells’s radiation effectiveness then rises steadily with frequency until it is once more complete for signals above f c .

FIGURE 14 Spectrum transformation function converting the internal spectrum envelope into the room-average one. Nonclarinet woodwinds, solid curve; brasses, dotted line; clarinet, a generally rising dashed line for the odd-numbered tonal components and a horizontal dashed line for the even components. For clarity both clarinet curves have been displaced downward 15 dB from the other curves.

All this explains why there are no resonance peaks above f c , and why those having frequencies just below f c are not very high: the energy loss associated with radiation acts to provide a frequency-dependent damping on the resonance, a damping that becomes complete above f c . In other words, the same phenomena that increase the emission of high-frequency sound from the interior of a wind instrument to the room around it also lead to a progressively falling ability of the instrument to generate high-frequency sounds within itself (this is shown by the dotted curve in Fig. 13). The solid line in Fig. 14 shows the behavior of nonclarinet woodwinds in transferring their internal sounds into the room. Note that the transfer becomes essentially complete for components having frequencies above the cutoff frequency. The dotted and the dashed curves of this figure show the analogous spectrum transformation function for the brass instruments and for the odd and even components of the clarinet spectrum. G. Overall Spectrum Envelopes of Wind Instruments in a Room Figure 15 illustrates the nature of the room-average soundspectrum envelopes of the main reed instrument classes, as calculated from the curves for their mouthpiece spectra and their transformation functions. Once again the solid curve pertains to the nonclarinet woodwinds, the dotted line to the brasses, and the pair of dashed line to the clarinets. The essential correctness of these diagrammatic

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FIGURE 15 External spectrum envelopes. Nonclarinet woodwinds, solid line; brasses, dotted line; clarinet, a pair of dashed lines: one for the odd and one for the even components of the tone.

representations is clearly shown in Fig. 16a–c, which presents the room-average spectrum envelopes measured for the oboe (C4 to C6 ), trumpet (E3 to F#5 ), and clarinet (E3 to C6 ). These were obtained using room-averaging techniques quite similar to those used to obtain the spectrum envelope of a piano (see Fig. 6). Exactly as is the case for the purely heterodyned evenharmonic components of the clarinet tone, the (1, 2), (4, 5) (7, 8), . . . components of the trumpet F2 pedal note are weakly generated but strongly radiated. As a result, in the measured room-averaged spectrum of this note these components essentially fit the envelope belonging to the directly generated 3, 6, 9 . . . components (being only about 3 dB weaker). Study of a large variety of instruments (including soprano, alto, and bass representatives of each family) shows that the basic curves change very little from one example to another. In all cases the general trend at high frequencies is for the envelope to fall away as 1/ f 3 , with a breakpoint close to 1500 Hz for the soprano instruments (oboe, trumpet, clarinet). For the alto instruments (English horn, alto saxophone, alto clarinet), the breakpoint is around 1000 Hz, paralleling the fact that their playing range lies a musical fifth below the soprano instruments. For the next lower range of instruments (trombone, tenor saxophone, bass clarinet), the break lies around 1500/2 = 750 Hz, while the bassoon (whose playing range lies at one-third the frequency of the oboe) has its break near 1500/3 = 500 Hz. Almost nothing has been said so far about the flute family of woodwinds beyond a description of its associated flow controller and the basic nature of its usable air column. Despite the flute’s apparent mechanical simplicity, it is in many ways dynamically more sub-

FIGURE 16 Measured room-average spectra (a) oboe; (b) trumpet; (c) clarinet, for clarity the even-component data have been displaced downward by 20 dB.

tle than the other woodwinds and somewhat less well understood. Because the flute’s flow controller is velocity operated rather than pressure operated, a somewhat roundabout proof shows that it is the peaks in the admittance curve (flow/pressure) rather than those of its reciprocal, the impedance curve, that cooperate with and instruct the

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flute’s air reed. The room-average spectrum of a flute may be expected a priori to be quite different from that of the other woodwinds for two reasons. First, the oscillation dynamics of the primary energy production mechanism are drastically different. Second, there are two sources of sound radiation into the room: one is the familiar one associated with the tone hole lattice ( f c 2000 Hz), while the other is the oscillatory flow at the embouchure hole across which the player blows. It is somewhat as though the room were supplied simultaneously with the internal and the external spectrum of a normal woodwind! The room-average spectrum envelope for a flute has a curiously simple form that is well represented by E( f ) = e− f / fa .

(16)

Here f a is near 800 Hz for the ordinary concert flute, close to 530 Hz for the alto, and 1600 Hz for the piccolo (as might be expected for such systematically scaled instruments).

V. THE BOWED STRING INSTRUMENTS The bowed string family of musical instruments shares many features with the instruments that have been discussed so far. For this reason the present section can serve both as a review and elaboration of the earlier material and as an introduction to another major class of instruments. As indicated in Fig. 17, the violin, like the guitar, has a boxlike structure with a rigid neck and a set of strings whose vibrating length can be controlled by the player’s

fingers. To a first approximation then, the two instrumental types have similar dynamical processes that convert the driving-force spectrum (transferred from the strings via the bridge) to the sound spectrum that is measured in the concert hall. On the other hand, the excitation mechanism of the self-sustained string oscillation of violin family instruments proves to be essentially the same as that which generates sound in the woodwind and brass instruments. A. The Excitatory Functioning of the Bow The mechanism used by the violin family to keep a string in vibration is easily sketched: The frictional force exerted at the contact point between bow and string is smaller when there is a fast slipping of the bow hair over the string and larger when the sliding rate is slower. Thus, during those parts of its oscillation cycle when the contact point of the string chances to be swinging in the same direction as that of the rapidly moving bow (so that the slipping velocity is small), there is a strong frictional force urging the string forward in the direction of its motion. During the other half of the cycle, the string is moving in a direction opposite to that of the bow. Under these conditions the slipping velocity is large, making the frictional force quite small. Notice that this frictional drag is still exerted in the direction of the (forward) bow motion, and it therefore acts to somewhat retard the (backward) vibrational motion of the string. In short, during part of each vibratory cycle a strong force acts to augment the oscillation, and during the remainder of the cycle there is a weaker depleting action. The oscillation builds up until the bow’s energy augmentation process exactly offsets all forms of energy dissipation that may take place at the bowing point and everywhere else. It is clear that the vigor of the oscillation is ultimately limited by the fact that during its “forward” swing the string velocity at the bowing point can equal, but not exceed, the velocity of the bow, otherwise the friction would reverse itself and pull the string velocity back down to match that of the bow. In the earliest version of the formal theory of the bowed string excitation process, it was assumed that the string and bow hair stick and move together during the forward motion of the string, while during the return swing the friction is negligible. Such a theory is remarkably successful in predicting many of the most obvious features of the oscillation but is powerles to give a dependable account of the actual driving-force spectrum envelope as it might appear at the bridge. B. The Frequency-Domain Formulation Reapplied

FIGURE 17 Structural parts of the violin, and their names.

In the framework of the resonance-curve/excitationcontroller theory of self-sustained oscillators, it is easy

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erties that together play the role of the reed resonance frequency f r in limiting the production of energy at high frequencies. C. Spectrum Systematics

FIGURE 18 Violin bowing-point admittance (velocity/force) curve f for F the way along the string from the bridge to the player’s left-hand fingers.

to see that the bow/string interaction serves as a velocityoperated force controller; this is in contrast to the pressureoperated flow controllers found in the woodwinds and brasses. To be consistent then, in our theory we replac the pressure-response curves of flow-driven air columns (as measured in the mouthpiece) by the velocity-response curve of a force-driven string (measured at the bowing point). Figure 18 shows a velocity-response (driving-point admittance) curve calculated for a violin D string fingered to play F4 . In this example, it has been assumed that the bow crosses the string at a point one-tenth of the string length away from the bridge (about 25 mm). Notice the remarkable similarity of this resonance curve to the one shown in Fig. 11 for the air column of a trumpet. For the trumpet, the increasing height of the resonance peaks and the initial falling away beyond their maximum is determined chiefly by the design of the mouthpiece cup (a cavity) and back bore (a constriction). The ultimate disappearance of the peaks is, as already explained, due to the radiation loss to the room suffered by the air column. In the case of a violin however, it is the distance of the bowing point from the bridge that determines the frequency region (around 1750 Hz in the example) where the admittance peaks are tallest. The subsequent weakening of the higher frequency peaks is controlled jointly by the rising influence of frictional and radiation damping and by some bow physics that is related to that which produces “notches” in the plucked string’s E a ( f ) function [see Eq. (5)]. There exists a force-control function representing the bow/string interaction that is analogous to the wind instrument F( f ) function [see Eq. (15)]. While this analog to F is not shown in Fig. 18, it may be taken to be quite similar to the one shown for woodwinds in Fig. 10. There is unfortunately no simple description for the bow prop-

The small height of the lowest few resonance peaks in Fig. 18 shows that the major part of the total energy production comes via the tall response peaks that lie at higher frequencies. Not surprisingly, the nonlinear nature of the bow/string stick-slip force results in heterodyne effects that lead to a bowing point spectrum very similar to that of a trumpet mouthpiece pressure spectrum. The systematic transfer of high-frequency energy into lowfrequency vibrational components is as effective for the violin as for the trumpet, so that (as in the trumpet) the driving-point spectrum ends up with the first component strongest and higher ones becoming progressively weaker. The simplest stick-slip theory of the bow/string oscillation gives a reasonably accurate initial picture of the spectrum envelope E v ( f ) for the string velocity at the bowing point: E v ( f ) = [sin(π f / f β )]/(π f / f β ).

(17)

Here f β is defined in terms of the point of application of the bow on the string in exactly the same manner as f a was defined via the plucking point in Eq. (5). The shape of E v ( f ) is shown exactly by the curve for E a ( f ) in Fig. 4. The actual velocity spectrum envelope of a bowed string is quite similar to that implied by Eq. (17), except that (a) the spectral notches do not go all the way to zero, and (b) at high frequencies the effects of dampings, etc., reduce the spectral amplitudes very considerably. So far the discussion of the spectral properties of the string velocity at the bowing point serves as a means for clarifying the fundamental energy production processes of the bowed string. What actually leads to the radiation of sound in the room, however, is the force exerted by the string on the bridge and the consequent emission of sound by the vibrating violin body. The spectrum transformation function relating the “internal” (bowing-point) spectrum to the “external” (room-average) spectrum must thus be considered in two parts. The first part relates the bowingpoint velocity to the bridge force, while the second part converts this force spectrum into the one measured in the room. The bowing-point velocity/bridge-force spectrum transformation function TvF ( f ) turns out, according to the simplest theory, to be 1/ sin(π f / f β ). The most striking consequence of this fact is that it exactly cancels out the notches in the simple formula for E words, at those frequencies for which there is supposedly no velocity signal at all at the bowing point, the

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FIGURE 19 Simplified transformation function connecting the violin bridge drive-force spectrum envelope with the room-average spectrum envelope.

transformation function is so enormously effective that it apparently “creates” a drive force at the bridge! Curiously enough, although the stick-slip versions of both the velocity-spectrum and force-transformation function have been common knowledge for over a century, serious attempts to resolve the paradox have been made only recently. Much of the necessary information is currently available, but it has not been fitted together into a coherent whole. For present purposes, it will suffice to remark that the general trend of the force spectrum envelope of the bridge drive force is roughly constant at low frequencies, and it falls away fairly quickly for frequencies above a breakpoint that is determined in part by the distance of the bowing point from the bridge. Figure 19 outlines the main behavior of the bridgeforce-to-room transformation function. This must of course be evaluated for a drive force whose direction lies roughly parallel to the plane of the violin’s top plate and tangent to the curved paths of the string anchorages on the bridge (see Fig. 17). To a first approximation the trend line is horizontal, in agreement with the general assertions made about the force-to-room transformation in connection with the plucked and struck string instruments. However, there is a very rapid weakening in the radiating ability of the body in the low-frequency region below a strong peak near 260 Hz. This radiative transformation peak and associated loss of low-frequency efficacy (whose cognate on the guitar falls at 85 Hz and below) is associated with a joint vibrational mode of the elastic-walled body cavity and the Helmholtz resonator formed by this air cavity and the apertures in it provided by the f holes (see Fig. 17 for details and terminology). There is a second radiativity peak just below 500 Hz on a violin. This one is associated with the resonant response of a body mode in which the top plate vibrates (chiefly on the bass-bar side) in a sort of twisting motion having a quasi-fulcrum at the position of the sound post. The back plate is also in vigorous motion, being coupled to the top plate by the sound post.

(In the guitar, this radiativity peak is relatively unimportant. While it too has a body mode in which the bridge rocks strongly, the lack of a bass bar and sound post makes for a vibrational symmetry that gives a very small radiation of sound. Furthermore, the lowness of a guitar bridge means that this mode is only very weakly driven by a vibrating string.) The violin has two more strong radiativity peaks. One is found near 3000 Hz, and the other near 6000 Hz, beyond which the radiativity falls as 1/ f 2 or faster. The first of these peaks is determined by a bridge-plus-body mode in which the predominant motion is a rocking of the top part of the bridge about its waist. The second peak belongs to a mode in which there is a sort of bouncing motion (normal to the plane of the top plate) of the upper part of the bridge on the bent “legs” connecting its waist to its feet. Analogs to these peaks do not exist on the guitar. There are many additional resonance-related peaks and dips in the transformation function besides those described previously and indicated in Fig. 19. For a violin these are spaced (on thge) only about 35 Hz apart, and they are proportionally closer on larger members of the family. We have dealt explicitly here only with those of major acoustical and musical importance whose positions along the frequency axis are well established for each family of bowed instruments. It is an important part of a fiddle maker’s skill to place these selected peaks in their correct frequency positions. He must also properly proportion the interactions of the various parts of an instrument (e.g., by suitably dimensioning and placing the soundpost). D. The Violin’s Measured Room-Average Spectrum and Its Implications Figure 20 shows the room-average spectra of all chromatic notes between a violin’s bottom G3 and the A#4 that lies

FIGURE 20 Measured room-average spectra of violin notes of the chromatic scale between G3 and A#4 .

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264 somewhat more than an octave above. One feature calls instant attention to itself in the spectrum envelope implied by these data: This measured envelope is similar to that of soprano wind instruments and also the piano in that the envelope is roughly uninform at low frequencies and falls away at the rate of about 1/ f 3 at high frequencies. The similarity is closest between the violin and the wind instruments, because the breakpoint between the low- and high-frequency regions lies in all cases near 1500 Hz! Comparison of the envelope shape from Fig. 20 with the transformation function of Fig. 19 causes an initial feeling of surprise. Figure 19 shows a strong radiativity peak around 3000 Hz and another one near 450 Hz. Neither one of these shows up in the spectral envelope of the radiated sound. While no one seems to have worked out the details yet, the basic explanation has already been met among the wind instruments. Efficacy of radiation generally means a lowering of the resonance peaks that operate the primary excitation controller (reed or bow). As a result, there is less energy produced at the radiativity peaks than elsewhere, thus offsetting the increased emission of the enegy that is produced. However, the fact that all parts of the generated spectrum are strongly interconnected by heterodyne action makes it impossible to make detailed predictions of what will happen from general principles alone. More detailed comparison of the radiativity curve and the spectrum envelope shows further evidence that their relationship is not simple: The strong radiativity peak at 260 Hz differs from the others in that it does appear to stregthen the radiated tonal components that coincide with it. Furthermore, the rapid decrease in radiativity below 250 Hz is reflected in a rapid loss of power in the corresponding components of the violin’s tone. We also see (Fig. 20) hints of a strong emission of sound in the regions around 400 Hz and clear indications of even stronger emission around 550 Hz, despite the fact that there are no prominent resonances to be found at these frequencies in the violin’s modal collection. Other hints of systematic fine structure in the observed spectrum are tantalizingly visible in the present data, hints that strengthen and weaken surprisingly when the data are displayed in different ways. As remarked earlier, much more remains to be done to elucidate the detailed origins of the violin’s spectral envelope, as is the case with many other features of its acoustical behavior. Meanwhile, clues as to what sorts of phenomena are to be expected may be looked for in the spectral relations among the components of the brass instrument pedal tones. The musical interpretability of the bowed string sound is made yet more difficult by the fact that the human auditory system is readily able to recognize the tonal influences of all the resonances displayed in Fig. 19, along with several

Musical Acoustics

other less well-marked or invariable ones belonging to this complex system. This is despite the fact that they are not visible in the measured spectrum. Physicists must always remember in cases like this that while the ear does not analyze sounds in the ways most readily chosen by laboratory scientists, it must in the final analysis act upon whatever pieces of physical data offer themselves, many of which can be at least listed for the scientist’s serious consideration, even though they may be difficult for him to measure. Claims are made from time to time that “the secret of Stradivari” has been discovered. Such claims arise in part because of a sometimes unrecognized conflict between the remarkably effective but subliminal routines of musical listening and the highly intellectualized activities of a laboratory researcher, and in part because of everyone’s romantic desire to create a better instrument. Each discover proclaims some “truth” that he has found. If the “scientific” discoverer is often less guarded in his claims than in his craftsman or musician counterpart, it is because he often knows only one aspect of the primary oscillation problem or of the vibration/radiation aspects of the net sound production process. Moreover, he is not subjected to the discipline of successful practice in the real-world fields of instrument making or musical performance, where partial success is often equivalent to failure!

VI. THE APTNESS OF INSTRUMENTAL SOUNDS IN ROOMS The diverse musical instruments that we have studied share a remarkable number of properties. Let us list some of these and attempt to relate them to the ways in which they provide useful data to the auditory processor. All of the standard orchestral instruments generate sounds that are (note by note) made up of groups of sinusoids whose frequencies are whole-number multiples of some repetition rate. We might ask, at least for the plucked or struck string instruments, whether it is an accident that this should be so, since it comes about (and only approximately at that) via the choice of thin, elongated, uniform wires as the primary vibrating object. Why should such vibratiors take precedence over vibrating plates or membranes, or even over wires of nonuniform cross section? In the case of the wind and bowed instruments (including the singing voice), self-sustained oscillations are possible only under conditions where the resulting spectrum is of the strictly harmonic type. Here, then, the traditional instrument maker has no choice: It is impossible for him to provide inharmonic sound sources. For the moment, the question remains partly open as to why the harmonic-type instruments are dominant. We

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have however been given a strong hint by the observation that the auditory processor treats such aggregations of spectral components in a special way—it perceives each such grouping as an individual, compact tone. It can also distinguish several such tones at the same time, and even recognize well-marked relationships between them (such as the octave or the musical fifth). The problem remains, however, as to whether the recognizability of individual harmonic groups can survive the room’s transmission path. Regardless of the complexity of transmission of amplitude or phase, the frequencies of the components radiated from an instrument arrive unaltered at the listener’s ear. It is an easily verified fact that the pitch of a harmonic complex is almost rigorously established by the harmonic pattern of its component frequencies (which determine in a mathematically unique way its repetition rate), rather than by the amplitudes of these components. In other words, as long as even a few of the partials of each instrument’s tone detectably arrive at the listener’s ear, the pitch (music’s most important attribute) is well established. Great emphasis has been laid throughout this article on the fact that each instrument is constructed in such a way that all of its notes share a common spectrum envelope. It has also been pointed out that for the keyboard instruments at least, it is a matter of considerable difficulty to achieve such an envelope. The structure of the brass instruments, on the other hand, almost guarantees a well-defined envelope; and even though it is possible to build woodwind instruments that lack an envelope, many things become easier if one is arranged for them. Finally, the guitarist and the violinist were found to have instruments that inherently tend to produce a spectral envelope, but one whose breakpoint and high-frequency slope can be influenced by the player via his choice of plucking or bowing point. What are the perceptual reasons for these instruments to have evolved to produce a well-defined spectral envelope? This question can be answered at least in part by the facts of radiation acoustics and musical perception in rooms. It takes only a very limited collection of auditory samples of transmission-distorted data from an instrument for the listener to form an impression of the breakpoint and highfrequency slope, and so (even for a single note in a musical passage) to permit him to decide which of the instruments before him has produced it. A question that is less easy to decipher is why the instruments seem to have very nearly the same envelopes. A partial explanation is to be found in the observation that the bulk of the available acoustical energy in a tone is allocated to the first four or five partials, which puts these energy packages into a set of independent critical bands,

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265 thus maximizing the net loudness. A less obvious explanation is that the high-frequency rolloff may be a way of preventing excessive tonal roughness of the sort that comes about when too many harmonics find themselves in the same critical band. For example, harmonics 7, 8, and 9 will contribute some roughness because they all lie within the 25% bandwidth of significant mutual interaction. This explanation is not adequate, however. It turns out that critical-band-induced roughnesses of this sort are not strongly active, whereas (for the soprano instruments at least) an insufficient rolloff rate tends to produce a quite unacceptable tone color. While much remains to be done to fully clarify questions of the sort raised in the preceding paragraph, we can find hints as to where the answers may be sought. Psychoacousticians have shown the existence of a tonal attribute known as sharpness (which is what its German originator calls it in English, but edginess, or harshness, would be a better term). This attribute may be calculated dependably from the spectral envelope of a sound, its power level, and the frequency range in which its components are found. We may summarize the calculation method thus. The total loudness N perceived by the listener is given by the integral of a loudness density function n(z) that is a perceptual cognate of the product of the physicist’s spectral envelope E( f ) and the level of the acoustical signals received by the listener:

N = n(z) dz. (18) It also takes into account varying amounts of interaction between spectral components that are not widely separated in frequency. Here the variable z is the transformation of the ordinary frequency axis into a perceptual coordinate such that increments of one unit of z correspond to the width of one critical band. The sharpness, S, is then calculated as an overlap integral of n(z) and a sharpness weighting function g(x), given by

const S= n(z)g(z) dz. (19) ln(N /20 + 1) The sharpness weighting function g(z) is small at low values of z, and it rises rapidly for values of z that correspond to frequencies above 1000 Hz. It should probably not be taken as accidental that the most important contributions to the sharpness integral arise above 1000 Hz, in the region where the primary receptors are beginning to randomly misfire relative to their mechanical stimuli. Figure 21a shows the function n(z) calculated for a harmonic tone having a fundamental frequency f 0 of 200 Hz, a spectral envelope with breakpoint frequency 1500 Hz,

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FIGURE 21 (a) Loudness density function n (z ) calculated for harmonic tones based on 200 Hz, and having a spectral envelope with 1500 Hz break frequency and 1/ f 3 high frequency rolloff. Also shown is the sharpness function g (z ). (b) Similar curves, for a tone having the same spectral envelope but with an 800 Hz fundamental frequency.

and 1/ f 3 high-frequency rolloff. Also plotted is the sharpness function g(z). Qualitatively speaking, we may understand the net sharpness as being related to the area of the shaded region lying between the z axis and the two curves. Figure 21b similarly illustrates the case of a tone belonging to the same spectral envelope and fundamental frequency 800 Hz. We find by direct electronic synthesis (or by sounding a specially made laboratory wind instrument) that an unpleasant harshness is attributed to tones having a raised breakpoint or reduced rate of high-frequency falloff relative to the one we described previously. Furthermore, the tone is generally pronounced to lack piquancy and to be somewhat dull and muffled when the envelope has a lowered breakpoint or steepened falloff. It is not difficult to expect from the general nature of Fig. 21 that instruments built to play in the alto, tenor, and bass ranges will be very little influenced by the sharpness phenomenon, freeing them (in accordance with observation and experiment) from the constraints that appear to hold for the soprano instruments. On the other hand, treble instruments (having breakpoint frequencies of 2000 Hz or above) are found to have a great deal of sharpness regardless of the high-frequency envelope slope. In the orchestra these instruments are rarely used, and then only for special purposes. Quantitative study of the relation of sharpness,

Musical Acoustics

loudness, and spectrum envelope for instruments in various pitch ranges is in its infancy, but already a considerable amount of consistency is apparent. One flaw is to be noticed in the apparently coherent picture that has been sketched above: A most important musical instrument, the piano, seems not to provide the otherwise universal spectral envelope. Here the break frequency turns out to lie near 800 Hz rather than 1500 Hz. However, it is a simple matter to electronically refilter a set of recorded piano tones to move the breakpoint to 1500 Hz without changing anything else. Listening tests on such a modified (normalized) sound show at once that the tone does not so much become harsh as that the pounding of the hammers becomes obtrusive. Readers who have listened to the actual sound of the early-nineteenth-century pianoforte will have heard a mild form of the same kind of hammer heard a mild form of the same kind of hammer clang. (Do not count on a recording to inform you, because many recordings have been so tampered with that nothing can be learned from them.) Apparently, pianos have evolved away from an original design based on the harpsichord (where the continuous-spectrum impulsive hammer sound is not produced, and the spectrum envelope is essentially of the familiar type) to one in which one tonal virtue is sacrificed to avoid a serious flaw. The perceptual symbiosis that exists between a musical instrument and the concert hall in which it is played can be illustrated further by considering the details of the primary radiation processes whose signals are compiled in making a room-average spectrum. For every instrument family, the spectrum envelope of the sounds radiated in some particular direction (in reflection-free surroundings) differs significantly from that radiated in some other direction. In many cases smoothly varying discrepancies between tow such envelopes can amount to as much as 40 dB. We also find that the signals from microphones placed at various positions close to an instrument have peculiar and highly irregular spectra that have no easily recognizable relationship with more thoughtfully obtained spectra. We have already seen what happens when many individual samples of the more distant version of the sound are combined into a room average: a reliable picture of it emerges. Our hearing mechanism and the surrounding hall join to accomplish just this task. The concert halls in which we normally listen to music offer many reflections, which means that data concerning all aspects (literally!) of the emitted sound are made available to our auditory processors. The chief reason (from the point of view of music) why the room-average spectrum is important is that the ear actually can assemble the equivalent information by means of early-reflection processing and/or multiple-sample averaging via use of two-ear, moving-listener, moving-source

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data collected over the span of several seconds. Almost none of this multiplicity of data is available for processing in reflection-free surroundings, which provides a significant hint as to why serious performers and listeners alike tend to dislike open-air music: It subjects them to auditory deprivation. Despite the noise-reduction and harmonic distortionfree techniques of digital recording and the use of compact disks, many modern attempts at musical recording are frequently quite unsatisfactory. Recording engineers sometimes misuse their technical resources in an attempt to remove the confusion from the recorded sound by the use of reflection-free studios, partitions between instruments, and the “mixing down” and “filter enhancement” of signals from numerous highly directional microphones (each placed very close to its own instrument). These actions (which are increasingly resented by performing classical musicians) produce distortion of the primary musical data when they do not eliminate them altogether. On the other hand, recordings of the sort made in the 1950s and 1960s using two or three microphones properly placed in a good concert hall have never been surpassed, at least in the informed judgement of those listeners to classical music whose experience has been gained largely by actual concert-going. In short, for music we need and enjoy all of the data from our instruments, instruments that have evolved over several centuries to communicate their voices effectively in the environment of a concert hall.

SEE ALSO THE FOLLOWING ARTICLES ACOUSTICAL MEASUREMENT • ACOUSTICS, LINEAR • SIGNAL PROCESSING, ACOUSTIC • SIGNAL PROCESSING, GENERAL • ULTRASONICS AND ACOUSTICS

BIBLIOGRAPHY Benade, A. H. (1976). “Fundamentals of Musical Acoustics,” Oxford Univ. Press, London and New York. Benade, A. H. (1985). From instrument to ear in a room: Direct, or via recording. J. Audio Eng. Soc. 33, 218–233. Benade, A. H., and Kouzoupis, S. N. (1988). The clarinet spectrum: Theory and experiment. J. Acoust. Soc. Am. 83, 292–304. Benade, A. H., and Larson, C. O. (1985). Requirements and techniques for measuring the musical spectrum of a clarinet. J. Acoust. Soc. Am. 78, 1475–1497. Benade, A. H., and Lutgen, S. J. (1988). The saxophone spectrum. J. Acoust. Soc. Am. 83, 1900–1907. Causs´e, R., Kergomard, J., and Lurton, X. (1984). Input impedance of brass musical instruments—Comparison between experiment and numerical models. J. Acoust. Soc. Am. 75, 241–254. Cremer, L. (1984). “The Physics of Violins” (J. S. Allen, translator). MIT Press, Cambridge, Massachusetts. De Poli, A. (1991). “Representations of Musical Signals,” MIT Press, Cambridge, MA. Griffith, N., and Todd, P. M. (1999). Musical Networks: Parallel Distributed Perception and Performance, MIT Press, Cambridge, MA. Hall, D. E. (1986). Piano string excitation, I. J. Acoust. Soc. Am. 79, 141–147. Hall, D. E. (1987). Piano string excitation: The question of missing modes. J. Acoust. Soc. Am. 82, 1913–1918. Hall, D. E. (1988). Piano string excitation: Spectra for real hammers and strings. J. Acoust. Soc. Am. 83, 1627–1638. Hutchins, C. M. (1983). A history of violin research. J. Acoust. Soc. Am. 73, 1421–1440. Marshall, K. D. (1985). Modal analysis of a violin. J. Acoust. Soc. Am. 77, 695–709. McIntyre, M. E., Schumacher, R. T., and Woodhouse, J. (1983). On the oscillations of musical instruments. J. Acoust. Soc. Am. 74, 1345– 1375. Pierce, J. R. (1992). “Science of Musical Sound, Rev. Ed.” Holt, New York. Rossing, T. D., and Fletcher, N. H. (1998). “The Physics of Musical Instruments, 2nd Ed,” Springer-Verlag, New York. Sadie, S. (ed.) (1980). “The New Grove Dictionary of Music and Musicians, Macmillan, London, England. Weinreich, G., and Kergomard, J. (1996). “Mechanics of Musical Instruments,” Springer-Verlag, New York.

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Nonlinear Dynamics F. C. Moon Cornell University I. II. III. IV. V. VI. VII.

Introduction The Undamped Pendulum Nonlinear Resonance Self-Excited Oscillations: Limit Cycles Stability and Bifurcations Flows and Maps: Poincare´ Sections One-Dimensional Maps, Bifurcations, and Chaos

GLOSSARY Bifurcation Denotes the change in the type of long-time dynamical motion when some parameter or set of parameters is varied (e.g., as when a rod under a compressive load buckles—one equilibrium state changes to two stable equilibrium states). Chaotic motion Denotes a type of motion that is sensitive to changes in initial conditions. A motion for which trajectories starting from slightly different initial conditions diverge exponentially. A motion with positive Lyapunov exponent. Controlling chaos The ability to use the parameter sensitivity of chaotic attractors to stabilize any unstable, periodic orbit in a strange attractor. Duffing’s equation Second-order differential equation with a cubic nonlinearity and harmonic forcing x¨ + c x˙ + bx + ax 3 = f 0 cos ωt. Feigenbaum number Property of a dynamical system related to the period-doubling sequence. The ratio of successive differences between period-doubling bifurcation parameters approaches the number 4.669. . . .

VIII. IX. X. XI.

Fractals and Chaotic Vibrations Fractal Dimension Lyapunov Exponents and Chaotic Dynamics The Lorenz Equations: A Model for Convection Dynamics XII. Spatiotemporal Dynamics: Solitons XIII. Controlling Chaos XIV. Conclusion

This property and the Feigenbaum number have been discovered in many physical systems in the prechaotic regime. Fractal dimension Fractal dimension is a quantitative property of a set of points in an n-dimensional space that measures the extent to which the points fill a subspace as the number of points becomes very large. Hopf bifurcation Emergence of a limit cycle oscillation from an equilibrium state as some system parameter is varied. Limit cycle In engineering literature, a periodic motion that arises from a self-excited or autonomous system as in aeroelastic flutter or electrical oscillations. In dynamical systems literature, it also includes forced periodic motions (see also Hopf bifurcation). Linear operator Denotes a mathematical operation (e.g., differentiation, multiplication by a constant) in which the action on the sum of two functions is the sum of the action of the operation on each function, similar to the principle of superposition. Lorenz equations Set of three first-order autonomous differential equations that exhibit chaotic solutions.

523

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524 This set of equations is one of the principal paradigms for chaotic dynamics. Lyapunov exponents Numbers that measure the exponential attraction or separation in time of two adjacent trajectories in phase space with different initial conditions. A positive Lyapanov exponent indicates a chaotic motion in a dynamical system with bounded trajectories. (Sometimes spelled Liapunov). Nonlinearity Property of an input–output system or mathematical operation for which the output is not linearly proportional to the input. For example, y = cx n (n = 1), or y = x d x/dt, or y = c(d x/dt)2 . Period doubling Sequence of periodic vibrations in which the period doubles as some parameter in the problem is varied. In the classic model, these frequency halving bifurcations occur at smaller and smaller intervals of the control parameter. Beyond a critical accumulation parameter value, chaotic vibrations occur. This scenario to chaos has been observed in may physical systems but is not the only route to chaos (see Feigenbaum number). Phase space In mechanics, an abstract mathematical space with coordinates that are generalized coordinates and generalized momenta. In dynamical systems, governed by a set of first-order evolution equations; the coordinates are the state variables or components of the state vector. Poincar´e section (map) Sequence of points in phase space generated by the penetration of a continuous evolution trajectory through a generalized surface or plane in the space. For a periodically forced second-order nonlinear oscillator, a Poincar´e map can be obtained by stroboscopically observing the position and velocity at a particular phase of the forcing function. Quasi-periodic Vibration motion consisting of two or more incommensurate frequencies. Saddle point In the geometric theory of ordinary differential equations, an equilibrium point with real eigenvalues with at least one positive and one negative eigenvalue. Solitons Nonlinear wave-like solutions that can occur in a chain of coupled nonlinear oscillators. Strange attractor Attracting set in phase space on which chaotic orbits move; an attractor that is not an equilibrium point or a limit cycle, or a quasi-periodic attractor. An attractor in phase space with fractal dimension. Van der Pol equation Second-order differential equation with linear restoring force and nonlinear damping, which exhibits a limit cycle behavior. The classic mathematical paradigm for self-excited oscillations.

Nonlinear Dynamics

DYNAMICS is the mathematical study of the way systems change in time. The models that measure this change include differential equations and difference equations, as well as symbol dynamics. The subject involves techniques for deriving mathematical models as well as the development of methods for finding solutions to the equations of motion. Such techniques involve both analytic methods, such as perturbation techniques, and numerical methods.

I. INTRODUCTION In the classical physical sciences, such as mechanics or electromagnetics, the methods to derive mathematical models are classified as dynamics, advanced dynamics, Lagrangian mechanics, or Hamiltonian mechanics. In this review, we discuss neither techniques for deriving equations nor the specific solution methods. Instead, we describe some of the phenomena that characterize how nonlinear systems change in time, such as nonlinear resonance, limit cycles, coupled motions, and chaotic dynamics. An important class of problems in this subject consists of those problems for which energy is conserved. Systems in which all the active forces can be derived from a force potential are sometimes called conservative. A branch of dynamics that deals with such systems is called Hamiltonian mechanics. The qualifier nonlinear implies that the forces (or voltages, etc.) that produce change in physical problems are not linearly proportional to the variables that describe the state of the system, such as position and velocity in mechanical systems (or charges and currents in electrical systems). Mathematically, the term linear refers to the action of certain mathematical operators L, such as are used in multiplication by a constant, taking a derivative, or an indefinite integral. A linear operator is one that can be distributed among a sum of functions without interaction, that is, L[a f (z) + bg(t)] = a L[ f (t)] + bL[g(t)]. Nonlinear operators, such as those that square or cube a function, do not obey this property. Dynamical systems that have nonlinear mathematical models behave very differently from ones that have linear models. In the following, we describe some of the unique features of nonlinear dynamical systems. Another distinction is whether the motion is bounded or not. Thus, for a mass on an elastic spring, the restoring forces act to constrain the motion, whereas in the case of a rocket, the distance from some fixed reference can grow

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without bound. In this review, we discuss only bounded problems typically involving vibrating phenomena. Mathematical models in dynamical systems generally take one of three forms: differential equations (or flows), difference equation (called maps), and symbol dynamic equations. Although the physical laws from which the models are derived are often second-order differential equations, the theory of nonlinear dynamics is best studied by rewriting these equations in the form of first-order equations. For example, Newton’s law of conservation of momentum for a unit mass with one degree of freedom is usually written as a second-order differential equation:

with respect to some reference. We can also label states with colors, such as red (R), yellow (Y), or blue (B). The evolution of a system is then expressed in the form

x¨ = F(x, x˙ , t).

An equilibrium solution might be LLLL. . . , whereas a periodic motion has the form RRLR-RLRRL. . . , or LRLRLR. . . . For a given physical system, one can use all three types of models.

(1)

In nonlinear dynamics one often rewrites this in the form x˙ = y,

y˙ = F(x, y, t).

(2)

The motion is then viewed in phase space with vector components (x, y) corresponding to position and velocity. (In advanced dynamics, phase space is sometimes defined in terms of generalized position coordinates and generalized momentum coordinates.) For more complex problems, one studies dynamical models with differential equations in an N -dimensional phase space with N components {x1 (t), x2 (t), . . . , xi (t), . . . , xn (t)}, where the equation of motion takes the form x˙ = F(x, t) x1 = x1

(3) x2 ≡ x˙ = y

using Eq. (2). Difference equations or maps are also used in nonlinear dynamics and are sometimes derived or related to continuous flows in phase space by observing the motion or state of the system at discrete times, that is, xn ≡ x(tn ). In distinction to Eq. (3), the subscript refers to different times or different events in the history of the system. First- and second-order maps have the following forms: xn+1 = f (xn )

an+1 = h(an ).

(5)

Here, however, h(an ) may not be an explicit algebraic expression but a rule that may incorporate inequalities. For example, suppose that x(tn ) is the position of some particle at time tn . Then one could have an+1 = L

if xn < 0

an+1 = R

if xn ≥ 0.

II. THE UNDAMPED PENDULUM A. Free Vibrations A classical paradigm in nonlinear dynamics is the circular motion of a mass under the force of gravity (Fig. 1). A balance equation between the gravitational torque and the rate of change of angular momentum yields the nonlinear ordinary differential equation θ¨ + (g/L) sin θ = 0,

(6)

where g is the gravitational constant and L the length of the pendulum. A standard approach to understanding the dynamics of this system is to analyze the stability of motion of the linearized equations about equilibrium positions.

(4a)

or xn+1 = f (xn , yn ) (4b) yn+1 = g(xn , yn ) Examples are given later in this article. Another model is obtained when the variable X n is restricted to a finite set of values, say (0, 1, 2). In this case, there is no need to think in terms of numbers because one can make a correspondence between (0, 1, 2) and any set of symbols such as (a1 , a2 , a3 ) = (L, C, R) or (R, Y, B). Thus, in some systems we may be interested only in whether the particle is to the left (L), right (R), or in the center (C)

FIGURE 1 (a) The classical pendulum under the force of gravity. (b) Phase plane sketch of motions of the pendulum showing solutions near the origin (center) and solution near θ = ±π (saddle point).

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Using the form of Eq. (2) or (3) one has θ˙ = ,

˙ = −ω02 sin θ

(7)

where ω02 ≡ g /L . Equilibrium points of Eq. (3) are defined by F(xe ) = 0. In the example of the pendulum, x = (θ, ) and θe = ±m π, e = 0. Because the torque is periodic in θ , we can restrict θ to −π < θ ≤ π . In a linearized analysis, we define a perturbation variable ϕ = θ − θe so that sin θ is replaced by ±ϕ, depending on whether θe = 0 or π. About θe = 0, one finds that the linearized motion is oscillatory (i.e., θ (t) = A sin(ω0 t + B), where A and B are determined from initial conditions). The motion in the phase plane (θ, ) takes the form of an elliptic orbit with clockwise rotation (Fig. 1). Such motion is known as a center. The motion about θe = ±π can be shown to be an unstable equilibrium point, known as a saddle, with trajectories that are also shown in Figure 1. (One should note that the saddles at θe = ±π are physically the same.) Using the conservation of linearized system qualitatively represent those of the nonlinear system. These local qualitative pictures of the nonlinear phase plane motion can often be pieced together to form a global picture in Figure 1. The trajectory separating the inner orbits (libration) from the outer or rotary orbit is known as a separatrix. For small 1 motions the period of oscillation is 2π/ω0 or 2π (L /g) 2 . However, the period of libration increases with increasing amplitude and approaches infinity as the orbit approaches the separatrix. The dependence of the free oscillation period or frequency on the amplitude is characteristic of nonlinear systems.

III. NONLINEAR RESONANCE A classical model for nonlinear effects in elastic mechanical systems is a mass on a spring with nonlinear stiffness. This model is represented by the differential equation (known as Duffing’s equation) x¨ + 2γ x˙ + αx + βx 3 = f (t).

FIGURE 2 Phase plane motions for an oscillator with a nonlinear restoring force [Duffing’s equation (8)]. (a) Hard spring problem, α, β > 0. (b) Soft spring problem, α > 0, β < 0. (c) Two-well potential problem, α < 0, β > 0.

with amplitude (i.e., the period increases as in the pendulum) and the motion is unbounded outside the separatrix. For α < 0 and β > 0, there are three equilibria: two stable and one unstable (a saddle), as in Figure 2c. Such motions represent the dynamics of a particle in a two-well potential. Forced vibration of the damped system [Eq. (8)] represents an important class of problems in engineering. If the input force is oscillatory (i.e., f = f 0 cos ωt), the response of the system x(t) can exhibit periodic, subharmonic, or chaotic functions of time. A periodic output has the same frequency as the input, whereas a subharmonic motion includes motions of multiple periods of the input frequency 2π/ω: x(t) ∼ A cos[(n /m)ωt + B].

(9)

where n and m are integers. When the motion is periodic, the classic phenomenon of hysteretic nonlinear resonance occurs as in Figure 3. The output of the system has a different response for increasing versus decreasing forcing frequency in the vicinity of the lin√ ear natural frequency α. Also, the dotted curves in

(8)

This equation can also be used to describe certain nonlinear electrical circuits. When the linear damping term and external forcing are zero (i.e., γ = f = 0), the system is conservative and the nonlinear dynamics in the (x , x˙ = y) phase plane can exhibit a number of different patterns of behavior, depending on the signs of α and β. When α, β > 0. The system has a single equilibrium point, a center, where the frequency of oscillation increases with amplitude. For α > 0 and β < 0, the frequency decreases

FIGURE 3 Nonlinear resonance for the hard spring problem: response amplitude versus driving frequency.

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Figure 3 represent unstable motions that result in jumps in the response as frequency is increased or decreased. However, the output motion may not always be periodic, as Figure 3 implies, and may change to a subharmonic or chaotic motion depending on the parameters (γ , α, β, f 0 , ω). The multiplicity of possible solutions is not often pointed out in more classical treatments of nonlinear oscillations. Chaotic vibrations are discussed in the following.

IV. SELF-EXCITED OSCILLATIONS: LIMIT CYCLES Dynamic systems with both sources and sinks for energy comprise an important class of nonlinear phenomena. These include mechanical systems with relative motion between parts, fluid flow around solid objects, biochemical and chemical reactions, and circuits with negative resistance (created by active electronic devices such as operational amplifiers or feedback circuits), as shown in Figure 4. The source of energy may create an unstable spiral equilibrium point while the source of dissipation may limit the oscillation motion to a steady motion or closed orbit in the phase space, as shown in Figure 5. The classical model for this limit cycle phenomena is the so-called Van der Pol equation given by x¨ − γ x˙ (1 − βx 2 ) + ω02 x = f (t).

(10)

When f (t) = 0, the system is called autonomous, and the origin is the only equilibrium point in the phase plane, that is, (x , x˙ = γ ) = (0, 0). This point can be shown to

FIGURE 5 Phase plane portrait for a limit cycle oscillation. (a) Small γ [Eq. (10)]. (b) Relaxation oscillations, large γ [Eq. (10)].

be an unstable spiral when γ > 0. When γ is small, the limiting orbit in a set of normalized coordinates (β = ω02 = 1) is a circle of radius 2. As shown in Figure 5a, solutions inside the circle spiral out and onto the limit cycle while those outside spiral inward and onto the limit orbit. The frequency of the resulting periodic motion for β = ω = 1 is one radian per nondimensional time unit. When γ is larger (e.g., γ ∼ 10), the motion takes a special form known as a relaxation oscillation, as shown in Figure 5b. It is periodic but is not sinusoidal, that is, it includes higher harmonics. The system exhibits sudden periodic shifts in motion. If periodic forcing is added to a self-excited system such as Eq. (10) (i.e., f (t) = f 0 cos ω1 t), then more complicated motions can occur. Note that when a nonlinear system is forced, superposition of free and forced motion is not valid. Two important phenomena in forced, selfexcited systems are mentioned here: entrained oscillation and combination or quasi-periodic oscillations. When the driving frequency is close to the limit cycle frequency, the output x(t) may become entrained at the driving frequency. For larger differences between driving and limit cycle frequencies, the output may be a combination of the two frequencies in the form x = A1 cos ω0 t + A2 cos ω1 t.

(11)

When ω0 and ω1 are incommensurate (i.e., ω0 /ω1 is an irrational number), the motion is said to be quasi-periodic, or almost periodic. Phase plane orbits of Eq. (11) are not closed when ω0 and ω1 are incommensurate.

V. STABILITY AND BIFURCATIONS FIGURE 4 Sources of self-excited oscillations. (a) Dry friction between a mass and a moving belt. (b) Aeroelastic forces on a vibrating airfoil. (c) Negative resistance in an active circuit element.

The existence of equilibria or steady periodic solutions is not sufficient to determine if a system will actually behave

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FIGURE 6 Bifurcation diagrams. (a) Pitchfork bifurcation, the transition from one to two stable equilibrium positions. (b) Hopf bifurcation, the transition from stable spiral to limit cycle oscillation.

that way. The stability of these solutions must also be checked. As parameters are changed, a stable motion can become unstable and new solutions may appear. The study of the changes in the dynamic behavior of systems as parameters are varied is the subject of bifurcation theory. Values of the parameters at which the qualitative or topological nature of the motion changes are known as critical or bifurcation values. An example of a simple bifurcation is the equation for motion in a two-well potential [Eq. (8)]. Suppose we view α as a control parameter. Then in Eq. (8), the topology of the phase space flow depends critically on whether α < 0 or α > 0, as shown in Figure 2a and c, for zero damping and forcing. Thus α = 0 is known as the critical or bifurcation value. A standard bifurcation diagram plots the values of the equilibrium solution as a function of α (Fig. 6a) and is known as a pitchfork bifurcation. When damping is present, the diagram is still valid. In this case, one stable spiral is transformed into two stable spirals and a saddle as α decreases from positive to negative values. A bifurcation for the emergence of a limit cycle in a physical system is shown in Figure 6b. This is sometimes known as a Hopf bifurcation. Here, the equilibrium point changes from a stable spiral or focus to an unstable spiral that limits onto a periodic orbit.

Nonlinear Dynamics

Eq. (4b), known in modern parlance as a map. The study of maps obtained from Poincar´e sections of flows is based on the theory that certain topological features of the motion in time are preserved in the discrete time dynamics of maps. To illustrate how a Poincar´e section is obtained, imagine that a system of three first-order differential equations of the form of Eq. (3) has solutions that can be represented by continuous trajectories in the Cartesian space (x, y, z), where x1 (t) = x , x2 (t) = y, and x3 (t) = z (Fig. 7). If the solutions are bounded, then the solution curve is contained within some finite volume in this space. We then choose some surface through which the orbits of the motion pierce. If a coordinate system is set up on this two-dimensional surface with coordinates (ξ, η), then the position of the (n + 1)th orbit penetration (ξn+1 , ηn+1 ) is a function of the nth orbit penetration through the solution of the original set of differential equations. A period-one orbit means that ξn+1 = ξn , ηn+1 = ηn . A period-m orbit is defined such that ξn+m = ξn , ηn+m = ηn . Such orbits in the map correspond to periodic and subharmonic motions in the original continuous motion. On the other hand, if the sequence of points in the map seem to lie on a closed curve in the Poincar´e surface, the motion is termed quasi-periodic and corresponds to the sum of two time-periodic functions of different incommensurate frequencies, as in Eq. (9). Motions whose Poincar´e maps have either a finite set of points (periodic or subharmonic motion) or a closed curve of points are known as classical attractors. A motion with a set of Poincar´e points that is not a classical attractor and that has certain fractal properties is known as a strange attractor. Strange attractor motions are related to chaotic motions and are defined as follows.

VI. FLOWS AND MAPS: POINCARE´ SECTIONS An old technique for analyzing solutions to differential equations, developed by Poincar´e around the turn of the 20th century, has now assumed greater importance in the modern study of dynamical systems. The Poincar´e section is a method to transform a continuous dynamical process in time into a set of difference equations of the form of

FIGURE 7 Poincare´ section. Construction of a difference equation model (map) from a continuous dynamic model.

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FIGURE 8 Experimental Poincare´ map for chaotic motions of a particle in a two-well potential with periodic forcing and damping [Eq. (10)].

In certain periodically forced problems, there is a natural way to obtain a Poincar´e section or map. Consider the damped mass with a nonlinear spring and time periodic force x˙ = y (12) y˙ = −γ y − F(x) + f 0 cos ωt . A Poincar´e section can be obtained in this system by defining a third variable z = ωt, where 0 ≤ z < 2π, so that the system is converted to a autonomous system of equations using z˙ = ω. We also connect the planes defined by z = 0 and z = 2π so that the motion takes place in a toroidal volume (Fig. 7). The Poincar´e map is obtained by observing (x , y) at a particular phase of the forcing function. This represents a stroboscopic picture of the motion. Experimentally, one can perform the phase plane trace at a particular phase z = z 0 on a storage oscilloscope (Fig. 8).

FIGURE 9 Graphical solution to a first-order difference equation. The example shown is the parabolic or logistic map.

This is a nonlinear difference equation that has equilibrium points x = 0, 1. One can examine the stability of nonlinear maps in the same way as for flows by linearizing the righthand side of Eq. (14) about the equilibrium or fixed points. The orbits of a solution to one-dimensional maps can be solved graphically by reference to Figure 9, in which the (n + 1)th value is reflected about the identity orbit (straight line). An orbit consists of a sequence of points {xn } that can exhibit transient, periodic, or chaotic behavior, as shown in Figure 10. These properties of solutions can be represented by the bifurcation diagram in Figure 11, where λ is a control parameter. As λ is varied, periodic solutions change character to subharmonic orbits of twice the period of the

VII. ONE-DIMENSIONAL MAPS, BIFURCATIONS, AND CHAOS A simple linear difference equation has the form xn +1 = λxn .

(13)

This equation can be solved explicitly to obtain xn = Aλn , as the reader can check. The solution is stable (i.e., |xn | → 0 as n → ∞) if |λ| < 1 and unstable if |λ| > 1. The linear equation [Eq. (13)] is often used as a model for population growth in chemistry and biology. A more realistic model, which accounts for a limitation of resources in a given species population, is the so-called logistic equation xn +1 = λxn (1 − xn ).

(14)

FIGURE 10 Possible solutions to the quadratic or parabolic map [Eq. (14)]. (a) Steady or period-one motion. (b) Period-two and period-four motions. (c) Chaotic motions.

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FIGURE 11 Period-doubling bifurcation diagram for a first-order nonlinear difference equation [Eq. (14)].

table. When the system is conservative, ε = 0, the Eqs. (16) are essentially a Poincar´e map of the continuous motion obtained by observing the time of phase and velocity of impact when the ball hits the table. The first equation is a momentum balance relation before and after impact, whereas the second equation is found by integrating the free flight motion of the ball between impacts. These equations have also been used to model an electron in an electromagnetic field. This map is sometimes known as the standard map. In this problem, one can compare the difference between chaos in a conservative system (ε = 0) and chaos in a system for which there is dissipation. When ε = 0 (Fig. 12c), the map shows there are periodic orbits (fixed points in the map) and quasi-periodic motions, as evidenced by the closed map orbits. Islands of chaos exist for initial conditions starting near the saddle points of the

previous orbit. The bifurcation values of λ, {λn } accumulate at a critical value at which non-periodic orbits appear. The sequences of values of λ at which period-doubling occurs has been shown by Feigenbaum to satisfy the relation lim[(λn − λn −1 )/(λn +1 − λn )] → 4.6692 . . . .

(15)

These results have assumed great importance in the study of dynamic models in classical physics for two reasons. First, in many experiments, Poincar´e sections of the dynamics often (but not always) reveal the qualities of a onedimensional map. Second, Feigenbaum and others have shown that this period-doubling phenomenon is not only a prelude to chaos but is also universal when the one- dimensional map x → f (x) has at least one maximum or hump. Universal means that no matter what physical variable is controlled, it shows the same scaling properties as Eq. (15). This has been confirmed by many experiments in physics in solid- and fluid-state problems. However, when the underlying dynamics reveals a two-dimensional map, then the period-doubling route to chaos may not be unique. A two-dimensional map that can be calculated directly from the principles of the dynamics of a ball bouncing on a vibrating platform under the force of gravity is shown in Figure 12a and b (Guckenheimer and Holmes have given a derivation). The difference equations are given by xn +1 = (1 − ε)xn + κ sin yn

(16)

and yn +1 = yn + xn +1 , where xn is the velocity before impact, yn the time of impact normalized by the frequency of the vibrating table (i.e., y = ωt, modulo 2π ), and κ proportional to the amplitude of the vibrating table in Figure 12a. The parameter ε is proportional to the energy lost at each impact with the

FIGURE 12 Dynamics of the second-order standard maps [Eq. (16)]. (a) Physical model of a ball bouncing in a vibrating table. (b) Iterations of the map with dissipation ε = 0.4, κ ∼ 6 [Eq. (16)], showing fractal structure characteristic of strange attractors. (c) Iteration of the map for many different initial conditions showing regular and stochastic motions (no dissipation ε = 0, κ ∼ 1).

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map. In the dissipative case (Fig. 12b), the chaotic orbit shows a characteristic fractal structure but requires a much larger force amplitude κ. However, the forcing amplitude κ needed to obtain chaotic motion in the dissipative case is much larger than that required for chaos in the conservative case ε = 0.

VIII. FRACTALS AND CHAOTIC VIBRATIONS One of the remarkable discoveries in nonlinear dynamics in recent years is the existence of randomlike solutions to deterministic differential equations and maps. Stochastic motions in nondissipative or conservative systems were known around the time of Poincar´e. However, the discovery of such motions in problems with damping or dissipation was a surprise to many theorists and has led to experimental observations of chaotic phenomena in many areas of classical physics. Technically, a chaotic motion is one in which the solution is extremely sensitive to initial conditions, so much so that trajectories in phase space starting from neighboring initial conditions diverge exponentially from one another on the average. In a flow, this divergence of trajectories can take place only in a three-dimensional phase space. In a map, however, one can have chaotic behavior in a first-order nonlinear difference equation, as described in the logistic map example of Eq. (14). In the dissipative standard map for the bouncing ball [Eq. (16)], chaotic solutions exist when impact energy is lost (ε > 0) for κ ∼ 6. A typical long iterative map of such a solution is shown in Figure 12b. The iterates appear to occur randomly along the sets of parallel curves. If this solution is looked at with a finer grid, this parallel structure continues to appear. The occurrence of self-similar structure at finer and finer scales in this set of points is called fractal. Fractal structure in the Poincar´e map is typical of chaotic attractors.

IX. FRACTAL DIMENSION A quantitative measure of the fractal property of strange attractors is the fractal dimension. This quantity is a measure of the degree to which a set of points covers some integer n-dimensional subspace of phase space. There are many definitions of this measure. An elementary definition is called the capacity dimension. One considers a large number of points in an n-dimensional space and tries to cover these points with a set of N hypercubes of size ε. If the points were uniformly distributed along a linear curve, the number of points required to cover the set would vary as N ∼ 1/ε (Fig. 13). If the points were distributed on a two-

FIGURE 13 Definition of fractal dimension of a set of points in terms of the number of covering cubes N(ε).

dimensional surface, then N ∼ ε −2 . When the points are not uniformly distributed, one might find that N ∼ ε −d . If this behavior continues as ε → 0 and the number of points increases, then we define the capacity dimension as d = lim [log N (ε)/ log(1/ε)]. ε→0

(17)

Other definitions exist that attempt to measure fractal properties, such as the information dimension and the correlation dimension. In the latter, one chooses a sphere or hypersphere of size ε and counts the number of points of the set in the sphere. When this is repeated for every point in the set, the sum is called the correlation function C(ε). If the set is fractal, then C ∼ ε d c or dc = lim(log C / log ε), as ε → 0. It has been found that dc ≤ d, where d is the capacity dimension. As an example, consider the chaotic dynamics of a particle in a two-well potential. The equation of motion is given by 1 x¨ + γ x˙ − x(1 − x 2 ) = f 0 cos ωt . (18) 2 This is a version of the Duffing equation [Eq. (8)]. It has been shown by Holmes of Cornell University that for chaos to exist, the amplitude of the forcing function must be greater a critical value, that is, √ √ f 0 > [γ 2 cosh(π ω/ 2)]/3π ω. (19) The regions of chaos in the parameter plane ( f 0 , ω) are shown in Figure 14, as determined by numerical experiments. The above criterion gives a good lower bound. Equation (18) is a model for the vibrations of a buckled beam. Experiments with chaotic vibrations of a buckled beam show good agreement with this criterion [Eq. (19)]. The fractal dimension of the Poincar´e map of the chaotic motions of Eq. (18) depends on the damping γ . When γ is large (γ ∼ 0.5), the dimension dc ∼ 1.1, and when γ is small (γ ∼ 0.01), dc ∼ 1.9. The fractal dimension of the

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(t) = 0 2λt .

FIGURE 14 Regions of chaotic and regular motion for the twowell potential problem [Eq. (18)].

set of points in Figure 15a is close to dc = 1.5. This means that the points do not cover the two-dimensional plane. This is evidenced by the voidlike structure of this chaotic map as it is viewed at finer and finer scales.

X. LYAPUNOV EXPONENTS AND CHAOTIC DYNAMICS A measure of the sensitivity of dynamical motion to changes in initial conditions is the Lyapunov exponent. Thus, if two trajectories start close to one another, the distance between the two orbits (t) increases exponentially in time for small times, that is,

(20)

When λ is averaged over many points along the chaotic trajectory, λ is called the Lyapunov exponent. In a chaotic motion λ > 0, whereas for regular motion λ ≤ 0. In a two-dimensional map, one imagines a small circle of initial conditions about some point on the attractor (Fig. 16). If the radius of this circle is ε, then after several iterations of the map (say n), the circle may be mapped into an ellipse with principal axis of dimension (2εµn1 , 2εµn2 ); µ1 , µ2 are called Lyapunov numbers, and the exponents are found from λi = log µi . If the system is dissipative, the area decreases after each iteration of the map (i.e., λ1 + λ2 < 0). If λ1 > λ2 , then in a chaotic motion λ1 > 0 and λ2 < 0. Thus, regions of phase space are stretched in one direction and contracted in another direction (Fig. 16). Iterating this stretching and contraction through the map eventually produces the fractal structure seen in Figure 15. Of importance to the understanding of such motions are concepts such as horseshoe maps and Cantor sets. Space does not permit a discussion of these ideas, but they may be found in several of the modern references on the subject. The Lyapunov exponents (λ1 , λ2 ) can be used to calculate another measure of fractal dimension called the Lyapunov dimension. For points in a plane, such as a twodimensional Poincar´e map, this measure is given by d L = 1 − (λ1 /λ2 ) = 1 + [log µ1 / log(1/µ2 )].

(21)

where λ1 > 0, λ2 < 0. This relation can be extended to higher dimensional maps.

XI. THE LORENZ EQUATIONS: A MODEL FOR CONVECTION DYNAMICS As a final illustration of the new concepts in nonlinear dynamics, we consider a set of three equations proposed by Lorenz of MIT in 1963 as a crude model for thermal gradient induced fluid convection under the force of gravity. Such motions occur in oceans, atmosphere, home heating, and many engineering devices. In this model the variable x represents the amplitude of the fluid velocity stream function and y and z measure the time history of the temperature distribution in the fluid (a derivation has been

FIGURE 15 Poincare´ maps of chaotic motions of the two well potential problem with fractal dimensions of (a) d = 1.5 and (b) d = 1.1 for two different damping ratios.

FIGURE 16 Sketch of sensitivity of motion to initial conditions as measured by Lyapunov exponents.

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FIGURE 18 Trajectories of chaotic solution to the Lorenz equations for thermo-fluid convection. FIGURE 17 Sketch of local motion near the three equilibria for the Lorenz equations [Eq. (22)]. (a) As a model for thermo-fluid convection (b).

given by Lichtenberg and Lieberman). In nondimensional form, these equations become x˙ = σ (y − x) y˙ = r x − y − x z

(22)

z˙ = x y − bz These equations would be linear were it not for the two terms x z and x y in the second and third equations. For those familiar with fluid mechanics, σ is a Prandtl number and r is similar to a Rayleigh number. The parameter b is a geometric factor. If (x˙ , y˙ , z˙ ) ≡ v were to represent a velocity vector in phase space, then the divergence ∇ · v = −(σ + b + 1) < 0. This implies that a volume of initial conditions decreases as the motion moves in time. For σ = 10, b = 83 (a favorite set of parameters for experts in the subject), there are three equilibia for r > 1 with the origin an unstable saddle (Fig. 17a). When r > ∼25, the other two equilibria become unstable spirals (Fig. 17a) and a complex chaotic trajectory moves between regions near all three equilibria as shown in Figure 18. An appropriate Poincar´e map of this flow shows it to be nearly a one-dimensional map with period-doubling behavior as r is increased close to the critical value of r = ∼25. The fractal dimension of this attractor has been found to be 2.06, which indicates that the motion lies close to a two-dimensional surface.

problems have been modeled by using a linear chain of discrete oscillators with nearest neighbor coupling as shown in Figure 19. Of course, the limit of such models is continuum physics, such as acoustics or fluid mechanics, for which one uses partial differential equations in space and time. When the coupling between the oscillators is nonlinear, then a phenomena known as soliton wave dynamics can occur. Solitons are pulse-like waves that can propagate along the linear chain. Left- and right-moving waves can intersect and emerge as left and right waves without distortion (see Fig. 20). One example is the so-called Toda-lattice, in which the inter particle force is assumed to vary exponentially. (See Toda, 1989.) x¨ j + F(x j − x j −1 ) + F(x j − x j +1 ) = 0 F(x) = β[(exp(bx) − 1)]. A classic problem of dynamics on a finite particle chain was posed by the famous physicist Eurico Fermi and two colleagues at Los Alamos in the 1950s. They had expected that if energy were placed in one spatial mode, then the nonlinear coupling would disperse the waves into the N classic vibration modes. However to their surprise, most of the energy stayed in the first spatial mode and eventually, after a finite time, all the energy returned to the original initial energy mode. This phenomenon is known as the recurrence problem in finite degree of freedom systems and is known as the Fermi–Pasta–Ulam problem. (See Gapanov-Grekhov and Rubinovich, 1992.)

XII. SPATIOTEMPORAL DYNAMICS: SOLITONS Nonlinear dynamics models can be used to study spatially extended systems such as acoustic waves, electrical transmission problems, plasma waves, and so forth. These

FIGURE 19 Chain of coupled nonlinear oscillators.

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used as a source of controlled periodic motions. This idea originated in the work of three University of Maryland researchers in 1990; E. Ott, C. Gregogi, and J. Yorke (OGY). (See Kapitaniak, 1996 for a review of this subject.) This example of nonlinear thinking has resulted in the design of systems to control chaotic modulation of losers, systems to control heart arythymias, and circuits to encrypt and decode information. The idea is based on several premises.

FIGURE 20 Soliton wave dynamics along a chain of nonlinear oscillator.

However, it is also now recognized that a different set of initial conditions could result in spatiotemporal stochasticity or spatiotemporal chaos. (See e.g., Moon, 1992.)

XIII. CONTROLLING CHAOS One of the most inventive ideas to come out of modern nonlinear dynamics is the control of chaos. It is based on the concept that a system with a chaotic attractor may be

1. The nonlinear system has a chaotic or strange attractor. 2. That the strange attractor is robust in some variation of a control parameter. 3. That there exists an infinite number of unstable periodic orbits in the strange attractor. 4. There exists a control law that will locally stabilize the unstable motion in the vicinity of the saddle points of the orbit map. There are many variations of this OGY method, some based on the analysis of the underlying nonlinear map and some based on experimental techniques. An example of controlled dynamics is shown in Figure 21 from the author’s laboratory. The vertical scale shows the Poincar´e

FIGURE 21 Poincare´ map amplitude versus time for chaotic and controlled chaos of a period-four orbit for a two-well potential nonlinear oscillator.

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map sampled output of a vibrating nonlinear elastic beam. The horizontal scale shows the time. The figure shows first a chaotic signal, then when control in initiated, a period four orbit appears, then chaos returns when the control is switched off. The control consists of a pulsed magnetic force on the beam. The control force is only active for a fraction of the period. The method uses the inherent parameter sensitivity of the underlying chaotic attractor to achieve control.

XIV. CONCLUSION Dynamics is the oldest branch of physics. Yet, 300 years after the publication of Newton’s Principia (1687), new discoveries are still emerging. The ideas of Newton, Euler, Lagrange, Hamilton, and Poincar´e, once conceived in the context of orbital mechanics of the planets, have now transcended all areas of physics and even biology. Just as the new science of dynamics in the seventeenth century gave birth to a new mathematics (namely, the calculus), so have the recent discoveries in chaotic dynamics ushered in modern concepts in geometry and topology (such as fractals), which the 21st century novitiate in dynamics must master to grasp the subject fully. The bibliography lists only a small sample of the literature in dynamics. However, the author hopes that they will give the interested reader some place to start the exciting journey into the field of nonlinear dynamics.

SEE ALSO THE FOLLOWING ARTICLES CHAOS • DYNAMICS OF ELEMENTARY CHEMICAL REACTIONS • FLUID DYNAMICS • FRACTALS • MATHEMATICAL MODELING • MECHANICS, CLASSICAL • NONLINEAR PROGRAMMING • VIBRATION, MECHANICAL

BIBLIOGRAPHY Abraham, R. H., and Shaw, C. D. (1983). “Dynamics: The Geometry of Behavior,” Parts 1–3. Aerial Press, Santa Cruz, CA. Gaponov-Grekhov, A. V., and Rabinovich, M. I. (1992). “Nonlinearities in Action,” Springer-Verlag, New York. Guckenheimer, J., and Holmes, P. J. (1983). “Nonlinear Oscillations; Dynamical Systems and Bifurcations of Vector Fields.” SpringerVerlag, New York. Jackson, A. (1989). “Perspectives in Nonlinear Dynamics,” Vol. 1. Cambridge Univ. Press, New York. Kapitaniak, T. (1996). “Controlling Chaos,” Academic Press, London. Lichtenberg, A. J., and Liebermann, M. A. (1983). “Regular and Stochastic Motion,” Springer-Verlag, New York. Minorsky, N. (1962). “Nonlinear Oscillations,” Van Nostrand, Princeton, NJ. Moon, F. C. (1992). “Chaotic and Fractal Dynamics,” Wiley, New York. Nayfeh, A. H., and Balachandran, B. (1993). “Nonlinear Dynamics,” Wiley, New York. Schuster, H. G. (1984). “Deterministic Chaos,” Physik-Verlag GmbH, Weinheim, Federal Republic of Germany. Strogatz, S. H. (1994). “Nonlinear Dynamics and Chaos,” Addison Wesley, Reading, MA. Toda, M. (1989). “Theory of Nonlinear Lattices,” Springer-Verlag, Berlin.

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I. II. III. IV. V.

Polarization States Polarizers Retarders Mathematical Representations Polarimetry

GLOSSARY Birefringence The property of optically anisotropic materials, such as crystals, of having the phase velocity of propagation dependent on the direction of propagation and polarization. Numerically, birefringence is the refractive index difference between eigenpolarizations. Diattenuation The property of having optical transmittance depend on the incident polarization state. In diattenuators, the eigenpolarizations will have principal transmittances Tmax and Tmin , and diattenuation is quantified as (Tmax − Tmin )/(Tmax + Tmin ). Diattenuation may occur during propagation when absorption coefficients depend on polarization (also called dichroism) or at interfaces. Eigenpolarization A polarization state that propagates unchanged through optically anisotropic materials. Eigenpolarizations are orthogonal in homogeneous polarization elements. Jones calculus A mathematical treatment for describing fully polarized light. Light is represented by 2 × 1 complex Jones vectors and polarization components as 2 × 2 complex Jones matrices. Mueller calculus A mathematical treatment for describ-

ing completely, partially, or unpolarized light. Light is represented by the 4 × 1 real Stokes vector and polarization components as 4 × 4 real Mueller matrices. Polarimetry The measurement of the polarization state of light or the polarization properties (retardance, diattenuation, and depolarization) of materials. Polarized light A light wave whose electric field vector traces a generally elliptical path. Linear and circular polarizations are special cases of elliptical polarization. In general, light is partially polarized, and is a mixture of polarized light and unpolarized light. Polarizer A device with diattenuation approaching 1 that transmits one unique polarization state regardless of incident polarization. Retardance The optical phase shift between two eigenpolarizations. Unpolarized light Light of finite spectral width whose instantaneous polarization randomly varies over all states during the detection time. Not strictly a polarization state of light.

THE POLARIZATION state is one of the fundamental characteristics (along with intensity, wavelength,

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and coherence) required to describe light. The earliest recorded observation of polarization effects was reported by Bartholinus, who observed double refraction in calcite in 1669. Huygens demonstrated the concept of polarization by passing light through two calcite crystals in 1690. Today, the measurement, manipulation, and control of polarization plays an important role in optical sciences.

I. POLARIZATION STATES Light can be represented as an electromagnetic wave that satisfies Maxwell’s equations. A transverse electromagnetic wave has electric and magnetic field components that are orthogonal to the direction of propagation. As the wave propagates, the strengths of these transverse fields oscillate in space and time, and the polarization state is defined by the direction of the electric field vector E. For our discussion, we will use a right-handed Cartesian coordinate system with orthogonal unit vectors xˆ , yˆ , and zˆ . A monochromatic plane wave E(z , t) traveling in vacuum along the zˆ direction with time t can be written as E(z , t) = Re{ˆx E x exp[i(ωt − k0 z + φx )] + yˆ E y exp[i(ωt − k0 z + φ y )]}

(1a)

or E(z , t) = xˆ E x cos(ωt − k0 z + φx ) + yˆ E y cos(ωt − k0 z + φ y ),

(1b)

where ω is the angular optical frequency and E x and E y are the electric field amplitudes along the xˆ and yˆ axes, respectively. The free-space wavenumber is k0 = 2π/λ for wavelength λ, and φx and φ y are absolute phases. The difference in phase between the two component fields is then φ = φ y − φx . The direction of E and the polarization of the wave depend on the field amplitudes E x and E y and the phases φx and φ y .

FIGURE 1 Two linear polarized waves. The electric field vector of x-polarized light oscillates in the xz plane. The shaded wave is y-polarized light in the yz plane.

vector sum of these orthogonal fields yields a wave polarized at 45◦ from the x axis. If E x = −E y (or if E x = E y and φ = π ), the light is linearly polarized at −45◦ . For in-phase component fields (φ = 0), the linear polarization is oriented at an angle α = tan−1 (E y /E x ) with respect to the x axis. In general, linear polarization states are often defined by an orientation angle, though descriptive terms such as x- or y-polarized, or vertical or horizontal, may be used. However, when a wave is incident upon a boundary two specific linearly polarized states are defined. The plane of incidence (Fig. 2) is the plane containing the incident ray and the boundary normal. The linear polarization in the plane of incidence is called p-polarization and the field component perpendicular to the plane is s-polarized. This convention is used with the Fresnel equations (Section II.A) to determine the transmittance, reflectance, and phase shift when light encounters a boundary.

A. Linear Polarization A wave is linearly polarized if an observer looking along the propagation axis sees the tip of the oscillating electric field vector confined to a straight line. Figure 1 depicts the wave propagation for two different linear polarizations when Eq. (1b) is plotted for φx = φ y = 0. In Fig. 1, E y = 0 and light is linearly polarized along the x axis; in the other example, light is polarized along the y axis when E x = 0. For a field represented by Eqs. (1a) and (1b), light will be linearly polarized whenever φ = m π, where m is an integer; the direction of linear polarization depends on the magnitudes of E x and E y . For example, if E x = E y , the

FIGURE 2 Light waves at a boundary. The plane of incidence coincides with the plane of the page. Incident, reflected, and transmitted p-polarized waves are in the plane of incidence. The corresponding s -polarizations (not shown) would be perpendicular to the plane of incidence.

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are combined to create circular polarization. Adding equal amounts of right- and left-circularly polarized light will yield a linearly polarized state. For example, 1 rcp E 2

+ 12 Elcp = xˆ E 0 cos(ωt − k0 z).

(4)

In contrast, adding equal quantities of left- and rightcircular polarization that are out of phase [by adding an additional π phase to both component fields in Eq. (2)] yields − 12 Ercp + 12 Elcp = yˆ E 0 sin(ωt − k0 z).

(5)

FIGURE 3 The electric field propagation for right-circular polarization, Eq. (2), when t = 0. At a fixed time, the tip of the electric field vector traces a right-handed corkscrew as the wave propagates along the +z direction.

In general, equal amounts of left- and right-circular polarization combine to produce a linear polarization with an azimuthal angle equal to half the phase difference.

B. Circular Polarization

C. Elliptical Polarization

Another special case occurs when E x = E y = E 0 and the field components have a 90◦ relative phase difference [φ = (m + 1/2)π]. If φ = π/2, Eq. (1b) becomes

For elliptically polarized light the electric field vector rotates at ω but varies in amplitude so that the tip traces out an ellipse in time at a fixed position z. Elliptical polarization is the most general state and linear and circular polarizations are simply special degenerate forms of elliptically polarized light. Because of this generality, attributes of this state can be applied to all polarization states. The polarization ellipse (Fig. 4) can provide useful quantities for describing the polarization state. The azimuthal angle α of the semi-major ellipse axis from the x axis is given by

Ercp = E 0 [ˆx cos(ωt − k0 z) + yˆ cos(ωt − k0 z + π/2)] = E 0 [ˆx cos(ωt − k0 z) − yˆ sin(ωt − k0 z)].

(2)

As the wave advances through space the magnitude of Ercp is constant but the tip of this electric field vector traces a circular path about the propagation axis at a frequency ω. A wave with this behavior is said to be right-circularly polarized. Figure 3 shows the electric field vector for right-circular polarization when viewed at a fixed time (t = 0); here the field will trace a right-handed spiral in space. An observer looking toward the origin from a distant point (z > 0) would see the vector tip rotating counterclockwise as the field travels along z. In contrast, the same observer looking at a right-circularly polarized field at a fixed position (for example, z = 0) would see the vector rotation trace out a clockwise circle in the x y plane as time advances. This difference in the sense of rotation between space and time is often a source of confusion, and depends on notation (see Section I.F). When light is left-circularly polarized the field traces out a left-handed spiral in space at a fixed time and a counterclockwise circle in time at a fixed position. Equation (1b) describes left-circular polarization when E x = E y = E 0 and φ = −π/2:

tan(2α) = tan(β) cos(φ),

(6)

where tan(β) = E y /E x and 0 ≤ β ≤ π/2. The ellipticity tan |ε| = b/a, the ratio of the semi-minor and semi-major axes, is calculated from the amplitudes and phases of Eq. (1) as tan() = tan[sin−1 (sin 2β sin φ)/2].

(7)

Elcp = E 0 [ˆx cos(ωt − k0 z) + yˆ cos(ωt − k0 z − π/2)] = E 0 [ˆx cos(ωt − k0 z) + yˆ sin(ωt − k0 z)].

(3)

Right- and left-circular polarizations are orthogonal states and can be used as a basis pair for representing other polarization states, much as orthogonal linear states

FIGURE 4 The polarization ellipse showing fields Ex and E y , ellipticity tan || = b/a, and azimuthal angle α. The tip of the electric field E traces this elliptical path in the transverse plane as the field propagates down the z axis.

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Polarization is right-elliptical when 0◦ < φ < 180◦ and tan() > 0◦ and left-elliptical when −180◦ < φ < 0◦ and tan() < 0◦ . D. Unpolarized Light Monochromatic, or single-frequency, light must necessarily be in some polarization state. Light that contains a band of wavelengths does not share this requirement. Quasi-monochromatic light can be represented by modifying Eq. (1b) as E(z, t) = Re(ˆx E x (t) exp{i[ωm t + φx (t)]} + yˆ E y (t) exp{i[ωm t + φ y (t)]})

(8)

where ωm is the mean frequency of an electric field with bandwidth ω < ωm . Taking the real part of this complex analytic representation yields the true field. Whereas the field amplitudes E i (t) and phases φi (t) are constants for strictly monochromatic light, these quantities fluctuate irregularly when the light has finite bandwidth. The pairs of functions E i (t) and φi (t) have statistical correlations that depend on the spectral bandwidth of the light source. The coherence time τ ∼ 2π/ω describes the time scale during which the pairs of functions show similar time response. For some brief time t τ , E i (t) and φi (t) are essentially constant, and E(t) possesses some elliptical polarization state, but a later field E(t + τ ) will have a different elliptical polarization. Light is described as unpolarized, or natural, if the time evolutions of the pairs of functions are totally uncorrelated within the detection time, and any polarization state is equally likely during these successive time intervals. While strictly monochromatic light cannot be unpolarized, natural light can be polarized into any desired elliptical state by passing it through the appropriate polarizer. Indeed, when unpolarized light is incident on a polarizer, the detected output intensity is independent of the polarization state transmitted by the polarizer. This occurs because a unique polarization exists for an infinitesimal time t τ and the average projection of these arbitrary states on a given polarizer is 12 over the relatively long integration time of the detector. In the absence of dispersive effects, unpolarized light, when totally polarized by an ideal polarizer, will behave much like monochromatic polarized light. It is often desirable to have unpolarized light, especially when the undesired polarization dependence of components degrades optical system performance. For example, the responsivity of photodetectors can exhibit polarization dependence and cause measurements of optical power to vary with the polarization even when intensity is constant. In some cases, pseudo-depolarizers are useful for modi-

fying polarization to produce light that approximates unpolarized light (Section III.F). For quasi-monochromatic light, the orthogonal field components can be differentially delayed, or retarded, longer than τ , so that the fields become uncorrelated. Alternatively, repeatedly varying the polarization state over a time shorter than the detector response causes the measurement to include the influence of many polarization states. This method, known as polarization scrambling, can reduce some undesirable polarization effects by averaging polarizations. The previous discussion implicitly assumes that the light has uniform properties over the wavefront. However, the polarization can be varied over the spatial extent of the beam using a spatially varying retardance. Further description of these methods and their limitations is found in the discussion on optical retarders. E. Degree of Polarization Light that is neither polarized nor unpolarized is partially polarized. The fraction of the intensity that is polarized for a time much longer than the optical period is called the degree of polarization P and ranges from P = 0 for unpolarized light to P = 1 when a light beam is completely polarized in any elliptical state. Light is partially polarized when 0 < P < 1. Partially polarized light occurs when E i (t) and φi (t) are not completely uncorrelated, and the instantaneous polarization states are limited to a subset of possible states. Partially polarized light may also be represented as a sum of completely polarized and unpolarized components. We can also define a degree of linear polarization (the fraction of light intensity that is linearly polarized) or a degree of circular polarization (the fraction that is circularly polarized). Degrees of polarization can be described formally using the coherency matrix or Stokes vector formalism described in Section IV. F. Notation The choice of coordinate system and the form of the field in Eqs. (1a) and (1b) is not unique. We have chosen a right-handed coordinate system such that the crossproduct is xˆ × yˆ = zˆ and used fields with a time dependence exp[i(ωt − kz)] rather than the complex conjugate exp[−i(ωt − kz)]. Both choices are equally valid, but may result in different descriptions of the same polarization states. Descriptions of circular polarization in particular are often contradictory because of the confusion arising from the use of varied conventions. In this article we follow the “Nebraska Convention” adopted in 1968 by the participants of the Conference on Ellipsometry at the University of Nebraska.

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Also, the choice of the Cartesian basis set for describing the electric field is common but not obligatory. Any polarization state can be decomposed into a combination of any pair of orthogonal polarizations. Thus Eqs. (1a) and (1b) could be written in terms of right- and left-circular states or orthogonal elliptical states.

II. POLARIZERS An ideal polarizer transmits only one unique state of polarization regardless of the state of the incident light. Polarizers may be delineated as linear, circular, or elliptical, depending on the state that is produced. Linear polarizers that transmit a linear state are the most common and are often simply called “polarizers.” The transmission axis of a linear polarizer corresponds to the direction of the output light’s electric field oscillation. This axis is fixed by the device, though polarizers can be oriented (rotated normal to the incident light) to select the azimuthal orientation of the output state. When linearly polarized light is incident on a linear polarizer, the transmittance T from the polarizer follows Malus’s law, T = cos2 θ,

(9)

where θ is the angle between the input polarization’s azimuth and the polarizer’s transmission axis. When the incident light is formed by a linear polarizer, Eq. (9) describes the transmission through two polarizers with angle θ between transmission axes. In this configuration the second polarizer is often called an analyzer, and the polarizer and analyzer are said to be crossed when the transmittance is minimized (θ =90◦ ). Since an ideal polarizer transmits only one polarization state it must block all others. In practice polarizers are not ideal, and imperfect polarizers do not exclude all other states. For an imperfect polarizer Malus’s law becomes T = (Tmax − Tmin ) cos2 θ + Tmin ,

(10)

where Tmax and Tmin are called the principal transmittances, and transmittance T varies between these values. The extinction ratio Tmin /Tmax provides a useful measure of polarizer performance. Diattenuation is the dependence of transmittance on incident polarization, and can be quantified as (Tmax − Tmin )/(Tmax + Tmin ), where the maximum and minimum transmittances occur for orthogonal polarizations in homogeneous elements. (Homogeneous polarization elements have eigenpolarizations that are orthogonal and we consider such elements exclusively in this article.) Polarizers are optical elements that have a diattenuation approaching 1. Most interfaces with nonnormal optical incidence will exhibit some linear diattenuation since the Fresnel reflec-

tion and transmission coefficients depend on the polarization. High-performance polarizers exploit these effects to achieve very high diattenuations by differentially reflecting and transmitting orthogonal polarizations. In contrast, dichroism is a material property in which diattenuation occurs as light travels through the medium. Most commercial polarizers exploit dichroism, polarizationdependent reflection or refraction in birefringent crystals, or polarization-dependent reflectance and transmittance in dielectric thin-film structures. A. Fresnel Equations Maxwell’s equations applied to a plane wave at an interface between two dielectric media provide the relationship among incident, transmitted, and reflected wave amplitudes and phases. Figure 2 shows the electric fields and wavevectors for a wave incident upon the interface between two lossless, isotropic dielectric media. The plane of incidence contains all three wavevectors and is used to define two specific linear polarization states; p-polarized light has its electric field vector within the plane of incidence, and s-polarized light is perpendicular to this plane. The law of reflection, θi = θr , provides the direction of the reflected wave. The refraction angle is given by Snell’s law, n i sin θi = n t sin θt .

(11)

Fresnel’s equations yield the amplitudes of the transmitted field E t and reflected field E r as fractions of the incident field E i . For p-polarized light in isotropic, homogeneous, dielectric media, the amplitude reflectance r p is Er n t cos θi − n i cos θt rp = = (12) Ei p n i cos θt + n t cos θi and amplitude transmittance t p is Et 2n i cos θi = . tp = Ei p n i cos θt + n t cos θi

(13)

For s-polarized light, the corresponding Fresnel equations are Er n i cos θi − n t cos θt rs = = (14) Ei s n i cos θi + n t cos θt and

ts =

Et Ei

= s

2n i cos θi . n i cos θi + n t cos θt

(15)

The Fresnel reflectance for cases n i /n t = 1.5 and n t /n i = 1.5 is shown in Fig. 5. At an incidence angle θB = tan−1 (n t /n i ) (for n t > n i ), known as the Brewster angle, r p = 0 and p-polarized light is totally transmitted. In a pile-of-plates polarizer, plates of glass are oriented at the

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FIGURE 5 Fresnel reflectances for p -polarized (solid curve) and s -polarized (dashed) light for cases ni /nt = 1.5 and nt /ni = 1.5. The amplitude reflectance is 0 for p -polarized light at the Brewster angle θB , and is one for all polarizations when the incidence angle is θ ≥ θ C .

Brewster angle so that only s-polarized light is reflected from each plate, and the successive diattenuations from each plate increase the degree of polarization of transmitted light. When n i > n t , both polarizations may be completely reflected if the incidence angle is larger than the critical angle θc , nt θc = sin−1 . (16) ni When θi ≥ θc , the light undergoes total internal reflection (TIR). For these incidence angles no net energy is transmitted beyond the interface and an evanescent field propagates along the direction θt . The reflectance can be reduced from 1 if the medium beyond the interface is thinner than a few wavelengths and followed by a higher refractive index material. The resulting frustrated total internal reflection allows energy to flow across the interface, leading to nonzero transmittance. For this reason, TIR devices using glass–air interfaces must be kept free of contaminants that may frustrate the TIR. Birefringent crystal polarizers obtain very high extinction ratios by transmitting one linear polarization while forcing the orthogonal polarization to undergo TIR.

Polarization and Polarimetry

beamsplitter transmits two distinct orthogonally polarized beams that are angularly separated or displaced. In birefringent materials, the incident polarization is decomposed into two orthogonal states called principal polarizations or eigenpolarizations. When the eigenpolarizations travel at the same velocity (and see the same refractive index), the direction of propagation is called an optic axis (see Section III.A). When light does not travel along an optic axis, the eigenpolarizations see different refractive indices and thus propagate at different velocities through the material. When light enters or exits a birefringent material at a nonnormal angle θ that is not along an optic axis, the eigenpolarizations refract at different angles, undergoing what is termed double refraction. Also, each eigenpolarization may encounter different reflectance or transmittance at interfaces (since Fresnel coefficients depend on the refractive indices), and diattenuation results. Complete diattenuation occurs if one eigenpolarization undergoes total internal reflection while the other eigenpolarization is transmitted. Most birefringent polarizers are made from calcite, a naturally occurring mineral. Calcite is abundant in its polycrystalline form, but optical-grade calcite required for polarizers is rare, which makes birefringent polarizers more costly than most other types. Calcite transmits from below 250 nm to above 2 µm and is used for visible and nearinfrared applications. Other birefringent crystals, such as magnesium fluoride (with transmittance from 140 nm to 7 µm), can be used at some wavelengths for which calcite is opaque. Prism polarizers are composed of two birefringent prisms cut at an internal incidence angle that transmits only one eigenpolarization while totally internally reflecting the other (Fig. 6). The prisms are held together by a thin cement layer or may be separated by an air gap and externally held in place for use with higher power laser beams. The transmitted beam contains only one eigenpolarization since the orthogonal polarization is completely reflected. The prisms are aligned with parallel optic axes, so that this transmitted beam undergoes very small deviations, usually less than 5 min of arc. Often the reflected beam also

B. Birefringent Crystal Polarizers Birefringent polarizers spatially separate an incident beam into two orthogonally polarized beams. In a conventional polarizer, the undesired polarization is eliminated by directing one beam into an optical absorber so that a single polarization is transmitted. Alternatively, a polarizing

FIGURE 6 Glan–Thompson prism polarizer. At the interface, p polarized light reflects (and is typically absorbed by a coating at the side of the prism) and s -polarized light is transmitted. The optic axes (shown as dots) are perpendicular to the page.

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contains a small amount of the transmitted eigenpolarization since nonzero reflectance results if the refractive indices of the cement and transmitted eigenpolarization are not exactly equal. Because the reflected beam has poorer extinction, it is usually eliminated by placing an indexmatched absorbing layer on the side face toward which light is reflected. Glan prism polarizers are the most common birefringent crystal polarizer. They exhibit superior extinction; extinction ratios of 10−5 –10−6 are typical, and extinctions below 10−7 are possible. The small residual transmittance can arise from material imperfection, scattering at the prism faces, or misalignment of the optic axes in each prism of the polarizer. Because total internal reflection requires incidence angles larger than θc , the polarizer operates over a limited range of input angles that is often asymmetric about normal incidence. The semi-field angle is the maximum angle for which output light is completely polarized regardless of the rotational orientation of the polarizer (that is, for any azimuthal angle of output polarization). The field angle is twice the semi-field angle. The field angle depends on the refractive index of the intermediate layer (cement or air) and the internal angle of the contacted prisms. Since the incidence angle at the contacting interface depends in part on the refractive index when light is nonnormally incident on the polarizer, the field angle is wavelength dependent. Birefringent crystal polarizing beamsplitters transmit two orthogonal polarizations. Glan prism polarizers can act as beamsplitters if the reflected beam exits through a polished surface, though extinction is degraded. Polarizing beamsplitters with better extinction separate the beams through refraction at the interface. In Rochon prisms, light linearly polarized in the plane normal to the prism is transmitted undeviated, while the orthogonal polarization is deviated by an angle dependent on the prism wedge angle and birefringence (Fig. 7a). S´enarmont polarizing beam splitters are similar, but the polarizations of the deviated and undeviated beams are interchanged. Wollaston polarizers (Fig. 7b) deviate both output eigenpolarizations with nearly equal but opposite angles when the input beam is normally incident. For all these polarizers, the deviation angle depends on the wedge angle and varies with wavelength. C. Interference Polarizers The Fresnel equations show that the transmittance and reflectance of obliquely incident light will depend on the polarization. Dielectric stacks made of alternating highand low-refractive index layers with quarter-wave optical thickness can be tailored to provide reflectances and

FIGURE 7 (a) Rochon and (b) Wollaston polarizers. The directions of the optic axes are shown in each prism (as dots for axes perpendicular to page and as a two-arrow line for axes in the plane of the page).

transmittances with large diattenuation. Optical thickness depends on incidence angle, and polarizers based on quarter-wave layers are sensitive to incidence angle and wavelength. Designs that increase the wavelength range do so at the expense of input angle range, and vice versa. Polarizing beamsplitter cubes are made by depositing the stack on the hypotenuse of a right-angle prism and cementing the coated side to the hypotenuse of a second prism. The extinction of these devices is limited by the defects in the coating layers or the optical quality of the optical substrate material through which light must pass. The state of polarization may also be altered by the birefringence in the substrate. Commercial thin-film polarizers are available with an extinction of about 10−5 . D. Dichroic Polarizers Some molecules are optically anisotropic, and light polarized along one molecular direction may undergo greater absorption than perpendicularly polarized light. When these molecules are randomly oriented, this molecularlevel diattenuation will average out as the light propagates through the thickness, and bulk diattenuation may not be observed. However, linear polarizers can be made by orienting dichroic molecules or crystals in a plastic or glass matrix that maintains a desired alignment of the transmission axes. Extinction ratios between 10−2 and 10−5 are possible in oriented dichroics in the visible and nearinfrared regions. Dichroic sheet polarizers are available with larger areas and at lower cost than other polarizer types. Also, the acceptance angle, or maximum input angle from normal incidence that does not result in degraded extinction, is typically large in dichroics because diattenuation occurs during bulk propagation rather than at interfaces. However, the maximum transmittance of these polarizers may be significantly less than unity since the transmission axis

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III. RETARDERS Retarders are devices that induce a phase difference, or retardation, between orthogonally polarized components of a light wave. Linear retarders are the most common and produce a retardance φ = φ y − φx [using the notation of Eqs. (1a) and (1b)] between orthogonal linear polarizations. Circular retarders cause a phase shift between rightand left-circular polarizations and are often called rotators because circular retardance changes the azimuthal angle of linearly polarized light. Because the polarization state of light is determined by the relative amplitudes and phase shifts between orthogonal components, retarders are useful for altering and controlling a wave’s polarization. In fact, an arbitrary polarization state can be converted to any other state using an appropriate retarder. A. Linear Birefringence In optically anisotropic materials, such as crystals, the phase velocity of propagation generally depends on the direction of propagation and polarization. The optic axes are propagation directions for which the phase velocity is independent of the azimuth of linear polarization. For other propagation directions, two orthogonal eigenaxes perpendicular to the propagation define the linear polarizations of waves that propagate through the crystal with constant phase velocity. These eigenpolarizations are linear states whose refractive indices are determined by the crystal’s dielectric tensor and propagation direction. Light polarized in an eigenpolarization will propagate through an optically anisotropic material with unchanging polarization, while light in other polarization states will change with distance as the beam propagates. Uniaxial crystals and materials that behave uniaxially are commonly used in birefringent retarders and polarizers. These crystals have a single optic axis, two principal refractive indices n o and n e , and a linear birefringence n = n e − n o . When light travels parallel to the optic axis, the eigenpolarizations are degenerate, and all polarizations propagate with index n o . For light traveling in other directions, one eigenpolarization has refractive index n o and the other’s varies with direction between n o and n e (and equals n e when the propagation is perpendicular to the optic axis). B. Waveplates Waveplates are linear retarders made from birefringent materials. Rewriting Eq. (1a) for propagation through a

Polarization and Polarimetry

birefringent medium of length L yields E(z = L , t) = Re{E x exp[i(ωt − k0 n x L)] +E y exp[i(ωt − k0 n y L)]},

(17)

where the x and y directions coincide with eigenpolarizations and the absolute phases are initially equal (at z = 0, φx = φ y = 0). The retardance φ = k0 (n x − n y )L is the relative phase shift between eigenpolarizations and depends on the wavelength, the propagation distance, and the difference between the refractive indices of the eigenpolarizations. If the z axis is an optic axis, then n x = n y = n o , and there is no retardance; if zˆ is perpendicular to an optic axis, the retardance is φ = ±k0 (n o − n e )L. In general, the retardance over a path of length L in a material with birefringence n is given by φ = 2π n L/λ.

(18)

Retardance may be specified in radians, degrees [φ = 360◦ · (n o − n e )L/λ0 ], or length [φ = (n o − n e )L]. A waveplate that introduces a π -radian or 180◦ phase shift between the eigenpolarizations is called a half-wave plate. Upon exiting the plate, the two eigenpolarizations have a λ/2 relative delay and are exactly out of phase. A half-wave plate requires a birefringent material with thickness given by L λ/2 =

λ0 (2m + 1) , 2 |n o − n e |

(19)

where the waveplate order m is a positive integer that need not equal 0 since additional retardances of 360◦ do not affect the phase relationship. Quarter-wave plates are another common component and provide phase shifts of 90◦ or π/2. The eigenaxis with the lower refractive index (n o in positive uniaxial crystals such as quartz, and n e in negative uniaxial crystals such as calcite) is called the fast axis of the retarder due to the faster phase velocity and is often marked by the manufacturer. The eigenaxes can be identified by rotating the retarder between crossed polarizers until the transmittance is minimized. When the polarizer transmission axis coincides with the retarder eigenaxis, the input polarization matches the eigenpolarization, and the light travels through the crystal unchanged until blocked by the analyzer. An input different from the eigenpolarization will exit the crystal in a different polarization state and will not be completely blocked by the analyzer. Waveplates are commonly made using quartz, mica, or plastic sheets that are stretched to produce an anisotropy that gives rise to birefringence. At visible wavelengths, n e − n o ∼0.009 for quartz, and the corresponding zerothorder (m = 0) quarter-wave plate thickness of ∼40 µm

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poses a severe manufacturing challenge. Mica can be cleaved into thin sections to obtain zeroth-order retardance, but the resulting waveplate usually has poorer spatial uniformity. Polymeric materials often have lower birefringence and can be most easily fabricated into zeroth-order waveplates. In many applications, retardance of integral multiples of 2π is unimportant, and multiple-order (m ≥ 0) waveplates are often lower in cost because the increased thickness eases fabrication. However, this approach can result in increased retardance errors. For example, retardance depends on the wavelength [explicitly in Eq. (18) or through dispersion]. Also, retardance can change with temperature or with nonnormal incidence angles that vary the optical thickness and propagation direction. Retardance errors arising from changes in wavelength, temperature, or incidence angle linearly increase with thickness and make multiple-order waveplates unadvisable in applications that demand accurate retardance. Compound zeroth-order waveplates represent a compromise between manufacturability and performance when true zeroth-order waveplates are not easily obtained. When two similar waveplates are aligned with orthogonal optic axes, the phase shifts in each waveplate have opposite sign and the combined retardance will be the difference between the two retardances. Compound zerothorder retarders are made by combining two multipleorder waveplates in this way so that the net retardance is less than 2π . For example, two multiple-order waveplates with retardance φ1 = 20π + π/2 and φ2 = −20π can be combined to yield a compound zeroth-order quarterwave plate. Compound zeroth-order waveplates exhibit the same wavelength and temperature dependence as zeroth-order waveplates since retardance errors are proportional to the difference of plate thicknesses. However, input angle dependence is the same as in a multiple-order waveplate with equivalent total thickness. C. Compensators A compensator is a variable linear retarder that can be adjusted over a continuous range of values (Fig. 8). In a Babinet compensator, two wedged plates of birefringent material are oriented with their optic axes perpendicular. In this arrangement, the individual wedges impart opposite signs of retardance, and the net retardance is the difference between the individual magnitudes. The magnitudes depend on the thickness of each wedge traversed by the optical beam. Typically one wedge is fixed and the other translated by a micrometer drive so that this moving wedge presents a variable thickness in the beam path, and the net retardance depends on the micrometer adjustment. The use of two wedges eliminates

FIGURE 8 (a) Babinet and (b) Soleil–Babinet compensators. One wedge moves in the direction of the vertical arrow to adjust the retardance. The direction of the optic axes are shown using notation from Fig. 7.

the beam deviation and the output beam is collinear to the input. The Babinet compensator has the disadvantage that the retardance varies across the optical beam because the relative thicknesses of each wedge and corresponding net retardance vary over the beam in the direction of wedge travel. This can be overcome using a Soleil (or Babinet– Soleil) compensator. In this device the two wedged pieces have coincident optic axes and translation of the moving wedge changes the total thickness and retardance of the combined retarder. The total thickness of this two-wedge piece is now constant over the useful aperture. A parallel plate of fixed retardance is placed after the wedge, in the same manner as a compound zeroth-order retarder, to improve performance. D. Rhombs Retarders can also be fabricated of materials that do not exhibit birefringence. The phase shift between s- and ppolarized waves that occurs at a total internal reflection (Section II, Fresnel equations) can be exploited to obtain a linear retarder. When light is incident at angles larger than the critical angle, the retardance at the reflection is 2 2 −1 cos θi sin θi − (n i /n t ) φ = φ p − φs = 2 tan sin2 θi (20) and depends on the incidence angle and refractive indices. A Fresnel rhomb is a solid parallelogram fabricated so that a beam at normal incidence at the entrance face totally reflects twice within the rhomb to provide a net retardance of π/2. This retarder is, however, very sensitive to the incidence angle and laterally displaces the beam. Concatenating two Fresnel rhombs (Fig. 9) provides collinear output and can greatly reduce the sensitivity of retardance to incident angle since retardance changes at the first pair of reflections are partially canceled by the second pair.

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FIGURE 9 Two Fresnel rhombs concatenated to form a Fresnel double rhomb.

Total-internal-reflection retarders are less sensitive to wavelength variation than waveplates whose retardance increases with L/λ since the rhomb retardance does not depend on the optical path length. Wavelength dependence is limited only by the material dispersion dn/dλ, which contributes small retardance changes. Thus, rhomb devices are more nearly achromatic than waveplates and can be operated over ranges of 100 nm or more. Rhomb devices are much larger than waveplates, and the clear aperture has practical limits since increasing cross section requires a proportional increase in length. Performance can also be compromised by the presence of birefringence in the bulk glass. Birefringence, arising from stresses in material production or optical fabrication, can lead to spatial variations and path-length dependence, and limit retardance stability to several degrees if not mitigated. E. Circular Retarders Some materials can exhibit circular birefringence, or optical activity, in which the eigenpolarizations are right- and left-circular and the retardance is a phase shift between these two circular states. Circular retarders are often called rotators because incident linear polarization will generally exit at a different azimuthal angle that depends on the rotary power (circular retardance per unit length) and thickness. A material that rotates linearly polarized light clockwise (as viewed by an observer facing the light source) is termed dextrorotary or right-handed, while counterclockwise rotation occurs in levorotary, or left-handed, materials. The sense of rotation is fixed with respect to the propagation direction; if the beam exiting an optically active material is reflected back through the material, the polarization will be restored to the initial azimuth. Thus a double pass through an optically active material will cause no net rotation of linear polarization. Crystalline quartz exhibits optical activity that is most evident when propagation is along the optic axis and retardance is absent. The property is not limited to crystalline materials, however; molecules that are chiral (that lack plane or centrosymmetry and are not superposable on their mirror image) can yield optical activity. Enantiomers are chiral molecules that share common molecular formulas and ordering of atoms but differ in the three-dimensional arrangement of atoms; separate enantiomers have equal rotary powers but differ in the sense of rotation. Liquids

Polarization and Polarimetry

and solutions of chiral molecules such as sugars may be optically active if an excess of one enantiomer is present. In solution, each enatiomeric form will rotate light, and the net rotation depends on the relative quantities of dextrorotary and levorotary enantiomers. Mixtures with equal quantities of enantiomers present are called racemic and the net rotation is zero. Most naturally synthesized organic chiral molecules, for example, sugars and carbohydrates, occur in only one enatiomeric form. Saccharimetry, the measurement of the optical rotary power of sugar solutions, is used to determine the concentration of sugar in single-enantiomer solutions. F. Electrooptic and Magnetooptic Effects In some materials, retardance can be induced by an electric or magnetic field. These effects are exploited to create active devices that produce an electrically controllable retardance. Crystals that are not centrosymmetric may exhibit a linear birefringence proportional to an applied electric field called the linear electrooptic effect or Pockels effect. In these materials, applied fields cause an otherwise isotropic crystal to behave uniaxially (and uniaxial crystals to become biaxial). Crystal symmetry determines the direction of the optic axes and the form of the electrooptic tensor. The magnitude of the induced birefringence thus depends on the polarization direction, the applied field strength and direction, and the material. The electrically induced birefringence can be appreciable in some materials, and the Pockels effect is widely used in retardance modulators, phase modulators, and amplitude modulators. Modulators are often characterized by their half-wave voltage Vπ , or the voltage needed to cause a 180◦ phase shift or retardance. Vπ can vary from ∼10 V in waveguide modulators to hundreds or thousands of volts in bulk modulators. The Kerr, or quadratic, electrooptic effect occurs in solids, liquids, or gases and has no symmetry requirements. In this effect, the linear birefringence magnitude is proportional to the square of the applied electric field and the induced optic axis is parallel to the field direction. The effect is typically smaller than the Pockels effect and is often negligible in Pockels materials. The Faraday effect is an induced circular birefringence that is proportional to an applied magnetic field. It is often called Faraday rotation because the circular birefringence rotates linearly polarized light by an angle proportional to the field. The Faraday effect can occur in all materials, though the magnitude is decreased by birefringence. In contrast to optical activity, the sense of Faraday rotation is determined by the direction of the magnetic field. Thus, a double-pass configuration in which light exiting a

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Faraday rotator reflects and propagates back through the material will yield twice the rotation of a single pass. This property is exploited in optical isolators, or components that transmit light in only one direction. In the simplest isolators, a 45◦ Faraday rotator is placed between polarizers with transmission axes at 0◦ and 45◦ . In the forward direction, light linearly polarized at 0◦ is azimuthally rotated 45◦ to coincide with the analyzer axis and is fully transmitted; backward light input at 45◦ rotates to 90◦ and is completely blocked by the polarizer at 0◦ . Faraday mirrors, made by combining a 45◦ Faraday rotator with a plane mirror, have the extraordinary property of “unwinding” polarization changes caused by propagation. Polarized light that passes through an arbitrary retarder, reflects off a Faraday mirror, and retraces the input path will exit with a fixed polarization for all magnitudes or orientations of the retarder so long as the retardance is unchanged during the round-trip time. When the input light is linearly polarized, the return light is always orthogonally polarized for all intervening retardances. These devices find applications in fiber optic systems since bendinduced retardance is difficult to control in an ordinary optical fiber. G. Pseudo-Depolarizers Conversion of a polarized, collimated light beam into a beam that is truly unpolarized is difficult. Methods for obtaining truly unpolarized light rely on diffuse scattering, such as passing light through ground glass plates or an integrating sphere. These methods result in light propagating over a large range of solid angles and decrease the irradiance, or power per unit area, away from the depolarizer. The loss is often unacceptable when a collimated beam is needed. Approximations to the unpolarized state can be created using pseudo-depolarizers that produce a large variety of states over time, wavelength, or the beam cross section. As described in Section I, temporal decorrelation requires that the beam propagate through a retardance that is much larger than the light’s coherence length L c = cτ ≈ 2π c/ω. If nonmonochromatic, linearly polarized light bisects the axes of a waveplate with sufficiently large retardance, the two linear eigenpolarizations will emerge with a relative phase shift that rapidly and arbitrarily changes on the order of the coherence time. At any moment the instantaneous output state will be restricted to a point on the Poincar´e sphere (see Section IV) along the great circle connecting the ±45◦ and circular polarization states. When the detector is slower than τ , the averaged response will include the influence of all these states. Lyot depolarizers are configurations of two retarders that perform this temporal decorrelation for any input

polarization state. These are commonly made by concatenating thick birefringent plates that act as high-order waveplates or by connecting lengths of polarizationmaintaining (PM) fiber. PM fiber has about one wavelength of retardance every few millimeters, and can be obtained in lengths sufficient to decorrelate multimode laser light. A polarized light beam can also be converted to a beam with a spatial distribution of states to approximate unpolarized light, without the requirements on spectral bandwidth. For example, the retardance across a wedged waveplate is not spatially uniform, and an incident beam will exit with a spatially varying polarization. When detected by a single photodetector, the influence of all the states will be averaged in the output response. These methods often satisfy needs for unpolarized light, but clearly depend on the details and requirements of the application.

IV. MATHEMATICAL REPRESENTATIONS Several methods have been developed to facilitate the representation of polarization states, polarization elements, and the evolution of polarization states as light passes through components. Using quasimonochromatic fields, the 2×2 coherency matrix can be used to represent polarizations and determine the degree of polarization of light. The four-element Stokes vector describes the state of light using readily measurable intensities and can be related to the coherency matrix. Mueller calculus represents optical components as real 4×4 matrices; when combined with Stokes vectors it provides a quantitative description of the interaction of light and optical components. In contrast, Jones calculus represents components using complex 2×2 matrices and represents light using two-element electric field vectors. Jones calculus cannot describe partially polarized or unpolarized light, but retains phase information so that coherent beams can be properly combined. Finally, the Poincar´e sphere is a pictorial representation that is useful for conceptually understanding the interaction between retarders and polarization states. A brief discussion introduces each of these methods.

A. Coherency Matrix Using Eq. (8), we can define orthogonal field components of a quasi-monochromatic plane wave E x = E x (t) exp[i(ωt − k0 z + φx (t))] and likewise for E y . The coherency matrix J is given by E x E x∗ E x E y∗ Jx x Jx y J= , (21) = E y E x∗ E y E y∗ Jyx Jyy

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532 where the angle brackets denote a time average and the asterisk denotes the complex conjugate. The total irradiance I is given by the trace of the matrix, Tr(J) = Jx x + Jyy , and the degree of polarization is 4|J| P = 1− , (22) (Jx x + Jyy )2 where |J| is the determinant of the matrix. Recalling the notation for elliptical light, one can find the azimuthal angle α of the semi-major ellipse axis from the x axis and the ellipticity angle of the polarized component as 1 −1 Jx x + J yy α = tan 2 Jx x − Jyy (23) −i(Jx y − Jyx ) 1 = tan−1 . 2 P(Jx x − Jyy ) Partially polarized light can be decomposed into polarized and unpolarized components and expressed using coherency matrices as J = Jp + Ju . Thus the state of the polarized portion of light can be extracted from the coherency matrix even when light is partially polarized. The coherency matrix representation of several states is provided in Table I. B. Mueller Calculus In Mueller calculus the polarization state of light is represented by a four-element Stokes vector S. The Stokes parameters s0 , s1 , s2 , and s3 are related to the coherency matrix elements or the quasi-monochromatic field representation through

s0 = Jx x + Jyy = E x (t)2 + E y (t)2

s1 = Jx x − Jyy = E x (t)2 − E y (t)2

(24) s2 = Jx y + Jyx = 2 E x (t)E y (t) cos(φ)

s3 = i(Jyx − Jx y ) = 2 E x (t)E y (t) sin(φ) , where the angle brackets denote a time averaging required for nonmonochromatic light. Each Stokes parameter is related to the difference between light intensities of specified orthogonal pairs of polarization states. Thus, the Stokes vector is easily found by measuring the power Pt transmitted through six different polarizers. Specifically,     s0 P0◦ + P90◦ s   P ◦ − P ◦  0 90  1   S= = (25) ,  s2   P+45◦ − P−45◦  s3 Prcp − Plcp so that s0 is the total power or irradiance of the light beam, s1 is the difference of the powers that pass through horizontal (along xˆ ) and vertical (along yˆ ) linear polarizers, s2

Polarization and Polarimetry

is the difference between +45◦ and −45◦ linearly polarized powers, and s3 is the difference between right- and left-circularly polarized powers. The values of the Stokes parameters are limited to s02 ≥ s12 + s22 + s32 and are often normalized so that s0 = 1 and −1 ≤ s1 , s2 , s3 ≤ 1. Table I lists normalized Stokes vectors for several polarization states. The degree of polarization [Eq. (22)] can be written in terms of Stokes parameters as s12 + s22 + s32 P= . (26) s02 Additionally, we can define the degree of linear polarization (the fraction of light in a linearly polarized state) by replacing the numerator of Eq. (26) with s12 + s22 , or the degree of circular polarization by replacing the numerator with s3 . An optical component that changes the incident polarization state from S to some output state S (through reflection, transmission, or scattering) can be described by a 4 × 4 Mueller matrix M. This transformation is given by      m 00 m 01 m 02 m 03 s0 s0 m     s   10 m 11 m 12 m 13   s1    S =  1  = MS =   ,  m 20 m 21 m 22 m 23   s2   s2  s3 m 30 m 31 m 32 m 33 s3 (27) where M can be a product of n cascaded components Mi using n M= Mi . (28) i=1

Matrix multiplication is not commutative and the product must be formed in the order that light reaches each component. For a system of three components in which the light is first incident on component 1 and ultimately exits component 3, S = M3 M2 M1 S, for example. Examples of Mueller matrices for several homogeneous polarization components are given in Table II. The Mueller matrix for a component can be experimentally obtained by measuring S for at least 16 judiciously selected S inputs, and procedures for measurement and data reduction are well developed. C. Jones Calculus In Jones calculus a two-element vector represents the amplitude and phase of the orthogonal electric field components and the phase information is preserved during calculation. This allows the coherent superposition of waves and is useful for describing the polarization state in systems such as interferometers that combine beams. Since this

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Polarization and Polarimetry TABLE I Matrix Representations of Selected Polarization Statesa State

Coherency matrix

Linear along xˆ (α = β = 0◦ ; tan = 0)

I

1

Stokes vector

Jones vector

  1   1     0

1

 Linear along yˆ (α = β =

90◦ ; tan

= 0)

I

1

 1    −1       0 

0 1

Linear

at +45◦

(α = β =

45◦ ; tan

= 0)

1 2I

1

1

1

1

  1   0     1

General linear (−90◦ < α < 90◦ ; tan = 0)

1 2I

cos(α)2

sin α cos α

sin α cos α

sin(α)2

Right circular (tan = 1; φ = 90◦ ; β = 45◦ )

1 2I

1

i

i

1

 1    cos 2α       sin 2α 

 Left circular (tan = −1; φ =

−90◦ ; β

=

45◦ )

1 2I

1 i



1 1

1 2I

1

1

cos(α)

√1 2

1 i

 √1 2

1

−i

−1

 1    cos 2 cos 2β       cos 2 sin 2β 

Unpolarized

1

sin(α)

   0       0 

1

General elliptical

0   1   0     0

1



√1 2

sin 2   1   0     0

√1 2

cos βe−iφ/2

sin βeiφ/2

None

0 a The parameters α, β, , and φ are defined corresponding to elliptical light as discussed in Section I. Extensive lists of Stokes and Jones vectors are available in several texts.

method is based on coherent waves, however, the Jones vector describes only fully polarized states, and partially or unpolarized states and depolarizing components cannot be represented. Recalling Eqs. (1a) and (1b), one can write a vector formulation of complex representation for a fully coherent field E x eiφx , E = eiωt E y eiφ y

(29)

where the space-dependent term kz has been omitted. When the time dependence is also omitted, this vector is known as the full Jones vector. For generality, the Jones vector J is often written in a normalized form cos β cos βe−iφ/2 , = J= sin βeiφ sin βeiφ/2

(30)

where φ = φ y − φx and tan(β) = E y /E x . The Jones vector can also be found from the polarization azimuthal angle

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TABLE II Matrix Representation of Optical Components Component

Mueller matrix

Jones matrix 



0 0 p1 + p2 p1 − p2     p1 − p2 p1 + p2 0 0   √   0 0 2 p p 0 1 2   √ 0 0 0 2 p1 p2   1 1 0 0   1 1 0 0 1    2 0 0 0 0  0 0 0 0   1 cos 2θ sin 2θ 0   2 cos 2θ cos 2θ cos 2θ sin 2θ 0  1   2 2 sin 2θ 0   sin 2θ cos 2θ sin 2θ 0 0 0 0   1 0 0 0   0 1 0 0     0 0 −1 0    0 0 0 −1   1 0 0 0   0 1 0 0   0 0 0 1   0 0 −1 0

Linear diattenuator with maximum (minimum) transmission p12 ( p22 ) or absorber (if p = p1 = p2 )

1 2

Linear polarizer at 0◦

Linear polarizer at an angle θ

Half-wave (δ = 180◦ ) linear retarder with the fast axis at 0◦

Quarter-wave (δ = 90◦ ) linear retarder with the fast axis at 0◦

0 p2

Right circular retardance δ or rotator with θ = δ/2

Mirror

Faraday mirror

Depolarizer

sin 2β sin δ

− cos 2β sin δ   1 0 0 0   0 cos δ/2 sin δ/2 0  1    2  0 − sin δ/2 cos δ/2 0   0 0 0 0   1 0 0 0   0 1 0 0     0 0 −1 0    0 0 0 −1   1 0 0 0    0 −1 0 0    0 0 1 0    0 0 0 −1   1 0 0 0   0 0 0 0   0 0 0 0   0 0 0 0

α and ellipticity tan(ε) of the polarization ellipse using −1 tan(2) φ = tan sin(2α) 1 (31) cos−1 [cos(2) cos(2α)]. 2 Table I provides examples of Jones vectors for several polarization states. β=

cos2 θ

sin θ cos θ

sin θ cos θ

sin2 θ

1 0 0 −1

 1 0 0 0 2 2 2  sin 4β sin δ/2 − sin 2β sin δ  1  0 cos 4β sin δ/2 + cos δ/2    sin 4β sin2 δ/2 − cos 4β sin2 δ/2 + cos2 δ/2 cos 2β sin δ  2 0 0

1 0 0 0



General linear retarder: retardance δ, fast axis at angle β from x axis

p1 0

ei π/4

−i π/4

e

ei δ/2 cos2 β + e−i δ/2 sin2 β

i sin 2β sin δ/2

i sin 2β sin δ/2

e−i δ/2 cos2 β + ei δ/2 sin2 β

cos δ

cos δ/2 sin δ/2

− sin δ/2 cos δ/2

1 0 0 −1

0 −1 −1 0

None

The polarization properties of optical components can be represented as 2×2 Jones matrices (Table II). The output polarization state is J = MJ, where the Jones matrix M may be constructed from a cascade of components Mi using Eq. (28). In general the matrices are not commutative and require the same ordering as in Mueller calculus, with the rightmost matrix representing the first element the light is incident upon, and so on.

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Jones used this calculus to establish three theorems that describe the minimum number of optical elements needed to describe a cascade of many elements at a given wavelength: 1. A system of any number of linear retarders and rotators (circular retarders) can be reduced to a system composed of only one retarder and one rotator. 2. A system of any number of partial polarizers and rotators can be reduced to a system composed of only one partial polarizer and one rotator. 3. A system of any number of retarders, partial polarizers, and rotators can be reduced to a system composed of only two retarders, one partial polarizer, and, at most, one rotator. The Jones matrices in Table II assume forward propagation. In some cases, for example, with nonreciprocal components such as Faraday rotators, backward propagation must be explicitly described. Furthermore, since fields are used to represent polarization states, the phase shift arising from normal-incidence reflection may be important. For propagation in reciprocal media, the transformation from the forward Jones matrix to the backward case is given by a b a −c → . (32) c d forward −b d backward For nonreciprocal behavior, such as the Faraday effect, the transformation is instead a b a −b → . (33) c d forward −c d backward When M is composed of a cascade of Mi that include both reciprocal and nonreciprocal polarization elements, each matrix must be transformed and a new combined matrix calculated. Upon reflection, the light is now backward propagating and the Jones matrix can be transformed to the forward-propagating form (for direct comparison with the input vector, for example) by changing the sign of the second element; in other words, 1 0 Jforward = Jbackward (34) 0 −1 The calculi discussed above are applicable to problems when the polarization properties are lumped, that is, the system consists of simple components such as ideal waveplates, rotators, and polarizers, etc. Because the Jones (or Mueller) matrix from a cascade of matrices depends on the order of multiplication, an optical component with intermixed polarization properties cannot generally be represented by the simple multiplication matrices representing each individual property. For example, a component in

which both linear retardance (represented by Jones matrix ML ) and circular retardance (MC ) are both distributed throughout the element is not properly represented by either ML MC or MC ML . A method known as the Jones N -matrix formulation can be used to find a single Jones matrix that properly describes the distribution of multiple polarization properties. The N -matrix represents the desired property over a vanishingly small optical path. The differential N -matrices for each desired property can be summed and the combined properties found by an integration along the optical path. Tables of N -matrices and algorithms for calculating corresponding Jones matrices can be found in several references. Jones and Mueller matrices can be related to each other under certain conditions. Jones matrices differing only in absolute phase (in other words, a phase common to both orthogonal eigenpolarizations) can be transformed into a unique Mueller matrix that will have up to seven independent elements, though the phase information will be lost. Thus Mueller matrices for distributed polarization properties can be derived from Jones matrices calculated using N -matrices. Conversely, nondepolarizing Mueller matrices [which satisfy the condition Tr(MMT ) = 4m 00 , where MT is the transpose of M] can be transformed into a Jones matrix. D. Poincare´ Sphere The Poincar´e sphere provides a visual method for representing polarization states and calculating the effects of polarizing components. Each state of polarization is represented by a unique point on the sphere defined by its azimuthal angle α, the ellipticity tan |ε|, and the handedness. Orthogonal polarizations occupy points at opposite ends of a sphere diameter. Propagation through retarders is represented by a sphere rotation that translates the polarization state from an initial point to a final polarization. Figure 10 shows a Poincar´e sphere with several polarizations labeled. Point x represents linear polarization along the x axis and point y represents y-polarized light. Right-circular polarization (tan = 1) lies at the north pole, and all polarizations above the equator are rightelliptical. Similarly, the south pole represents left-circular polarization (tan = −1), and states below the equator are left-elliptically polarized. (In many texts the locations of the circular states are reversed; while a source of confusion, this change is valid so long as other conventions are observed.) In Fig. 10, a general polarization state with azimuthal angle α and ellipticity angle is represented by the point p with longitude 2α and latitude 2. Linear polarizations have zero ellipticity (tan|| = 0) and are located along the

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FIGURE 10 The Poincare´ sphere. The polarization represented by point p is located using the azimuthal angle α (in the equatorial plane measured from point x) and the ellipticity angle (a meridional angle measured from the equator toward the north pole). Linear polarization along the x axis is located at point x, linear polarization along the y axis is represented by point y, and rcp and lcp denote right- and left-circularly polarized states, respectively. The origin represents unpolarized light.

equator. A linear polarization with azimuthal angle α from the x axis is located at a longitudinal angle 2α along the equator from point x. Polarization states that lie upon a circle parallel to the equator have the same ellipticity but different orientations. Polarizations at opposite diameters have the same ellipticity, perpendicular azimuthal angles, and opposite handedness. The Poincar´e sphere can also be used to show the effect of a retarder on an incident polarization state. A retarder oriented with a fast axis at α and an ellipticity and handedness given by tan can be represented by a point R on the sphere located at angles 2α and 2. For a given input polarization represented by point p, a circle centered at point R that includes point p is the locus of the output polarization states possible for all retardance magnitudes. A specific retardance magnitude δ is represented by a clockwise arc of angle δ along the circle from the point p. The endpoint of this arc represents the polarization state output from the retarder. Consider x-polarized light incident on a quarter-wave linear retarder oriented with its fast axis at +45◦ from horizontal; using Jones calculus, we find that right circular polarization should exit the waveplate. To show this graphically using the Poincar´e sphere, we locate the point +45◦ , which represents the retarder orientation. The initial polarization is at point x; for a retardance δ = 90◦ , we trace a clockwise arc centered at the point +45◦ that subtends 90◦ from point x. This arc ends at the north pole, so the resulting output is right-circular polarization. If the retardance was δ = 180◦ , the arc would subtend 180◦ , and the

Polarization and Polarimetry

output light would be y-polarized. Similarly, left-circular polarization results if δ = 270◦ (or if δ = 90◦ and the fast axis is oriented at −45◦ ). The evolution of the polarization through additional components can be traced by locating each retarder’s representation on the sphere, defining a circle centered by this point and the polarization output from the previous retarder, and tracing a new arc through an angle equal to the retardance. Comparing the Poincar´e sphere definitions to Eq. (25) shows that for normalized Stokes vectors (s0 = 1), each vector element corresponds to a point along Cartesian axes centered at the sphere’s origin. Stokes element s1 (= cos 2 cos 2α) falls along the axis between x- and y-polarized; s1 = 1 corresponds to point x and s1 = −1 corresponds to point y. Values of s2 (= cos 2 sin 2α) correspond to points along the diameter connecting the ±45◦ linear polarization points; s2 = −1 corresponds to the −45◦ point. Element s3 (= sin 2) is along the axis between the north and south poles. These projections on the Poincar´e sphere can be equivalently represented by rewriting Eq. (25) and normalizing to obtain     s0 1  s   cos(2) cos(2α)   1   (35)  = .  s2   cos(2) sin(2α)  sin(2) s3 Any fully polarized state on the surface of the sphere can be found using these Cartesian coordinates. Partially polarized states will map to a point within the sphere, and unpolarized light is represented by the origin.

V. POLARIMETRY Polarimetry is the measurement of a light wave’s polarization state, or the characterization of an optical component’s or material’s polarization properties. Complete polarimeters measure the full Stokes vector of an optical beam or measure the full Mueller matrix of a sample. In many cases, however, some characteristics can be neglected and the measurement of all Stokes or Mueller elements is not necessary. Incomplete polarimeters measure a subset of characteristics and may be used when simplifying assumptions about the light wave (for example, that the degree of polarization is 1) or sample (for example, a retarder exhibits negligible diattenuation or depolarization) are appropriate. In this section, a few techniques are briefly described for illustration. A. Light Measurement A polarization analyzer, or light-measuring polarimeter, characterizes the polarization properties of an optical beam. An optical beam’s Stokes vector can be completely

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characterized by measuring the six optical powers listed in Eq. (25) using ideal polarizers. When the optical beam’s properties are time invariant, the measurements can be performed sequentially by measuring the power transmitted through four orientations of a linear polarizer and two additional measurements with a quarter-wave retarder (oriented ±45◦ with respect to the polarizers axis) placed before the polarizer. In practice, as few as four measurements are required since s2 = 2P+45◦ − s0 and s3 = 2Prcp − s0 . The Stokes vector can alternatively be measured with a single circular polarizer made by combining a quarterwave plate (with the fast axis at 45◦ ) with a linear polarizer. Prcp is measured when the retarder side faces the source. Flipping so that the retarder faces the detector allows measurement of P0◦ , P90◦ , and P±45◦ . The Stokes vector elements can be measured simultaneously with multiple detector configurations. Division of amplitude polarimeters use beamsplitters to direct fractions of the power to appropriate polarization analyzers. Using division of wavefront polarization analyzers, we assume that the polarization is uniform over the optical beam and subdivisions of the beam’s cross section are directed to appropriate analyzers. Incomplete light-measuring polarimeters are useful when the light is fully polarized (degree of polarization approaches 1). For example, the ellipticity magnitude and azimuth can be found by analyzing the light with a rotating linear polarizer and measuring the minimum and maximum transmitted powers. Linear polarization yields a detected signal with maximum modulation, while minimum modulation occurs for circular polarization. The handedness of the ellipticity can be found using a right(or left-) circular polarizer. These methods are photometric, and accurate optical power measurements are required to determine the light characteristics. Before the availability of photodetectors, null methods that rely on adjusting system settings until light transmission is minimized were developed, and these are still useful today. For example, an incomplete polarimetric null system for analyzing polarized light uses a calibrated Babinet–Soleil compensator followed by a linear polarizer. Adjusting both the retardance δ and angle θ between the fast axis and polarizer axis until the transmitted power is zero yields the ellipticity angle ω (using sin 2ω = sin 2θ sin δ) and azimuthal angle α (using tan α = tan 2θ cos δ). When unpolarized light is present, the minimum transmission is not zero, and photometric measurement of this power can be used to obtain the degree of polarization. B. Sample Measurement A polarization generator is used to illuminate the sample with known states of polarization to measure the sample’s

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537 polarization properties. The reflected or transmitted light is then characterized by a polarization analyzer, and the properties of the sample are inferred from changes between the input and output states. A common configuration for determining the Mueller matrix combines a fixed linear polarizer and a rotating quarter-wave retarder for polarization generation with a rotating quarter-wave retarder followed by a fixed linear polarizer for analysis. Power is measured as the two retarders are rotated at different rates (one rotates five times faster than the other) and the Mueller matrix elements are found from Fourier analysis of the resulting time series. Alternatively, measurements can be taken at 16 (or more) specific combinations of generator and analyzer states, typically with the polarizers fixed and at specified retarder orientations. Data reduction techniques have been developed for efficiently determining the Mueller matrix from such measurements. Several methods include measurements at additional generator/analyzer combinations to overdetermine the matrix; least-squares techniques are then applied to reduce the influence of nonideal system components and decrease measurement error. Because of the simplicity and reduction of variables, incomplete polarizers can often provide a more accurate measurement of a single polarization property when other characteristics are negligible. For example, there are many methods for measuring linear retardance in samples with negligible circular retardance, diattenuation, and depolarization, and these are often applicable to measurements of high-quality waveplates. In a rotating analyzer system, the retarder is placed between two linear polarizers so that the input polarization bisects the retarder’s birefringence axes. Linear retardance is calculated from measurements of the transmitted power when the analyzer is parallel (P0◦ ) and perpendicular (P90◦ ) to the input polarizer using |δ| = cos−1 [(P0◦ − P90◦ )/(P0◦ + P90◦ )]. In this measurement, retardance is limited to two quadrants (for example, measurements of 90◦ and 270◦ = −90◦ retarders will both yield δ = 90◦ ). If a biasing quarter-wave retarder is placed between the input polarizer and retarder and both retarders are aligned with the fast axis at 45◦ , retardance in quadrants 1 and 4 (|δ| ≤ 90◦ ) can be measured from δ = sin−1 [(P90◦ − P0◦ )/(P90◦ + P0◦ )]. There are several null methods, including those that use a variable compensator aligned with the retarder at 45◦ between crossed polarizers (retardance is measured by adjusting a calibrated compensator until no light is detected) or that use a fixed quarter-wave-biasing retarder and rotate the polarizer and/or analyzer until a null is obtained. Ellipsometry is a related technique that allows the measurement of isotropic optical properties of surfaces

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538 and thin films from the polarization change induced upon reflection. Linearly polarized light is directed toward the sample at known incidence angles, and the reflected light is analyzed to determine its polarization ellipse. Application of electromagnetic models to the configuration (for example, via Fresnel equations) allows one to calculate the refractive index, extinction coefficient, and film thickness from the measured ellipticities. Ellipsometry can be extended to other configurations using various incident polarizations and polarization analyzers to measure polarimetric quantities, blurring any distinction between ellipsometry and polarimetry.

SEE ALSO THE FOLLOWING ARTICLES ELECTROMAGNETICS • LIGHT SOURCES • OPTICAL DIFFRACTION • WAVE PHENOMENA

Polarization and Polarimetry

BIBLIOGRAPHY Anonymous (1984). “Polarization: Definitions and Nomenclature, Instrument Polarization,” International Commission on Illumination, Paris. Azzam, R. M. A., and Bashara, N. M. (1997). “Ellipsometry and Polarized Light,” North Holland, Amsterdam. Bennett, J. M. (1995). “Polarization.” In “Handbook of Optics,” Vol. 1, pp. 5.1–5.30 (Bass, M., ed.), McGraw-Hill, New York. Bennett, J. M. (1995). “Polarizers.” In “Handbook of Optics,” Vol. 2, pp. 3.1–3.70 (Bass, M., ed.), McGraw-Hill, New York. Born, M., and Wolf, E. (1980). “Principles of Optics,” Pergamon Press, Oxford. Chipman, R. A. (1995). “Polarimetry.” In “Handbook of Optics,” Vol. 1, pp. 22.1–22.37 (Bass, M., ed.), McGraw-Hill, New York. Collet, E. (1993). “Polarized Light: Fundamentals and Applications,” Marcel Dekker, New York. Hecht, E., and Zajac, A. (1979). “Optics,” Addison-Wesley, Reading, MA. Kilger, D. S., Lewis, J. W., and Randall, C. E. (1990). “Polarized Light in Optics and Spectroscopy,” Academic Press, San Diego, CA. Yariv, A., and Yeh, P. (1984). “Optical Waves in Crystals,” Wiley, New York.

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Radiometry and Photometry Ross McCluney Florida Solar Energy Center

I. II. III. IV. V. VI. VII. VIII. IX.

Background Radiometry Photometry Commonly Used Geometric Relationships Principles of Flux Transfer Sources Optical Properties of Materials The Detection of Radiation Radiometers and Photometers, Spectroradiometers, and Spectrophotometers X. Calibration of Radiometers and Photometers

GLOSSARY Illuminance, E v The area density of luminous flux, the luminous flux per unit area at a specified point in a specified surface that is incident on, passing through, or emerging from that point in the surface (unit: lm · m−2 = lux). Irradianc, E e The area density of radiant flux, the radiant flux per unit area at a specified point in a specified surface that is incident on, passing through, or emerging from that point in the surface (unit: watt · m−2 ). Luminance, Lv The area and solid angle density of luminous flux, the luminous flux per unit projected area and per unit solid angle incident on, passing through, or emerging from a specified point in a specified surface, and in a specified direction in space (units: lumen · m−2 · sr−1 = cd · m−2 ). Luminous efficacy, Kr The ratio of luminous flux in lu-

mens to radiant flux (total radiation) in watts in a beam of radiation (units: lumen/watt). Luminous flux, Φv The V (λ)-weighted integral of the spectral flux λ over the visible spectrum (unit: lumen). Luminous intensity, I v The solid angle density of luminous flux, the luminous flux per unit solid angle incident on, passing through, or emerging from a point in space and propagating in a specified direction (units: lm · sr−1 = cd). Photopic spectral luminous efficiency function, V(λ) The standardized relative spectral response of a human observer under photopic (cone vision) conditions over the wavelength range of visible radiation. Projected area, Ao Unidirectional projection of the area bounded by a closed curve in a plane onto another plane making some angle θ to the first plane. Radiance, Le The area and solid angle density of radiant flux, the radiant flux per unit projected area and per unit

731

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732 solid angle incident on, passing through, or emerging from a specified point in a specified surface, and in a specified direction in space (units: watt · m−2 · sr−1 ). Radiant flux, Φe The time rate of flow of radiant energy (unit: watt). Radiant intensity, I e The solid angle density of radiant flux, the radiant flux per unit solid angle incident on, passing through, or emerging from a point in space and propagating in a specified direction (units: watt · sr−1 ). Solid angle, Ω The area A on a sphere of the radial projection of a closed curve in space onto that sphere, divided by the square r 2 of the radius of that sphere. Spectral radiometric quantities The spectral “concentration” of quantity Q, denoted Q λ , is the derivative d Q/dλ of the quantity with respect to wavelength λ, where “Q” is any one of: radiant flux, irradiance, radiant intensity, or radiance.

RADIOMETRY is a system of language, mathematical formulations, and instrumental methodologies used to describe and measure the propagation of radiation through space and materials. The radiation so studied normally is confined to the ultraviolet (UV), visible (VIS), and infrared (IR) parts of the spectrum, but the principles are applicable to radiant energy of any form that propagates in space and interacts with matter in known ways, similar to those of electromagnetic radiation. This includes other parts of the electromagnetic spectrum and to radiation composed of the flow of particles where the trajectories of these particles follow known laws of ray optics, through space and through materials. Radiometric principles are applied to beams of radiation at a single wavelength or those composed of a broad range of wavelengths. They can also be applied to radiation diffusely scattered from a surface or volume of material. Application of these principles to radiation propagating through absorbing and scattering media generally leads to mathematically sophisticated and complex treatments when high precision is required. That important topic called radiative transfer, is not treated in this article. Photometry is a subset of radiometry, and deals only with radiation in the visible portion of the spectrum. Photometric quantities are defined in such a way that they incorporate the variations in spectral sensitivity of the human eye over the visible spectrum, as a spectral weighting function built into their definition. In determining spectrally broadband radiometric quantities, no spectral weighting function is used (or one may consider that a weighting “function” of unity (1.0) is applied at all wavelengths). The scope of this treatment is limited to definitions of the primary quantities in radiometry and photometry, the

Radiometry and Photometry

derivations of several useful relationships between them, the rudiments of setting up problems in radiation transfer, short discussions of material properties in a radiometric context, and a very brief discussion of electronic detectors of electromagnetic radiation. The basic design of radiometers and photometers and the principles of their calibration are described as well. Until the latter third of the 20th century, the fields of radiometry and photometry developed somewhat independently. Photometry was beset with a large variety of different quantities, names of those quantities, and units of measurement. In the 1960s and 1970s several authors contributed articles aimed at bringing order to the apparent confusion. Also, the International Lighting Commission (CIE, Commission International de l’Eclairage) and the International Electrotechnical Commission (CEI, Commision Electrotechnic International ) worked to standardize a consistent set of symbols, units, and nomenclature, culminating in the International Lighting Vocabulary, jointly published by the CIE and the CEI. The recommendations of that publication are followed here. The CIE has become the primary international authority on terminology and basic concepts in radiometry and photometry.

I. BACKGROUND A. Units and Nomenclature Radiant flux is defined as the time rate of flow of energy through space. It is given the Greek symbol and the metric unit watt (a joule of energy per second). An important characteristic of radiant flux is its distribution over the electromagnetic spectrum, called a spectral distribution or spectrum. The Greek symbol λ is used to symbolize the wavelength of monochromatic radiation, radiation having only one frequency and wavelength. The unit of wavelength is the meter, or a submultiple of the meter, according to the rules of System International, the international system of units (the metric system). The unit of frequency is the hertz (abbreviated Hz), defined to be a cycle (or period) per second. The symbol for frequency is the Greek ν. The relationship between frequency ν and wavelength λ is shown in the equation λν = c,

(1)

where c is the speed of propagation in the medium (called the “speed of light” more familarly). The spectral concentration of radiant flux at (or around) a given wavelength λ is given the symbol λ , the name spectral radiant flux, and the units watts per unit wavelength. An example of this is the watt per nanometer (abbreviated W/nm). The names, definitions, and units of additional radiometric quantities are provided in Section II.

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The electromagnetic spectrum is diagramed in Fig. 1. The solar and visible spectral regions are expanded to the right of the scale. Though sound waves are not electromagnetic waves, the range of human-audible sound is shown in Fig. 1 for comparison. The term “light” can only be applied in principle to electromagnetic radiation over the range of visible wavelengths. Radiation outside this range is invisible to the human eye and therefore cannot be called light. Infrared and ultraviolet radiation cannot be termed “light.” Names and spectral ranges have been standardized for the ultraviolet, visible, and infrared portions of the spectrum. These are shown in Table I. B. Symbols and Naming Conventions When the wavelength symbol λ is used as a subscript on a radiometric quantity, the result denotes the concentration of the quantity at a specific wavelength, as if one were dealing with a monochromatic beam of radiation at this wavelength only. This means that the range λ of wavelengths in the beam, around the wavelength λ of definition, is infinitesemally small, and can therefore be defined in terms of the mathematical derivative as follows. Let Q be a radiometric quantity, such as flux, and Q be the amount of this quantity over a wavelength interval λ centered at wavelength λ. The spectral version of quantity Q, at wavelength λ, is the derivative of Q with respect to wavelength, defined to be the limit as λ goes to zero of the ratio Q /λ. dQ Qλ = . (2) dλ This notation refers to the “concentration” of the radiometric quantity Q, at wavelength λ, rather than to its functional dependence on wavelength. The latter would be notated as Q λ (λ). Though seemingly redundant, this notation is

FIGURE 1 Wavelength and frequency ranges over the electromagnetic spectrum.

TABLE I CIE Vocabulary for Spectral Regions Name UV-C UV-B UV-A VIS IR-Aa IR-B IR-Cb a b

Wavelength range 100 to 280 nm 280 to 315 nm 315 to 400 Approx. 360–400 to 760–800 nm 780 to 1400 nm 1.4 to 3.0 µm 3 µm to 1 mm

Also called “near IR” or NIR. Also called “far IR” or FIR.

correct within the naming convention established for the field of radiometry. When dealing with the optical properties of materials rather than with concentrations of flux at a given wavelength, the subscripting convention is not used. Instead, the functional dependence on wavelength is notated directly, as with the spectral transmittance: T (λ). Spectral optical properties such as this one are spectral weighting functions, not flux distributions, and their functional dependence on wavelength is shown in the conventional manner. C. Geometric Concepts In radiometry and photometry one is concerned with several geometrical constructs helpful in defining the spatial characteristics of radiation. The most useful are areas, plane angles, and solid angles. The areas of interest are planar ones (including small differential elements of area used in definitions and derivations), nonplanar ones (areas on curved surfaces), and what are called projected areas. The latter are areas resulting when an original area is projected at some angle θ, as viewed from an infinite distance away. Projected areas are unidirectional projections of the area bounded by a closed curve in a plane onto another plane, one making angle θ to the first, as illustrated in Fig. 2. A plane angle is defined by two straight lines intersecting at a point. The space between these lines in the plane defined by them is the plane angle. It is measured in radians (2π radians in a circle) or degrees (360 degrees to a circle). In preparation for defining solid angle it is pointed out that the plane angle can also be defined in terms of the radial projection of a line segment in a plane onto a point, as illustrated in Fig. 3. A plane angle is the quotient of the arc length s and the radius r of a radial projection of segment C of a curve in a plane onto a circle of radius r lying in that plane and centered at the vertex point P about which the angle is being defined.

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FIGURE 2 Illustration of the definition of projected areas.

If θ is the angle and s is the arc length of the projection onto a circle of radius r , then the defining equation is s θ= . (3) r According to Eq. (3), the plane angle is a dimensionless quantity. However, to aid in communication, it has been given the unit radian, abbreviated rad. The radian measure of a plane angle can be converted to degree measure with the multiplication of a conversion constant, 180/π . A similar approach can be used to define solid angle. A solid angle is defined by a closed curve in space and a point, as illustrated in Fig. 4. A solid angle is the quotient of the area A and square of the radius r of a radial projection of a closed curve C in space onto a sphere of radius r centered at the vertex point P relative to which the angle is being defined.

If is the solid angle being defined, A is the area on the sphere enclosed by the projection of the curve onto that sphere, and r is the sphere’s radius, then the defining equation is

FIGURE 4 Definition of the solid angle.

=

A r2

(4)

According to Eq. (4), the solid angle is dimensionless. However, to aid in communication, it has been given the unit steradian, abbreviated sr. Since the area of a sphere is 4π times the square of its radius, for a unit radius sphere the area is 4π and the solid angle subtended by it is 4π sr. The solid angle subtended by a hemisphere is 2π sr . It is important to note that the area A in Eq. (4) is the area on the sphere of the projection of the curve C. It is not the area of a plane cut through the sphere and containing the projection of curve C. Indeed, the projections of some curves in space onto a sphere do not lie in a plane. One which does is of particular interest—the projection of a circle in a plane perpendicular to a radius of the sphere, as illustrated in Fig. 5, which also shows a hemispherical solid angle. Let α be the plane angle subtended by the radius of the circle at the center of the sphere, called the “half-angle” of the cone. It can be shown that the solid angle subtended by the circle is given by = 2π (1 − cos α).

(5)

If α = 0 then = 0 and if α = 90◦ then = 2π sr , as required. A derivation of Eq. (5) is provided on pp. 28–30 of McCluney (1994).

FIGURE 3 Definition of the plane angle.

FIGURE 5 (a) Geometry for determining the solid angle of a right circular cone. α is the “half-angle” of the cone. (b) Geometry of a hemispherical solid angle.

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r Derived units: joule (= kg · m2 · s−2 = N · m), watt

D. The Metric System To clarify the symbols, units, and nomenclature of radiometry and photometry the international system of units and related standards known as the metric system was embraced. There have been several versions of the metric system over the last couple of centuries. The current modernized one is named Le System International d’Unites (SI). It was established in 1960 by international agreement. The Bureau International des Poids et Mesures (BIPM) regularly publishes a document containing revisions and new recommendations on terminology and units. The International Standards Organization (ISO) publishes standards on the practical uses of the SI system in a variety of fields. Many national standards organizations around the world publish their own standards governing the use of this system, or translations of the BIPM documents, into the languages of their countries. In the United States the units metre and litre are spelled meter and liter, respectively. The SI system calls for adherence to standard prefixes for standard orders of magnitude, listed in Table II. There are some simple rules governing the use of these prefixes. The prefix symbols are to be printed in roman type without spacing between the prefix symbol and the unit symbol. The grouped unit symbol plus its prefix is inseparable but may be raised to a positive or negative power and combined with other unit symbols. Examples: cm2 , nm, µm, klx, 1 cm2 = 10−4 m2 . No more than one prefix can be used at a time. A prefix should never be used alone, except in descriptions of systems of units. There are now two classes of units in the SI system: r Base units and symbols: meter (m), kilogram (kg),

second (s), ampere (A), kelvin (K), mole (mol), and candela (cd). Note that the abbreviations of units named for a person are capitalized, but the full unit name is not. (For example, the watt was named for James Watt and is abbreviated “W.”)

TABLE II SI Prefixes Factor

Prefix

Symbol

Factor

Prefix

Symbol

1024 1021

yotta

Y

10−1

deci

zetta

Z

10−2

centi

c

1018

exa peta

E P

10−3 10−6

milli micro

m µ

1015 1012

d

tera

T

10−9

nano

n

109

giga

G

10−12

pico

p

106

mega kilo

M k

10−15 10−18

femto atto

f a

hecto

h

10−21

zepto

z

decka, deca

da

10−24

yocto

y

103 102 101

(= J · s−1 ), lumen (= cd · sr), and lux (= lm · m−2 ). These are formed by combining base units according to algebraic relations linking the corresponding physical quantities. The laws of chemistry and physics are used to determine the algebraic combinations resulting in the derived units. Also included are the units of angle (radian, rad), and solid angle (steradian, sr).

A previously separate third class called supplementary units, combinations of the above units and units for plane and solid angle, was eliminated by the General Conference on Weights and Measures (CGPM, Conference Generale des Poids et Mesures) during its 9–12 October 1995 meeting. The radian and steradian were moved into the SI class of derived units. Some derived units are given their own names, to avoid having to express every unit in terms of its base units. The symbol “·” is used to denote multiplication and “/” denotes division. Both are used to separate units in combinations. It is permissible to replace “·” with a space, but some standards require it to be included. In 1969 the following additional non-SI units were accepted by the International Committee for Weights and Measures for use with SI units: day, hour, and minute of time, degree, minute and second of angle, the litre (10−3 m3 ), and the tonne (103 kg). In the United States the latter two are spelled “liter” and “metric ton,” respectively. The worldwide web of the internet contains many sites describing and explaining the SI system. A search on “The Metric System” with any search engine should yield several. The United States government site at http://physics.nist.gov/cuu/Units/ is comprehensive and provides links to other web pages of importance. E. The I-P System The most prominent alternative to the metric system is the inch-pound or the so-called “English” system of units. In this system the foot and pound are units for length and mass. The British thermal unit (Btu) is the unit of energy. This system is used little for radiometry and photometry around the world today, with the possible exception of the United States, where many illumination engineers still work with a mixed metric/IP unit, the foot-candle (lumen · ft−2 ) as their unit of illuminance. There are about 10.76 square feet in a square meter. So one foot-candle equals about 10.76 lux. The I-P system is being deprecated. However, in order to read older texts in radiometry and photometry using the I-P system, some familiarity with its units is advised. Tables 10.3 and 10.4 of McCluney (1994) provide conversion factors for many non-SI units.

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II. RADIOMETRY A. Definitions of Fundamental Quantities There are five fundamental quantities of radiometry: radiant energy, radiant flux, radiant intensity, irradiance, and radiance. Each has a photometric counterpart, described in the next section. Radiant energy, Q, is the quantity of energy propagating into, through, or emerging from a specified surface area in a specified period of time (unit: joule). Radiant energy is of interest in applications involving pulses of radiation, or exposure of a receiving surface to temporally continuous radiant energy over a specific period of time. An equivalent unit is the watt · sec. Radiant flux (power), , is the time rate of flow of radiant energy (unit: watt). One watt is 1 J sec−1 . The defining equation is the derivative of the radiant energy Q with respect to time t. dQ . (6) dt Radiant flux is the quantity of energy passing through a surface or region of space per unit time. When specifying a radiant flux value, the spatial extent of the radiation field included in the specification should be described. Irradiance, E, is the area density of radiant flux, the radiant flux per unit area at a specified point in a specified surface that is incident on, passing through, or emerging from that point in the surface (unit: watt · m−2 ). All directions in the hemispherical solid angle producing the radiation at that point are to be included. The defining equation is =

E=

d , dso

(7)

where d is an infinitesimal element of radiant flux and dso is an element of area in the surface. (The subscript “o” is used to indicate that this area is in an actual surface and is not a projected area.) The flux incident on a point in a surface can come from any direction in the hemispherical solid angle of incidence, or all of them, with any directional distribution. The flux can also be that leaving the surface in any direction in the hemispherical solid angle of emergence from the surface. The irradiance leaving a surface can be called the exitance and can be given the symbol M, to distinguish it from the irradiance incident on the surface, but it has the same units and defining equation as irradiance. (The term emittance, related to the emissivity, is reserved for use in describing a dimensionless optical property of a material’s surface and cannot be used for emitted irradiance.) Since there is no mathematical or physical distinction between flux incident upon, passing through, or leaving a

surface, the term irradiance is used throughout this article to describe the flux per unit area in all three cases. Irradiance is a function of position in the surface specified for its definition. When speaking of irradiance, one should be careful both to describe the surface and to indicate at which point on the surface the irradiance is being evaluated, unless this is very clear in the context of the discussion, or if the irradiance is known or assumed to be constant over the whole surface. Radiant intensity, I , is the solid angle density of radiant flux, the radiant flux per unit solid angle incident on, passing through, or emerging from a point in space and propagating in a specified direction (units: watt · sr−1 ). The defining equation is d , (8) dω where d is an element of flux incident on or emerging from a point within element d ω of solid angle in the specified direction. The representation of d ω in spherical coordinates is illustrated in Fig. 6. Radiant intensity is a function of direction from its point of specification, and may be written as I (θ, φ) to indicate its dependence upon the spherical coordinates (θ, φ) specifying a direction in space. Its definition is illustrated in Fig. 7. Intensity is a useful concept for describing the directional distribution of radiation from a point source (or a source very small compared with the distance from it to the observer or detector of that radiation). The concept can be applied to extended sources having the same intensity at all points, in which case it refers to that subset of the radiation emanating from the entire source of finite and known area which flows into the same infinitesimal solid angle direction for each point in that area. (The next quantity I =

FIGURE 6 Representation of the element of solid angle dω in Spherical coordinates.

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FIGURE 7 Geometry for the definition of Intensity.

to be described, radiance, is generally a more appropriate quantity for describing the directional distribution of radiation from nonpoint sources.) When speaking of intensity, one should be careful to describe the point of definition and the direction of radiation from that point for clarity of discourse, unless this is obvious in the context of the discussion, or if it is known that the intensity is constant for all directions. The word “intensity” is frequently used in optical physics. Most often the radiometric quantity being described is not intensity but irradiance. Radiance, L, is the area and solid angle density of radiant flux, the radiant flux per unit projected area and per unit solid angle incident on, passing through, or emerging from a specified point in a specified surface, and in a specified direction (units: watt · m−2 · sr−1 ). The defining equation is L=

d 2 d ω ds

L=

d , d ω dso cos θ

or

(9) 2

where ds = dso cos θ is the projected area, the area of the projection of elemental area dso along the direction of propagation to a plane perpendicular to this direction, d ω is an element of solid angle in the specified direction and θ is the angle this direction makes with the normal (perpendicular) to the surface at the point of definition, as illustrated in Fig. 8. Radiance is a function of both position and direction. For many real sources, it is a strongly varying function of direction. It is the most general quantity for describing the propagation of radiation through space and transparent or semitransparent materials. The radiant flux, radiant intensity, and irradiance can be derived from the radiance by the mathematical process of integration over a finite surface area and/or over a finite solid angle, as demonstrated in Section IV.B.

FIGURE 8 Geometry for the definition of radiance.

Since radiance is a function of position in a defined surface as well as direction from it, it is important when speaking of radiance to specify the surface, the point in it, and the direction from it. All three pieces of information are important for the proper specification of radiance. For example, we may wish to speak of the radiance emanating from a point on the ground and traveling upward toward the lens of a camera in an airplane or satellite traveling overhead. We specify the location of the point, the surface from which the flux emanates, and the direction of its travel toward the center of the lens. Since the words “radiance” and “irradiance” can sound very similar in rapidly spoken or slurred English, one can avoid confusion by speaking of the point and the surface that is common to both concepts, and then to clearly specify the direction when talking about radiance. B. Definitions of Spectral Quantities The spectral or wavelength composition of the five fundamental quantities of radiometry is often of interest. We speak of the “spectral distribution” of the quantities and by this is meant the possibly varying magnitudes of them at different wavelengths or frequencies over whatever spectral range is of interest. As before, if we let Q represent any one of the five radiometric quantities, we define the spectral “concentration” of that quantity, denoted Q λ , to be the derivative of the quantity with respect to wavelength λ. (The derivative with respect to frequency ν or wavenumber (1/ν) is also possible but less used.) dQ . (10) dλ This defines the radiometric “quantity” per unit wavelength interval and can also be called the spectral power density. It has the same units as those of the quantity Q Qλ =

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Radiometry and Photometry TABLE III Symbols and Units of the Five Spectral Radiometric Quantities Quantity Symbol

Spectral radiant energy Qλ

Units

J · nm−1

Spectral radiant flux

Spectral irradiance

Spectral intensity

Spectral radiance

λ W · nm−1

Eλ W · m−2 · nm−1

Iλ W · sr−1 · nm−1

Lλ W · m−2 · sr−1 · nm−1

divided by wavelength. The spectral radiometric quantity Q λ is in one respect the more fundamental of the two, since it contains more information, the spectral distribution of Q, rather than just its total magnitude. The two are related by the integral ∞ Q= Q λ d λ. (11) 0

If Q λ is zero outside some wavelength range, (λ1 , λ2 ) then the integral of Eq. (11) can be replaced by λ2 Q= Q λ dλ. (12) λ1

The symbols and units of the spectral radiant quantities are listed in Table III.

wavelength.) The infrared portion of the spectrum lies beyond the red, having frequencies below and wavelengths above those of red light. Since the eye is very insensitive to light at wavelengths between 360 and about 410 nm and between about 720 and 830 nm, at the edges of the visible spectrum, many people cannot see radiation in portions of these ranges. Thus, the visible edges of the UV and IR spectra are as uncertain as the edges of the VIS spectrum. The term “light” should only be applied to electromagnetic radiation in the visible portion of the spectrum, lying between 380 and 770 nm. With this terminology, there is no such thing as “ultraviolet light,” nor does the term “infrared light” make any sense either. Radiation outside these wavelength limits is radiation—not light—and should not be referred to as light.

III. PHOTOMETRY B. The Sensation of Vision A. Introduction Photometry is a system of language, mathematical formulations, and instrumental methodologies used to describe and measure the propagation of light through space and materials. In consequence, the radiation so studied is confined to the visible (VIS) portion of the spectrum. Only light is visible radiation. In photometry, all the radiant quantities defined in Section II are adapted or specialized to indicate the human eye’s response to them. This response is built into the definitions. Familiarity with the five basic radiometric quantities introduced in that section makes much easier the study of the corresponding quantities in photometry, a subset of radiometry. The human eye responds only to light having wavelengths between about 360 and 800 nm. Radiometry deals with electromagnetic radiation at all wavelengths and frequencies, while photometry deals only with visible light— that portion of the electromagnetic spectrum which stimulates vision in the human eye. Radiation having wavelengths below 360 nm, down to about 100 nm, is called ultraviolet, or UV, meaning “beyond the violet.” Radiation having wavelengths greater than 830 nm, up to about 1 mm, is called infrared, or IR, meaning “below the red.” “Below” in this case refers to the frequency of the radiation, not to its wavelength. (Solving (1) for frequency yields the equation ν = c/λ, showing the inverse relationship between frequency and

After passing through the cornea, the aqueous humor, the iris and lens, and the vitreous humor, light entering the eye is received by the retina, which contains two general classes of receptors: rods and cones. Photopigments in the outer segments of the rods and cones absorb radiation and the absorbed energy is converted within the receptors, into neural electrochemical signals which are then transmitted to subsequent neurons, the optic nerve, and the brain. The cones are primarily responsible for day vision and the seeing of color. Cone vision is called photopic vision. The rods come into play mostly for night vision, when illumination levels entering the eye are very low. Rod vision is called scotopic vision. An individual’s relative sensitivity to various wavelengths is strongly influenced by the absorption spectra of the photoreceptors, combined with the spectral transmittance of the preretinal optics of the eye. The relative spectral sensitivity depends on light level and this sensitivity shifts toward the blue (shorter wavelength) portion of the spectrum as the light level drops, due to the shift in spectral sensitivity when going from cones to rods. The spectral response of a human observer under photopic (cone vision) conditions was standardized by the International Lighting Commission the International de l’Eclairage (CIE), in 1924. Although the actual spectral response of humans varies somewhat from person to person, an agreed standard response curve has been adopted,

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as shown graphically in Fig. 9 and listed numerically in Table IV. The values in Table IV are taken from the Lighting Handbook of the Illuminating Engineering Society of North America (IESNA). Since the symbol V (λ) is normally used to represent this spectral response, the curve in Fig. 9 is often called the “V -lambda curve.” The 1924 CIE spectral luminous efficiency function for photopic vision defines what is called “the CIE 1924 Standard Photopic Photometric Observer.” The official values were originally given for the wavelength range from 380 to 780 nm at 10-nm intervals but were then “completed by interpolation, extrapolation, and smoothing from earlier values adopted by the CIE in 1924 and 1931” to the wavelength range from 360 to 830 nm on 1-nm intervals and these were then recommended by the International Committee of Weights and Measures (CIPM) in 1976. The values below 380 and above 769 are so small to be of little value for most photometric calculations and are therefore not included in Table IV. Any individual’s eye may depart somewhat from the response shown in Fig. 9, and when light levels are moderately low, the other set of retinal receptors (rods) comes into use. This regime is called “scotopic vision” and is characterized by a different relative spectral response. The relative spectral response curve for scotopic vision is similar in shape to the one shown in Fig. 9, but the peak is shifted from 555 to about 510 nm. The lower wavelength cutoff in sensitivity remains at about 380 nm, however, while the upper limit drops to about 640 nm. More information about scotopic vision can be found in various books on vision as well as in the IESNA Lighting Handbook. The latter contains both plotted and tabulated values for the scotopic spectral luminous efficiency function.

FIGURE 9 Human photopic spectral luminous efficiency.

C. Definitions of Fundamental Quantities Five fundamental quantities in radiometry were defined in Section II.A. The photometric ones corresponding to the last four are easily defined in terms of their radiometric counterparts as follows. Let Q λ (λ) be one of the following: spectral radiant flux λ , spectral irradiance E λ , spectral intensity Iλ , or spectral radiance L λ . The corresponding photometric quantity, Q v is defined as follows: 770 Q v = 683 Q λ (λ)V (λ) d λ (13) 380

with wavelength λ having the units of nanometers. The subscript v (standing for “visible” or “visual”) is placed on photometric quantities to distinguish them from radiometric quantities, which are given the subscript e (standing for “energy”). These subscripts may be dropped, as they were in previous sections, when the meaning is clear and no ambiguity results. Four fundamental radiometric quantities, and the corresponding photometric ones, are listed in Table V, along with the units for each. To illustrate the use of (13), the conversion from spectral irradiance to illuminance is given by 770 E v = 683 E λ (λ) V (λ) d λ. (14) 380

The basic unit of luminous flux, the lumen, is like a “lightwatt.” It is the luminous equivalent of the radiant flux or power. Similarly, luminous intensity is the photometric equivalent of radiant intensity. It gives the luminous flux in lumens emanating from a point, per unit solid angle in a specified direction, and therefore has the units of lumens per steradian or lm/sr, given the name candela. This unit is one of the seven base units of the metric system. More information about the metric system as it relates to radiometry and photometry can be found in Chapter 10 of McCluney (1994). Luminous intensity is a function of direction from its point of specification, and may be written as Iv (θ, φ) to indicate its dependence upon the spherical coordinates (θ, φ) specifying a direction in space, illustrated in Fig. 6. Illuminance is the photometric equivalent of irradiance and is like a “light-watt per unit area.” Illuminance is a function of position (x, y) in the surface on which it is defined and may therefore be written as E v (x, y). Most light meters measure illuminance and are calibrated to read in lux. The lux is an equivalent term for the lumen per square meter and is abbreviated lx. In the inch-pound (I-P) system of units, the unit for illuminance is the lumen per square foot, or lumen · ft−2 , which also has the odd name “foot-candle,” abbreviated “fc,” even though connection with candles and the candela

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TABLE IV Photopic Spectral Luminous Efficiency V (λ) Values interpolated at intervals of 1 nm

Wavelength λ, nm

Standard values

1

2

3

4

5

6

7

8

9

380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610

.00004 .00012 .0004 .0012 .0040 .0116 .023 .038 .060 .091 .139 .208 .323 .503 .710 .862 .954 .995 .995 .952 .870 .757 .631 .503

.000045 .000138 .00045 .00138 .00455 .01257 .0243 .0399 .0627 .0950 .1448 .2173 .3382 .5229 .7277 .8739 .9604 .9969 .9926 .9455 .8600 .7449 .6182 .4905

.000049 .000155 .00049 .00156 .00515 .01358 .0257 .0418 .0654 .0992 .1507 .2270 .3544 .5436 .7449 .8851 .9961 .9983 .9898 .9386 .8496 .7327 .6054 .4781

.000054 .000173 .00054 .00174 .00581 .01463 .0270 .0438 .0681 .1035 .1567 .2371 .3714 .5648 .7615 .8956 .9713 .9994 .9865 .9312 .8388 .7202 .5926 .4568

.000058 .000193 .00059 .00195 .00651 .01571 .0284 .0459 .0709 .1080 .1629 .2476 .3890 .5865 .7776 .9056 .9760 1.0000 .9828 .9235 .8277 .7076 .5797 .4535

.000064 .000215 .00064 .00218 .00726 .01684 .0298 .0480 .0739 .1126 .1693 .2586 .4073 .6082 .7932 .9149 .9083 1.0002 .9786 .9154 .8163 .6949 .5668 .4412

.000071 .000241 .00071 .00244 .00806 .01800 .0313 .0502 .0769 .1175 .1761 .2701 .4259 .6299 .8082 .9238 .9480 1.0001 .9741 .9069 .8046 .6822 .5539 .4291

.000080 .000272 .00080 .00274 000889 .01920 .0329 .0525 .0802 .1225 .1833 .2823 .4450 .6511 .8225 .9320 .9873 .9995 .9691 .8981 .7928 .6694 .5410 .4170

.000090 .000308 .00090 .00310 .00976 .02043 .0345 .0549 .0836 .1278 .1909 .2951 .4642 .6717 .8363 .9398 .9902 .9984 .9638 .8890 .7809 .6565 .5282 .4049

.000104 .000350 .00104 .00352 .01066 .02170 .0362 .0574 .0872 .1333 .1991 .3087 .4836 .6914 .8495 .9471 .9928 .9969 .9581 .8796 .7690 .6437 .5156 .3929

620 630 640 650 660 670 680 690 700 710 720 730 740 750 760

.381 .265 .175 .107 .061 .032 .017 .0082 .0041 .0021 .00105 .00052 .00025 .00012 .00006

.3690 .2548 .1672 .1014 .0574 .0299 .01585 .00759 .00381 .001954 .000975 .000482 .000231 .000111 .000056

.3575 .2450 .1596 .0961 .0539 .0280 .01477 .00705 .00355 .001821 .000907 .000447 .000214 .000103 .000052

.3449 .2354 .1523 .0910 .0506 .0263 .01376 .00656 .00332 .001699 .000845 .000415 .000198 .000096 .000048

.3329 .2261 .1452 .0862 .0475 .0247 .01281 .00612 .00310 .001587 .000788 .000387 .000185 .000090 .000045

.3210 .2170 .1382 .0816 .0446 .0232 .011,92 .00572 .00291 .001483 .000736 .000360 .000172 .000084 .000042

.3092 .2082 .1316 .0771 .0418 .0219 .01108 .00536 .00273 .001387 .000668 .000335 .000160 .000078 .000039

.2977 .1996 .1251 .0729 .0391 .0206 .01030 .00503 .00256 .001297 .000644 .000313 .000149 .000074 .000037

.2864 .1912 .1188 .0688 .0366 .0194 .00956 .00471 .00241 .001212 .000601 .000291 .000139 .000069 .000035

.2755 .1830 .1128 .0648 .0343 .0182 .00886 .00440 .00225 .001130 .000560 .000270 .000130 .000064 .000032

is mainly historical and indirect. The I-P system is being discontinued in photometry, to be replaced by the metric system, used exclusively in this treatment. For more information on the connections between modern metric photometry and the antiquated and deprecated units, the reader is directed to Chapter 10 of McCluney (1994). As with radiant exitance, illuminance leaving a surface can be called luminous exitance.

Luminance can be thought of as “photometric brightness,” meaning that it comes relatively close to describing physically the subjective perception of “brightness.” Luminance is the quantity of light flux passing through a point in a specified surface in a specified direction, per unit projected area at the point in the surface and per unit solid angle in the given direction. The units for luminance are therefore lm · m−2 · sr−1 . A more common unit for

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Radiometry and Photometry TABLE V Basic Quantities of Radiometry and Photometry Radiometric quantity

Symbol

Photometric quantity

Radiant flux Radiant intensity

e Ie

watt (W)

Luminous flux

v

lumen (lm)

W/sr

Luminous intensity

Iv

lumen/sr = candela (cd)

Irradiance

Ee

Radiance

Le

W/m2 W · m−2 · sr−1

Illuminance Luminance

Ev Lv

lumen/m2 = lux (lx) lm · m−2 · sr−1 = cd/m2

Units

Symbol

Units

luminance is the cd · m−2 , which is the same as the lumen per steradian and per square meter.

from the absolute radiometric scale using any of a variety of absolute detection methods discussed in Section X.)

D. Luminous Efficacy of Radiation

IV. COMMONLY USED GEOMETRIC RELATIONSHIPS

Radiation luminous efficiacy, K r , is the ratio of luminous flux (light) in lumens to radiant flux (total radiation) in watts in a beam of radiation. It is an important concept for converting between radiometric and photometric quantities. Its units are the lumen per watt, lm/W. Luminous efficacy is not an efficiency since it is not a dimensionless ratio of energy input to energy output—it is a measure of the effectiveness of a beam of radiation in stimulating the perception of light in the human eye. If Q v is any of the four photometric quantities (v , E v , Iv , or L v ) defined previously and Q e is the corresponding radiometric quantity, then the luminous efficacy associated with these quantities has the following defining equation: Kr =

Qv [lm · W −1 ] Qe

(15)

Q e is an integral over all wavelengths for which Q λ is nonzero, while Q v depends on an integral (13) over only the visible portion of the spectrum, where V (λ) is nonzero. The luminous efficacy of a beam of infrared-only radiation is zero since none of the flux in the beam is in the visible portion of the spectrum. The same can be said of ultraviolet-only radiation. The International Committee for Weights and Measures (CPIM), meeting at the International Bureau of Weights and Measures near Paris, France, in 1977 set the value 683 lm/W for the spectral luminous efficacy (K r ) of monochromatic radiation having a wavelength of 555 nm in standard air. In 1979 the candela was redefined to be the luminous intensity in a given direction, of a source emitting monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 W/sr . The candela is one of the seven fundamental units of the metric system. As a result of the redefinition of the candela, the value 683 shown in Eq. (13) is not a recommended good value for K r but instead follows from the definition of the candela in SI units. (Prior to 1979, the candela was realized by a platinum approximation to a blackbody. After the 1979 redefinition of the candela, it can be realized

There are several important spatial integrals which can be developed from the definitions of the principal radiometric and photometric quantities. This discussion of some of them will use radiometric terminology, with the understanding that the same derivations and relationships apply to the corresponding photometric quantities. A. Lambertian Sources and the Cosine Law To simplify some derivations, an important property, approximately exhibited by some sources and surfaces, is useful. Any surface, real or imaginary, whose radiance is independent of direction is said to be a Lambertian radiator. The surface can be self-luminous, as in the case of a source, or it can be a reflecting or transmitting one. If the radiance emanating from it is independent of direction, this radiation is considered to be Lambertian. A Lambertian radiator can be thought of as a window onto an isotropic radiant flux field. Finite Lambertian radiators obey Lambert’s cosine law, which is that the flux in a given direction leaving an element of area in the surface varies as the cosine of the angle θ between that direction and the perpendicular to the surface element: d(θ) = d(0) cos θ. This is because the projected area in the direction θ decreases with the cosine of that angle. In the limit, when θ = 90 degrees, the flux drops to zero because the projected area is zero. There is another version of the cosine law. It has to do not with the radiance leaving a surface but with how radiation from a uniform and collimated beam (a beam with all rays parallel to each other and equal in strength) incident on a plane surface is distributed over that surface as the angle of incidence changes. This is illustrated as follows: A horizontal rectangle of length L and width W receives flux from a homogeneous beam of collimated radiation of irradiance E, making an angle θ with the normal (perpendicular) to the plane of

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the rectangle, as shown in Fig. 2. If is the flux over the projected area A, given by E times A, this same flux o will be falling on the larger horizontal area Ao = L · W , producing horizontal irradiance E o = o /A. The flux is the same on the two areas ( = o ). Equating them gives EA = E o Ao .

(16)

But A = Ao cos θ so that E o = E cos θ.

(17)

This is another way of looking at the cosine law. Although it deals with the irradiance falling on a surface, if the surface is perfectly transparent, or even imaginary, it will also describe the irradiances (or exitances) emerging from the other side of the surface. B. Flux Relationships Radiance and irradiance are quite different quantities. Radiance describes the angular distribution of radiation while irradiance adds up all this angular distribution over a specified solid angle and lumps it together. The fundamental relationship between them is embodied in the equation E= L(θ, φ) cos θ dω (18)

for a point in the surface on which they are defined. In this and subsequent equations, the lower case dω is used to identify an element of solid angle dω and the upper case to identify a finite solid angle. If = 0 in Eq. (18), there is no solid angle and there can be no irradiance! When we speak of a collimated beam of some given irradiance, say E o , we are talking about the irradiance contained in a beam of nearly parallel rays, but which necessarily have some small angular spread to them, filling a small but finite solid angle , so that (18) can be nonzero. A perfectly collimated beam contains no irradiance, because there is no directional spread to its radiation—the solid angle is zero. Perfect collimation is a useful concept for theoretical discussions, however, and it is encountered frequently in optics. When speaking of collimation in experimental situations, what is usually meant is “quasi-collimation,” nearly perfect collimation. If the radiance L(θ, φ) in Eq. (18) is constant over the range of integration (over the hemispherical solid angle), then it can be removed from the integral and the result is E = πL .

(19)

This result is obtained from Eq. (18) by replacing dω with its equivalence in spherical coordinates, sin θ dθ dφ, and integrating the result over the angular ranges of 0 to 2π for φ and 0 to π/2 for θ .

A constant radiance surface is called a Lambertian surface so that (19) applies only to such surfaces. It is instructive to show how Eq. (18) can be derived from the definition of radiance. Eq. (9) is solved for d 2 and the result divided by dso . Since d 2 /dso = d E by Eq. (7), we have d E = L cos θ dω.

(20)

Integrating (20) yields (18). Similarly, one can replace the quotient d 2 /dω with the differential d I [from Eq. (8)] in Eq. (9) and solve for d I . Integrating the result over the source area So yields I = L cos θ dso . (21) So

Intensity is normally applied only to point sources, or to sources whose area So is small compared with the distance to them. However, Eq. (21) is valid, even for large sources, though it is not often used this way. Solving (8) for d = I dω, writing dω as dao /R 2 , and dividing both sides by dao yields the expression E=

I dω I dao I = = 2 dao dao R 2 R

(22)

for the irradiance E a distance R from a point source of intensity I , on a surface perpendicular to the line between the point source and the surface where E is measured. This is an explicit form for what is known as “the inverse square law” for the decrease in irradiance with distance from a point source. The inverse square law is a consequence of the definition of solid angle and the “filling” of that solid angle with flux emanating from a point source. Next comes the conversion from radiance L to flux . Let the dependence of the radiance on position in a surface on which it is defined be indicated by generalized coordinates (u, v) in the surface of interest. Let the directional dependence be denoted by (θ, φ), so that L may be written as a function L(u, v, θ, φ) of position and direction. Solve (9), the definition of radiance, for d 2 . The result is d 2 = L cos θ dso dω.

(23)

Integrating (23) over both the area So of the surface and the solid angle of interest yields = L(u, v, θ, φ) cos θ dω dso . (24) So

In spherical coordinates, dω is given by sin θ dθ dφ. Letting the solid angle over which (23) is integrated extend to the full 2π sr of the hemisphere, we have the total flux emitted by the surface in all directions. 2π π 2 = L(u, v, θ, φ) cos θ sin θ dθ dφ dso . So

(25)

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B. Fundamental Equations of Flux Transfer

V. PRINCIPLES OF FLUX TRANSFER Only the geometrical aspects of flux transfer through a lossless and nonscattering medium are of interest in this section. The effects of absorption and scattering of radiation as it propagates through a transparent or semitransparent medium from a source to a receiver are outside the scope of this article. The effects of changes in the refractive index of the medium, however, are dealt with in Section V.E. All uses of flux quantities in this section refer to both their radiant (subscript e) and luminous (subscript v) versions. The subscripts are left off for simplicity. When the terms radiance and irradiance are mentioned in this section, the discussion applies equally to luminance and illuminance, respectively. A. Source/Receiver Geometry The discussion begins with the drawing of Fig. 10 and the definition of radiance L in (9): L=

d 2 , d ω dso cos θ

(26)

where θ is the angle made by the direction of emerging flux with respect to the normal to the surface of the source, dso is an infinitesimally small element of area at the point of definition in the source, and d ω is an element of solid angle from the point of definition in the direction of interest. In Fig. 10 are shown an infinitesimally small element dso of area at a point in a source, an infinitesimal element dao of area at point P on a receiving surface, the distance R between these points, and the angles θ and ψ between the line of length R between the points and the normals to the surfaces at the points of intersection, respectively.

FIGURE 10 Source/receiver geometry.

The element dω of solid angle subtended by element of projected receiver area da = dao cos ψ at distance R from the source is da dao cos ψ dω = 2 = (27) R R2 so that, solving (26) for d 2 and using (27), the element of flux received at point P from the element dso of area of the source is given by dso cos θ dao cos ψ (28) R2 with the total flux received by area Ao from source area So being given by dso cos θ dao cos ψ = L . (29) R2 So Ao d 2 = L

This is the fundamental (and very general within the assumptions of this section) equation describing the transfer of radiation from a source surface of finite area to a receiving surface of finite area. Most problems of flux transfer involve this integration (or a related version shown later, giving the irradiance E instead of the flux). For complex or difficult geometries the problem can be quite complex analytically because in such cases L , θ, ψ, and R will be possibly complicated functions of position in both the source and the receiver surfaces. The general dependency of L on direction is also embodied in this equation, since the direction from a point in the source to a point in the receiver generally changes as the point in the receiver moves over the receiving surface. The evaluation of (29) involves setting up diagrams of the geometry and using them to determine trigonometric and other analytic relationships between the geometric variables in (29). If the problem is expressed in cartesian coordinates, for example, then the dependences of L , θ, ψ, R, dso , and dao upon those coordinates must be determined so that the integrals in (29) can be evaluated. Two important simplifications allow us to address a large class of problems in radiometry and photometry with ease, by simplifying the mathematical analysis. The first results when the source of radiance is known to be Lambertian and to have the same value for all points in the source surface. This makes L constant over all ranges of integration, both the integration over the source area and the one over the solid angle of emerging directions from each point on the surface. In such a case, the radiance can be removed from all integrals over these variables. The remaining integrals are seen to be purely geometric in character. The second simplification arises when one doesn’t want the total flux over the whole receiving surface— only the flux per unit area at a point on that surface, the

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irradiance E at point P in Fig. 10. In this case, we can divide both sides of (28) by the element of area in the receiving surface, dao , to get dE = L

cos θ cos ψ dso . R2

(30)

This equation is the counterpart of (28) when it is the irradiance E of the receiving surface that is desired. For the total irradiance at point P, one must integrate this equation over the portion So of the source surface contributing to the flux at P. cos θ cos ψ E= L dso . (31) R2 So When L is constant over direction, it can be removed from this integral and one is left with a simpler integration to perform. Equation (31) is the counterpart to (29) when it is the irradiance E at the receiving point that is of interest rather than the total flux over area Ao . C. Simplified Source/Receiver Geometries If the source area So is small with respect to the distance R to the point P of interest (i.e., if the maximum dimension of the source is small compared with R), then R 2 , cos ψ, and cos θ do not vary much over the range of integration shown in (31) and they can be removed from the integral. If L does not vary over So , then it also can be removed from the integral, even if L is direction dependent, because the range of integration over direction is so small; that is, only the one direction from the source to point P in the receiver is of interest. We are left with an approximate version of (30) for small homogeneous sources some distance from the point of reception: So cos θ cos ψ E≈L . R2

(32)

This equation contains within it both the cosine law and the inverse square law. If the source and receiving surfaces face each other directly, so that θ and ψ are zero, both of the cosines in this equation have values of unity and the equation is still simpler in form.

receiver or from Surface 1 to Surface 2. In essence, it indicates the details of how flux is transferred from a source area of some known form to a reception area. Its value is most evident when the source radiance is of such a nature that it can be taken from the integrals, leaving integrals over only geometric variables. The geometry can still be quite complex, making analytical expressions for F1−2 difficult to determine and calculate. Many important geometries have already been analyzed, however, and the resulting configuration factors published. In many problems, one is most concerned with the magnitude and spectral distribution of the source radiance and the corresponding spectral irradiance in a receiving surface, rather than with the geometrical aspects of the problem expressed by the shape factor. It is very convenient in such cases to separate the spectral variations from the geometrical ones. Once the configuration factor has been determined for a situation with nonchanging geometry, it remains constant and attention can be focused on the variable portion of the solution. A general expression for the configuration factor results from dividing (29) for the flux r on the receiver by (24) for the total flux s , emitted by the source.

Fs−r

r = = s

So

L cos θ cos ψ dso dao R2 . So 2π L cos θ dω dso Ao

(33)

This is the most general expression for the configuration factor. If the source is Lambertian and homogeneous, or if So and Ao are small in relation to R 2 then L can be removed from the integrals, resulting in

Fs−r

cos θ cos ψ dso dao So Ao R2 = π So

(34)

a more conventional form for the configuration factor. As desired, it is purely geometric and has no radiation components. For homogeneous Lambertian sources of radiance L, the flux to a receiver, s−r is given by s−r = π So L Fs−r

(35)

D. Configuration Factor In analyzing complicated radiation transfer problems, it is frequently helpful to introduce what is called the configuration factor. Alternate names for this factor are the view, angle, shape, interchange, or exchange factor. It is defined to be the fraction of total flux from the source surface that is received by the receiving surface. It is given the symbol Fs−r or F1−2 , indicating flux transfer from source to

E. Effect of Refractive Index Changes For a ray propagating through an otherwise homogeneous medium without losses, it can be shown that the quantity L/n 2 is invariant along the ray. L is the radiance and n is the refractive index of the medium. If the refractive index is constant, the radiance L is constant along the ray. This is known as the invariance of radiance.

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Suppose this ray passes through a (specular) interface between two isotropic and homogeneous media of different refractive indices, n 1 and n 2 , and suppose there is neither absorption nor reflection at the interface. In this case it can be shown that L1 L2 = 2. 2 n1 n2

portions of those distributions. The matching of a proper detector/filter combination to a given radiation source is one of the most important tasks facing the designer. Section VIII deals with detectors. B. Blackbody Radiation

(36)

Equation 36 shows how radiance invariance is modified for rays passing through interfaces between two media with different refractive indices. A consequence of (36) is that a ray entering a medium of different refractive index will have its radiance altered, but upon emerging back into the original medium the original radiance will be restored, neglecting absorption, scattering, and reflection losses. This is what happens to rays passing through the lens of an imaging system. The radiance associated with every ray contributing to a point in an image is the same as when that ray left the object on the other side of the lens (ignoring reflection and transmission losses in the lens). Since this is true of all rays making up an image point, the radiance of an image formed by a perfect, lossless lens equals the radiance of the object (the source). This may seem paradoxical. Consider the case of a focusing lens, one producing a greater irradiance in the image than in the object. How can a much brighter image have the same radiance as that of the object? The answer is that the increased flux per unit area in the image is balanced by an equal reduction in the flux per unit solid angle incident on the image. This trading of flux per unit area for flux per unit solid angle is what allows the radiance to remain essentially unchanged.

VI. SOURCES A. Introduction The starting point in solving most problems of radiation transfer is determining the magnitude and the angular and spectral distributions of emission from the source. The optical properties of any materials on which that radiation is incident are also important, especially their spectral and directional properties. This section provides comparative information about a variety of sources commonly found in radiometric and photometric problems within the UV, VIS, and IR parts of the spectrum. Definitions used in radiometry and photometry for the reflection, transmission, and absorption properties of materials are provided in Section VII. Spectral distributions are probably the most important characteristics of sources that must be considered in the design of radiometric systems intended to measure all or

All material objects above a temperature of absolute zero emit radiation. The hotter they are, the more they emit. The constant agitation of the atoms and molecules making up all objects involves accelerated motion of electrical charges (the electrons and protons of the constituent atoms). The fundamental laws of electricity and magnetism, as embodied in Maxwell’s equations, predict that any accelerated motion of charges will produce radiation. The constant jostling of atoms and molecules in material substances above a temperature of absolute zero produces electromagnetic radiation over a broad range of wavelengths and frequencies. 1. Stefan–Boltzmann Law The total radiant flux emitted from the surface of an object at temperature T is expressed by the Stefan–Boltzmann law, in the form Mbb = σT 4 ,

(37)

where Mbb is the exitance of (irradiance leaving) the surface in a vacuum, σ is the Stefan–Boltzmann constant (5.67031 × 10−8 W · m−2 · K−4 ), and T is the temperature in degrees kelvin. The units for Mbb in (37) are W · m−2 . Using (37), a blackbody at 27o C, (27 + 273 = 300 K), emits at the rate of 460 W/m2 . At 100o C this rate increases to 1097 W/m2 . Equation (37) applies to what is called a perfect or full emitter, one emitting the maximum quantity of radiation possible for a surface at temperature T . Such an emitter is called a blackbody, and its emitted radiation is called blackbody radiation. A blackbody is defined as an ideal body that allows all incident radiation to pass into it (zero reflectance) and that absorbs internally all the incident radiation (zero transmittance). This must be true for all wavelengths and all angles of incidence. According to this definition, a blackbody is a perfect absorber, having absorptance 1.0 at all wavelengths and directions. Due to the law of the conservation of energy, the sum of the reflectance R and absorptance A of an opaque surface must be unity, A + R = 1.0. Thus, if a blackbody has an absorptance of 1.0, its reflectance must be zero. Accordingly, a perfect blackbody at room temperature would appear totally black to the eye, hence the origin of the name. Only a few surfaces, such as carbon black, carborundum, and gold black, approach a blackbody in these optical properties.

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The radiation emitted by a surface is in general distributed over a range of angles filling the hemisphere and over a range of wavelengths. The angular distribution of radiance from a blackbody is constant; that is, the radiance is independent of direction; it is Lambertian. Specifically, this means that L λ (θ, φ) = L λ (0, 0) = L λ . Thus, the relationship between the spectral radiance L bbλ and spectral exitance Mbbλ of a blackbody is given by (19), repeated here as Mbbλ = π L bbλ . −2

If L bbλ is in W · m · sr will be W · m−2 · nm−1 .

−1

· nm

−1

(38)

then the units of Mbbλ

2. Greybodies Imperfect emitters, which emit less than a blackbody at any given temperature, can be called greybodies if their spectral shape matches that of a blackbody. If that shape differs from that of a blackbody the emitter is called a nonblackbody. The Stefan–Boltzmann law still applies to greybodies, but an optical property factor must be included in (37) and (38) for them to be correct for greybodies. That is the emissivity of the surface, defined and discussed in Section VII.C.3.

FIGURE 11 Exitance spectra for blackbodies at various temperatures from 300 to 20,000 K, calculated using Eq. (47).

tion lie in the visible portion of the spectrum for temperatures below about 1000 K. With increasing temperatures, blackbody radiation first appears red, then white, and at very high temperatures it has a bluish appearance. From (38), the spectral exitance Mbbλ of a blackbody at temperature T is just the spectral radiance L bbλ given in (39) multiplied by π . Mbbλ =

2π hc2 hc

λ5 (e λ kT − 1)

.

(40)

3. Planck’s Law As the temperature changes, the spectral distribution of the radiation emitted by a blackbody shifts. In 1901, Max Planck made a radical new assumption—that radiant energy is quantized—and used it to derive an equation for the spectral radiant energy density in a cavity at thermal equilibrium (a good theoretical approximation of a blackbody). By assuming a small opening in the side of the cavity and examining the spectral distribution of the emerging radiation, he derived an equation for the spectrum emitted by a blackbody. The equation, now called Planck’s blackbody spectral radiation law, accurately predicts the spectral radiance of blackbodies in a vacuum at any temperature. Using the notation of this text the equation is L bbλ =

2hc2 hc

λ5 (e λkT − 1)

,

(39)

where h = 6.626176 × 10−34 J · s is Planck’s constant, c = 2.9979246 × 108 m/s is the speed of light in a vacuum, and k = 1.380662 × 10−23 J · K−1 is Boltzmann’s constant. Using these values, the units of L bbλ will be W · m−2 · µm−1 · sr−1 . Plots of the spectral distribution of a blackbody for different temperatures are illustrated in Fig. 11. Each curve is labeled with its temperature in degrees Kelvin. Insignificant quantities of blackbody radia-

4. Luminous Efficacy of Blackbody Radiation Substituting (40) for the hemispherical spectral exitance of a blackbody into (11) for E e and (14) for E v , for each of several different temperatures T , one can calculate the radiation luminous efficacy K bb of blackbody radiation as a function of temperature. Some numerical results are given in Table VI, where it can be seen that, as expected, the luminous efficacy increases as the body heats up to white hot temperatures. At very high temperatures K bb declines, since the radiation is then strongest in the ultraviolet, outside of the visible portion of the spectrum. 5. Experimental Approximation of a Blackbody The angular and spectral characteristics of a blackbody can be approximated with an arrangement similar to the one shown in Fig. 12. A metal cylinder is hollowed out to form a cavity with a small opening in one end. At the opposite end is placed a conically shaped “light trap,” whose purpose is to multiply reflect incoming rays, with maximum absorption at each reflection, in such a manner that a very large number of reflections must take place before any incident ray can emerge back out the opening. With the absorption high on each reflection, a vanishingly small

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Radiometry and Photometry TABLE VI Blackbody Luminous Efficacy Values Temperature in degrees K 500 1,000 1,500 2,000 2,500 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 15,000 20,000 30,000 40,000 50,000

Luminous efficacy in lm/W 7.6 × 10−13 2.0 × 10−4 0.103 1.83 8.71 21.97 56.125 81.75 92.9 92.8 87.3 79.2 70.6 37.1 20.4 7.8 3.7 2.0

by the multiple reflections taking place inside (at least over the useful solid angle indicated in Fig. 12, for which the apparatus is designed). C. Electrically Powered Sources

fraction of incident flux, after being multiply reflected and scattered, emerges from the opening. In consequence, only a very tiny portion of the radiation passing into the cavity through the opening is reflected back out of the cavity. The temperature of the entire cavity is controlled by heating elements and thick outside insulation so that all surfaces of the interior are at precisely the same (known) temperature and any radiation escaping from the cavity will be that emitted from the surfaces within the cavity. The emerging radiation will be rendered very nearly isotropic

Modern tungsten halogen lamps in quartz envelopes produce output spectra that are somewhat similar in shape to those of blackbody distributions. A representative spectral distribution is shown in Fig. 13. This lamp covers a wide spectral range, including the near UV, the visible, and much of the infrared portion of the spectrum. Only the region from about 240 to 2500 nm is shown in Fig. 13. Although quartz halogen lamps produce usable outputs in the ultraviolet region, at least down to 200 nm, the output at these short wavelengths is quite low and declines rapidly with decreasing wavelength. Deuterium arc lamps overcome the limitations of quartz halogen lamps in this spectral region, and they do so with little output above a wavelength of 500 nm except for a strong but narrow emission line at about 660 nm. The spectral irradiance from a deuterium lamp is plotted in Fig. 14. Xenon arc lamps have a more balanced output over the visible but exhibit strong spectral “spikes” that pose problems in some applications. Short arc lamps, such as those using xenon gas, are the brightest manufactured sources, with the exception of lasers. Because of the nature of the arc discharges, these lamps emit a continuum of output over wavelengths covering the ultraviolet and visible portions of the spectrum. The spectral irradiance outputs of the three sources just mentioned cover the near UV, the visible, and the near IR. They are plotted along with the spectrum of a 50-W mercury-vapor arc lamp in Fig. 14. Mercury lamps emit strong UV and visible radiation, with strong spectral

FIGURE 12 Schematic diagram of an approximation to a blackbody.

FIGURE 13 Spectral irradiance from a quartz halogen lamp 50 cm from the filament.

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FIGURE 14 Spectral irradiance distributions for four sources of radiation.

lines in the ultraviolet superimposed over continuous spectra. Tungsten halogen lamps have substantial output in the near infrared. For sources with better coverage of IR-A and IR-B, different sources are more commonly used. Typical outputs from several infrared laboratory sources are shown in Fig. 15. The sources are basically electrical resistance heaters, ceramic and other substances that carry electrical current, which become hot due to ohmic heating, and which emit broadband infrared radiation with moderately high radiance. In addition to these relatively broadband sources, there are numerous others that emit over more restricted spectral ranges. The light-emitting diode (LED) is one example. It is made of a semiconductor diode with a P-N junction designed so that electrical current through the junction in the forward bias direction produces the emission of optical radiation. The spectral range of emission is limited, but not so much to be considered truly monochromatic. A sample

FIGURE 15 Spectral irradiance distributions from four sources of infrared radiation.

Radiometry and Photometry

LED spectrum is shown in Fig. 16. LEDs are efficient converters of electrical energy into radiant flux. Lasers deserve special mention. An important characteristic of lasers is their extremely narrow spectral output distribution, effectively monochromatic. A consequence of this is high optical coherence, whereby the phases of the oscillations in electric and magnetic field strength vectors are preserved to some degree over time and space. Another characteristic is the high spectral irradiance they can produce. For more information the reader is referred to modern textbooks on lasers and optics. Most gas discharge lasers exhibit a high degree of collimation, an attribute with many useful optical applications. A problem with highly coherent sources in radiometry and photometry is that not all of the relationships developed so far in this article governing flux levels are strictly correct. The reason is the possibility for constructive and destructive interference when two coherent beams of the same wavelength overlap. The superposition of two or more coherent monochromatic beams will produce a combined irradiance at a point that is not always a simple sum of the irradiances of the two beams at the point of superposition. A combined irradiance level can be more and can be less than the sum of the individual beam irradiances, since it depends strongly on the phase difference between the two beams at the point of interest. The predictions of radiometry and photometry can be preserved whenever they are averaged over many variations in the phase difference between the two overlapping beams. D. Solar Radiation and Daylight Following its passage through the atmosphere, direct beam solar radiation exhibits the spectral distribution shown in Fig. 17. The fluctuations at wavelengths over 700 nm are the result of absorption by various gaseous constituents of the atmosphere, the most noticeable of which are water vapor and CO2 . The V -lambda curve is also shown in Fig. 17 for comparison.

FIGURE 16 Relative spectral exitance of a red light-emitting diode.

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749 now be identified for the processes of reflection, transmission, and emission of radiant flux by or through material media. Although symbols have been standardized for most of these properties, there are a few exceptions. To begin, the CIE definitions for reflectance, transmittance, and absorptance are provided:

FIGURE 17 Spectral irradiance of terrestrial clear sky direct beam solar radiation.

The spectral distribution of blue sky radiation is similar to that shown in Fig. 17, but the shape is skewed by what is called Rayleigh scattering, the scattering of radiation by molecular-sized particles in the atmosphere. Rayleigh scattering is proportional to the inverse fourth power of the wavelength. Thus, blue light is scattered more prominently than red light. This is responsible for the blue appearance of sky light. (The accompanying removal of light at short wavelengths shifts the apparent color of beam sunlight toward the red end of the spectrum, responsible for the orange-red appearance of the sun at sunrise and sunset.) The spectral distribution of daylight is important in the field of colorimetery and for many other applications, including the daylight illumination of building interiors.

VII. OPTICAL PROPERTIES OF MATERIALS A. Introduction Central to radiometry and photometry is the interaction of radiation with matter. This section provides a discussion of the properties of real materials and their abilities to emit, reflect, refract, absorb, transmit, and scatter radiation. Only the rudiments can be addressed here, dealing mostly with terminology and basic concepts. For more information on the optical properties of matter, the reader is directed to available texts on optics and optical engineering, as well as other literature on material properties. B. Terminology The improved uniformity in symbols, units, and nomenclature in radiometry and photometry has been extended to the optical properties of materials. Proper terminology can

1. Reflectance (for incident radiation of a given spectral composition, polarization and geometrical distribution) (ρ): Ratio of the reflected radiant or luminous flux to the incident flux in the given conditions (unit: 1) 2. Transmittance (for incident radiation of given spectral composition, polarization and geometrical distribution) (τ ): Ratio of the transmitted radiant or luminous flux to the incident flux in the given conditions (unit: 1) 3. Absorptance: Ratio of the absorbed radiant or luminous flux to the incident flux under specified conditions (unit: 1) These definitions make explicit the point that radiation incident upon a surface can have nonconstant distributions over the directions of incidence, over polarization state, and over wavelength (or frequency). Thus, when one wishes to measure these optical properties, it must be specified how the incident radiation is distributed in wavelength and direction and how the emergent detected radiation is so distributed if the measurement is to have meaning. Polarization effects are not dealt with here. The wavelength dependence of radiometric properties of materials is indicated with a functional lambda λ thus: τ (λ), ρ(λ), and α(λ). The directional dependencies are indicated by specifying the spherical angular coordinates (θ, φ) of the incident and emergent beams. In other fields it is common to assign the ending -ivity to intensive, inherent, or bulk properties of materials. The ending -ance is reserved for the extensive properties of a fixed quantity of substance, for example a portion of the substance having a certain length or thickness. (Sometimes the term intrinsic is used instead of intensive and extrinsic is used instead of extensive.) Figure 18 illustrates the difference between intrinsic and extrinsic reflection properties and introduces the concept of interface reflectivity. An example from electronics is the 30 ohm electrical resistance of a 3 cm length of a conductor having a resistivity of 10 ohms/cm. According to this usage in radiometry, reflectance is reserved for the fraction of incident flux reflected (under defined conditions of irradiation and reception) from a finite and specified portion of material, such as a 1-cm-thick plate of fused silica glass having parallel, roughened

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This terminology is summarized as follows:

FIGURE 18 (a) Intrinsic versus (b) extrinsic reflection properties of a material. (c) Interface reflectivity.

surfaces in air. The reflectivity of a material, such as BK7 glass, would refer to the ratio of reflected to incident flux for the perfectly smooth (polished) interface between an optically infinite thickness of the material and some other material, such as air or vacuum. The infinite thickness is specified to ensure that reflected flux from no other interface can contribute to that reflected by the interface of interest, and to ensure the collection of subsurface flux scattered by molecules of the material. CIE definitions for the intrinsic optical properties of matter read as follows: 1. Reflectivity (of a material) (ρ∞ ): Reflectance of a layer of the material of such a thickness that there is no change of reflectance with increase in thickness (unit: 1) 2. Spectral transmissivity (of an absorbing material) (τi,o (λ)): Spectral internal transmittance of a layer of the material such that the path of the radiation is of unit length, and under conditions in which the boundary of the material has no influence (unit: 1) 3. Spectral absorptivity (of an absorbing material) (αi,o (λ)): Spectral internal absorptance of a layer of the material such that the path of the radiation is of unit length, and under conditions in which the boundary of the material has no influence (unit: 1) One can further split the reflectivity ρ∞ into interfaceonly and bulk property components. We use the symbol ρ, ¯ rho with a bar over it, to indicate the interface contribution to the reflectivity. The interface transmissivity τ¯ is included in this notational custom. Since there is presumed to be no absorption when radiation passes through or reflects from an interface, τ¯ + ρ¯ = 1.0. To denote the optical properties of whole objects, such as parallel sided plates of a material of specific thickness, we use upper case Roman font characters, as with the symbol R for reflectance, and the “-ance” suffix.

ρ¯ Reflectivity of an interface ρ Reflectivity of a pure substance, including both bulk and interface processes R Reflectance of an object τ¯ Transmissivity of an interface τ (Internal) linear transmissivity of (a unit length of) a transparent or partially transparent substance, away from interfaces; unit: m−1 T Transmittance of an object α (Internal) linear absorptivity of (a unit length of) a transparent or partially transparent substance, away from interfaces; unit: m−1 A Absorptance of an object C. Surface and Interface Optical Properties 1. Conductor Optical Properties A perfect conductor, characterized by infinitely great conductivity, has an infinite refractive index and penetration of electromagnetic radiation to any depth is prohibited. This produces perfect reflectivity. Real conductors such as aluminum and silver do not have perfect conductivities nor do they have perfect reflectivities. Their reflectivities are quite high, however, over broad spectral ranges. They are therefore useful in radiometric and photometric applications. Unprotected mirrors made of these materials, unfortunately, tend to degrade with exposure to air over time and they are seldom used without protective overcoatings. The normal incidence spectral reflectances of optical quality glass mirrors coated with aluminum, with aluminum having a magnesium fluoride protective overcoat, with al