SOUND WAVES IN ROOMS

Philip M. Morse and Richard H. Bolt *)

CONTENTS

I. Introduction. 1. Historical survey. 2. Geometrical and wave acoustics. 3. Reverberation time and acoustic criteria: a) Criteria for evaluating the intelligibility of speech; b) Criteria for evaluating the audibility of music; c) Modern tendencies . . . . . . . . . . . . . . . . . . 185

II. Geometrical acoustics of rooms. 4. Absorption coefficient. 5. Sabine’s approximation. 6. Geometrical theory. 7. Other formulas of geometrical acoustics. 8. Measurements of reverberation. 9. Critique of the geometrical theory . . . . . . . . . . . . . . 194

III. General principles of wave acoustics. 10. Acoustic impedance. 11. Properties of rooms in the steady-state and transient regimes. 12. Ergodic vibrations. 13. Effect of irregularities. 14. Classification of vibrations. 15. Distribution of the frequencies of natural vibrations . . . . . . . . . . . . . . . . . . . 205

IV. Acoustic impedance. 16. Impedance and absorption. 17. Mechanical properties of porous materials. 18. Equivalent circuit for long waves. 19. Short waves. 20. Effect of an air gap. 21. Vibrations of panels. 22. Types of sound-absorbing materials. 23. Measurements of acoustic impedance. 24. Determinations of effective porosity, density, and flow resistance . . . . . . . . . . . . . . . . . . . 215

V. Steady-state sound regime in rectangular rooms.
VI. Steady-state regime in rectangular rooms.
VII. Application of perturbation theory to the calculation of rooms of varied shape.
VIII. Method of free plane waves for disordered vibrations. Literature.

INTRODUCTION

1. Historical survey

Architectural acoustics is a comparatively narrow branch of physics and attracts the attention of relatively few physicists. Nevertheless, during the last decade it has achieved notable successes, and some of its new results may find application in other fields. To Sabine’s work S1**), which appeared in

*) Reviews of Modern Physics 16, No. 2, 69 (1944). Translated by N. N. Osipenko under the editorship of G. A. Ostroumov and M. Antokolsky.
**) The list of references will be placed at the end of the article (UFN, vol. XXXII, issue 4).

...the turn of the nineteenth and twentieth centuries and, having a pioneering character, over a considerable period very little scientific data were added. It seemed already that room acoustics had become merely a branch of engineering art S5. During the last decade scientific interest in architectural acoustics has revived. This occurred chiefly as a result of the theoretical discussion between Schuster and Weitzmann S10, on the one hand, and Strett S15, on the other, as well as of the instructive experimental results of Knudsen, Hunt K3, K5, H7, H9 and others C2, H5, M10, P7, P9, S3, W8. At present substantial progress has been achieved in understanding the foundations of the subject.

Ten years ago Knudsen K7 wrote a review describing the state of the question up to that time. He pointed out the incorrectness of many notions that had previously been widely current, and indicated the direction of new efforts. In the time that has elapsed, the investigations which he then outlined have provided sufficiently extensive and interesting material to justify the appearance of a new review.

Room acoustics studies the behavior of sound waves in an enclosed volume in transient and steady-state regimes. From the scientific point of view this field is in many respects intertwined with other fields of physics. The interrelation between experimental acoustics and the development of electron-tube technology is obvious, just as obvious is the connection between theoretical acoustics and the theory of other kinds of wave motion. Rayleigh’s law for the number of electromagnetic waves in a given frequency range was first established for the acoustic case R3, R4. A more precise determination of this number was also originally carried out for acoustic purposes B6, M1, R8, and recently these expressions have been used H10 in the corpuscular statistics of Bose–Einstein. Such a connection also exists in the opposite direction: certain theoretical methods of wave mechanics have been applied to theoretical acoustics M11, R8. Many of the results cited in this review, in turn, may find application in other fields of physics.

Sound waves have special significance for the study of wave motion in general. First, sound waves have a suitable wavelength. They are not as long as ordinary radio waves, and not as short as light waves or matter waves, which makes it possible to study the details of wave motion by direct and visual methods. This advantage is of greater importance than the fact that the disadvantage of sound waves is that their power is very small. Second, objects which usually reflect and absorb sound have dimensions approximately the same as sound waves S7. It is precisely under these conditions that the most complex phenomena arise. Many questions in this field have remained unresolved. The fact is that the necessity of studying them in other fields is not so great, and the efforts required for their resolution would not pay off. In acoustics, however, these questions cannot be avoided if the aim is to obtain any scientific result at all.

2. Geometrical and Wave Acoustics

One curious circumstance should be noted: the ideas that were developed by the first investigators in the field of room acoustics, and that are still used in engineering practice, completely neglect the wave properties of sound. Here, in a field in which the wave properties of the phenomenon are continuously manifested, the ray picture, using geometrical reflection, has found wide application. The reason is that analysis of the wave process presents extreme difficulties, and without considerable schematization it would be impossible to obtain any preliminary result. Sabine^S1, of course, observed phenomena arising from the wave nature of sound; he pointed out a certain influence of diffraction and interference, but he neglected these complicating circumstances. They are still neglected by the majority of acoustical engineers.

It is therefore not surprising that the acoustical engineer must, in order to make his formulas practically useful, season them with large doses of “common sense.” However, despite the dubiousness of the formulas, the modern acoustical engineer, with the aid of common sense, successfully carries out the calculation of auditoriums and service rooms. Knudsen says^K7: “Approximate theories solve practical problems satisfactorily when they are used with caution and with an understanding of the matter. They lose their significance when they are replaced by more exact theories.”

Such a state of affairs may satisfy the engineer, but it is completely unsatisfactory from the point of view of the physicist. In order for room acoustics to develop fruitfully for physics, it is necessary to pay more attention to the wave nature of the question^K5, S4. It is necessary to study the forms of the natural oscillations of the air in such rooms as do not have simple outlines. It is necessary to study the influence of the distribution of absorbing material and of irregularities in the outline of the walls on the distribution of sound in the steady state. It is important to study exact solutions of the equations of non-steady sound oscillations in rooms with absorbing walls. In this case the boundary conditions depend on the frequency of the oscillations, the characteristic numbers (frequencies) are complex, and the fundamental functions do not form an orthogonal system. It is necessary to study the case when several forms of natural oscillations exist simultaneously, each form having its own damping exponent, while in the overall damping interference processes are manifested. It is necessary to investigate the phenomena of sound absorption by various materials, to formulate this absorption in the form of boundary conditions for the wave equations, and to choose the best method for the quantitative measurement of the effects obtained. In the last ten years many of these questions have been developed, and below a survey of the results achieved is given.

After a brief summary of the results obtained by geometrical acoustics, we shall consider the principles on which wave acoustics rests, as well as the additions and clarifications that it introduces into the results obtained by geometrical methods. At the end of the survey we shall present the latest results for certain problems of wave acoustics and note questions that still require resolution.

3. Reverberation time and acoustic criteria

Although the present survey is devoted mainly to the above-mentioned physical problems, it is necessary to devote some space to the consideration of technical questions, since it is precisely they that determine the relative importance of the physical problems. However, in connection with technical questions it is also necessary to touch upon aesthetic and psychophysiological questions. Before undertaking the acoustic calculation of an auditorium, one must know by what the “good” qualities of an auditorium are determined. Sabine S1 partially answered this question. He showed that one of the important criteria is the behavior of the auditorium under steady-state conditions. He distinguished between incorrect processes producing echoes and the proper decay of sound after the source has ceased to act, which he called reverberation. For a numerical characterization of this important concept he defined the reverberation time as follows: it is that interval of time during which the mean-square pressure of a suitably selected distribution of sound waves decreases to one millionth of its initial value. By comparing opinions, he established what values of the duration of reverberation in various rooms may be considered satisfactory. Other investigators expanded these data. In the following sections some criteria are considered by which modern engineering practice is guided.

a) Criteria for evaluating the audibility of speech. The intelligibility of speech in rooms was quantitatively investigated by Knudsen K3 and Lifshits L3. They used the methods of “articulation tests” F3, developed in the study of telephone lines. In this case the articulation percentage is determined statistically, as the average percentage of correctly understood speech sounds in observations among a certain number of listeners and speakers, using specially prepared lists of typical syllables*).

*) The generally accepted scale for determining the articulation percentage is as follows:
Articulation percentage = 96% — “perfect intelligibility”: certain sounds are perceived incorrectly even under ideal conditions, but their meaning is clarified by the context.
85—96%. Very satisfactory intelligibility.
75—85%. Satisfactory intelligibility.
65—75%. Speech is intelligible with normal hearing and concentrated attention.
Below 65%. Unsatisfactory intelligibility.

SOUND WAVES IN ROOMS

The percentage of articulation in a room is computed by means of a series of empirically determined factors:

\[ \text{Percentage of articulation}=96\, k_l k_r k_n k_s . \tag{1.1} \]

Here \(k_l\) determines the decrease in intelligibility as a function of loudness, \(k_r\)—of reverberation, \(k_n\)—of noise, \(k_s\)—of the shape of the room. The values of \(k_l\), \(k_r\), and \(k_n\) are given in Figs. 1a, 1b, and 1c according to Knudsen’s data \({}^{3}\).

As is seen from Fig. 1c, \(k_l\) falls rapidly when the average loudness drops below approximately 40 decibels. This circumstance is due above all to psychological causes, for example, the strain of attention, and, secondly, to physiological causes; among the latter is the change of the ear’s sensitivity with frequency, as a result of which certain weaker sounds (\(b, v, th\), etc.) fall below the threshold of audibility earlier than other sounds. On the other hand, we see that \(k_r\) decreases with increasing reverberation time. This circumstance is due to the mutual superposition of sounds.

Fig. 1a. Factors determining the percentage of articulation in rooms (according to Knudsen, reference \({}^{3}\)). Decrease in articulation as a function of noise.

Fig. 1b. Decrease in articulation as a function of reverberation.

Fig. 1c. Changes in articulation as a function of loudness.

In Fig. 1b are shown the experimentally found values of the function \(k_r\) for a frequency of 512 hertz. We shall return later to the changes of reverberation with frequency.

The reverberation time \(T\) and the sound intensity \(I\) in the steady state are connected by the following (asymptotic) relation:

\[ I=\frac{\Pi}{KV}\,T . \tag{1.2} \]

Here \(\Pi\) denotes the emitted sound power, \(K\)—a constant depending on the choice of units, \(V\)—the volume of the room. Subjective ...

quantity—loudness—is a monotonic, but not linear, function of the objective quantity—the sound intensity. This function depends on frequency, as is shown in Fig. 2.

As a result of the opposite influence of the quantities \(k_l\) and \(k_r\), connected by equation (1.2), and of the functional dependence depicted in Fig. 2, there is obtained a certain optimum reverberation time for a room of given volume. The optimum reverberation time is that value of \(T\) for which the articulation percentage becomes maximal; this value varies with the volume \(V\). In Fig. 3a the lower curve, marked by the word “speech,” represents the optimum reverberation time as a function of volume, according to Knudsen’s data. This curve takes into account only the influence of the factors \(k_l\) and \(k_r\) mentioned above; it does not take into account the role of the factors \(k_n\) and \(k_s\).

Fig. 2. Lines of equal loudness levels in the plane of variables: sound-intensity level, frequency.

As is seen from the curve for \(k_n\) in Fig. 1a, noise always reduces the articulation percentage, since it masks the sounds of speech. However, this reduction depends to a certain extent on the ratio of the loudness of the noise to the loudness of the speech. Therefore the influence of noise can be reduced by increasing the loudness of the speech up to a limiting value of about 80 decibels, above which \(k_l\) begins to decrease.

The influence of the form of the room has not been so well clarified and requires further quantitative investigations. In ordinary rooms of rectangular form, \(k_s\), apparently, does not differ noticeably from unity. In very large halls or in unsuccessfully designed

form of the room, it may decrease to 0.9. In small rooms with properly selected reflecting surfaces \(k_s\) may increase to 1.05. Some of the new studies reviewed below have substantially broadened our knowledge of the influence of room shape on the physical properties of sound. It is to be hoped that further study will make it possible to clarify the question of the relation of room shape to the conditions of audibility in it.

Fig. 3a. Criteria for reverberation duration (\(K^3\)). Optimum reverberation duration for auditoriums and concert halls at 512 Hz.

New studies in the field of room acoustics apparently, on the whole, confirm the material presented in the figures given, in particular with respect to the relation between the percentage of articulation and \(k_r\). First of all, there is a growing tendency to regard the decay coefficient*) as a more important quantity than the reverberation time. The point is that the decay coefficient has meaning even in the case of nonlinear decay, whereas \(T\) is defined for linear decay and corresponds to a decrease in loudness by 60 decibels. As will be indicated elsewhere, strictly linear decay is more the exception than the rule. In practice one encounters both sharp breaks in the curve and small superposed oscillations. There are indications that the first 30 or 40 decibels in the decay process have the greatest influence on the quality of sound. Thus it is quite probable that a room possessing considerable initial decay and a long tail of the decay curve, where its slope is small, will have a higher percentage of articulation than a room with linear reverberation having everywhere, up to 60 decibels, the same decay coefficient. Yet, on the basis of the considerations set forth above, we would have to assign \(k_r\) the same value in both cases. It is also possible that changes in the decay affect audibility as well—

*) In this review the term “decay curve” means the curve of the dependence of the natural logarithm of the mean quadratic value of pressure on time. The “decay coefficient” means the mean steepness of this curve. In some cases the decay curve is plotted in decibels (a tenfold decimal logarithm), but the decay coefficient always corresponds to the steepness for the natural logarithm.

speech, although, apparently, this effect is of greater significance for the perception of music, as will be indicated below. In addition to the direct influence of the shape of the decay curve on the percentage of articulation, one may expect that a detailed study of the decay curve will reveal new acoustic features which, in turn, affect the conditions of audibility. For example, strong interference phenomena, as well as the degree of mixing of sound energy, influence the shape of the decay curve. The steepness of this curve and its oscillations are determined by the quantity and placement of absorbing materials, as well as by the shape of the room and by deviations from the regularity of that shape. This question will be discussed in Chapter VII.

Fig. 36. Optimal frequency characteristic of reverberation, referred to the characteristic for the frequency 512 Hz, taken as unity.

The foregoing considerations show that \(k_r\) and \(k_s\) are not entirely independent of one another. In fact, it is necessary to change the functional form of equation (1.1) in order to express more precisely the dependence of the percentage of articulation on the physical properties of the room. At present, however, this problem has only been posed. A great deal of experimental work must be done before equation (1.1) can be replaced by a more perfect one. This equation, when used consciously, is very useful in the calculation of auditoriums.

b) Criteria for evaluating the audibility of music. The evaluation of the audibility of speech in rooms has been reduced to a fairly precise engineering calculation to a considerable extent because an objective measurement of the percentage of articulation is applicable here. The evaluation of musical audibility is more difficult, and no simple, well-substantiated criteria have yet been established for it. It is difficult here to apply an objective evaluation. Subjective evaluations differ considerably from one another and depend on the experience of the expert and on his musical habits. Nevertheless, it may be assumed that the results of the latest investigations will be of greatest use precisely in the design of concert halls.

The basic requirements here are similar to those for speech: a) sufficient loudness, b) absence of extraneous noises, c) absence of frequency and amplitude distortions, and d) separation in time of successive sounds to the extent dictated by aesthetic requirements.

Of greatest importance are the last two requirements, especially the very last one. It brings the reverberation time to the foreground. For the perception of music, the same factors are essential as for the perception of speech. One may expect that for music as well these factors will lead to the establishment of an optimum reverberation time and that this optimum time will depend on volume. But here the analogy ends. Here we are so closely bound up with considerations of musical taste that we must beware of “quantitative” measures. In practice at present the conditions shown in Fig. 3a have been adopted. As we see, here the optimum values form hatched areas, whereas for speech they are represented by curves. This is due to the fact that in a given room the reverberation time most pleasant to the ear depends on the kind of musical work. Lively, light, fast music generally requires a shorter \(T\), whereas broad, flowing music sounds better with longer reverberation times. No kind of music requires a \(T\) as short as that for speech. Fig. 3a refers to a frequency of 512 hertz.

The dependence of the optimum reverberation time on frequency has been investigated by many authors \(^{W2, L2, L4, S10, L3}\). Knudsen \(^{K1,K3}\) assumes that \(T\) should be chosen so that all frequency components of the sound decay simultaneously to the threshold of audibility. MacNair \(^{W4}\) assumes that the attenuation index for all components should be the same. Other criteria have also been proposed. Many of them proceed rather from the convenience of acoustic treatment than from any rational basis. In Fig. 3b the frequency variation of some of these criteria is shown. In engineering practice any of them is successfully applied. Recently there has been a tendency to prefer an almost horizontal frequency characteristic of reverberation, especially in rooms with good sound diffusion \(^{H2}\). This relation between the degree of diffuseness of sound and optimum reverberation will be examined in Chapter VII. It is also observed that the ear readily reconciles itself to a long reverberation time in well-planned rooms, with surfaces “broken up” to increase the scattering of sound \(^{MS}\). In these cases the “liveliness” of sounds increases without detriment to clarity or intelligibility*).

c) Modern tendencies. At the present time it is considered that reverberation time is not always a sufficient measure of the quality of an auditorium. It is also necessary that the mean-square pressure be as nearly as possible the same over the entire area occupied by the listeners. It is also necessary that not less than a certain percentage

*) Maxfield arbitrarily defines the “liveliness” of a sound \(L\) by the equation:

\[ L = KT^2 d^2 / V. \]

Here \(K\) denotes an empirical constant, \(T\) is the reverberation time, \(d\) is the distance between the sound source and the ear, and \(V\) is the volume of the room. He found that in some cases the optimum curve in the coordinates \(T, V\) (Fig. 3a) runs as though the ear preferred equal livelinesses of sound.

sound energy should reach the listener directly from the speaker, and no more than a certain percentage of the sound energy should reach the listener after reflection from the walls. These requirements form part of the concept of the “liveliness of sound.” It is now well known that sound in many rooms decays nonlinearly, so that the term “reverberation time” becomes not entirely definite. Therefore, in this survey we shall consider the decay index, i.e., the slope of the decay curve—a concept that is meaningful also in the case of nonlinear decay. If one excludes the superposition of small fluctuations, which make the decay of sound in a room more pleasant to the ear, then for good acoustics it is apparently necessary that the decay index remain constant over the first 30–40 decibels. These auxiliary criteria have not yet been reduced to quantitative statements, but it may be hoped that studies will soon be carried out to fill this gap. For this purpose, articulation experiments are needed in rooms with specially adjustable acoustic properties, as well as quantitative experiments characterizing the conditions for listening to music.

II. GEOMETRICAL ACOUSTICS OF ROOMS

Before the twentieth century, information on room acoustics was scanty and qualitative in character^A55, S7, K3, W1^. Perhaps the most remarkable observations were made (1854–1856) by Joseph Henry^H3^, who investigated questions of echo, reverberation, resonance, and the shape of rooms, insofar as these affect acoustic properties. Many of his conclusions, although qualitative in nature, were based on experimental observations. Other physicists of that time, in particular Tyndall and Rayleigh, studied questions concerning the media of action on sound in rooms. However, no systematic investigations had been carried out at that time.

4. Absorption coefficient

The first quantitative investigations in room acoustics were begun in 1896 by Wallace Sabine^S1^. His immediate task was to study the conditions of audibility in the newly built auditorium of the Fogg Art Museum at Harvard University. From this particular problem he began the general study of the sound properties of rooms. By means of cleverly designed experiments and inductive reasoning, Sabine arrived at the now well-known theory of reverberation and at the formula

$$ T = KV / \sum a_j S_j . $$

Here \(T\) denotes the reverberation time in seconds (as it was defined above), \(K\) is a constant depending on the choice of units, \(V\) is the volume,

$S$ is the surface bounding the room. The summation in the denominator extends over the various kinds of materials covering the walls, floor, and ceiling. $S_j$ is the area of each type of material, and $\alpha_j$ is a constant characterizing each material. Sabine called this constant the absorption coefficient and defined it as the mean value of the ratio of the sound energy absorbed in the material to the sound energy incident upon it.

The details of Sabine’s experimental work have been discussed by many authors K3, W2, S5, and we shall not touch upon them here. Instead, we must review his most important theoretical concepts and the conditions for their applicability, since they underlie geometrical theory and its essential shortcomings. From the modern point of view, the technical part of Sabine’s apparatus was very primitive. Organ pipes were used as sound sources. The ear served as the sound receiver, which was considered quite acceptable, since the entire series of measurements was carried out by one and the same experimenter. The duration of the decay of sound was determined by means of a chronograph. Absorption was produced by a set of a large number of small, uniform absorbers. These remarks are made not at all in order to diminish the great importance of Sabine’s work, which served as the guiding basis for acoustic calculations of rooms for forty years. They are made in order to emphasize that the approximate geometrical approximation, which was suitable for describing the experimental results obtained by Sabine in one particular case, is by no means always applicable.

5. Sabine’s Approximation

Before undertaking a detailed study of reverberation, Sabine investigated possible sources of experimental error. He arrived at the following conclusions: 1) The duration of the residual audibility of sound is almost the same at all points of the auditorium. 2) This duration is almost independent of the position of the source. 3) The role of the absorbing material in shortening the period of audibility under ordinary conditions is almost independent of its location. These propositions, as we shall see, are necessary conditions for the applicability of geometrical methods in room acoustics. They were due, on the one hand, to the experimental method and, on the other, to the properties of the room that was subjected to investigation. If Sabine had had at his disposal, for measuring sound intensity, fast-acting instrumental means instead of the ear, he would have observed significant differences in sound intensity of both a spatial and a temporal character. The averaging properties of the ear, which smoothed out variations of sound intensity in space and in time, allowed him to adopt his simplified scheme. The rooms he examined were all of moderate size and sufficiently resonant; the reverberation time usually exceeded

... was 1.5 sec. If more varied rooms had been examined, the simple conclusions listed above would not have been obtained.

Of great interest is Sabine’s choice of the standard unit of absorption. He took an open window as a perfect absorber of sound and assigned to it an absorption coefficient \(\alpha = 1\). Subsequently, numerous absorbing materials were studied, and the area of open windows was determined that would give the same reverberation time as an absorbing material of a specified size. Sabine accurately established the limits of applicability of the open-window standard; in particular, the effect of diffraction was manifested in the fact that a small window absorbed sound comparatively more strongly than a large one. In reality this limitation applies to all absorbing materials of small area in general, the influence of diffraction being greater for strongly absorbing materials than for materials with a small absorption coefficient. This drawback, however, does not appear strongly so long as the areas of the windows or absorbing materials are not too small. Sabine came to the conclusion that, with the same degree of accuracy as is attainable when other factors that play a role in reverberation measurements are taken into account, the effectiveness of an absorbing material, under ordinary conditions, does not depend on its area. This enabled him to use the simple equation (2.3).

6. Geometrical theory

Soon after Sabine’s empirical substantiation \(^{S2}\) of the theory of reverberation, Eger arrived at the same equation by arguments similar to those used in the classical kinetic theory of gases \(^{J1, A1, F5}\). Subsequently, other methods of derivation were given by Eckhardt \(^{E1}\), Buckingham \(^{B14}\), and others \(^{C5, C6, D2, F4, S15, S16}\). These derivations are based on certain simplifying assumptions (necessary for the application of statistical methods), which rest on Sabine’s experimental observations. In the main these assumptions are as follows: 1) a homogeneous diffuse distribution of sound energy throughout the room at any instant; 2) equal probability of sound propagation in any direction; 3) continuous absorption of sound energy at the boundaries of the room. It is evident that this scheme is strictly geometrical: sound energy propagates by rays, and wave phenomena play no role. These assumptions are valid with the same degree of accuracy as Sabine’s experiments, in which the averaging properties of the ear had an effect.

The stated assumptions lead to a simple differential equation following from the law of conservation of energy:

\[ \left[ \begin{array}{c} \text{Increase of sound}\\ \text{energy in the}\\ \text{room.} \end{array} \right] = \left[ \begin{array}{c} \text{Amount of energy}\\ \text{radiated by the}\\ \text{source.} \end{array} \right] - \left[ \begin{array}{c} \text{Amount of energy}\\ \text{absorbed by the}\\ \text{walls.} \end{array} \right] \]

Let \(W\) denote the energy density (assumed everywhere the same), \(V\) the volume of the room, and \(\Pi\) the power of the source. Then \(V \dfrac{dW}{dt}\) represents the rate of increase of energy. The total amount of absorbed energy is computed by determining the absorbed fraction of the energy incident per second on a unit surface from one definite direction, and integrating this quantity over all angles of incidence. Then the amount of energy incident on a unit area is equal to

\[ \frac{W}{4\pi}\int_{0}^{c} dr \int_{0}^{2\pi} d\theta \int_{0}^{\frac{\pi}{2}} \cos \varphi \sin \varphi\, d\varphi = \frac{Wc}{4}. \]

If \(\alpha\) denotes the fraction of the incident energy that is absorbed by a surface of area \(S\), then the absorbed energy will be equal to \(\dfrac{Wc}{4}\alpha S\).

We obtain the following differential equation:

\[ V\frac{dW}{dt}=\Pi-\frac{c\alpha S}{4}W. \]

If the sound source is switched on at the moment \(t=0\), then the solution of the equation will be

\[ W=\frac{4\Pi}{c\alpha S}\left[1-e^{-\frac{c\alpha S}{4V}t}\right], \tag{2.1} \]

and if the source is switched off at the moment \(t=0\), then

\[ W=\frac{4\Pi}{c\alpha S}e^{-\frac{c\alpha S}{4V}t}. \]

As a measure of the rate of establishment of the stationary regime one may take the attenuation constant

\[ k=\frac{c\alpha S}{8V}. \tag{2.2} \]

However, more often the reverberation time is chosen as the measure of the time of establishment of the regime. This is the time during which the energy decreases to \(10^{-6}\) of its initial value:

\[ T=\frac{4V}{c\alpha S}\ln(10^{6})=\frac{KV}{\alpha S}. \tag{2.3} \]

In English units \(K=0.049\), in metric units \(K=0.161\). This is the same equation as that obtained experimentally by Sabine. It again emphasizes the fact that Sabine’s scheme has a geometrical and statistical character.

7. Other formulas of geometrical acoustics

The first important modification of Sabine’s theory was the replacement of the assumption of “continuous absorption” by another assumption, in which a sudden decrease of the sound intensity was allowed at the moment of absorption by the wall.

Eyring E3 obtained the following final formula:

\[ T=\frac{KV}{-S\ln(1-\alpha)}. \tag{2.4} \]

Sabine’s assumption of a uniform distribution of sound energy and of the random character of the process is retained here; however, the assumption is introduced that sound energy propagates without attenuation over a certain mean free path and then suddenly decreases by a definite fraction depending on the absorption coefficient of the wall. The form of the equation obtained is similar to Sabine’s, but the simple absorption coefficient is here replaced by a logarithmic function. A graph of this function is given in Fig. 4. For large absorptions, equation (2.4) gives values differing from those given by equation (2.3) by more than 100%. Measurements carried out in highly damped rooms show that for these cases Eyring’s formula is far more accurate than Sabine’s simple formula.

Fig. 4. Ratio between the Sabine and Eyring formulas for the duration of reverberation.

Eyring’s formula can be derived very simply by the method proposed by Norris N1. On the average, every time the wave front meets a wall, some fraction of the incident energy \(\alpha\) is absorbed, and \(1-\alpha\) is reflected. The mean free path is equal to \(4V/S\), as follows from Sabine’s analysis S1. The mean number of reflections of the wave during the time \(t\) is equal to \(Sct/4V\), where \(c\) is the speed of sound. After the time \(t\), the sound intensity will be equal to

\[ I=I_0(1-\alpha)(1-\alpha)(1-\alpha)\ldots =I_0(1-\alpha)^{Sct/4V} =I_0\exp\left\{\frac{Sc\cdot\ln(1-\alpha)}{4V}t\right\}. \]

Putting \(I/I_0=10^{-6}\), we obtain the reverberation time according to equation (2.4). In paragraph 53 of the present review we shall discuss the inaccuracies inherent in this method of analysis.

The Eyring formula includes the “mean absorption coefficient.” It is assumed that the walls of the room are uniformly covered with a material of identical absorption, or that different absorbing materials are distributed sufficiently uniformly over all the walls, so that the mean value of the absorption may be taken. In this case

\[ a=\frac{\sum a_i S_i}{\sum S_i}. \]

For a very nonuniform distribution of the absorber—for example, when one wall absorbs strongly while the others reflect well—the Eyring formula gives significant errors. This contradiction led Millington^M7 and Sette^S12 to another method of averaging the absorption coefficient. Their equation is

\[ T=\frac{KV}{-\sum S_i \ln(1-a_i)} . \tag{2.5} \]

The main divergence arises in the following point: Eyring’s theory assumes that the sound energy in the room preserves a uniform distribution after each encounter with a wall throughout the entire process of decay; Millington and Sette trace the history of a bundle of sound rays over the course of many reflections and assume that, on average, each ray encounters some surface a number of times proportional to its area. Both theories adopt the Sabine geometrical conditions, but they use different methods of averaging. Eyring uses an arithmetic averaging over the absorbing surface; Millington and Sette use a geometric one. Since the geometric mean is always less than the arithmetic mean, the reverberation times and absorption coefficients obtained from the Eyring and Millington–Sette formulas are related as follows:

\[ T_{\mathrm{MS}}>T_E,\qquad a_{\mathrm{MS}}<a_E. \]

An important shortcoming of the Millington–Sette formula is that it gives \(T=0\) if any arbitrarily small surface has a significant absorption coefficient.

Thus, we see that each of the three equations indicated may, in particular cases, lead to gross errors. This confirms the correctness of Knudsen’s remark concerning “caution and understanding,” noted in the preceding section.

All the theories of reverberation discussed up to this point assume that energy losses occur only at the walls. In reality, energy is also partly dissipated in the air of the room. Below 1000 hertz, absorption of sound energy in air may be neglected, but it increases progressively with increasing frequency. In some cases absorption in air at frequencies above 4000 hertz may exceed many times the total absorption of sound by the walls of the room. This

absorption in detail was studied by Knudsen K2, K6. He incorporated his results into the theory of reverberation, making use of Eyring’s formula as having wider applicability:

\[ T=\frac{KV}{-S\ln(1-\alpha)+4mV}. \]

If \(V\) is expressed in \(m^3\), \(S\) in \(m^2\), then \(K=0.161\); \(m\) is the absorption coefficient for plane waves in air (in \(m^{-1}\)), according to the formula \(I=I_0 e^{-mx}\). The coefficient \(m\) depends on frequency, humidity, and temperature K3.

For the proper application of these geometrical formulas it is often necessary to trace the “rays” of sound in a room and establish how they are reflected from the surfaces bounding the room. In this way it is possible to reveal the focusing action of curved walls, and also to determine how strictly each individual room satisfies the requirements of a uniform distribution of sound energy. These requirements determine the applicability of the geometrical formulas. For this purpose many model experimental studies were carried out with capillary waves in vessels, photographed by means of a spark A. The tracing of sound rays was also used to determine the ratio of reflected sound energy to the energy directly reaching the listener at various places in an auditorium A, K3.

8. Measurements of reverberation

It is necessary to give a brief survey of the experimental methods of room acoustics in order to note the role that measurement technique played in revealing the imperfections of the geometrical theory. The experimental details are given in full in numerous articles A, A1, C1, H5, H6, H7, O1, S13, W5, W6, W7 and are concisely set forth in some books A.

Generally speaking, measurements in architectural acoustics pursue two aims: (a) the determination of the absorptive properties of materials and (b) the measurement of the acoustic properties of the rooms themselves. In many cases the same methods are used for both purposes.

The first measurements were the Sabine measurements S1 described above. In them, the ear and a stopwatch were used for recording and measuring the duration of sound decay. This method was improved by many investigators and led to Hunt’s automatic apparatus H5, which contains many refinements. The sound source is a warble-tone generator (periodic variation of frequency within a narrow range). It excites in the room a whole band of frequencies and thus smooths the decay process. In addition, to reduce experimental scatter the following measures were adopted: (a) switching off the loudspeaker in one and the same phase at each...

measurements; b) rectification of the output microphone current and filtering out the envelope of rapid variations superposed on the mean decay; c) carrying out measurements at different microphone positions; and d) averaging a large number (40 or more) of separate trials for each case. In Hunt’s method an automatic device is used which switches off the sound source at the moment when the loudness level in the room reaches a specified value, switches it on again when the loudness level has fallen to a specified value, and records the duration of the decay process. These operations are repeated cyclically several times, and in this way the average duration of the decay process is obtained directly. An example of a decay curve obtained in this way is shown in Fig. 5.

Fig. 5. Decay curve, as the mean time of decrease of loudness within specified limits. Room unfurnished, volume \(28\ \text{m}^3\). Four microphone positions. Frequency 200 cycles. Slope of the straight-line portion \(23.6\ \text{db/sec}\). Mean deviation from linearity \(0.006\ \text{sec}\).

An entirely different method of observing sound decay by means of an oscillographic recording was used by Knudsen and others\(^{K3}\). In this case the entire oscillatory process is reproduced as a whole, and the decay of the sound is clearly represented by the fall of the envelope of the oscillatory curve. In these recordings, shown, for example, in Fig. 25, details of changes in sound intensity and frequency stand out. If it is necessary to obtain an “average” value of the decay, it is determined graphically. Both mechanical and cathode-ray oscillographs are used, and a high recording speed can be obtained.

The third type of instruments, developed by Wente, Bedell, and others\(^{S13, W6, W7}\), is a fast-acting apparatus for recording the loudness level. In its most improved form it contains a light lever with a stylus; the latter traces a curve on a moving strip of smoked paper, the position of the lever and stylus being controlled by the amplitude of the signal at the input of the apparatus. With such an apparatus, decays of up to 600 decibels per second can be recorded. Although this apparatus is capable, at the greatest speed of paper motion,

to record rapid fluctuations of sound intensity; nevertheless, it cannot reproduce all the details of the sound oscillatory process in the way an oscillograph can. By changing the recording speed, one can record the oscillations with a greater or lesser degree of detail; in other words, one can obtain a certain degree of smoothing of the curve W7. Thus the decay curve of the sound, which in an oscillographic recording has a very tortuous appearance, is obtained here in the form of a slightly wavy line marking only the largest fluctuations (see Fig. 6).

Fig. 6. Decay curve at 500 cycles, traced by a fast-acting recorder at four different recording speeds. All the curves have the same mean decay coefficient W6.

The same methods of measuring reverberation in rooms can be used to determine the absorption coefficient of acoustical materials, according to equation (2.4). This procedure was standardized in many laboratories, and in all cases the same area of absorbing material was used (6.5 m²), always placed in one and the same position in the given room. In order to obtain the random distribution of sound required by geometrical theory, sound-scattering devices were used, for example large rotating plywood stirrers.

The experimental techniques of the period of dominance of geometrical acoustics were not limited to the study of steady-state regimes. Measurement of the steady-state sound intensity was applied by Knudsen K4 to determine the absorption coefficient of materials. His method (the “intensity method”) is based on that term of equation (2.1) which corresponds to the steady-state regime. It follows from this equation that the mean square pressure in the steady-state regime is equal to

\[ \overline{(p^2)}_{\mathrm{av}}=\frac{4\rho c}{aS}\,\Pi . \tag{2.6} \]

Here \(\rho\) denotes the density of air, \(\Pi\) is the power at the output of the sound source (ergs per second). In addition, it is assumed that the density of sound energy in the room is equal to \(p_{\mathrm{av}}^{2}/\rho c^{2}\). The application of this equation is subject to the same restrictions as equation (2.3).

In the experimental use of equation (2.6), a comparative method is employed. It is necessary first to know the absorption of the empty room or the absorption of a “standard specimen.” Then the absorption of another specimen is determined directly from the ratio of sound intensities. Although this method has not come into wide use, it

has its experimental advantages. They consist in the fact that high accuracy can easily be achieved, and also in the fact that measurements can be made at a high loudness of sound, considerably exceeding parasitic noises*).

However, this method is subject to all the same essential limitations as any method based on geometrical acoustics. This circumstance will be considered in detail in Chapter V.

Another method of measurements in the steady-state regime was proposed by Wente W8,H7. He regards the room as an acoustic “transmission line” and measures the transmission constant as a function of frequency with the aid of a loudspeaker placed at one point and a microphone at another. From the geometrical point of view, only a qualitative interpretation can be given of oscillations in the value of the transmission constant. However, the results obtained in this way are very instructive from the wave point of view, as will be shown in Chapter V.

Measurements of the absorption coefficient were also carried out by various variants of the “tube method,” in which standing waves and a small specimen are used D3, L5, P3, T1, T2, M9, P1. This method requires only a simple measurement of the ratio of the maximum pressure to the minimum in the standing wave. Here it is comparatively easy to achieve good accuracy and reproducibility of the results. However, the absorption coefficients measured by this method are not in good agreement with the results of reverberation measurements, except in some cases. The reasons for this discrepancy have now been clarified. In the following chapters we shall see that wave theory, as applied to the tube method, leads to very useful techniques for measuring acoustic impedance. This impedance, in turn, can be used for calculating various coefficients characterizing absorption and permitting calculations of the acoustic process in rooms of large dimensions.

9. Critique of the geometrical theory

The wide popularity of the theory of reverberation as applied to the acoustic correction of auditoriums and halls, and the growing demand for acoustic calculations for new buildings, clearly show the full importance of the field whose beginning was laid by Sabine. Applying the theory of reverberation, which rests on a few simple conditions fully confirmed by broad practice, it is now possible to predict accurately the acoustic properties of large auditoriums, music halls, assembly halls, theaters, and studios. Taking greater—

*) In reverberation methods it is very difficult to achieve sufficient elimination of parasitic noises, especially in those cases where measurements over a wide range of decays are involved. These difficulties led to the development of costly constructions for the sound insulation of reverberation chambers.

...number of constraints and empirical rules, one can also calculate meeting rooms and small music halls, radio studios, etc.; however, these cases require a certain intuition, which an engineer can acquire only as the result of long practical experience in the calculation, design, and testing of rooms.

As for the struggle against noise in rooms, reverberation theory is applicable in most cases even to large offices, industrial enterprises, and also other rooms of corresponding dimensions. In this case it is assumed that the absorbing material is distributed “in the proper manner.”

However, numerous discrepancies are observed between the conclusions of reverberation theory and the results of measurements. The calculated reverberation times sometimes differ from reality by 20%–50%, regardless of the formula used. In individual cases, for example in very small rooms at low frequencies, the errors may reach several hundred percent.

In rooms of complex outline—for example, in halls with a strongly reverberant stage, or in large round buildings with a domed ceiling—the reverberation curve may consist of two parts with very different decay exponents. Reverberation in coupled volumes has also been studied C5, but ideas here are still far from complete.

The greatest difficulties in applying reverberation theory in rooms arise when the simple geometrical assumptions are not satisfied. Often sound waves are distributed not in random directions and not uniformly over the volume of the room. Therefore, in these cases the decay of sound does not proceed according to a logarithmic law; its course is distorted both by small irregularities and by sharp breaks, and its magnitude is different in different parts of the room. From the geometrical point of view, the most important observed cases are those in which the effective absorption of the acoustic material depends on its area and its location in the room. Numerous measurements have been devoted to these effects—of “area” and “placement”—A2, C2, D4, E4, P7, R2, S3, but a numerical calculation of these effects can be expected only from the development of wave acoustics.

Perhaps the most serious shortcoming of the geometrical approximation was its inability to provide reliably comparable, standard methods for measuring the absorption coefficient of materials. Hunt, in a recent survey of this topic, speaks of the “absorption-coefficient problem” H8 and summarizes the answers received from many acousticians to a circulated questionnaire in the following propositions: 1) The coefficients of the same materials, measured in different laboratories, do not always coincide. 2) Measurements in a sound field give smaller values of the coefficients than laboratory measurements. 3) An increase...

the size of the specimen lead to a smaller value of the coefficient; however, this circumstance is insufficient to explain the discrepancies noted earlier.

P. E. Sabine^S7, Eyring^E5, and others^S14 discussed the problem of measuring absorption from various points of view. If any arbitrary method is adopted as a standard for different laboratories, then the discrepancies in the coefficients are considerably reduced, especially at high frequencies. But even when all proper precautions are taken, different laboratories obtain absorption coefficients that sometimes differ by 20 percent.

III. GENERAL PRINCIPLES OF WAVE ACOUSTICS

The data presented show that some problems of room acoustics can be solved only with allowance for the wave nature of sound. The fact that wave properties play an essential role in acoustics has long been known^S1, S10, S15. However, the importance of this role has been clarified only recently. Knudsen^K5 was the first to show experimentally that reverberation sounds do not have the frequency of the sound source that excited the reverberation phenomenon, but rather frequencies corresponding to the natural frequencies of rooms. In some cases his measurements revealed several characteristic frequencies. The resulting beats caused a substantial deviation of the sound-decay curve from the usual exponential form. Wente^W8 investigated the loudness of sound in a room in the steady state as a function of the frequency of the source and observed sharp resonance peaks. Many other experimentalists and theorists^B5, B6, B8, H9, M2, M11, M12, S3, M1, M6 studied questions of wave acoustics from other points of view, and their work has considerably enriched this field, although many questions still remain unresolved.

In the entire problem of wave acoustics, which constitutes the main content of the present review, three questions stand out: the interaction between a sound wave and the boundaries—the walls of the room; the steady sound regime in a room; and the character of transient processes, in particular, reverberation. Each of these questions will be illuminated to some extent in the present review. It is useful to begin with the preliminary establishment of certain general propositions.

10. Acoustic impedance

It has become clear that the absorption coefficient is not an unambiguous characteristic of the acoustic properties of a wall surface. The experimental work of Hunt and others^B3, B5, B12, C7, H9 has shown that a more fundamental quantitative characteristic of a surface is its acoustic impedance \(Z\). This impedance is defined

as the complex ratio*) of the sound pressure at the surface to the normal component of the air velocity in the immediate vicinity of the surface. This normal component of the velocity is due either to the motion of the wall itself or to the motion of the air in its pores. In both cases we may speak of a small acoustic impedance for a “soft”—yielding—wall, or of a large acoustic impedance for a “hard” wall. If the impedance has a real component, then the wall absorbs sound energy; a purely reactive impedance means that only a phase jump occurs upon reflection.

The natural unit for \(Z\) is the acoustic resistance \(\rho c\) of air for free plane waves, which is approximately equal to \(42\ \mathrm{g}/\mathrm{cm}^2\) per sec. Let us denote the impedance expressed in these units by \(\zeta=(Z/\rho c)=(1/\rho c)(R-iX)\), and call it the specific impedance of the material. The absorbing properties of a surface are measured by the specific admittance \(\beta=(\rho c/Z)=\gamma-i\sigma\), where \(\gamma\) denotes the specific active conductance, and \(\sigma\) the specific reactive conductance. In particular, the quantity which in some cases most closely corresponds to the absorption coefficient is the so-called normal coefficient of the wall \(a_p\), defined as the specific active conductance \(\gamma\) multiplied by 8. If \(Z\) has phase angle \(\varphi\), then these quantities are related by the following relations:

\[ \left. \begin{aligned} \zeta&=(|Z|/\rho c)e^{-i\varphi}, \qquad \beta=(\rho c/|Z|)e^{i\varphi},\\ \gamma&=(\rho c/|Z|)\cos\varphi, \qquad \sigma=-(\rho c/|Z|)\sin\varphi,\\ a_p&=8\gamma . \end{aligned} \right\} \tag{3.1} \]

As will be shown below, the acoustic impedance usually depends on the frequency of the sound. Sometimes it also depends on the angle of incidence of the sound wave. These questions will be considered in Chapter IV. Since the normal component of the air velocity at the wall is proportional to the normal pressure gradient, the boundary condition for sound waves in a room will be the proportionality between the pressure at the wall and its normal gradient. This condition is more complicated than that for a room with rigid walls, although it is linear and homogeneous. The ratio of the pressure to its normal gradient generally depends on frequency, so that the boundary conditions, in essence—

*) We shall assume for simple harmonic motion a time dependence of the form \(e^{-i\omega t}\), as is customary in the description of wave motions. To compare the concept of impedance with the electrotechnical one, it should be borne in mind that the quantity \(i\) in the present survey corresponds in electrical engineering to \(-i\). Therefore we put \(Z=R-iX=|Z|e^{-i\varphi}\). The reactance \(X\) and the phase angle \(\varphi\) correspond to the usual definitions adopted in electrical engineering, with \(X\) and \(\varphi\) negative for elastic (capacitive) reactance and positive for inertial (inductive) reactance. These notations are retained below, except in Chapters V and VI, where the established regimes are investigated by operational-calculus methods, when both signs before \(i\) are required.

nesses, different for each standing wave in the room. Thus, free sound oscillations do not form an orthogonal system of eigenfunctions, as a result of which the usual methods based on the application of orthogonal systems of normalized eigenfunctions cannot be used here. This difficulty cannot be avoided, since absorption of sound at the boundaries is an essential part of the problem. Treating an actual room as differing only slightly from a room with rigid walls would be too crude an approximation, so that the method of small perturbations usually proves unsuitable.

11. Properties of rooms in steady-state and transient regimes.

The difficulties mentioned above disappear in the case of a steady-state regime, when the sound source brings the air in the room into simple harmonic oscillation. This oscillatory motion can be expanded in an orthogonal system of normalized eigenfunctions, each eigenoscillation having the frequency of the source. Therefore the boundary conditions are the same for all components. The case of transient regimes can also be studied by the methods of operational calculus, similarly to how this is done in the theory of electrical circuits. This question will be discussed in Chapter VI.

Generally speaking, the transient regime consists of an aggregate of “standing” waves, each with its own characteristic frequency, which is complex, corresponding to its exponential attenuation. Generally speaking, all oscillations have different attenuations. However, we shall see later that in some cases whole groups of oscillations have almost the same attenuation exponent, and in many cases all exponents are almost identical. However, in a simple rectangular room (Chapter V) there are several different attenuation exponents.

In this connection it is necessary to note one important difference in the results of geometrical acoustics and wave acoustics. In the geometrical theory, all the sound in a room behaves as one simple oscillator; then the decay curve on a logarithmic scale has a rectilinear course. From the wave point of view, it is obvious that if some standing waves forming the transient process have different attenuation exponents, then the total attenuation curve cannot be rectilinear. Moreover, since many waves with nearly identical “eigenfrequencies” are usually excited together, the resulting beats and interference phenomena cause a further deviation of the attenuation curve from a straight line. Knudsen’s work\(^5\), which played a pioneering role, did indeed reveal the presence of these phenomena in certain cases, and many-

subsequent investigators ^{B5, H9, M3, W8} confirmed and extended his observations.

As we shall show in Chapter VII, rooms with simple regular outlines give the greatest scatter of damping indices for different standing waves. Those oscillations which propagate parallel to walls with maximum absorption, in most cases, decay more slowly than waves reflected normally from this wall. This applies to a plane or convex smooth surface; for a smooth concave surface the opposite is true. If the wall is concave, then tangentially incident waves, traveling along the wall, usually decay much more rapidly than normally incident waves, which are focused at a great distance from the wall.

In both cases the damping curve on a logarithmic scale falls more steeply at the beginning of the damping process than at the end. In the end there remain only those natural oscillations which have small damping. In both cases some oscillations decay more rapidly than is predicted by the formulas of geometrical acoustics, while others decay much more slowly. Such rooms, apparently, possess unsatisfactory acoustical qualities ^{B11, M5, V1}.

12. Ergodic oscillations.

On the basis of the principle of correspondence, one might expect that, for sufficiently high frequencies, the behavior of sound in rooms with regular outlines tends to coincide with the behavior predicted by geometrical theory. In reality this is not so. In rooms with smooth regular outlines, the difference in damping indices for different natural frequencies is more sharply expressed at high frequencies than at low frequencies, and therefore the damping curves deviate much more from a straight line. Such a result is, at first sight, in direct contradiction with the principle of correspondence, which is usually observed in these cases. The reason for this lies in the circumstance noted above: the smooth, regular shape of the walls. Standing waves in rooms of regular outline have a corresponding regular and symmetric character, and this circumstance accounts for the differences in the damping indices. The forms of the natural oscillations of the air in rooms of irregular outline do not possess such symmetry. Not one of the standing waves propagates “parallel” to a curved wall (if it is sufficiently curved), and not one of them is perpendicular to it at all points. We shall see subsequently that the introduction of an irregularity into the outline of a room decreases the damping index for the more rapidly decaying oscillations and increases it for the more slowly decaying ones, so that all the values tend toward the value predicted by geometrical acoustics.

The questions touched upon here are very close to the ideas of statistical mechanics. Any system can be studied by the methods of statistical mechanics only when that system is so complex that all signs of symmetry completely disappear, i.e., when the only parameter characterizing the process is its energy. It is almost impossible to find a system that could be fully investigated by the methods of both dynamical mechanics and statistical mechanics. If the motion is sufficiently simple for dynamical methods to give an exact solution for it, then it is usually not ergodic. Conversely, most systems possessing ergodic motion are too complex for analysis by any methods other than statistical ones. The formulas of geometrical acoustics are statistical formulas. They apply only to such rooms in which “ergodic” processes take place. From this point of view, the practical role of wave acoustics reduces to the fact that it can indicate how to design such a room for which geometrical acoustics would be applicable and there would be no need for wave acoustics!

However, even from this point of view, room acoustics lies in a difficult intermediate region. There exist many rooms that have such regular outlines that the statistical formulas are unsuitable for a large part of the practically important frequency range, although one can find many other rooms so irregular in outline that the formulas of geometrical acoustics are fully applicable in the required frequency range. In general, wave acoustics has to be applied to small rooms of regular shape, whereas geometrical acoustics is usually sufficient for calculating large halls. This is the reason for the annoying circumstance that the “absorption coefficient” measured in laboratory reverberation chambers differs from the coefficient measured in the sound field of large auditoria S¹³, S¹⁴, W¹⁰.

13. Influence of irregularities

Both from the theoretical point of view and from the practical one, it is very important to study the intermediate cases in detail, in order to establish what degree of irregularity must be created to obtain ergodic oscillations. The scanty materials obtained so far show that tilting one wall through a small angle is insufficient for this, since the wall remains plane M⁶. They also show that irregularities having the same order of magnitude as the wavelength are the most effective for establishing the ergodic character of the oscillations, and that an irregular random arrangement of absorbing material on curved walls can create sufficient diffraction and scattering of sound for the formulas of geometrical acoustics to become valid. However, this method is not as effective as introducing irregularities into the outlines of the walls M⁵.

The correspondence principle is valid for most of these intermediate cases. When the wavelength is greater than, approximately, one fifth of the length of the room, the scattering of sound by comparatively small irregularities in the outlines of the walls is insufficient to give the oscillations an ergodic character, and the deviation from rectilinearity of the decay curves, typical of wave acoustics, is clearly manifested. For shorter waves, the same irregularities are more effective, and the decay curves turn into straight lines, typical of the statistical results of geometrical acoustics.

For large halls, geometrical acoustics is usually sufficient for the study of sound reverberation. Here too, however, wave acoustics is necessary for an exact calculation of the acoustic properties of samples of absorbing materials and for determining the ratio of the sound intensity arriving directly from the speaker to the intensity of sound that has undergone a single reflection. For these calculations it may be assumed that sound consists of free traveling waves, since the results of the calculation will be interpreted from the point of view of the geometrical scheme. These cases will be considered in Chapter VIII. For rooms of regular shape it is necessary to apply wave acoustics; moreover, if possible, one must seek the exact solution of the boundary-value problem. For a rectangular room, the steady-state regime will be analyzed in Chapter V, and the transient regime in Chapter VI.

Although the perturbation method is usually not sufficiently accurate for solving acoustic problems, it is nevertheless at present the only method applicable to those intermediate cases between purely wave acoustics and purely geometrical acoustics in which the walls are only slightly irregular. It proves suitable for determining what degree of irregularity is sufficient for the applicability of geometrical acoustics. The perturbation method also makes it possible to ascertain the principal cause of the difference in the decay indices of oscillations in rooms of regular shape, although the magnitudes of these indices can be determined only by more accurate methods. Chapter VII is devoted to the perturbation method.

14. Classification of Oscillations

Before proceeding to the consideration of these questions, we must derive formulas that would determine how many sound waves of different kinds (if different kinds exist) are excited by a given source; it is also necessary to investigate in detail the nature of the boundary conditions on the walls of the room. The boundary conditions will be analyzed in Chapter IV. The end of the present chapter is devoted to the first question: the determination of the number of different oscillations in a given room whose “natural frequencies” lie in a specified range.

This number can be determined if it is known how many different oscillations exist whose frequencies are less than a given value.

SOUND WAVES IN ROOMS

...frequency \(\nu\). The number \(n(\nu)\) of oscillations whose frequencies are less than \(\nu\) was investigated by Rayleigh, Weyl, and others \(^{J2,L1,M11,R3,S15,W9}\), and the first term of the asymptotic series for \(n(\nu)\) was determined.

In room acoustics the length of sound waves is fairly large in comparison with the dimensions of the room. Therefore the first term in the expansion of \(n(\nu)\) is insufficient, and it is necessary to determine the following terms.* This was done by Maa, Bolt, Xeshimi, and Ro \(^{B6,H10,M1,R8}\) for a rectangular room. These solutions are given below.

First of all it is necessary to classify the various types of oscillations that can arise in a rectangular room. If the walls of the room are absolutely rigid and \(L_x, L_y, L_z\) are the lengths of the edges of the room, then the velocity potential \(\psi\), the pressure \(p\), and the acoustic velocity \(\mathbf{u}\) for each proper oscillation will be expressed as follows:

\[ \begin{gathered} \psi=A\cos(\pi n_x x/L_x)\cos(\pi n_y y/L_y)\cos(\pi n_z z/L_z)e^{-2\pi i\nu t},\\ p=\rho(\partial\psi/\partial t),\quad \mathbf{u}=-\operatorname{grad}\psi,\\ \nu^2=(c/2)^2\left[(n_x/L_x)^2+(n_y/L_y)^2+(n_z/L_z)^2\right]. \end{gathered} \tag{3.2} \]

These expressions are sufficiently accurate for our purpose even in the case where there is some sound absorption at the walls.

Standing waves described by these formulas may be divided into three classes: first, those for which none of the \(n\)’s is zero; such waves we shall call oblique; second, those for which one of the \(n\)’s is zero—tangential waves (where waves for which \(n_x\) is zero are called \(yz\)-tangential, etc.); third, those for which two \(n\)’s are equal to zero—axial waves (waves for which \(n_y\) and \(n_z\) are zero are called \(x\)-axial, etc.). The grounds for this terminology are obvious. Axial waves are formed from two waves traveling parallel to one axis and incident only on two faces of the room. Tangential waves are formed from four waves traveling parallel to two faces and reflected by four faces of the room. Oblique waves are formed from eight traveling waves reflected by all six faces.

Each of these three classes of waves has different properties. In Chapter VI we shall see that they have different damping indices. They also differ in the energy associated with them. For each standing wave the total store of sound energy in the room is determined by the formula

\[ E=\frac{\rho}{2}\iiint\left[(\operatorname{grad}\psi)^2+\frac{1}{c^2}\left(\frac{\partial\psi}{\partial t}\right)^2\right]\,dv =\frac{2\pi^2\nu^2\rho}{c^2}L_xL_yL_zA^2\varepsilon. \tag{3.3} \]

* For example, for a room \((3.0\times4.5\times9\ \mathrm{m}^3)\) the first term for \(n\) is less than the true value by approximately 50% for a frequency of 100 cycles, approximately 10% for 1000 cycles, and approximately 1% for 10,000 cycles (see Fig. 7).

Here the factor \(\varepsilon\) has the value \(\frac{1}{8}\) for oblique waves, \(\frac{1}{4}\) for tangential waves, and \(\frac{1}{2}\) for axial waves. Thus, for a given amplitude of acoustic pressure, the axial wave contains four times more energy than the oblique wave. The consequences of this fact will be explained below in the present paper.

The fact that, in this simple case, the traveling waves are plane waves should not give rise to the idea that in rooms with curved walls there will be no tangential or axial waves. In Chapter VII it will be shown that axial waves also occur in cylindrical rooms. Some of them propagate parallel to the cylindrical walls and are reflected from the plane end faces. Others are reflected normally from the cylindrical walls and are focused at the center. A third type circles around the room parallel both to the end faces and to the curved walls. These waves may be called, respectively, \(z\)-, \(\rho\)-, and \(\varphi\)-axial waves. We shall see subsequently that each of these types of waves has different attenuation exponents \(K^8\).

Both cylindrical and rectangular rooms have walls corresponding to a coordinate system for which the wave equation can be separated, and it might be supposed that the above classification is applicable only to such cases. In fact, only for the indicated coordinates can there exist standing waves parallel to one of the coordinate axes of the system. It may be assumed, however, that in rooms having smooth walls with dimensions of several wavelengths, there exist waves propagating parallel to these walls, and waves reflected from these walls. Both of these kinds will have different absorption exponents, depending on the absorbing material on the given wall. This proves to be true for a triangular room (one of the few cases with nonseparable variables for which a solution has been obtained), as will be shown in Chapter VII. It may be hoped that other nonseparable cases will also be investigated and that the limits of applicability of the above classification will be clarified.

15. Distribution of the frequencies of natural oscillations

Equation (3.2) shows that the natural frequencies of a rectangular room have the properties of a vector with components \(c n_x/2L_x\), and so on.\(^{M11}\) Therefore each natural frequency may be represented by a point in frequency space. These points are located at the nodes of a rectangular space lattice, and in the direction of the \(x\)-axis the distance between the layers of the lattice is equal to \(c/2L_x\), and so on. Thus the volume occupied by one natural frequency in “frequency space” is a rectangular parallelepiped of volume \(c^3/8V\), surrounding the given point. Here \(V = L_x L_y L_z\) is the volume of the room.

The points fill the entire first octant in frequency space, so that the volume occupied by them is greater than the first octant, owing to the fact that the indicated parallelepipeds protrude, for example, above the plane \(xz\) by a length \(c/4L_y\), etc.*)

Thus the volume occupied by points with frequencies less than \(\nu\) is equal to:

\[ \frac{\pi}{6}\nu^3 +\left(\frac{c}{4L_x}+\frac{c}{4L_y}+\frac{c}{4L_z}\right)\frac{\pi\nu^2}{4} +\left(\frac{c^2}{16L_xL_y}+\frac{c^2}{16L_xL_z}+\frac{c^2}{16L_yL_z}\right)\nu+\cdots . \]

To determine the number of natural frequencies less than \(\nu\), we divide this volume by the volume occupied by one natural frequency in frequency space, and obtain \(^{M1,\,B6,\,E8}\)

\[ n(\nu)=\frac{4\pi V}{3c^3}\nu^3+\frac{\pi S}{4c^2}\nu^2+\frac{L}{8c}\nu+O(\nu). \tag{3.4} \]

Here

\[ V=L_xL_yL_z;\quad S=2(L_xL_y+L_xL_z+L_yL_z);\quad L=4(L_x+L_y+L_z), \]

and \(O(\nu)\) represents an irregular step function of order unity. The quantity \(S\) denotes the area of all the faces, and \(L\) the total length of the edges of the rectangular room. The first term of this expansion is obtained predominantly from oblique waves, the second from tangential waves, and the third from axial waves. The term \(O(\nu)\) reflects the circumstance that \(n\) is, in essence, a step function increasing by unity when the octant of increasing radius \(\nu\) encloses one more point. In Fig. 7 are shown the calculated functions: \(N_C\)—the step curve \(n(\nu)\) corresponding to the full expression (3.4), \(N_B\)—the smoothed curve obtained if the term \(O(\nu)\) is discarded, and \(N_A\)—the first term alone. As we see, the last curve lies considerably below the exact curve \(^{B6}\).

Fig. 7. Distribution of natural frequencies in a chamber of dimensions \((3\times4.5\times9\ \text{m})\). The true function \(n(\nu)\) in comparison with the continuous part of equation (3.4), and also with only its first term \(^{B6}\).

Weyl \(^{W9}\) showed that the first term in this asymptotic expansion for \(n(\nu)\) has the same form for all rooms with volume

) In addition, it is necessary to count the number of points lying directly on the coordinate axes. These points form the third term of the above sum. Ed. note.*

$V$, independently of their outlines. Such a generalization with respect to the second term has not so far been obtained. However, for triangular and cylindrical rooms[^8] the second term has the same form as in equation (3.4), with $S$ everywhere denoting the area of the walls. Thus it is possible that the expression for the second term also has general applicability. To study this question, experimental measurements of the natural frequencies were made on several small models[^88]. The chief difficulty here proved to be the circumstance that it is impossible to keep count of the resonant frequencies in the region where these frequencies come very close together, since the resonance peaks merge as a result of the influence of damping. The results show that it is always necessary to introduce a correction to the asymptotic term, and that its magnitude is estimated by the following rule: a) in the first term of equation (3.4) one should take the actual volume of the room $V$; b) in the second term, for $S$ one must choose a smoothed “mean” surface of the room such that the volume bounded by it is equal to $V$. This means that, in rooms with irregular outlines, the “effective” surface $S$ is often smaller than the actual surface of the walls.

The third term is still more uncertain. Apparently, for rooms bounded by plane walls, $L$ is the sum of the lengths of the edges (this, at any rate, is true for rectangular and triangular rooms). For a cylindrical room, however, it turned out that $L=4\pi R+4L_z$, where $R$ is the radius of the cylinder and $L_z$ is the distance between the flat end faces; as we see, in addition to the term $4\pi R$ for the two circular edges, there is here a term $4L_z$. However, most of our discussion will concern rectangular rooms, and for them the validity of equation (3.4) is reliable.

The number of waves of each of the three types mentioned above, having frequencies less than $\nu$, can be found by means of similar considerations. In particular, for a rectangular room the number of oblique standing waves with frequencies less than $\nu$ will be:

$$ n_p(\nu)=\frac{4\pi V}{3c^3}\nu^3-\frac{\pi S}{4c^2}\nu^2+\frac{L}{8c}\nu+O_p(\nu), \tag{3.5} $$

the number of $yz$-tangential waves:

$$ n_{tyz}(\nu)=\frac{\pi}{c^2}L_yL_z\nu^2-\frac{1}{c}(L_y+L_z)\nu+O_{tyz}(\nu), \tag{3.6} $$

and two analogous expressions for the remaining planes. The number of $x$-axial waves with frequency less than $\nu$:

$$ n_{ax}(\nu)=\frac{2}{c}L_x\nu+O_{ax}(\nu), \tag{3.7} $$

and two similar expressions for the other axes. If in the last four equations the terms $O(\nu)$ are omitted, then they give very accurate expressions for $n$ under the condition that the half-wavelength $(c/2\nu)$ is shorter than the smallest dimension of the room.

IV. ACOUSTIC IMPEDANCE

16. Impedance and Absorption

It has been shown in various ways that the absorption coefficient entering the formulas of geometrical acoustics is not a fundamental acoustic property of the surface of a wall A2, B3, B12, H9, M2, P1, S4, W10. Measurements of the magnitude of this coefficient give different results when the material is investigated in different rooms H8, P5, S14, W10. For some materials its variation with the angle of incidence of the sound ray has been observed B9, W10. Thus, the absorption coefficient is an averaged property, and the averaging refers to that particular case of sound distribution which in the preceding section we called “ergodic.” This quantity loses its meaning in cases where the sound distribution is not ergodic H8, B3.

It is important to emphasize this limitation, because an uncritical use of the concept “absorption coefficient” may lead to erroneous results. In this article we shall use the term “absorption coefficient” only on the condition that Sabine’s reverberation formula is applicable to the room under consideration, i.e., that the logarithmic decay curve is rectilinear, and that the decay constant for the mean-square pressure is equal to:

\[ k=\frac{c}{8V}\sum_n a_n S_n. \tag{4.1} \]

Here \(V\) denotes the volume of the room, and \(S_n\) the area of a definite kind of material on the walls. This equation is a definition of the coefficient \(a_n\). Thus, this concept loses its meaning when the decay curve is not rectilinear.

Recently there has been an increasing tendency to regard the acoustic impedance of the wall material as a more convenient measure of absorbing properties than the absorption coefficient B2, B12, N9, M11, M12, S8. To be sure, impedance is by no means a more “fundamental” physical property than the absorption coefficient. Its advantages consist in the fact that its measurement can be performed more simply and uniformly, and that its value for a given material does not depend on the distribution of sound in the room. The acoustic impedance of a material varies with frequency B2, B3, S8 and, in particular cases, with the angle of incidence of the sound ray B9, S8, W10. Nevertheless, with its aid one can determine the decay constant for rooms when Sabine’s formula is invalid and when the absorption coefficient has no meaning.

Among some acoustical engineers, the adoption of impedance as the principal acoustic property of a material has met with a certain resistance. This is due, on the one hand, to an understandable hesitation before changing concepts; on the other hand, to the impression that the relation between impedance and the absorption coefficient is ...

indeterminate. In reality, this impression arose from the fact that the indeterminacy and limited applicability of the very concept of the absorption coefficient were overlooked. It will be shown below that the relationship between the impedance of the material on the walls and the index of sound decay in a room is not single-valued; it depends on the distribution of sound energy in the room. In those cases where the room, the sound source, etc. are such that an ergodic sound process is established, there is a definite and unambiguous dependence between the slope of the resulting straight-line decay curve and the impedance of the material on the walls. Since this is the only case in which the concept of the absorption coefficient can be applied, we are justified in asserting that there exists a definite and unambiguous relationship between impedance and the absorption coefficient. This relationship will be discussed in Chapters VII and VIII. For those cases, however, in which the phenomenon is not ergodic and it is necessary to use wave acoustics, the theory developed in Chapter VI shows that the relationship between impedance and the index of decay may be different. Thus the indeterminacy of the concept of the absorption coefficient arises only when its application goes beyond its legitimate limits. The resulting discrepancy must be ascribed to the natural limitations on the applicability of geometrical acoustics, and they should be regarded as a new argument against the view of the absorption coefficient as a fundamental acoustic parameter of a material.

The experimental work of Hunt and his collaborators B3, H7, H9, M2, which is pioneering in character, has shown that the assertions of the preceding section are in general valid. However, a very large amount of detailed work still remains to be done before the interrelation between the acoustic impedance of a material and its physical properties, on the one hand, and between the impedance of walls and the reverberation of a room, on the other, becomes fully understood.

The present chapter describes the current state of the science with respect to the first question—the interrelation between the mechanical properties of a material and its acoustic impedance. Along the way, experimental methods for measuring impedances will be described, and the consequences that follow, in the light of the theory under discussion, from the few reliable experimental results will also be indicated.

17. Mechanical Properties of Porous Materials

The relationships between density, porosity, and various other mechanical properties of a material and its acoustic properties have been studied for a long time. Rayleigh R4 calculated the sound absorption at the surface of a solid porous material, taking into account the dissipation of sound energy into heat in capillary channels caused by the viscosity of air. Suppose that channels less than \(0.01\ \mathrm{cm}\) in diameter have

cylindrical shape, normal to the surface and so long that sound cannot be reflected from their bottoms*), but short in comparison with the wavelength. Then, according to Rayleigh’s calculations, the absorption coefficient will be

\[ a = 4M/(2M^2 + 2M + 1). \]

Here it is assumed that

\[ M = \frac{2(1+g)(r\eta)^{1/2}}{r\omega^{1/2}} . \]

\(g\) denotes the perforation coefficient (the ratio of the area of the holes to the remaining area), \(\eta\) the kinematic viscosity of the gas, \(\gamma\) the ratio of heat capacities, \(r\) the radius of the pores (assumed homogeneous and cylindrical), and \(\omega\) the angular frequency of the sound signal. Since the quantity \(g\) is here defined as surface porosity, it is analogous to the volume porosity which we shall discuss later (in the particular case considered here). This equation shows that \(a\) can have a maximum in the range of audible frequencies. Rayleigh also considered the case when sound is incident on the material at different angles and showed that in certain cases, at definite angles, the absorption may be complete.

Paris \(^{P2, P6}\) derived a formula determining the absorption coefficient for any angle of incidence, if the “acoustic conductivity” of the surface is known. In doing so he makes no assumptions concerning the physical nature of the absorbing material and introduces the single special assumption that sound does not propagate inside the material parallel to its surface. If Paris’s equation is expressed in terms of the reciprocal of conductivity—the acoustic impedance—then it takes the form:

\[ \alpha(\Theta) = 1 - \left|\frac{Z\cos\Theta - \rho c}{Z\cos\Theta + \rho c}\right|^2 . \]

Here \(Z\) denotes the acoustic impedance of the surface, in general a complex number, \(\rho\) is the density of air, and \(c\) is the speed of sound in air (\(\rho c \approx 42\) C.G.S. units). This equation, which determines the absorption coefficient for a given angle of incidence \(\Theta\), has acquired especially important significance in recent investigations of the relation between impedance and the absorption coefficient \(^{H7, M11, S8, W4}\).

Crandall \(^{C1}\) derived a formula relating the absorptive capacity of a layer of material to its thickness, assuming that it is applied to an absolutely rigid surface and that the sound ray is normally incident. His work was the first in which the possibility of reflection from the rear surface of the material is allowed, leading to the occurrence

*) More precisely, so that the wave reflected from the bottom would produce a negligibly small pressure amplitude at the surface. Ed. note.

interference maxima and minima of the quantity \(\alpha\) as a function of the frequency of the sound or of the thickness of the material. Crandall also showed that, as the thickness of the material is increased, the absorption coefficient tends to a definite limit. Crandall’s various conclusions were confirmed by measurements using the method of standing waves in tubes.

Further studies on sound absorption were carried out by Davis and Evans \(D^3\), who studied the effect of an air cavity behind a porous specimen. Meyer \(M^6\) investigated numerous types of flexible panels. Many other investigations were also performed.

In the present survey we shall distinguish two types of facing acoustic materials \(R^7\): the panel type, in which the reaction of the wall to a change in pressure is determined by the stiffness of the wall, while the normal component of the velocity at the wall is determined by the motion of the panel as a whole; and the porous type, in which the normal component of the velocity is determined by the penetration of air into the pores of the material, while the reaction is determined by the interaction of this air with the porous material. Of course, there are also intermediate cases, when a porous material is located behind the panel, or when a porous plate acts as a panel, but these cases should be considered on the basis of an analysis of both limiting cases.

In considering the porous type we shall, in the main, follow the work of Wintergerst, Gemant, and Rettinger \(G^1, R^6, R^7, W^{11}, M^8\). Elastic waves can propagate through a porous material in various directions. In some cases the speed of sound normal to the surface differs from the speed in the direction parallel to the surface \(W^{11}\), owing either to a layered structure or to the orientation of the pores within the material. In such cases the “index of refraction” of the material for waves traveling normal to the surface differs from that for waves traveling parallel to the surface. Such a material will be acoustically birefringent*.

First of all it is necessary to determine the principal mechanical properties of a porous material. The porosity \(P\) is the ratio of the volume of air in the pores to the entire volume of the material. Thus the specific volume velocity of air through the material \(\mathbf{u}\), in \(\text{cm}^3/\text{cm}^2\cdot\text{sec}\), is related to the mean velocity of the air in the pores \(\mathbf{v}\), in \(\text{cm}/\text{sec}\), by the following relation: \(\mathbf{u}=P\mathbf{v}\). Then the continuity equation takes the form

\[ \operatorname{div}\mathbf{u}=-(P/\rho c^2)(\partial p/\partial t). \tag{4.2} \]

It relates the pressure to the mean flux. By means of this equation the porosity can be defined in a new way, as the ratio of the stiffness of the air to the stiffness (bulk modulus of elasticity) of the same volume of material as a whole (including the pores). Let us note that the effective porosity which enters into this equation (it may

* The authors incautiously apply this term to longitudinal waves. Ed. note.

be called dynamic porosity), may differ from the geometrical porosity, since the material surrounding the pores may also be compressible. There is also another reason why the effective porosity will differ from the geometrical porosity. In equation (4.2) we assume that the stiffness of the air in the pores is expressed by the quantity \(\rho c^{2}\), corresponding to adiabatic expansion. Some works by Beranek \(^{\mathrm{B}4}\) indicate that the expansion of the air in the pores may be almost isothermal. The difference in stiffnesses leads to the appearance of a factor equal to the ratio of the heat capacities \(\dfrac{C_p}{C_v}\), included in the value of the effective porosity \(P\).

The equation of motion of the air in the pores is sufficiently complicated. First of all, one has to take into account both frictional resistance and inertial resistance. Secondly, the motion of the air need not coincide in direction with the acting force. This means that the effective density of the air in the pores, as well as its resistance, have the character not of scalars but of dyads transforming the force vector into the velocity or acceleration vector. In the equation

\[ \rho m(\partial \mathbf{u}/\partial t) + r\mathbf{u} = -\operatorname{grad} p, \]

both \(m\) and \(r\) are dyads, while \(\mathbf{u}\) is a vector. In the case of isotropy \(m\) and \(r\) reduce to scalars. Then \(r\) is the effective resistance of the material per unit volume, and \(m\) is the ratio of the effective density of the air in the pores to its density in free space. Since the material itself moves together with the air, \(m\) can often be considerably greater than unity.

These effective values are assumed to include the possible consequences of motion of the porous material, if the latter is a compliant structure. Thus it is not assumed that this effective resistance \(r\) necessarily coincides with the value obtained from measurements by blowing an air stream through the material \(^{\mathrm{B}13}\), since such measurements do not take into account the indicated motion of the material. As we shall see below, measurements show that for porous materials with a rigid structure the values of \(r\) obtained from dynamic measurements coincide with the results of blowing tests. Such coincidence is not observed in compliant structures.

In those cases where \(m\) and \(r\) are not scalars, it is usually found that their principal axes are normal to the surface. Then the matrices for \(m\) and \(r\) become diagonal, and the equations of motion take the form:

\[ \rho m_n(\partial u_x/\partial t) + r_n u_x = -(\partial p/\partial x), \]

\[ \rho m_t(\partial u_y/\partial t) + r_t u_y = -(\partial p/\partial y), \]

\[ \rho m_t(\partial u_z/\partial t) + r_t u_z = -(\partial p/\partial z). \]

Here \(x\) is assumed to be located normally, and \(y\) and \(z\) tangentially, to the surface. If the time dependence of the variables

of the quantity is simply harmonic, so that the time-dependent term has the form \(e^{-i\omega t}\), then the equations of motion can be combined with equation (4.2). Then we obtain the wave equation for sound in a porous material:

\[ \begin{gathered} (1/\varepsilon_n)\frac{\partial^2 p}{\partial x^2} + (1/\varepsilon_t)\left(\frac{\partial^2 p}{\partial y^2}+\frac{\partial^2 p}{\partial z^2}\right) + (\omega/c)^2p=0;\\ \varepsilon_n=(n_n+iq_n)^2=P\,[m_n+i(r_n/\rho\omega)];\\ \varepsilon_t=(n_t+iq_t)^2=P\,[m_t+i(r_t/\rho\omega)]. \end{gathered} \tag{4.3} \]

Here \(n_n\) and \(q_n\) are the real and imaginary parts of the refractive index for sound waves propagating in the material along the normal to the surface; \(n_t\) and \(q_t\) are the corresponding values for tangential waves.

If the pressure is determined, the air flow can be calculated from the above equations of motion. For a simple harmonic oscillation this gives

\[ u_x=\frac{P}{i\rho\omega\varepsilon_n}\cdot\frac{\partial p}{\partial x}; \qquad u_y=\frac{P}{i\rho\omega\varepsilon_t}\cdot\frac{\partial p}{\partial y}; \qquad u_z=\frac{P}{i\rho\omega\varepsilon_t}\cdot\frac{\partial p}{\partial z}. \tag{4.4} \]

By means of the indicated equations one can express the acoustic impedance of the material in terms of the “fundamental constants” \(m\), \(P\), and \(r\). Knowing the dependence of these properties on frequency, one can find the dependence of the impedance on the frequency and the angle of incidence of the sound ray.

Let the volume of the room occupied by air correspond to negative values of \(x\), the front face of the layer of material coincide with the plane \(x=0\), and the rear face of the material coincide with the plane \(x=L\). Let us further suppose that the sound pressure in the plane of the wall satisfies the following conditions:

\[ \begin{gathered} (\partial^2p/\partial x^2)=-\mu_n^2p,\\ \nabla_t^2p=(\partial^2p/\partial y^2)+(\partial^2p/\partial z^2)=-\mu_t^2p;\\ (\partial p/\partial t)=-i\mu c p\\ \mu_n^2+\mu_t^2=\mu^2;\qquad \mu=(\omega/c)=(2\pi/\lambda). \end{gathered} \tag{4.5} \]

Then the angle of incidence \(\varphi_i\) of the sound ray is determined from the relations

\[ \sin\varphi_i=(\mu_t/\mu);\qquad \cos\varphi_i=(\mu_n/\mu). \tag{4.6} \]

The dependence of the sound pressure on \(y\) and \(z\) is the same both inside the porous material and outside the material directly at the wall. Its dependence on \(x\) will have the form

\[ \operatorname{sh}\{\psi+i\mu(n_n+iq_n)x\cos\varphi_r\},\qquad 0<x<L. \]

Here \(\psi\) denotes the phase angle determined by the boundary conditions at the rear face of the specimen, and the angle of refraction \(\varphi_r\) is determined by Snell’s law:

\[ \begin{gathered} \sin\varphi_r=(n_t+iq_t)^{-1}\sin\varphi_i,\\ \cos^2\varphi_r=1-\varepsilon_t^{-1}\sin^2\varphi_i. \end{gathered} \tag{4.7} \]

If \(q_t\) is not equal to zero, then this angle is complex. But if the modulus \(n_t+iq_t\) is considerably greater than unity (and this is often the case), then \(\cos \psi_r\) has a small imaginary part. The real part of \(\cos \psi_r\), however, is always close to unity. In the following discussion this will be assumed, unless the contrary is expressly stated.

The ratio of the pressure at any point inside the material to the component of volume velocity along the \(x\)-axis is written as follows:

\[ \frac{p}{u_x}=Z_x=\frac{\rho c(n_n+iq_n)}{P\cos\psi_r}\cdot \operatorname{th}\pi\left[x+(2q_n/\lambda)(L-x)\cos\psi_r+\right. \]
\[ \left.+\,i\beta-i(2n_n/\lambda)(L-x)\cos\psi_r\right], \qquad (4.8) \]

\[ \psi=\pi(\alpha+i\beta)=\operatorname{Arth}\,[PZ_L\cos\psi_r/\rho c(n_n+iq_n)]. \]

Here \(Z_L\) is the acoustic impedance of the material which is placed behind the specimen in the plane \(x=L\). The acoustic impedance of the wall will be the value of \(Z\) in the plane \(x=0\). We shall investigate the properties of this quantity for long waves and, moreover, for several typical values of \(Z_L\) for waves of any length.

18. Equivalent circuit for long waves

The most instructive case is that of long waves, more precisely, of such frequencies for which \((2\pi L/c)(n_n+iq_n)\) is small compared with unity. In this case the properties of the porous material are analogous to the properties of an electrical circuit with lumped constants. The mass \(\rho m L\) of a unit surface is analogous to self-inductance; the resistance to the flow of air \(rL\) is analogous to active electrical resistance; the quantity reciprocal to the stiffness, \(\rho c^2/PL\), is analogous to capacitance. Expanding equation (4.8) in a power series with respect to the quantity \((2\pi L/c)(n_n+iq_n)\), and retaining terms of the third order, we obtain:

\[ Z_x \simeq \frac{ Z_L+\left[(1/Z)-\frac{1}{3}i\omega C\right]^{-1} }{ 1+Z_L\left[\frac{1}{-i\omega C}+\frac{1}{3}Z\right]^{-1} }. \]

Here \(Z_L\) is the impedance of the wall material on which the absorbing layer is placed, \(C=PL/\rho c^2\), \(Z=-i\omega \rho m_n L+r_nL\). This formula cannot describe a single circuit, just as an electrical line cannot be replaced exactly by a single circuit. Nevertheless, some limiting cases may be reduced to consideration of a single circuit. Usually the reactance \(1/\omega C\) at low frequencies is greater than \(Z\).

If the impedance \(Z_L\) of the backing material is of the same order of magnitude as \(Z\), then with a high degree of accuracy it may be assumed that \(Z\) is included in series with \(Z_L\), while \(C\) shunts them both. When \(Z_L\) is very small, then in the approximate equivalent circuit.

\(\frac{1}{3} C\) shunts only \(Z\). When the backing material is very rigid and \(Z_L\) is large, the equivalent capacitance shunts \(Z_L\), while \(\frac{1}{3} Z\) is connected in series. These simplified combinations are shown in Fig. 8.

Fig. 8. Equivalent circuits for a porous panel of thickness \(L\) with backing-material impedance \(Z_L\) for long waves. \(R_S, M_S\), and \(K_S\) are the constants corresponding to bending of the panel, equal to infinity for a rigid panel. The quantities \(r_m, m_m\), and \(P\) are constants corresponding to the porosity of the panel.

In the case when the porous material acts simultaneously also as a panel, with total mass per unit area \(M_S\) and effective flexural elastic constant \(K_S\), there appears a second circuit consisting of self-inductance, capacitance, and resistance, shunting the circuit associated with the motion of the air in the pores. The resistance \(R_S\) of this branch is due to the internal friction of the panel during its bending.

Variants of the complete equivalent circuit for long waves are shown in Fig. 8. The acoustic impedance at the surface of the material is equal to the impedance measured at the break \(Z_0\) of the equivalent circuits, provided only that the thickness of the material is small in comparison with the wavelength in it. A layered material is analogous to an electrical filter. It is possible so to choose the porous structure that it absorbs any desired frequency band. An air gap of thickness \(Z'\) (small in comparison with the wavelength) between the panel and the backing material corresponds to a capacitance \(L'/\rho c^2\), shunting \(Z_L\), where \(Z_L\) now denotes the acoustic impedance of the material behind the gap. This gap must be divided by partitions perpendicular to the surface of the panel in order to prevent the formation of standing waves in it \(^9\).

The acoustic transparency of materials is also derived from the circuits of Fig. 8, since the current through \(Z_L\) corresponds to the normal component of the air velocity immediately behind the panel. Therefore a stack of plates separated by air gaps corresponds to a filter transmitting low frequencies.

In those cases in which the wavelength is small in comparison with the thickness of the panel, the acoustic properties become analogous to elec-

tric line with leaks, and the need arises for an exact solution of equation (4.8). Some of these cases merit discussion.

19. Short waves

The simplest case is obtained when the backing material is absolutely rigid, \(Z_t\) is infinite, and \(\alpha+i\beta=\dfrac{1}{2}i\). The specific acoustic impedance of the wall surface \((Z_0/\rho c)\) is equal to

\[ \zeta_p=\frac{n_n+iq_n}{P\cos\varphi_r}\operatorname{th}\left[(2\pi L/\lambda)(q_n-in_n)\cos\varphi_r+\frac{1}{2}\pi\right]. \]

The subscript \(p\) means that this impedance is due to the motion of the air in the pores of the material. The subscript \(s\) will mean that the impedance is due to the motion of the panel.

Consideration of the quantities entering into this equation shows that the dependence of the wall impedance \(\zeta\) on frequency and on the parameters of the wall can be expressed by a family of curves with one parameter. We shall denote the dependent variable by \(\Gamma\), the independent variable by \(\sigma\), and the parameter by \(\gamma\), defining them by the following relations:

\[ \begin{aligned} \Gamma&=2\zeta(P/m_n)^{1/3}\cos\varphi_r,\\ \sigma&=(L/\lambda)(m_nP)^{1/2}\cos\varphi_r,\\ \gamma&=(r_nL/2\pi\rho c)(P/m_n)^{1/2}\cos\varphi_r. \end{aligned} \tag{4.9} \]

Then the equation for \(\Gamma\) takes the form:

\[ \left. \begin{aligned} \Gamma&=(a+ib)\operatorname{th}\pi\left[-i\sigma(a+ib)+\frac{1}{2}i\right]\\ &=(2/\rho c)(P/m_n)^{1/2}(\cos\varphi_r)(R_p-iX_p),\\ a&=2n_n(Pm_n)^{-1/2}=\left\{2\left[1+(\gamma/\sigma)^2\right]^{1/2}+2\right\}^{1/2},\\ b&=2q_n(Pm_n)^{-1/2}=\left\{2\left[1+(\gamma/\sigma)^2\right]^{1/2}-2\right\}^{1/2}. \end{aligned} \right\} \tag{4.10} \]

The variable \(\sigma\) is proportional to frequency, and the parameter \(\gamma\) is proportional to the resistance to blowing through the material, \(r_nL\). In the case when the numbers \(m_n\), \(r_n\), and \(P\) are independent of frequency, \(\sigma\) is proportional to frequency, while \(\gamma\) does not depend on frequency.

Curves for the real and imaginary parts of the quantity \(\Gamma\) are shown in Fig. 9 as functions of \(\sigma\) for four different values of \(\gamma\). In calculating these curves it was assumed that the imaginary part of \(\cos\varphi_r\) may be neglected. From these curves it is seen that for low frequencies (small values of \(\sigma\)) the reactive part of the impedance is very large, so that the elasticity of the air in the pores is of decisive importance. At these low frequencies the real and imaginary parts of the acoustic impedance of the walls \(Z\) tend to the following simple values:

\[ R_p\approx\frac{1}{3}r_nL,\qquad X_p\approx-(\rho c^2/\omega PL\cos^2\varphi_r), \tag{4.11} \]

for \(\sigma \ll \gamma < 1\) for a rigid backing material (see Fig. 8). In this limiting case the effective resistance is equal to one third of the static resistance to flow, \(r_a L\). The factor \(1/3\) is due to the circumstance that the rigid backing material prevents the air, after passing through the pores, from escaping outward, as occurs in static experiments for determining resistance to flow. The reactive resistance is due to the elasticity of the air in the pores.

Fig. 9. Curves of acoustic impedance for a porous material of thickness \(L\) on a rigid wall.

Such a result is obtained from consideration of the equivalent circuit of Fig. 8 for the case of infinite impedance \(Z_1\).

As the frequency is increased, interference phenomena arise owing to the reflection of waves from the backing material, and resonance peaks appear; as the curve in Fig. 9 shows, these are higher and sharper the smaller the values of \(\gamma\). For very large values of \(\gamma\) the waves do not reach the rear side of the layer of material, the acoustic impedance exhibits no resonance phenomena, and the following approximate formulas are applicable:

\[ \begin{aligned} R_p &\simeq (\rho c/P)n_a \sec \varphi_r,\\ X_p &\simeq -(\rho c/P)q_a \sec \varphi_r \end{aligned} \tag{4.12} \]

(if \(\gamma > 1,\ \gamma\sigma > 1\) for any backing material).

These quantities represent the wave resistance of an electrical line equivalent to our acoustic material. For high frequencies \(R_p\) tends to \(\rho c(m_a/P)^{1/2}\sec\varphi_r\), while the reactive term reduces to \(- (r_a c'/2\omega)(m_a P)^{-1/2}\sec\varphi_r\). For very low frequencies (when \(\sigma\) is considerably less than \(1/\gamma\)) equation (4.12) becomes unsuitable, and one must use equation (4.11).

In the next section we shall see that the qualities of a wall as a sound absorber are usually approximately proportional to the real part of the acoustic conductance \(1/Z\). Therefore a large value of the reactive acoustic conductance at some frequency diminishes the absorption at that frequency. Equation (4.11) shows that, in order to obtain a material that absorbs strongly at low frequencies, it must be thick (larger \(L\)), porous (\(P\)—almost unity), and its tangential index of refraction must be much greater than unity (so that \(\cos^2\varphi_r\) is as close as possible to unity at all angles of incidence). The last condition prevents the acoustic impedance from changing appreciably with the angle of incidence of the sound ray.

In the next section we shall see that a material absorbs somewhat better when it has not a purely active resistance, but contains a small negative reactive resistance (elastic).

20. Influence of an Air Gap

For low frequencies one can obtain a negative reactive resistance by arranging an air gap behind the material \(^{\mathrm{B4,\,B9}}\). In this case the constants \(\alpha\) and \(\beta\) are determined if \(Z_L\) is set equal to the impedance of the air gap. Suppose that the thickness of the gap \(L_a\) is small in comparison with the wavelength in air. Then \(Z_L\) will be an elastic reactive resistance. \(Z_L\) will be approximately equal to \((i\rho c^2/\omega L\cos^2\varphi_i)\). It should be noted that this formula includes the angle of incidence. This approximation is valid as long as sound waves propagating parallel to the wall cannot arise in the gap, i.e. as long as the cells of the frame separating the porous material from the main wall do not exceed the wavelength in size.

If these conditions are satisfied, then it turns out that the only consequence of the presence of the air gap is an increase in the effective thickness of the material. Equation (4.10) remains valid; only the values of \(\sigma\) and \(\gamma\) change: instead of the actual thickness of the layer of material \(L\), one must substitute the effective thickness

\[ L_e = L + (L_a/P)(\cos\varphi_i/\cos\varphi_r) \tag{4.13} \]

(the air gap, \(L_a < \lambda/4\), transverse waves may exist).

If the thickness of the air gap \(L_a\) is greater than a quarter of a wavelength, then for \(Z\) one must take a more exact expression, and equation (4.10) must be revised.

Generally speaking, the presence of an air gap increases the effective thickness of the layer of material and decreases the height of the resonance peaks on the impedance curve as a function of frequency. The reason is that the air gap shifts the surface of zero normal velocity, which previously coincided with the rear wall of the material, into the region of the air gap, as a result of which the flow of air through the pores is intensified. In this process a greater loss of energy is obtained \(^{84}\).

In those cases where propagation of tangential waves in the air gap is possible, the impedance proves to depend on the angle of incidence \(\varphi_i\), since the factor \(\cos \varphi_i\) enters into the formula for the effective thickness; this means that, at nearly grazing incidence of sound rays, the absorption does not depend on the presence or absence of the air gap. In order to reduce such a dependence on \(\varphi_i\), it is necessary to subdivide the air gap by a suitable device whose cells are small in dimensions compared with the half-wavelength \(^{89}\). In this case the effective thickness of the material, which must be substituted in equations (4.10) and (4.11) instead of \(L\), is equal to

\[ L_e = L + (L_a/P \cos \varphi_r) \tag{4.14} \]

(a narrow air gap, transverse waves eliminated). This expression depends less on \(\varphi_i\) than \(L_e\), which corresponds to the absence of a frame preventing tangential waves. Therefore the effective thickness of the material is increased even for grazing incidence of the sound rays.

When the thickness of the air gap is made greater than a quarter wavelength, or when the material is made of several layers with different properties, the analysis becomes more complicated. By applying equation (4.8) several times, one can calculate the impedance of the wall even for these complicated cases. Of course, for low frequencies, as we have already indicated, one may proceed from the analysis of equivalent electrical circuits (see Fig. 8).

21. Panel Vibrations

The preceding analysis took into account that part of the normal component of the velocity of the air at the wall which is due to the penetration of air into the porous material. In many cases the surface of the wall, under the action of acoustic pressures, moves as a whole, like a membrane \(^{87}\), and this motion must be taken into account. We may consider this phenomenon separately because, if some porous wall operates together with it as a membrane, then the normal component of the velocity of the air is simply the sum of the velocity of the material as a whole plus

the velocity of the air in its pores. Therefore it may be assumed that the impedances due to the two phenomena are connected in parallel with one another. In this case the more important role is played by the impedance which is smaller. Conversely, when the wall consists of a dense impermeable layer (for example, paint) covering a porous material, the impedances of both layers act as if connected in series. In this case the principal role is played by the impedance which is larger.

Just as in the case of an air gap, the influence of the supporting frame (assumed rigid) will be very different depending on whether the distance between the elements of the frame is greater or smaller than a half-wavelength. In the first case transverse vibrations arise in the panel\(^{11}\), and in order to calculate the impedance it is necessary to use the theory of vibrations of plates.

Let \(\rho_s\) denote the density of the material of the plate, \(s\) its Poisson coefficient, \(Q\) its Young’s modulus, and \(L\) its thickness. Then the equation for the normal displacements of the plate at the point \((y,z)\) of its surface will be as follows:

\[ \rho_s L \frac{d^2 \xi}{dt^2} + \left[\frac{Q L^3}{12(1-s^2)}\right]\nabla_t^4 \xi = p_0(y,z)-p_L(y,z). \tag{4.16} \]

Here \(p_0\) denotes the pressure acting on the front surface of the panel, and \(p_L\) on the rear surface.

It may be assumed that the pressure on the front surface \(p_0\) satisfies the same conditions as were adopted earlier in equation (4.5), the angle of incidence being determined by equation (4.6). Substituting these expressions into equation (4.16), we obtain the expression for the acoustic impedance of the panel wall:

\[ Z_s = Z_L - i\omega\rho_s L + \frac{i Q L^3 \omega^3}{192\,c^4(1-s^2)}\sin^4\varphi_i . \tag{4.17} \]

(the distance between the elements of the frame is much greater than the wavelength). Here \(Z_L\) is the impedance of the material (or of the air gap) behind the panel.

The second term in this expression represents the usual reactive inertial resistance. The third term is due to the elasticity of the plate, which, under oblique incidence of the sound ray, tends to adjust itself to the changes of pressure from point to point along its surface. This term has the positive sign of reactive elastic resistance (\(X\) is negative). Its magnitude increases rapidly with frequency, since the plate strongly resists bending in the form of a very short-wavelength corrugation. In view of the fact that the wavelength along the panel depends not only on the frequency but also on the angle of incidence, this part of the impedance depends very noticeably on the latter. It is small at small angles of incidence and has, for a given frequency, a maximum value at grazing incidence. Therefore

a wall with a very rigid outer covering has a large impedance for grazing sound waves and absorbs such waves poorly, especially at high frequencies.

At sufficiently low frequencies the wavelength becomes greater than the distance between the frame elements, and equation (4.17) loses its force. In this case each piece of the wall, located between the supporting elements of the frame, acts as an elastic membrane with effective mass per unit area \(M_s \approx \rho_s L\), effective elastic constant \(K_s\), and effective resistance \(R_s\) (as in the equivalent circuit). The values of these constants depend on the distribution of the supporting elements of the frame. In this case the wall impedance is

\[ Z_s = Z_L - i\omega M_s + i(K_s/\omega) + R_s \tag{4.18} \]

(the distance between the frame elements is less than one quarter of the wavelength). This impedance does not depend on the angle of incidence, which is an advantage, but the impedance itself at low frequencies becomes very large, so that the wall is a poor absorber in this range.

22. Types of sound-absorbing materials

We can now classify the various kinds of facing acoustic materials and give a survey of the equations determining the total acoustic impedance for each type.

1. A thin layer of porous material on a rigid wall. The impedance corresponds to the equivalent circuits of Fig. 8.

2. A thin layer of porous material, with an air gap behind it; the supporting elements of the frame are spaced by more than a wavelength from one another. The impedances determined by equations (4.15) and (4.17) are connected in parallel, where

\[ Z_L = i\rho c^2/\omega L \cos^2 \varphi_i . \]

1. A thin layer of porous material, with an air gap behind it; the supporting elements of the frame are spaced at a distance of less than a half-wavelength from one another. The impedance is determined by equations (4.15) and (4.18), with parallel connection, where

\[ Z_L = i\rho c^2/\omega L . \]

1. A thick layer of material on a rigid wall. The impedance is given by equations (4.9) and (4.10).

2. A thick layer of material, with an air gap; the supporting elements are spaced by more than a wavelength. The impedances according to equations (4.10), (4.13), and (4.17) are in parallel connection:

\[ Z_L = i\rho c^2/\omega L \cos^2 \varphi_i . \]

1. A thick layer of material, an air gap, and frame elements closer than a half-wavelength to one another. The impedances, according to equations (4.10), (4.14), and (4.18), are in parallel connection:

\[ Z_L=(i\rho c^2\omega L). \]

1. A very thick layer of porous material on a wall of any material. The impedance is according to equation (4.12).

2. A thick layer of porous material, a rigid wall, the front surface covered with a thin dense impermeable layer (for example, paint). The impedance is according to equation (4.17), with \(Z_L\) determined by equations (4.9) and (4.10).

For other, more complicated cases, the impedance may be obtained in an analogous way.

It is interesting to note that in four of these cases (including the most widely used materials) the acoustic impedance is almost independent of the angle of incidence of the sound ray, since \(\cos\varphi_i\) in most cases is almost independent of \(\varphi_i\). Only in the case where a thin porous material \((L<\lambda)\) has behind it an air gap with frame supports at large distances from one another, or in the case where the front surface is sufficiently impermeable, so that panel action predominates, is a dependence of the impedance on \(\varphi_i\) noticeable. The most usual cases, analysis of which will be most useful, are: the case where \(\zeta\) does not depend on \(\varphi_i\), and the case where \(\zeta\) is equal to the sum of two terms: a term independent of \(\varphi_i\), plus a purely reactive term \((i\rho c^2/\omega L\cos^2\varphi_i)\). The latter case represents the typical behavior of a wall with an air gap at low frequencies.

To verify the adequacy of the description given to the phenomenon of sound absorption by these theoretical schemes, many impedance measurements were carried out. The experimental results presented later in this chapter indicate that for many commercial materials the quantities \(P\), \(m\), and \(r\) are independent of frequency in the range from 100 to 6000 hertz. This means that these quantities may be regarded as constants characterizing the material, and that measurement of these three constants for a given material makes it possible to predict completely all its acoustic properties. This theory considerably clarifies the properties of absorbing materials. For low frequencies, for which the equivalent schemes are applicable, it makes it possible to calculate absorbers satisfying any prescribed acoustic requirements.

23. Measurements of Acoustic Impedance

Let us now turn to a brief account of the experimental methods for measuring acoustic impedances. Of course, the most direct method would be the measurement of pressure, air velocity, and displacement

phases between them at the surface of the material under test. Experimentally this presents extreme difficulties. In particular, it is difficult to construct a small but sensitive velocity microphone, and the velocity of the air particles at the wall surface is usually much smaller than the pressure amplitude*).

Nevertheless, attempts have been made to construct apparatus for these measurements. Klepp and Firestone \(^{63}\) developed an “acoustic wattmeter.” It consists of a miniature pressure microphone and a resonant velocity microphone mounted very close to one another. The acoustic pressure, the acoustic velocity, and the phase shift between them can be measured separately, and from them the impedance is calculated. It proved very difficult to produce and maintain the calibration of the microphones, especially because of their sensitivity to slight wind and changes in temperature. The instrument was used to measure the impedance in a tube at the end of which a specimen of the material under test was placed. In its essential features this is the “hyperbolic tangent method,” described below, with the difference that \(p\), \(u\), and \(\varphi\) are measured directly at one and the same point in the standing wave, after which their values are extrapolated to the location of the specimen.

Another approximation to the direct method was developed by Bolt and Petrauskas \(^{67}\). The pressure and pressure gradient were measured by means of two small pressure microphones separated from one another by a distance \(d\) and placed near the specimen on the normal to its surface. A stationary plane wave of wavelength \(\lambda\), produced by a loudspeaker, is incident normally on the specimen. Let the acoustic pressures in front of the two microphones be, respectively, \(p_1\) and \(p_2\). The currents at the microphone outputs are combined in a circuit which gives a voltage proportional to the sum \((p_1 + p_2)\) and the difference \((p_1 - p_2)\), and indicates the phase shift \(\varphi\) between these voltages. Then the acoustic impedance is determined by the formula

\[ \frac{Z}{\rho c} = \frac{\pi d}{\lambda} \left|\frac{p_1+p_2}{p_1-p_2}\right| e^{i(\varphi-\pi/2)} . \]

It has a number of inaccuracies: instead of the pressure gradient, the finite difference \((p_1-p_2)/d\) is taken; the midpoint between the microphones is in fact not on the absorbing surface, but at a distance \(h\) from it; the microphones have finite dimensions; likewise, the waves are assumed plane, and diffraction phenomena arising near a small specimen are not taken into account. By applying proper experimental control and correction factors, however, it is possible to measure impedances in the range from 100 to 700 hertz when the impedance is less than 5. These measurements are quite—

*) The authors omitted here the words: “relative to the measurement thresholds of each of these quantities.” Ed. note.

fall, to within a few percent, with those values obtained by the hyperbolic-tangent method. The advantage of this direct method is the fundamental possibility of applying it to large surfaces of absorbing material already installed in the place intended for them.

The next most direct method consists in observing and analyzing, near an absorbing surface, interference phenomena^T1^. In its simplest form this method consists in using a long straight tube of constant cross-section, at the end of which the sample under test is placed. If a wave of definite frequency propagates along the tube, being partially reflected from the end, then the heights and positions of the pressure maxima in the resulting “standing wave”*) are determined exclusively by the complex impedance of the material at the end of the tube, if one abstracts from the scattering of energy in the tube and other phenomena for which corrections can be introduced. The impedance \(Z\) of the material is determined by the equations^M11^

\[ Z/\rho c=R/\rho c-iX/\rho c=\operatorname{th}\pi(\alpha-i\beta), \]

\[ P_{\min}/P_{\max}=\operatorname{th}(\pi\alpha), \]

\[ d_{\min}=\frac{\lambda}{2}(\beta+n),\qquad n=0,\ 1,\ 2,\ 3,\ldots \]

In these equations \(P_{\min}\) and \(P_{\max}\) denote the minimum and maximum values of the acoustic pressure, and \(d_{\min}\) is the distance from the surface of the sample to the first, second, third, etc. minimum.

This “hyperbolic-tangent method,” using a straight tube, has a number of shortcomings that limit its application. In order to work with low frequencies, long tubes must be taken; thus, at a frequency of 100 hertz, the tube must be longer than three meters. On the high-frequency side, the limit of applicability is set by transverse natural vibrations in the tube, which disturb the simple picture of longitudinal standing waves on which this method is based. In a tube 10 cm in diameter, the lowest transverse natural vibration arises at approximately 2000 hertz. Further difficulties arise owing to diffraction phenomena caused by the presence of the microphone in the standing wave. In view of these difficulties, the method proves suitable only for measurements on samples of moderate size^B9, P2, P3, P4, S9, D3, L5, P1, W10, P8^.

Some of the shortcomings of the rectilinear tube were eliminated by Hall in his ingenious modification of this method^H1^. The “tube” is a ring-shaped groove of square cross-section, cut into a mas—

*) Strictly speaking, this is a pseudo-standing wave, since sound energy is continuously absorbed and replenished. Such an interpretation is often used in acoustics as applied to the normal modes of a room even when there is absorption at the walls.

...lead circular plate. From above this groove is covered by another circular plate. A miniature condenser microphone is mounted in the upper plate with its diaphragm flush with its lower surface. When the upper plate is rotated it moves along the groove. The acoustic line is supplemented by two rectilinear sections of square cross-section, milled in the plates so that they connect the diametrically opposite ends of the annular groove. These tubes terminate respectively in a loudspeaker and the specimen being tested. In this way diffraction from the microphone is eliminated and the dimensions of the apparatus required for measurements at low frequencies are reduced.

Impedance may also be measured by its effect on acoustic resonance in a closed tube or in a chamber. A tube of given length, for a specified value of the terminal impedance, has discrete natural frequencies at which resonance occurs. The “sharpness of resonance” may be determined by three different methods, by measuring pressure as a function of: (a) sound frequency, (b) tube length, and (c) microphone position. In each case the terminal impedance can be uniquely determined with the aid of definite formulas and with the introduction of various corrections.

The most accurate method of measuring impedance was proposed by Beranek B2, B3. This method, besides its high degree of accuracy, has a number of other advantages. Measurements with it have been carried out over a wide frequency range from 100 to 8000 cycles per second. The method is absolute and is not based on comparison with a “standard” impedance. All possible sources of error, such as temperature changes, scattering of sound energy along the tube, etc., can be taken into account in processing the observations. The quantities measured are: pressure ratio, length, and frequency, all three being determined with a high degree of accuracy.

The resonance chamber of Beranek’s apparatus consists of a piece of steel tube. To cover the entire necessary frequency range, two tubes of different diameters are used. Into one end of each tube a massive brass plug about one inch thick is fitted. The sound source is a loudspeaker to which is attached a sliding capillary tube, conducting sound into the resonance chamber through a channel in the brass plug. Such a sound source, with high (internal) impedance, is essentially not subject to changes in impedance in the resonance chamber H7. The sound receiver is a thin brass tube, conducting sound from any point near the source outward to a crystal microphone. From the material being measured a circular disk is cut out, which is introduced inside a thin cylindrical holder. This holder is fastened to a solid brass piston about three inches thick, which is pushed into the resonant tube from the side opposite the sound source. The structure carrying the acoustic material is moved by means of a precision...

... screw, whose position is read with an accuracy of up to 0.0005 cm. The source of the sound signals is a generator, whose frequency is kept constant with an accuracy of up to \(10^{-5}\) by comparison with a primary frequency standard. The output current of the microphone is subjected to amplification and filtering, and then passes through a combined attenuator and measuring instrument, thus making it possible to measure the oscillations of the acoustic pressure. In order to calculate the acoustic impedance of the material, it is necessary to plot the resonance curves of pressure as a function of length for the cases of the presence

Fig. 10. Acoustic impedance of materials as a function of frequency according to Beranek’s measurements.

and absence of material at the end of the tube, determine the frequency, and find the speed of sound in the tube. In addition, the constancy of the temperature must be maintained. Temperature fluctuations in the course of the measurements can be introduced into the formula.

Typical results obtained by this method are illustrated in Fig. 10. These and other results obtained by Beranek have been used in our review for comparison of experimental results with the conclusions of theory. The final accuracy of these measurements is determined by the reproducibility of the results to within \(\pm 2\%\) over most of the sound range. This accuracy is quite sufficient, since it corresponds to the accuracy of manufacture of commercial acoustic materials.

Experience has shown that the methods discussed here have proved useful for measuring the acoustic impedance of small specimens fastened in a simple way to a rigid backing. In practice, acoustic materials are used in large areas and are mounted in various ways. It has been found (from the theoretical considerations given above and from preliminary experimental measurements) that, for some methods of fastening specimens, the impedance varies strongly with the angle of incidence of the sound ray. Therefore there arose an urgent need to measure the impedance of large specimens—at least large enough to reproduce the methods of fastening used in practice. It is likewise necessary to study the dependence of impedance on the angle of incidence. Both of these problems are being studied by improving the usual acoustic-chamber method, and also by the direct methods indicated above.

24. Determination of effective porosity, density, and flow resistance

The results of experimental measurements carried out by the methods indicated above make it possible to verify the correctness of the theory set forth in this chapter. Beranek’s measurements were made on specimens with a rigid backing material, so that equation (4.10) and the corresponding family of curves in Fig. 9 apply to this case. By comparing a particular experimental curve with the impedance curves in Fig. 9, one can indicate an approximate value of the parameter \(\gamma\). The frequency scale is first adjusted to the low-frequency end of the corresponding resistance curve, after which a further refinement of the value of \(\gamma\) may be required. This method of fitting gives probable values for \(r\), \(m\), and \(P\), from which the entire curve can be constructed.

One cannot, of course, expect that a curve constructed in this way for one system of constant values will reproduce the experimental result over the entire frequency range. Many acoustic materials are not completely isotropic and homogeneous, as was assumed in deriving equation (4.10). Therefore there is no reason to require constancy of \(r\), \(m\), and \(P\) for all frequencies. Similarly, in electrical circuits it happens that self-inductance and capacitance vary somewhat with frequency. Therefore it may be regarded as a very favorable result that more than half of Beranek’s curves are well represented by equation (4.10) with constant values of \(r\), \(m\), and \(P\) B\(^{12, B4}\). Later measurements, made by various methods, show that \(r\), \(m\), and \(P\) retain satisfactory constancy for most acoustic materials in the range from 100 to 6000 cycles.

These results make it possible to expect that measurement of \(r\), \(m\), and \(P\), followed by calculation of the impedance, may be used instead of the much more difficult method of direct impedance measurement

for all frequencies and for different methods of mounting. In order to test this possibility for porous materials, measurements of flow resistance were made \(^{B12, B13}\). Here the same relations exist for \(r\), which enters into the impedance equation, as in electrical circuits for resistance to direct and alternating currents—they are not necessarily equal to one another. The flow resistance of porous materials was studied and measured by Gemant \(^{G1}\), Rettinger \(^{R6}\), and others \(^{B4}\). The flow of air blown through specimens of the material is measured, as is the pressure drop across the specimen. The flow resistance per cubic unit is defined as

\[ r=\frac{\delta p}{(V/t)}\,\frac{A}{L^{2}}\cdot \mathrm{cm}^{-1}\,\mathrm{sec}^{-1}. \]

Here \(\delta p\) denotes the pressure drop in bars, \(V\) the volume of air in \(\mathrm{cm}^{3}\) passing through the specimen in \(t\) seconds, \(A\) the area, and

Fig. 11. Comparison of measured and computed values of acoustic impedance. The measured values of the resistive component are denoted by circles, and those of the reactive component by crosses. The theoretical values for the constants \(r\), \(P\), and \(m\) are shown by solid curves.

\(L\) is the thickness of the specimen. The measurements concerned a large number of materials, ranging from very porous hair felt with \(r=10\), to building fiberboard with \(r=2\cdot 10^{4}\ \mathrm{g}/\mathrm{cm}^{3}\cdot \mathrm{sec}\) \(^{B13}\). Most commonly used acoustic materials give values of \(r\) in the range from 50 to 500, with rare exceptions. About twelve of the fifteen materials investigated by Beranek \(^{B3}\) give a resistance lying within these limits.

Returning now to the comparison of numerical impedance values, we shall consider two cases in which the agreement between experiment and theory proved very good, on the assumption that

the values of $m$, $r$, and $P$ do not depend on frequency. These are the materials “permacoustic” and “acoustex” (Figs. 11 and 12). Both of these materials are physically similar in that they are comparatively rigid, while their surfaces are cracked, so that air readily penetrates inside. The measured flow resistance of these materials is in excellent

Fig. 12. Comparison of measured and calculated values of acoustic impedance.

agreement with the values of $r$ obtained by fitting the equivalent constants. The agreement is especially good for permacoustic; the other material, however, gives a discrepancy of up to $30\%$ between the static and the effective value of $r$. These data apparently indicate that

Fig. 13. Comparison of measured and calculated values of acoustic impedance.

the absorption of sound energy in this material occurs chiefly by the penetration of air and losses due to viscous friction in the pores—as is assumed by the simplest scheme.

Other materials have parameters that do not remain constant. For the material “thermacoustic,” 1.25 cm thick (Fig. 13), the experimental points systematically deviate from the theoretical value, revealing a gradual decrease in \(\gamma\) with increasing frequency. This material has a much more “fine-grained” structure than both of those mentioned above, and has no cracks. It is somewhat compressible and has a spongy structure. Blowing this material revealed a resistance value approximately twice as large as the effective value adopted in Fig. 13. Apparently, in this material the sound energy passes into vibrations of compression and rarefaction and is absorbed, in addition to viscous friction in the pores, also at the expense of internal friction. There are indications that in some cases the resistance to blowing exceeds the effective dynamic resistance by a factor of one hundred.

Fig. 14. Comparison of measured and calculated values of acoustic impedance. The discrepancies are explained by changes in \(P\) and \(m\) with frequency.

The numerical data given here are preliminary in character and may still be subject to revision. However, further accumulation of data is proceeding rapidly, and there are indications that, for homogeneous materials, the more accurate the measurements, the more closely they agree with the theoretical conclusions of this chapter.

The material “Celotex C-4” has channels penetrating the slabs of material almost all the way through, and a porosity on one side much greater than on the other, painted side. Of course, it cannot be correctly described by the simple theory expressed by equation (4.10). Since the conductivity of the pores is greater the lower the frequency, at low frequencies the effective porosity of the material is equal to the porosity of the channels.

The porosity for high frequencies corresponds to the porosity of the perforated surface of the material, and its role continuously decreases as the frequency increases. As evidence of this, in Fig. 14

two pairs of theoretical curves are shown for two different constant values of porosity and effective mass. In all probability, a curve calculated for parameters varying continuously with frequency would best represent the experimental data.

In conclusion, one may state that, in general, there is good agreement between the theoretical impedance values and the measurements for homogeneous materials on a rigid backing. Many materials have physical parameters that are quite constant over different frequencies. Likewise, the effective resistance values are equal to the flow resistance for the more rigid porous materials; for certain compressible materials it is less than the static value.

(To be continued in the next issue.)