Contemporary Development of Architectural Acoustics*

Vern O. Knudsen.

Contents. § 1. Introduction. § 2. The early stage in the development of architectural acoustics. § 3. The present era of architectural acoustics. § 4. Growth and decay of sound in enclosed spaces. General considerations. § 5. Sabine’s and Eyring’s theory of reverberation. § 6. Recent modifications of the reverberation formula. § 7. Room resonance. § 8. Reverberation measurements in small rooms. Determination of the absorption coefficients of building materials. § 9. Determination of absorption coefficients by direct reflection. § 10. Absorption of sound by porous materials. § 11. Optimum reverberation time for speech and musical rooms. § 12. Variation of reverberation time with frequency for speech and musical rooms. § 13. Transmission of sound through building materials. § 14. Calculation of sound insulation and noise reduction in buildings.

§ 1. Introduction

Architectural acoustics is no longer shrouded in mystery: it is firmly grounded in physical facts and principles that have been developed through the work of a relatively small number of physicists and engineers. Without these works, carried out chiefly by physicists during the last thirty-five years, architectural acoustics would still be a purely empirical science, based on generalizations of geometrical acoustics and on a few qualitative observations concerning the reflective and absorptive properties of building materials and interior furnishings. The practical achievements obtained as a result of recent research in architectural acoustics provide a striking example of the value of bringing physicists and their methods to bear on the development of neglected and undeveloped branches of technology, even amid today’s heightened interest in atomic physics. Thirty-five years ago, the acoustical design of buildings, if it existed at all, amounted to no more than an attempt—often unsuccessful—to imitate known constructions that had earned public approval and to avoid others that had a bad reputation. At the present time, acoustical design is based on reliable formulas and numerical data concerning sound-absorbing and

* “Recent Developments in Architectural Acoustics,” Vern O. Knudsen, University of California of Los Angeles. — Reviews of Modern Physics, 1,1 January 1934. Translated from the English by Ya. A. Kopilovich.

sound-insulating properties of building materials, as well as the form of the room, in such a way that the acoustics of a building can be determined before construction; or, what is still more important, any given acoustical requirements can be practically realized in a completed building.

Architectural acoustics is thus approaching the state of an applied art and is on the way toward becoming an equal branch of engineering. Unfortunately, however, in our educational system acoustic engineering still does not receive the attention it deserves, and therefore, although a number of large organizations interested in the production and sale of acoustic instruments or materials have done a great deal, especially in developing instruments for studying the acoustic properties of rooms, the direct development of architectural acoustics depends mainly on the investigations of physicists. Until architectural acoustics becomes an established and encouraged discipline in our technical schools, it is highly desirable not only that the physicists already working at present in this field continue their investigations, but also that additional workers be drawn to the solution of the many unfinished and promising problems awaiting an interested and qualified investigator.

In view of the foregoing, the aim of the present article will be not only to review the latest achievements obtained as a result of applying the instruments and methods of modern physics to room acoustics, but also to set forth unfinished or as yet unsolved problems that promise to yield results important in their significance.

§ 2. Early Stage in the Development of Architectural Acoustics

The author has elsewhere[^1] given a survey of the evolution of the auditorium and of early beginnings in the field of architectural acoustics in the nineteenth century. Although a numerous series of fundamental problems in the acoustics of Greek and Roman theaters is conscientiously analyzed in Vitruvius’ well-known Ten Books on Architecture, it was only in the nineteenth century that serious attention began to be paid to the acoustic problems of architectural spaces, and these investigations were only qualitative in character. The most noteworthy among the investigators of the nineteenth century were: 1) J. B. Upham,[^2] who studied the reverberational properties of the main auditorium of the Boston Music Hall during the completion of its construction, finishing, and furnishing, and recommended the use of additional lining or even canvas on the walls as a means of reducing reverberation; 2) Joseph Henry,[^3] who clearly defined the factors affecting reverberation and echo, and who, on the basis of his own theories and experiments, designed a new lecture hall for the Smithsonian Uni-

sity, possessing extremely satisfactory acoustical properties, and 3) T. Roger Smith[^4], an English architect who strictly defined various acoustic requirements for speech and music rooms. It is also necessary to mention the works of Lord Rayleigh[^5], who recognized the possibility of controlling reverberation in rooms by means of carpets, draperies, and furniture. All this, however, was only qualitative in character.

§ 3. The Modern Era of Architectural Acoustics

The beginning of the quantitative development of architectural acoustics was laid by W. C. Sabine in 1895, when he undertook an investigation of the acoustics of the Fogg Lecture Hall at Harvard University[^6]. He established that reverberation is the most important factor influencing the acoustic quality of a room, and accordingly devoted a large part of the rest of his life (he died in 1918) to the quantitative study of the growth and decay of sound in an enclosed room. He derived an equation and determined the absorption coefficients of building materials, which made it possible to calculate the reverberation properties of a room both before and after construction, and gave the best explanation of the principal factors affecting the acoustic properties of rooms. Sabine is deservedly considered the founder of architectural acoustics; his work made possible not only the design, construction, or shaping of rooms in such a way as to provide them with good acoustic properties, but also attracted the attention of other investigators, who continued the work he had left unfinished, improved his reverberation formula, and made a number of other important discoveries concerning sound insulation, sound amplification, room resonance, and optimum reverberation for speech and music rooms. All these factors are of great importance for room acoustics, but the most fundamental of them, however, is the problem so simply and elegantly solved by W. C. Sabine—the problem of the growth and decay of sound in enclosed rooms.

§ 4. The Growth and Decay of Sound in Enclosed Rooms. General Considerations

It is obvious that a rigorous consideration of the growth and decay of sound in an enclosed room must be based on the general theory of vibrations in a bounded three-dimensional continuum. Historically, however, much simpler, approximate treatments were considerably more popular. But even these approximate theories, when applied with caution and understanding, have satisfactorily served the practical purposes of acoustic design and will continue to serve until they are replaced by more exact theories.

These approximate theories were based on an incorrect premise, equivalent to the assumption that sound arising at some point in a room propagates quanta or rays of vibrational energy uniformly in all directions; that these rays are partially reflected by the boundaries of the room; and that even after the action of the source has ceased these rays retain their original frequency, but become weaker after each reflection, like a very large number of billiard balls moving with constant velocity on a billiard table, but diminishing in size after each reflection until they become infinitely small. Thus, in these approximate theories, at least in the later ones, it is assumed that the sound energy continues to remain in rays or beams; that during the decay the sound energy in the room remains constant in these rays or beams for a short interval of time, equal to the time required for the ray to traverse the average distance between successive reflections, called the “mean free path,” and then suddenly falls by a known amount determined by the “mean” absorption coefficient of the room boundaries; and that this process of absorption in separate steps continues until all the sound energy has been converted into heat. The formulas to which these approximate theories may lead are sufficiently valuable for all practical purposes in rooms bounded by materials having the same absorption coefficient. In rooms, however, bounded by materials having widely different absorption coefficients—which is usually the case in reality—the formulas acquire practical significance only in those cases where the decaying sound in the room approaches the state of complete diffusion. It must therefore be precisely established that the formulas which we shall now derive must be applied with caution and understanding, especially with respect to the average absorption coefficient in rooms bounded partly by highly reflecting and partly by highly absorbing materials.

§ 5. Sabine’s and Eyring’s theory of reverberation

The early experiments of W. C. Sabine\(^7\) showed that the reverberation time in a room, i.e. the time required for the sound intensity to decrease by 60 db, or to one millionth of its initial value, during free decay, is directly proportional to the volume of the room and inversely proportional to the total absorption provided by the boundaries of the room, namely \(\sum \alpha s\), where \(\alpha\) is the sound-absorption coefficient of any part of the boundary having area \(s\), and where the product is taken over the entire boundary of the room. By a series of ingenious experiments in different rooms Sabine was able to determine the constant of proportionality \(k\)

between the reverberation time \(t_{60}\) and the volume \(V\), divided by the total absorption \(a\), and to give architects and builders a simple, but extremely valuable, equation:

\[ t_{60}=\frac{kV}{a} \tag{1} \]

The value of \(k\), determined experimentally by Sabine for a large number of rooms of various forms and sizes, at normal room temperature was \(0.05\) in English units and \(0.164\) when \(V\) is expressed in \(m^3\) and \(a\) in \(m^2\).

Several years later Eger obtained the same equation, proceeding from theoretical considerations on the basis of a method that had proved extremely fruitful as applied to the kinetic theory of gases. He obtained an equation for the rate at which the scattered sound energy in a closed room strikes a unit area of the boundary, namely \(\frac{\rho c}{4}\), where \(\rho\) is the mean volume density of sound energy in the room, and \(c\) is the speed of sound. He further assumed that the rate of absorption of sound energy by the boundaries of the room is equal to \(\frac{\rho c}{4}\sum as\)—a quite probable assumption under the condition that the sound energy is completely scattered during the decay and under the condition that \(\rho\) changes continuously*. (This will approximately be the case if the decay occurs slowly, i.e., if the boundaries of the room are so highly reflecting that a large number of reflections is required for the degradation of the sound energy into heat.) Eger then equates the rate of change of sound energy in the room \(V\frac{d\rho}{dt}\) to the difference between the rate of emission of sound energy from the source \(E\) and the rate of absorption of sound energy by the boundaries of the room, i.e.

\[ V\frac{d\rho}{dt}=E-\left(\frac{\rho c}{4}\right)\sum as. \]

Solving this equation by introducing the corresponding boundary conditions, and remembering that \(a=\sum as\), gives, for the growth of sound energy,

\[ \rho=\rho_0\left(1-e^{-\frac{act}{4V}}\right); \tag{2} \]

for the decay of sound energy:

\[ \rho=\rho_0 e^{-\frac{act}{4V}}; \tag{3} \]

* Compare with Clausius’s formula for the rate at which gas molecules in a closed vessel strike a unit area of the boundary, namely \(\frac{nc}{4}\), where \(n\) is the number of molecules per \(cm^3\), and \(c\) is their mean speed.

and for the mean value of the energy density of the equilibrium state \(\rho_0\):

\[ \rho_0=\frac{4E}{ca}. \tag{4} \]

If equation (3) is solved for \(t\) at \(\frac{\rho}{\rho_0}=10^{-6}\), the result gives the reverberation time \(t_{60}\), namely:

\[ t_{60}=\frac{kV}{a}, \tag{5} \]

and the value \(k\), which depends on the speed of sound, will be, at normal room temperature, 0.049 in English units and 0.161 in metric units, which is in very good agreement with the experimental values obtained by Sabine [see equation (1)]. This relation gave such convincing confirmation of Sabine’s empirical equation that it was used for almost 30 years for calculating the reverberation time of both rooms being designed and completed rooms. Even the false conclusion to which the equation leads for a room with absolutely absorbing surfaces, namely that \(t_{60}=\frac{kV}{S}\) instead of zero (where \(S\) is the total surface area of the room), was not taken into account and was not sufficient to shake belief in the value of the Sabine–Eyring equation until very recently. This equation is practically satisfactory for frequencies between approximately 200 and 1000 hertz in the vast majority of rooms in which the rate of decay of sound is small, i.e., the equation applies to “live” rooms, provided that the frequencies are sufficiently high, considerably above the fundamental resonant frequency of the room, but not high enough for attenuation in the medium to have to be taken into account.

§ 6. Recent modifications of the reverberation formula

A more satisfactory reverberation formula can be obtained by assuming that the attenuation of sound occurs discontinuously, at intervals of time equal to the time required for sound to traverse the mean free path. Obviously, this discontinuous process of absorption corresponds more accurately to actual conditions than a continuous process, since each sound ray travels a finite distance (usually equal to the mean free path) without absorption, and then undergoes a finite absorption at each reflection. According to the data of Sabine\(^6\) and Eyring\(^7\), the mean free path for a room is equal to \(4\frac{V}{S}\). Strictly speaking, the mean free path depends on the shape

of the room and the position of the source, especially during the first few reflections, but for most rooms of ordinary shape it does not differ from \(4\dfrac{V}{S}\) by more than \(\pm 8.0\%\).* Below we shall take the mean free path to be equal to \(4\dfrac{V}{S}\).

Let \(\Delta t\) be the time required for sound to traverse a distance equal to the mean free path, i.e. \(\Delta t=\dfrac{4V}{Sc}\), and suppose that the decay is sufficiently slow and uniform, so that the mean density of sound energy during each time interval \(\Delta t\) is the arithmetic mean of the initial and final densities—an assumption sufficiently justified for the rates of decay encountered in practice. At the beginning of the decay the density is equal to \(\rho_0\), i.e. to the mean steady-state value. After an interval \(\Delta t\), i.e. after the first reflection, taking all path lengths as equal to the mean free path, the density will decrease approximately to \(\rho_0(1-\bar{\alpha})\), where

\[ \bar{\alpha}=\frac{\sum \alpha s}{S} \]

is the arithmetic mean of the absorption coefficients of all the boundaries of the room. Then the mean density during the first interval \(\Delta t\) will be

\[ \frac{1}{2}\left[\rho_0+\rho_0(1-\bar{\alpha})\right]. \]

Therefore the quantity of sound energy absorbed during the first interval \(\Delta t\) will be approximately equal to

\[ \sum \alpha s \Delta t \cdot \frac{1}{2}\left[\rho_0+\rho_0(1-\bar{\alpha})\right]\frac{c}{4}. \]

Consequently, the more correct value of the energy density at the end of the first time interval will be equal to

\[ \rho_0-\frac{\left[\rho_0+\rho_0(1-\bar{\alpha})\right]c}{8V}\sum \alpha s \Delta t, \]

or

\[ \rho_0-\frac{1}{2}\left[\rho_0+\rho_0(1-\bar{\alpha})\right]\bar{\alpha}, \]

where \(c\Delta t\) is the mean free path, i.e. \(4\dfrac{V}{S}\).

The approximate value of the energy density after successive time intervals \(\Delta t\), \(2\Delta t\), etc., is given in Table 1.

After a time interval \(t\), \(n=\dfrac{t}{\Delta t}\), and therefore

* For a cruciform type of room with a high ceiling, the mean free path is equal to \(4.24\dfrac{V}{S}\); for a large rectangular office room with a low ceiling it is equal to \(3.75\dfrac{V}{S}\).

VERN O. KNUDSEN

TABLE 1

\(t\)	\(\rho\)
\(0\)	\(\rho_0\)
\(\Delta t\)	\(\rho_0-\left[\dfrac{\rho_0+\rho_0(1-\alpha)}{2}\right]\alpha\).
\(2\Delta t\)	\(\rho_0-\left[\dfrac{\rho_0+\rho_0(1-\alpha)}{2}+\dfrac{\rho_0(1-\alpha)+\rho_0(1-\alpha)^2}{2}\right]\alpha\).
\(\vdots\)	\(\vdots\)
\(h\Delta t\)	\(\rho_0-\left[\dfrac{\rho_0+\rho_0(1-\alpha)}{2}+\cdots+\dfrac{\rho_0(1-\alpha)^{h-1}+\rho_0(1-\alpha)^h}{2}\right]\alpha\).

the value of \(\rho\) at this time \(t\) is equal to:

\[ \rho=\rho_0\left\{1-\alpha\left[\frac{1}{2}+ \sum_{n=1}^{\,n=\frac{Sct}{4V}}(1-\alpha)^n -\frac{(1-\alpha)^{\frac{Sct}{4V}}}{2}\right]\right\}. \tag{6} \]

For the rate of decay encountered in most rooms, practically in all rooms with the exception of those bounded by materials with an absorption coefficient almost equal to unity, the first and third terms in the brackets are small in comparison with the second term, and they have significance only in the case when the decay consists of a very small number of reflections. Therefore, approximately:

\[ \rho=\rho_0\left\{1-\alpha \sum_{n=1}^{\,n=\frac{Sct}{4V}}(1-\alpha)^n\right\} = \]

\[ =\rho_0\left\{1-\alpha \frac{1-(1-\alpha)^{\frac{Sct}{4V}}}{1-(1-\alpha)}\right\} = \]

\[ =\rho_0(1-\alpha)^{\frac{Sct}{4V}}. \tag{7} \]

This same result can be obtained by a number of methods, the simplest of which assumes that the energy density decreases by the same fractional amount \(\alpha\) after each reflec-

tion, i.e., after each time interval \(\Delta t\), so that the energy density after times \(\Delta t, 2\Delta t \ldots n\Delta t\) is equal to

\[ \rho_0(1-\alpha),\ \rho_0(1-\alpha)^2\ldots \rho_0(1-\alpha)^n, \]

respectively*. Or, in general:

\[ \rho=\rho_0(1-\alpha)^{-\frac{Sct}{4V}} . \tag{7'} \]

If we take \((1-\alpha)=e\), equation (7) becomes similar to equation (3), namely:

\[ \rho=\rho_0 e^{S\ln(1-\alpha)\frac{ct}{4V}}, \tag{8} \]

and reduces to equation (3) when \(\alpha\) is very small, i.e., when the boundaries of the room are highly reflecting. Both equation (3) and equation (8) are only approximate formulas. Equation (3) is satisfactory only for very live rooms, whereas equation (8) meets practical requirements for both live and “dead” rooms, provided that the absorbing materials in the room are not concentrated on one or two surfaces of the room. For frequencies above 1000 hertz, both equation (3) and equation (8) are of no value.

Equations (3) and (8) are based on the assumption that all absorption of sound energy in the room takes place at the boundaries, i.e., that absorption in the air is so small that it may be neglected. Stokes, Kirchhoff, and Rayleigh calculated the absorption of sound in air as a result of viscosity and thermal conductivity and found that, for frequencies up to 6,000 or even 8,000 hertz, the absorption is so small in comparison with surface absorption in actual rooms that it may be entirely neglected. Experimentally, however, this has been refuted1. The absorption of sound in air is not only considerably greater than according to the classical theory of Stokes, Kirchhoff, and Rayleigh, but also depends on temperature and humidity in such a way that this dependence can be explained only by absorption resulting from collisions between gas molecules[^11][^12]. Recent experiments on the absorption of sound in air show that the coefficient of absorption is of the order of 10 to 100 times greater than predicted by the classical theory, so that for high frequencies (10,000 hertz, for example) absorption in the air in a room may be greater than surface absorption, especially in large rooms. The curves in Fig. 1

give the absorption coefficient \(m\) per foot (or per centimeter) for plane waves in air at \(20^\circ\mathrm{C}\) and at the relative humidities shown on the abscissa, for frequencies of 1,500, 3,000, 6,000, and 10,000 hertz. The coefficient \(m\) is determined by \(\rho_x=\rho_0 e^{-mx}\), where \(\rho_0\) is the energy density in the plane wave at \(x=0\), and \(\rho_x\) is the density after the wave has traveled the distance \(x\). From the curves of Fig. 1 it is evident that \(m\) is at a maximum at a definite concentration of water vapor, different for each frequency. Further, the magnitudes of these maxima are proportional to the first power of the frequency, and not to the second power, as required by the classical theory of absorption. It is noteworthy, for example, that at a relative humidity of 18% \(m\) is equal to \(0.020\ \mathrm{ft}^{-1}\) for a frequency of 10,000 hertz, whence

Fig. 1. Coefficient of absorption of sound in air containing different amounts of water vapor, for frequencies of 1,500, 3,000, 6,000, and 10,000 hertz.

the intensity (or energy density) of such a wave after traversing a distance \(\left(\dfrac{1}{0.020}\right.\), or 50 feet\()\) will be reduced to \(\dfrac{1}{e}\) of its initial value. This is equivalent to a rate of attenuation of \(96\ \mathrm{db/sec}\), which considerably exceeds the most desirable rate of attenuation for musical rooms.

Such excessive absorption of sound in air makes it necessary to introduce a corresponding attenuation factor into equation (8), and the attenuation equation takes the form:

\[ \rho=\rho_0 e^{-mct}\left[e^{\ln(1-\alpha)\cdot \frac{Sct}{4V}}\right]; \]

\[ \rho=\rho_0 e^{\left[\ln(1-\alpha)\cdot\frac{S}{4V}-m\right]ct}. \tag{8a} \]

The reverberation time \(t_{60}\) is obtained by solving equation (8a) for \(t\) when \(\rho_0/\rho = 10^{-6}\). Thus,

\[ t_{60}=\frac{55.3V}{c[4mV-S\ln(1-\alpha)]}. \tag{9} \]

At room temperature, \(21^\circ\mathrm{C}\), \(c=1125\) ft/sec, so that for most working conditions:

\[ t_{60}=\frac{0.049\,V}{4mV-S\cdot \ln[(1-\alpha)]} \tag{10} \]

in English units, or

\[ t_{60}=\frac{0.161\,V}{4mV-S\cdot \ln(1-\alpha)}, \tag{11} \]

where \(V\) and \(S\) are expressed in \(m^3\) and \(m^2\), respectively. For frequencies below approximately 1000 cycles, \(m\) is so small that the first term of the denominator in equations (9), (10), or (11) may be neglected, i.e., absorption in air is insignificant, whereas at high frequencies (above 10,000 cycles) this term may become larger than the second term (surface absorption). At sufficiently high frequencies (above the audible range), the second term will become so small that it may be neglected, in which case the rate of attenuation, and consequently the reverberation time, will not depend on the size of the room.

The preceding reverberation formula, equation (9), is sufficiently valuable for practical purposes provided that the sound in the room is completely diffuse during attenuation. This condition is fulfilled for frequencies above approximately 250 cycles in all rooms, except very small ones, provided that all the boundaries of the room have approximately the same absorptivity, or provided that appropriate rotating vanes or “vibrating tones” are used for “mixing” the sound in the room. The corresponding precautions, analogous to those just mentioned, may be taken when measurements are made in acoustic laboratories. In many rooms encountered in practice, the absorbing material may be concentrated on one surface, as in the case where a carpet, upholstered seats, and the audience are located on the floor, while all other surfaces in the room are highly reflecting. In such cases, especially if the opposite walls are parallel and not too far apart, the attenuation of sound will not correspond to the approximately exponential attenuation of equation (9), but will consist of: 1) a high rate of attenuation, when the sound is relatively diffuse, and 2) a considerably slower attenuation, composed chiefly of a horizontal flow

sound energy between parallel and highly reflecting walls. The reverberation time in such a room will be greater than that calculated by means of equation (9) with the arithmetic mean for $\alpha$. This is shown by certain oscillograms of the rate of decay, obtained in a small room $8' \times 8' \times 9.5'$ (high) with a floor covered with a material having an absorption coefficient of 0.60 at 512 hertz, and with walls and ceiling covered with painted concrete. The first part of the decay (15 to 17 db) proceeded relatively rapidly (55—100 db/sec). This was followed by a considerably slower decay of 38—40 db/sec. If the first part of the decay is used to calculate the reverberation time $t_{60}$ for the absorptivity of the floor material $m\alpha$, we obtain $t_{60}=0.61$ sec. and $\alpha=0.55$. When the latter part is used, $t_{60}=1.54$ sec. and $\alpha=0.22$. The first part of the decay, when the sound is relatively diffuse, gives a value for $\alpha$ which is in fairly good agreement with the theoretical value of 0.60 and, consequently, satisfies the requirements of equation (9) fairly well, whereas the latter part of the decay, consisting chiefly of the horizontal flux (or resonance) of sound energy between parallel highly reflecting surfaces, proceeds considerably more slowly.

Fortunately, however, for the best acoustic quality of a room the absorbing material must be placed on all surfaces of the room in such a way that the rate of decay is, at least, approximately the same in all directions; under such conditions equation (9) gives results which, as a rule, will not differ by more than 10% from the observed values. Moreover, in very large rooms, for example theaters and school auditoriums, there is a very small tendency toward two-dimensional reverberation, even when a large part of the absorbing material is concentrated on the floor and ceiling, first, because the dimensions of the room are large in comparison with the wavelengths of sound, and, second, because the architectural treatment of large rooms usually includes structural forms and ornamentation tending to scatter sound during free decay. In such rooms, provided there are no curved surfaces causing concentration of sound, the decay of the first 30 db (or a little more) corresponds very closely to equation (9), and this very part of the decay is the most essential for the acoustic quality of speech and music in rooms. In other words, the rate of decay after the first 30 db is of the greatest significance, since in articulate speech or music such residual sounds are so weak that they are completely masked by the primary (and considerably louder) sounds that follow them. It is obvious, therefore, that equation (9) represents a satisfactory value for practical calculations of reverberation in most rooms.

§ 7. Resonance of Rooms

It would seem that the theory of reverberation described in the two preceding sections cannot be correct for the case in which the length of the sound wave is not small in comparison with the dimensions of the room, since under such conditions one or more of the low-frequency modes of vibration of the room may be sharply expressed, and the presence of these vibrations would exclude the possibility of a diffuse state of sound. In reality, however, irrespective of whether the wavelength is large or small, reverberation in a room must be regarded as the free decaying vibrations of the column of air enclosed in the room, as was indicated by Strutt[^13] and by Wetzmann and Schuster[^14]. The author has shown that, at least at low frequencies, reverberation consists precisely of these free damped vibrations of the air in the room[^15]. The natural frequencies of a rectangular room are readily obtained by introducing the corresponding boundary conditions into the wave equations[^16]. The frequencies \(n\) are expressed by the equation:

\[ n = \frac{c}{2}\left[\frac{p^2}{l_1^2}+\frac{q^2}{l_2^2}+\frac{r^2}{l_3^2}\right]^{\frac{1}{2}}, \tag{12} \]

where \(l_1, l_2\), and \(l_3\) are the dimensions of the room and \(p, q, r\) are the integers \(0, 1, 2, 3 \ldots\) The fundamental type of vibration in the direction \(l_1\) is obtained when \(p=1\) and \(q=r=0\), i.e. \(n_{1,0,0}=c/2l_1\), which corresponds to the fundamental frequency of vibration of a pipe of length \(l_1\), open or closed at both ends. Thus, for a rectangular room in which \(l_1=12.5\) ft, \(n_{1,0,0}=1125/25=45\). If 12.5 ft is the greatest dimension of the room, then the lowest type of vibration will have a frequency of 45 cycles. In general, however, three infinite series of overtones will occur. The first twenty characteristic frequencies of a room \(8' \times 8' \times 9.5'\), calculated according to equation (12), were readily determined, first, by their resonant effect on the intensity of a steady tone of slowly varying frequency (the intensity of the tone increased considerably whenever the frequency of the tone coincided with one of the natural frequencies of the room), and, second, by an oscillographic investigation of the decay of low tones in the room[^17]. In the author’s study of resonance in rooms it was found that the frequency of the reverberating tone is never equal to the frequency of the exciting tone, except in the case when the exciting tone is tuned exactly to one of the characteristic frequencies of the room. The frequency (or frequencies) of the reverberating tone always consists of one or more free modes of vibration of the room. Thus, one of the most prominent of the free vibrations in a small room was the fundamental vibration in the horizontal direction with a frequency of 71 cycles. When a tone of any frequency between 66 and 76 cycles was excited in the room and then stopped, the decay—

...the beating oscillation proved to be a tone with a frequency of 71 hertz. Fig. 2 contains a series of oscillograms which clearly show the resonant character of reverberation in a small room. The frequency of the tone in the steady state is shown under each oscillogram. Although seven different simple tones with frequencies from 90.8 to 100.6 hertz were used, the decay in each individual case consists, as shown, chiefly of two characteristic frequencies of the room, namely 92.8 and 99.8 hertz, two natural frequencies,

Fig. 2. Oscillograms of sound decay in a small rectangular room, showing that sound decay consists of the decaying free oscillations of the room.

closest to the imposed frequency. In each case a beat frequency of 7 hertz is clearly distinguishable. A number of other oscillograms obtained in this and in other rooms confirm the conclusion that reverberation is always made up of the free decaying oscillations of the air in the room. These considerations confirm the theoretical assumptions of Wezmann and Schuster and of Strutt, and consequently the theory of reverberation must necessarily take into account the resonant oscillations of a three-dimensional bounded space. However, as Strutt has shown^15, the formal law determining the free decaying oscillations of a three-dimensional continuum approaches asymptotically the simple reverberation law of Sabine and Eyring when the wavelength of the exciting sound becomes small in comparison with the wavelength of the lowest-frequency...

natural oscillations of a room. Thus, in a room with the greatest dimension of 10 feet (which constitutes a room of minimum size in which acoustics is a significant factor) the wavelength of the lowest-frequency mode of oscillation is 20 feet. In such a room, a sound with a wavelength equal to one tenth of the fundamental, namely 2.0 feet, would be sufficiently short to satisfy Sabine’s law. In other words, if the room is filled with sound having a wavelength shorter than 2.0 feet, i.e. a frequency greater than approximately 560 cycles, the excited types of oscillation are so numerous, and the frequencies so close to one another, that the sound in the room is substantially diffuse; therefore the requirements of the approximate reverberation theories described in the two preceding paragraphs are fulfilled, especially if the absorbing material is distributed uniformly over all the boundaries of the room, or if certain devices are provided for mixing or scattering the sound during decay. Moreover, the pitch of the decaying tone is indistinguishable from the pitch of the tone in the continuation of the steady state— a condition not realized for the lowest-frequency modes of oscillation. In rooms, for example concert halls, school auditoriums, and theaters, the lowest oscillations are usually in the inaudible frequency range, so that the elementary theory of reverberation applies in them with sufficient accuracy to all frequencies above 100 cycles, and the influence of room resonance can usually be neglected.

§ 8. Reverberation measurements in small rooms. Determination of the absorption coefficients of building materials

Accurate measurements of reverberation in small rooms are of primary importance, because they have been used, and are used, almost exclusively since the time of the first work in this field by W. C. Sabine, as a practical method for determining the sound-absorption coefficients of building materials and finishes, and especially of such materials as acoustic felts, tile, and plaster. The total absorption of a room, for example a reverberation chamber, is determined by measuring the rate of decay or the reverberation time, first when the room contains a known area of acoustic material, and then when the acoustic material has been removed from the room. The value of $\bar{\alpha}$ and, consequently, of the total absorption $\bar{\alpha} S$ can be computed by means of equations (8a) and (9). The absorption of the acoustic material in the room is taken to be equal to the difference between the absorption of the room with the material in it and the absorption of the room after the material has been removed. This is equivalent to the assumption that $\bar{\alpha}$ is the arithmetic mean of all the absorbing surfaces in the room, which is justified provided that the sound in the room is maintained in a thoroughly diffuse state during the-

decay of the steady state and of attenuation. For studies with tones below 500 hertz, warble tones of a width of at least 100 hertz and a rate of vibration of not less than 4 or 5 per second should be used; moreover, for studies with tones of all frequencies, large rotating vanes should be used. The test tones should be simple. If these precautions are taken, the rate of decay will satisfactorily correspond to the theoretical value of equation (8a), and, if the test area is of the order of 72 sq. feet, the difference between the rates of decay with the acoustic material in the room and without it will be large enough to give absorption coefficients accurate to approximately $\pm 0.03$ for frequencies up to 2000 hertz.* At higher frequencies, absorption in the air, which may vary during the time required to complete the test, is so large a factor that errors of the order of $\pm 0.10$ are inevitable, unless the reverberation chamber is provided with special equipment maintaining a definite state of the air. Even in such a room the accuracy is unsatisfactory at frequencies above 4000 hertz, because absorption in the air is so large that the difference between the rates of decay in the presence of acoustic material in the room and without it is insignificant, unless the test area is not too large.

This source of error in measuring the absorptivity of acoustic materials at high frequencies is so serious that the author is preparing an experimental chamber which can be filled, instead of air, with a nonabsorbing gas, for example nitrogen. In nitrogen at room temperature the absorption is only slightly greater than the absorption due to viscosity and thermal conductivity and, consequently, can almost be neglected for frequencies up to 8000 hertz. Preliminary experiments show that this method will considerably increase the accuracy of absorption measurements for frequencies above 2000 hertz.

The rate of decay is measured by some kind of reverberometer, which usually consists mainly of: 1) a suitable source of continuous or warble tones, a cathode generator, an electric low-frequency filter, a high-frequency amplifier, and an electrodynamic loudspeaker; 2) a high-quality microphone and amplifier; 3) an electric attenuator for regulating the amplification; and 4) either a recorder continuously registering, on a moving paper chart or on a light-sensitive film, a graphical record of the decay, or some type of indicator, usually a relay and chronograph, by means of which the rate of decay can be determined.

The Bell Laboratories have developed an automatic lever re—

* Gund16 showed that the frequency of the warble tone should vary within approximately 20%; see also Meyer and Just17.

corder with a response proportional to the logarithm of the current actuating it, the instrument being adjusted so that the recording is made directly as the paper tape moves at a constant speed; the rate of decay of the sound is given in decibels per second. If the decay follows an exponential law, the curves will be straight lines. However, since the decay consists of several adjacent frequencies in the immediate vicinity of the frequency of the exciting tone, there will be some interference between these frequencies (each of which may decay exponentially), so that the resulting

Fig. 3. Curves showing the decay of sound in enclosed rooms, obtained on an automatic recorder developed by Bell Laboratories. Curves 9, 10, and 11 are for a pure tone with the recorder adjusted to rates of 240, 120, and 50 db/sec, respectively. Curves 12, 13, and 14 were obtained with the corresponding rates, but with warbling tones.

decay curve will in general be irregular. Typical decay curves obtained with this instrument in the reverberation chamber of Electrical Research Products in New York are reproduced in Fig. 3*. As these records show, the decay is not strictly exponential; with the exception of small oscillations, which may be attributed to the resonance or interference phenomena discussed in the preceding paragraph, the general course of the decay conforms quite satisfactorily to an exponential law within the range of 40 db; and if a straight line is applied to the recorded decay curve, the slope of this line gives the rate of decay with sufficient accuracy for practical purposes.

The curves marked 9, 10, and 11 were produced by a simple tone, picked up by one microphone, and recorded by a recorder adjusted so as to “follow” the maximum

* These records were provided to us by C. K. Wolff.

at decay rates of 240, 120, and 60 db/sec, respectively. In curve 9, for example, the recorder is able to follow the actual decay considerably more accurately than in 11, where only the slower changes of decay are recorded. Curves 12, 13, and 14 were recorded at the same recorder speeds, respectively, but instead of a single pure tone a warbling tone was used. The advantage of the warbling tone for reverberation measurements is obvious.

In the indicator type of reverberometer used in most acoustic laboratories in the U.S.A., the attenuator is successively adjusted for different amounts of attenuation in the ampli-

Fig. 4. Decay curves in a 6-ft. cubic chamber, showing that the decay is exponential, and that the rate of decay increases with frequency. The increase in the rate of decay at higher frequencies is explained by absorption in the medium, and not by any significant increase in boundary absorption.

fier circuit. Then, for each setting of the attenuator, the time required for the decay to reach a certain predetermined level—such as, for example, that required to operate a relay and produce a reading on the chronograph, or only to flash a neon lamp—is measured separately for each attenuator setting. Then, if the attenuator readings (which are usually calibrated in decibels) are plotted as a function of these observed time intervals, the resulting curve gives the typical decay curve. Fig. 4 shows a series of decay curves obtained by this method (with a neon-lamp indicator) in a cubic steel stud chamber at the University of California in Los Angeles.* The arrangement of the apparatus for obtaining these decay curves is shown in Fig. 5. The set-

* See note.¹¹

setting of the lever \(L\) on the dial determines the time after the beginning of decay at which the neon lamp at the amplifier output flashes. For each setting of the attenuator (which determines the ordinates in Fig. 4), the lever \(L\) is adjusted until the lamp flashes when the contacts at \(B\) are closed by the rotating brass insert. This determines the abscissas in Fig. 4. The large increase in the rate of decay at the higher frequencies (shown in Fig. 4) is a consequence almost exclusively of the increased absorption in air at the higher frequencies. If the chamber were filled with a nonabsorbing gas (nitrogen, for example), the rate of decay would be almost constant for all frequencies and

Fig. 5. Diagram of the arrangement of the apparatus at the University of California in Los Angeles for measuring the rate of decay of sound in small chambers.

no greater than the rate for a tone of 2,000 cycles, i.e., no greater than 18 db/sec. The decay curves of Fig. 4 were obtained with the aid of simple tones, but with the use of a large rotating blade to keep the sound in the chamber thoroughly diffuse.

Reverberometers can be used not only to determine the sound-absorption coefficients of acoustic materials in a reverberation chamber, but are equally useful also for determining the reverberation properties of all rooms. In general, a reverberometer should be capable of making measurements at all frequencies from approximately 128 to 4,096 cycles. In special cases, such as, for example, in music rooms and theaters, it may be desirable to make measurements at frequencies up to 8,192 cycles. In most rooms, however, it is entirely sufficient to make measurements only at low, medium, and high frequencies, for example, 128, 512, and 2,048 cycles.

§ 9. DETERMINATION OF ABSORPTION COEFFICIENTS BY DIRECT REFLECTION

The reverberation method for measuring sound-absorption coefficients is an indirect method, subject to the errors and limitations mentioned in the preceding sections. Moreover, the coefficients obtained in this way are valid for arbitrary angles of incidence; however, in some cases of both practical and theoretical interest it is desirable to know the coefficient at any given angle of incidence. Obviously, the simplest means of making such measurements would consist in directing a plane parallel beam of sound onto a specimen of acoustic material and measuring the intensities of the incident and reflected beams.* The practical difficulties in applying this method are as follows: 1) reflecting surfaces must be present that are large in comparison with the wavelength of the sound (even for a frequency of 512 cycles, the reflecting surface must be at least \(12' \times 12'\)); 2) there must be no reflections from other surfaces in the experimental room, i.e. all other surfaces in the room must be so distant or so non-reflecting that reflections from these surfaces would not add a significant additional amount of sound to the beam reflected from the specimen under test; and 3) the microphone or detector must be small in comparison with the wavelength of the sound, i.e. such that it does not introduce distortions into the sound field in which the instrument is placed. If these difficulties are adequately allowed for, the absorption coefficient \(\alpha\) is given by

\[ \alpha = 1 - \left(\bar{P}_{r}/\bar{P}_{i}\right)^{2}, \tag{13} \]

where \(\bar{P}_{i}\) is the measured amplitude of the mean effective pressure in the incident beam, and \(\bar{P}_{r}\) is that in the reflected beam.

If the acoustic impedance \(z\) of the reflecting material is known,** it is easy to derive a simple formula giving the reflection (or absorption) coefficient for any angle of incidence. Thus, let the reflecting surface be in the plane \(x = 0\), and let a plane parallel sound wave (from a parabolic mirror large in comparison with the wavelength of sound) be incident on this surface at an angle of incidence \(\theta\). For incident waves propagating in the \(+x\) direction, the velocity potential \(\Phi_i\) may be represented as

\[ \Phi_i = A_i e^{jk(ct - x\cos\theta + y\sin\theta)}, \tag{14} \]

* Watson\(^{18}\) applied this method to measure both the reflection and the transmission of sound by building materials and partitions. Recently Kohl and Meyer\(^{19}\) have developed this method for measuring sound-absorption coefficients.

** Several methods have been developed for measuring the acoustic impedance of a material. See, for example, Troendle\(^{20}\) and Kohl and Meyer\(^{19}\).

Let us establish the velocity potential of the reflected wave:

\[ \Phi_r = B e^{ik(x + z\cos\theta + y\sin\theta)} . \tag{15} \]

Here \(A\) and \(B\) are proportional to the pressure amplitudes (velocity amplitudes would serve in exactly the same way, but most microphones are pressure indicators), and \(k\) is a function of the wavelength,

\[ k=\frac{2\pi}{\lambda}, \]

where \(\lambda\) is the wavelength, \(c\) is the speed of sound in a free medium. The resultant velocity potential is related to the particle velocity by the equations

\[ u=\frac{\partial\Phi}{\partial x},\qquad v=\frac{\partial\Phi}{\partial y},\qquad w=\frac{\partial\Phi}{\partial z}, \tag{16} \]

where \(u\), \(v\), and \(w\) are the components of the particle velocity. We may also write the equation for the pressure:

\[ \delta p=-\rho_0\frac{\partial\Phi}{\partial t}, \tag{17} \]

where \(\rho_0\) is the normal density of air.

The acoustic impedance of the reflecting medium is defined as the ratio of the instantaneous pressure change \(\delta p\) to the velocity of volume displacement in the direction \(x\), i.e.

\[ z_x=\frac{\delta p}{\left(\dfrac{\partial \xi}{\partial t}\right)_x} =\frac{\delta p}{u_x} =-\frac{\rho_0}{u_x}\frac{\partial\Phi}{\partial t}. \tag{18} \]

Hence, at the surface of the reflecting medium,

\[ u=\frac{\partial(\Phi_1+\Phi_r)}{\partial x} =-\left(\frac{\rho_0}{z}\right) \frac{\partial(\Phi_1+\Phi_r)}{\partial t}. \tag{19} \]

Substituting \(\Phi_1\) and \(\Phi_r\), according to equations (14) and (15), respectively, into equation (19), and solving for \(z\), we obtain

\[ z=\frac{\rho_0 c(A+B)}{(A-B)\cos\theta}. \tag{20} \]

Whence

\[ \frac{B}{A}=\frac{P_r}{P_i} =\frac{z\cos\theta-\rho_0 c}{z\cos\theta+\rho_0 c}. \tag{21} \]

And, since

\[ \alpha=1-\left|\frac{P_r}{P_i}\right|^2, \]

we obtain

\[ \alpha=1-\left|\frac{z\cos\theta-\rho_0 c}{z\cos\theta+\rho_0 c}\right|^2. \tag{22} \]

The absorption coefficient thus depends on the angle of incidence in a characteristic way, and if \(z\) is mainly real, which is usually the case for many porous materials, the absorption coefficient has a maximum at \(\cos \theta = \dfrac{\rho_0 c}{z}\). For most acoustic materials, for example felts, tiles, or plasters, \(z\) is of the order of 60 to 200 c.g.s. units, and \(\rho_0 c\) is approximately equal to 41 at room temperature. Hence the absorption coefficient usually has a maximum and approaches unity for some angle of incidence between 60 and 80°, and decreases to zero at grazing incidence \(^{19,31}\). Usually, however, one must take into account both the real and the imaginary component of \(z\), which as a result shows that \(\alpha\) is affected by a number of factors, such as, for example, porosity, frequency, angle of incidence, thickness of the reflecting medium, and resilience. In general \(\alpha\) is very small (usually below 0.20 for most auxiliary acoustic materials) for frequencies of 100 cycles or less and increases to about 0.80 or 0.90 at frequencies above 2000 cycles; \(\alpha\) usually increases as the angle of incidence increases from 0 to 70 or 80°, and then decreases to 0 at incidence of 90°.

§ 10. Sound Absorption by Porous Materials

The theory of sound absorption by porous materials was first studied by Rayleigh \(^{22}\) and at the end of his scientific activity \(^{23}\). Rayleigh’s last work was continued and extended by Crandall, who obtained a working formula for a material with a cellular structure, with small pores or channels running perpendicularly inward from an unprotected surface \(^{24}\). This formula, applied to a structure such as hair felt, assuming the pores to be closely spaced and circular with a diameter of 0.02 cm, gives results that are in fairly good agreement with those measured for frequencies above 400 cycles. Kuhl and Meyer \(^{19}\), closely following Rayleigh’s analysis, obtained formulas for the absorption coefficients of both finite and infinitely thick porous media. For an infinitely thick porous medium the absorption is a simple function of the porosity \(P\) (defined as the ratio of the volume of voids to the total volume of the absorbing material), namely

\[ \alpha = 1 - \left(\frac{\cos \theta - P}{\cos \theta + P}\right). \tag{23} \]

Kuhl and Meyer tested the value of this equation by making measurements by the ray method on corrugated paper folded into an angle, the porosity of which was varied by applying different pressures across the grooves in the paper. Fig. 6 shows four curves calculated from equation (23) for four porosities, namely 0.01, 0.10, 0.20, and 0.50, and two experimental curves,

obtained by Kolem and Meyer for corrugated paper of porosity 0.20 and 0.68.

This theory was extended by Zwikker^25 and Cremer for application to porous materials not only of the type considered by Kolem and Meyer, but also to materials in which the pores are arranged in a greater or lesser disorder, as, for example, in acoustic tile, felt, or plaster. Cremer shows that for a medium with regularly arranged pores, i.e., pores extending only perpendicular to the surface of the absorbing medium, the absorption coefficient increases with increasing angle of incidence, reaches a maximum, and then decreases to zero at an angle of incidence of 90° (which corresponds to the conclusions of Rayleigh, Paris, Kol and Meyer). Further, for a regular arrangement of the pores

Fig. 6. The solid curves show the theoretical values of the absorption coefficient of porous materials for various angles of incidence and for porosities of 0.01, 0.10, 0.20, and 0.50. The dotted curves represent the experimental values obtained by Kolem and Meyer for corrugated paper of porosity 0.20 and 0.68.

the angle of incidence at which the maximum occurs increases with decreasing porosity, and also with decreasing frequency. The value of the maximum does not depend on the porosity and approaches 1.00 at high frequencies and decreases to 0.83 at low frequencies.

For an irregular arrangement of the pores Cremer shows that, at low frequencies, the absorption depends on the angle of incidence in a manner analogous to the regular arrangement, but at high frequencies the absorption coefficient does not depend on the angle of incidence; moreover, the absorption increases with porosity. Further, at low frequencies the coefficient for normal incidence is lower than the coefficient obtained by the reverberation method, which gives the coefficient for average or arbitrary incidence, whereas at high frequencies the coefficient for normal incidence is practically equal to that obtained by the reverberation method. A complete experimental verification of these conclu-

tions has not yet been carried out, although Paris, as well as Kuehl and Meyer, have obtained at least corroborating evidence.

Penman and Richardson^26 recently tested Rayleigh’s theory experimentally for absorption by porous materials at normal incidence and summarize their conclusions with the statement that the tests they carried out confirm the general character of Rayleigh’s theory, but that “before quantitative agreement can be expected, other sources of attenuation of sound waves in narrow tubes must be found.”

§ 11. Optimal reverberation time for speech and musical rooms

The reverberation properties of rooms are so essential in determining the acoustic quality of both speech and musical rooms that it has become customary to evaluate such rooms by the reverberation time in them. Historically, excessive importance was attached to the reverberation time at one frequency—512 hertz; in fact, when it is said that the reverberation time of a room is equal to so many seconds, it is always implied that this reverberation time was calculated or measured for a frequency of 512 hertz. It is obvious, however, that, since speech and music are made up of frequencies approximately between 30 and 150,000 hertz, it is necessary to consider the reverberation properties of a room over this entire very broad frequency range. But, as was mentioned in § 8, it is usually sufficient to confine oneself to the range between 128 and 4,096 hertz. This is especially true in speech rooms, since frequencies outside this range have only a very small influence on the quality of speech. Fortunately, if the reverberation time is adjusted by the most commonly used absorbers, including the audience, to the most favorable time for the frequency of 512 hertz, the reverberation time at other frequencies will be quite satisfactory. Hence, at least from a practical point of view, one may consider reverberation in rooms at this single frequency of 512 hertz, provided that the absorbing materials used are of such a nature as to ensure a proper balance of absorption among low, middle, and high frequencies. In the immediately following paragraphs we shall limit our consideration to the reverberation time at this one frequency; later we shall consider the most favorable relationship between reverberation time and frequency.

A certain amount of reverberation is desirable in speech rooms, first, because it increases loudness, which is a condition of primary importance in large rooms, and, second, because our hearing readily perceives the mixing effect of such a magnitude of reverberation, which unites the separate sounds of speech into an artistically articulated whole.

Let us consider this first factor, which has a physical nature and can be evaluated quantitatively. The intensity of sound in a room for a constant sound source is inversely proportional to the amount of absorption in the room [equation (4)], and therefore is almost proportional to the reverberation time [equation (9)]. The intensity of average speech in large rooms is approximately only one hundredth of that required for the clearest audibility of speech. Thus, measurements of the average intensity of speech in typical auditoriums show that the speech level for an average speaker does not exceed 50 db[^27], whereas this level should be approximately 70 db in order to ensure the most favorable conditions of audibility[^28] (Fig. 7). Any gain in the loudness of speech that can be obtained, for example, from an increase in the reverberation time is extremely desirable. It should be borne in mind, however, that with too great an increase in reverberation the harmful effects of the mixing of successive words of speech reduce this gain to zero, or even to a negative value.

Fig. 7. The dotted curve gives the percentage articulation of speech at various loudness levels, according to Fletcher and Steinberg. The solid curve gives the loudness factor for various sound levels.

If we can derive a functional dependence between the loudness and the clarity of speech and between the reverberation time and clarity, then it is quite simple to determine the optimum reverberation time for any given loudness or, what is the same thing, the optimum reverberation time for any given room size. These relations were found experimentally, and the results showed not only the optimum reverberation time for rooms for speech of a given size, but also quantitatively indicated how well average speech can be heard in a room of a given size and reverberation time.

The dotted line in Fig. 7 gives the results obtained by Fletcher and Steinberg in studying the influence of loudness on the intelligibility of speech. The solid curve in Fig. 7 is called the loudness attenuation factor curve; it is assigned the arbitrary value 1.0 at the optimum sound level of 70 db (when articulation is equal to 96%), and its value at any other sound level is the ratio of the ordinate of the dotted curve at that level to the ordinate at the level 70 db. Thus, in the absence of interfering—

of the action of the noise, echo, or reverberation, the percentage articulation of speech will simply be equal to \(96K_l\), where \(K_l\) is the loudness-attenuation factor given by the solid curve in Fig. 7.

Fig. 8 gives the probable speech power, in microwatts, of an average speaker in auditoriums ranging in size from 6,000 cubic feet to more than \(10^6\) cubic feet. This is correspondingly confirmed by practice in large rooms, where the speaker raises his voice in an effort to provide sufficient loudness for distinct audibility. In general, however, such a speaker does not achieve his goal. Table 2 presents data for 8 speakers in an auditorium of volume \(6\,790\ \text{m}^3\) (240,000 cubic feet). The average sound level* was measured by a sound-measuring installation (microphone, amplifier, and

Fig. 8. Probable speech power in microwatts for average speakers in auditoriums.

electrostatic voltmeter), the microphone being placed near the center of the auditorium. The average power of the speaker’s voice was calculated from the data on the threshold of audibility, the average speech level, and equation (4). It is interesting to note the wide divergence in output

TABLE 2

Speaker	Observed average sound level (db)	Total absorption in the room (\(m^2\))	Average power of the speaker’s voice (\(\mu\text{W}\))
1 male	49.4	335	65.5
2 ”	45.6	335	27.5
3 ”	46.1	413	37.8
4 female	43.0	632	28.4
5 male	43.5	531	26.8
6 ”	51.0	502	142.0
7 ”	42.7	560	23.4
8 female	44.3	629	38.2
Average	45.7		48.9

* A speech level of 50 db means that the intensity is 50 db above the intensity of barely audible speech. The terms “sound level,” “noise level,” etc. will be used by us in an analogous manner.

power of individual speakers. This explains the frequently observed unintelligibility of the speech of many speakers, especially in large auditoria. With the aid of Fig. 8 and data analogous to those of Table 2, one can calculate the probable sound level of an average speaker in a room of known size and reverberation time.

Experimental data on the influence of reverberation on the intelligibility of speech are summarized in Fig. 9. The dotted curve gives the percentage articulation for average speech, amplified without distortion to a sound level of 70 db, in large auditoria (about 300,000 cubic feet) with various reverberation times between 0.5 and 8.0 sec. Articulation decreases by approximately 7% for each additional second of reverberation between 1.0 and 5.0 sec. Since—

Fig. 9. The dotted curve gives the percentage articulation of speech for various reverberation times in large auditoria at a speech level of 70 db. The solid curve gives the value of the reverberation factor for various values of reverberation time.

since for satisfactory audibility an articulation of 75% is necessary, the reverberation should not exceed 4.0 sec even at a sound level reaching 70 db and in the absence of the interfering effect of noise. The solid curve in Fig. 9 gives the reverberation attenuation factor \(K_r\), obtained analogously to \(K_l\). The value of \(K_r\) is arbitrarily taken to be equal to 1.0 for a reverberation time of 0.5 sec, but this is justified from a practical point of view, since the interfering effect of such slight reverberation is almost negligibly small.

Now, if a sound level below 70 db and a reverberation longer than 0.5 sec exert their complex influence on speech, the percentage articulation is given by \(96 K_l K_r\), and both \(K_l\) and \(K_r\) can be determined with the aid of Figs. 7 and 9 for a room of known size and reverberation time. Owing to the presence of unavoidable noise in auditoria, there is also a noise attenuation factor of loudness, which can be determined by a method analogous only

that described for loudness and reverberation. For a relatively quiet room the noise factor of loudness attenuation \(K_n\) is approximately equal to 0.96, so that the percentage articulation in such rooms is \(92K_lK_r\). The curves in Fig. 10 were computed with the aid of this relation, and the values of \(K_l\) and \(K_r\) are given in Figs. 7 and 9, respectively. These curves give the probable percentage articulation for an average speaker in auditoria of various sizes and with different reverberation times and thus provide a means for quantitatively determining the acoustic qualities of projected or completed speech rooms. The limitations that must be imposed on the design of speech rooms with respect both to size and

Fig. 10. Curves showing the probable percentage articulation of speech in rooms of various sizes with different reverberation times.

to reverberation time are clearly shown by these curves. Thus, average unamplified speech can never be heard satisfactorily in a room with a volume, for example, of \(45\,200\ \text{m}^3\) \((1\,600\,000\) cubic feet), regardless of what reverberation time is provided for the room, simply because the average speaker does not possess a sufficient reserve of power. Corresponding amplification of speech in large auditoria is, therefore, an inevitable condition for good acoustics.

The point of inflection of each of the curves in Fig. 10 gives the optimum reverberation time for a speech room of the corresponding size. The optimum time for small rooms is slightly less than \(1.0\) sec and increases with the size of the room, reaching approximately \(1.5\) sec for very large rooms. The maxima in these curves, however, are fairly broad, so that a deviation of \(\pm 0.25\) sec from the optimum time has little

meaning. This gives a certain freedom for satisfying the second factor influencing reverberation time and mentioned at the beginning of this paragraph—namely, that our habit of listening to speech in rooms has accustomed us (or has developed in us a sense of preference) to a certain amount of reverberation, sufficient to maintain a pleasant continuity and smoothness in the flow of articulated speech. The available theoretical and experimental data[^29] would seem to show that cultivated taste prefers a reverberation time approximately 0.3 to 0.5 longer than the maximum indicated in Fig. 10. Fortunately, this aesthetic requirement can be satisfied without serious sacrifices in the optimal conditions for listening to the sounds of speech. If a compromise is made by adding 0.2 sec. to the values of the optimal time shown in Fig. 10, the resulting reverberation values will serve as a satisfactory criterion for designing the acoustics of speech rooms.

Fig. 11. The shaded area gives a summary of the values of the optimum reverberation time for musical rooms. The optimum curve for speech rooms is shown by a solid line.

It is generally recognized that music requires a longer reverberation than speech, and that different kinds of music require different amounts of reverberation. The values indicated in the graph of Fig. 11 summarize the observations and conclusions of investigators and may serve as a practical guide in the design of musical rooms2. Heinrich Bénéke[^31] determines the optimal absorption for rooms on the basis of the values of the optimal reverberation time recommended by Lifshits and Watson. He concludes that the absorption in a room must be such that the “build-up time” (the time necessary for—

of the sound build-up to 0.63 of the steady-state value) was 0.06 sec.

At the most recent meeting of the American Acoustical Society, on 4 December 1933, J. P. Maxfield read a paper on the acoustics of rooms for sound recording, in which he showed that the ratio of the intensities of the direct sound and of the total reflected or reverberant sound is a quantity of greater significance in determining the acoustic condition of a room.

§ 12. Variation of Reverberation Time with Frequency for Speech and Music Rooms

In the preceding section we presented the results of experiments and inductive reasoning determining the values of the optimum reverberation time for speech rooms at a frequency of 512 hertz. The process of obtaining these values is simple, direct, and limited in accuracy only by errors of experimental technique. Further, the correctness of these optimum values is confirmed by the approval of the audience of the acoustics of speech rooms with the corresponding reverberation time. In the present section we shall consider the more difficult problem of determining how the reverberation time should vary with frequency in order to provide the best acoustic condition for listening to speech or music. We shall regard this condition as the “optimum reverberation characteristic.” It is especially interesting to find the functional dependence that should exist between reverberation time and frequency in an ideal speech or music room.

Several criteria have been proposed for determining the “optimum reverberation characteristic”; two of them we shall now consider. The first criterion, proposed by MacNair**, states that the loudness level of all frequency components in speech and music rooms should decay at the same and constant rate; and the second criterion, proposed by the author**, according to which all frequency components of a complex sound should decay at such a rate that they all reach the threshold of audibility simultaneously. When the first criterion is fulfilled, the second is already approximately fulfilled, at least for frequencies up to 2,000 hertz. Both criteria require a reverberation time at low frequencies (of the order of 100 hertz) approximately twice as great as at 512 hertz. According

* Loudness depends on frequency to a very significant degree. Thus, a tone at 100 hertz, 30 db above threshold, sounds just as loud to the average person as a tone at 1,000 hertz, 60 db above threshold. See Fletcher and Munson³³.

** This is essentially the criterion proposed by Lifshitz²⁰ for determining the optimum reverberation time in rooms of various sizes.

according to the first criterion, the reverberation time should remain approximately constant for frequencies above 1,000 hertz; according to the second criterion, it should increase for frequencies above 2,000 hertz. The first criterion depends on the functional relation between loudness, sound level, and frequency, and is the same for speech, music, or any other sounds. The second depends on the “spectral distribution” of the vibrations that make up speech or music, and therefore differs for individual sounds. From a practical point of view, the author considers it desirable to give preference to the reverberation characteristic corresponding to the second criterion, based on the average spectral distribution of speech and music, especially since this criterion requires only a small increase in reverberation time for frequencies above 1,000 hertz. This favors the emphasis of the high frequencies, i.e., the frequencies most important for the correct recognition of speech sounds and for preserving the quality of music, and which are subject to the greatest probable excessive weakening when propagated through air.

Fig. 12. Distribution of the energy of the speech of men and women (after Fletcher). The curve on the left is for men, the curve on the right for women.

The data giving the approximate spectral distribution of male and female speech were obtained by Fletcher^35 and are shown in the corresponding form in Fig. 12. The sound level at the various frequencies is approximately what it would be for unamplified speech in a large auditorium. The sound level has a maximum at 500–1,000 hertz both for speech and for music, and falls off very rapidly at both low and high frequencies. If the rates of decay at the various frequencies are such that all components reach the threshold of audibility at the same instant, then, obviously, the rate of decay (in decibels per second) at a given frequency must be directly proportional to the sound level at that frequency, i.e., proportional to the ordinates given by such a distribution as is shown in Figs. 12 and 13. Or, since the reverberation time is inversely proportional to the rate of decay, the criterion proposed by us will be justified under the condition \(St = \mathrm{const}\), where \(S\) is the sound level at a given frequency

and \(t\) is the reverberation time at this frequency. This criterion should be regarded only as a temporary working rule, representing a satisfactory reverberation characteristic between frequencies of approximately 100 to 4,000 hertz, and not as a law based on physical and physiological principles. At very low and very high frequencies, for example, the proposed criterion would require an infinitely long reverberation time. In this respect the criterion proposed by Maxfield, namely that during decay the loudness level of all components should decrease at a constant rate, is more acceptable; but the criterion \(St = \mathrm{const}\) is preferred here chiefly because it provides for an increase in reverberation time at frequencies above approximately

Fig. 13. Approximate distribution of the energy of music performed on the piano or by an orchestra.

1,000 hertz. Wente\(^{36}\) also draws attention to well-known facts which should favor an increase in reverberation time at high frequencies. He especially emphasizes the necessity: 1) of preserving or strengthening the high-frequency components of speech because of their significance in consonants, and 2) of suppressing low-frequency components because of their masking effect on high frequencies. These arguments deserve attention from a practical point of view, especially since a large part of the materials used for sound absorption in rooms has considerably greater absorptivity for high frequencies than for low ones.

The criteria considered here assume that the optimal reverberation characteristic should be such that the reverberation time is twice as long at 128 hertz as at 512 hertz, decreasing uniformly from 128 to 512 hertz, remains constant,

minimum for frequencies from 1000 to 2000 hertz; and then again increases at higher frequencies. It is still premature to formulate precisely the optimal reverberation characteristic: it can be determined only by further serious and very long experiments, requiring the cooperation of acousticians, phoneticians, musicians, and, possibly, aestheticians.*

§ 13. Transmission of Sound through Building Materials

From the theory of reflection and absorption considered in § 9, it is evident [equation (21)] that a plane sound wave incident on a rigid, nonporous material, for example plaster, stone, or wood, is almost wholly reflected, and that only a very small part

Fig. 14. Sound insulation by solid, nonporous partitions. Loss in transmission in dB is directly proportional to the logarithm of the mass per sq. ft of the partition.

of the incident sound wave continues to propagate in the form of a refracted ray in the new medium. The acoustic impedance \(z\) of most rigid materials (given by the product of the density of the medium and the velocity of sound in it) is very large in comparison with the acoustic impedance of air, so that for rigid nonporous materials forming the boundaries of most rooms, no more than one millionth of the incident sound energy is refracted into the bounding material. This small amount of refracted energy does not correspond to the observed considerably greater intensity of sound transmitted through building partitions, and, indeed, other explanations must be sought. The amount of insulation (measured in loss in decibels) provided by a rigid wall proves to be very precisely proportional to the logarithm of the mass per unit area of the wall (Fig. 14), which shows—

* The author’s experience shows that, at least for the case of speech rooms, the reverberation characteristic is not so critical as some believe (see note 35). M. Strat believes that the distortion introduced by selective absorption does not seriously impair the perception of sound.

...shows that the wall reacts to the variable force of the impinging air vibrations essentially in the same way as a mass. In fact, Meyer^38 showed that the average quadratic acceleration of a rigid partition (from quarter-inch wooden panels to heavy brick or concrete walls) is directly proportional to the variable pressure of the incident sound, at least for frequencies above 100 hertz. The fundamental frequency of rigid panels of the type used in buildings is considerably below 100 hertz (usually of the order of 20–50 hertz), and it is therefore to be expected that the mass reaction, proportional to frequency, will be dominant at frequencies considerably above the fundamental frequency of the partition. In the case of thin, flexible panels, stiffness, internal damping, the size of the panel, and the method of fastening—all these affect the amount of vibrational energy imparted to the partition. In general, these factors need be taken into account only for low frequencies, below approximately 200 hertz, for the materials or constructions used in buildings.

An entirely different type of sound transmission takes place in very porous materials, for example, in loose or compressed wool or cotton, or in supported panels of pumice, slag, etc. In a porous medium of this type the refracted ray becomes significant, and the loss of sound energy results from the action of viscous forces inside the small pores, the friction of fibers or other ingredients against one another, and the internal damping of bending vibrations of the component parts of the construction. The refracted ray therefore undergoes considerable attenuation, and the total attenuation (or insulation) in decibels will be proportional to the thickness of the porous partition. The insulation provided by felt materials is of the order of 3–5 db per inch of thickness for a frequency of 512 hertz. As was to be expected, the attenuation, and therefore the insulation, increases rapidly with increasing frequency.

§ 14. Calculation of Sound Insulation and Noise Attenuation in Buildings

Sound insulation is a very important factor in the acoustic design of buildings, to which almost never is due attention paid. Although satisfactory sound insulation in buildings is often incompatible with ventilation by means of open windows, much can be done to reduce the excessive noise observed in most urban premises. It is possible, at least, to compute the insulating value of any proposed construction and thereby determine which elements of the proposed building are most responsible for poor or insufficient insulation.

If equation (4) is rewritten in terms of the average intensity \(I\), determined by the average velocity of the flow of sound energy...

per unit area of the boundary, namely \(\rho_0 \dfrac{c}{4}\) in the steady state, then

\[ I=\frac{E}{a}, \tag{24} \]

where \(E\) is the rate of supply of sound energy into the room and \(a\) is the total absorption in the room. Suppose that a single-room structure is arranged in such a way that its walls and ceiling are surrounded by a sound field of uniform intensity \(I'\). Then

\[ E=I'(\tau_1s_1+\tau_2s_2+\ldots)=I'\sum \tau s, \tag{25} \]

where \(s_1, s_2 \ldots\) are the areas of the different types of boundaries and \(\tau_1, \tau_2 \ldots\) are the corresponding coefficients of transmission (defined by the ratio of transmitted to incident sound energy). Substituting equation (25) into equation (24) and defining the noise-attenuation factor as \(\dfrac{I'}{I}\), where \(I\) is the resulting intensity in the room, we obtain

\[ \frac{I'}{I}=\frac{a}{\sum \tau s}, \tag{26} \]

i.e. the noise-attenuation factor, showing how many times the intensity of the external noise is reduced after transmission into the room, is directly proportional to the amount of absorption in the room and inversely proportional to the total transmission. Since the absorption and transmission coefficients of most building materials and structures are known, by means of equation (26) it is possible to calculate the noise-attenuation factor for almost all proposed or completed buildings or rooms. It is customary to express noise attenuation in decibels, in which case equation (26) takes the form:

\[ \text{noise attenuation (in decibels)}=10\lg_{10}\left(\frac{a}{\sum \tau s}\right). \tag{27} \]

This equation, with obvious modifications, is applicable not only to single-room structures, but also to all types of rooms encountered in practice. For example, if two adjoining rooms are separated by a solid partition, except for a door, \(\sum \tau s\) consists only of \(\tau_1s_1+\tau_2s_2\), where \(\tau_1\) and \(s_1\) refer to the partition, and \(\tau_2\) and \(s_2\) to the door. In most rooms \(\tau_2\) will exceed \(\tau_1\) by a considerably greater ratio than \(s_1\) exceeds \(s_2\), and, consequently, the greater part of the transmission will be through the door. In such a case it would be hopeless to improve the insulation of the partition without providing an even greater improvement in the insulation of the door. Other examples, such as transmission through windows, openings, ventilation ducts, etc., show the necessity of carrying out calculations of sound insulation in connection with the entire

construction of the building. Unfortunately, this is rarely, if ever, done. It should be acknowledged, however, that many architects, and especially manufacturers of sound-absorbing materials, have recognized the significance of the numerator of equation (27) and accordingly use absorbing materials for ceilings (or for both ceilings and walls) in institutions, hospitals, restaurants, and other public buildings for the purpose of reducing both external and internal noise. By this means it is often possible to reduce the noise in a room by 7 or 8 db, which, fortunately, under the conditions most often encountered in practice, is perceived by the average person as a corresponding decrease in loudness by almost half; and this often constitutes the difference between satisfactory and unsatisfactory acoustic conditions.

In conclusion, it should be noted that if appropriate noise measurements have been carried out in the area of the site proposed for construction, and if the magnitude of the permissible noise in the building has been precisely determined or agreed upon, the building can be designed in such a way that the specified necessary requirements will be fully met without a significant increase in the cost of the building.

LITERATURE

1. V. O. Knudsen, Architectural Acoustics, John Wiley a. Sons, 1932; Bagenala Wo., Planning for Good Acoustics, Metnuen, 1931; P. E. Sabine, Acoustics and Architecture, Mc Graw. Hill, 1931.

2. J. B. Upham, A Consideration of Some of the Phenomena and Laws of Sound, Their Application in the Construction of Buildings, Designed especially for Musical Effects. Am. J. of Science and Art 65, 215—226, 348—363; 66, 21—23, 1853.

3. Joseph Henry, Acoustics Applied to Public Buildings, Smitsonion Reports 1854—1856.

4. T. Roger Smith, Acoustics of Public Buildings, 1861.

5. Rayleigh, Theory of Sound, V. II, p. 128, Mc Millan, 1926.

6. Collected Papers on Acoustics, Harvard University Press, 1922.

7. A. Jaeger. Zur Theorie des Nachhalls, Sitzungsber. d. Kais. Akad. d. Wiss. in Wien, Math.-Natur. Klasse, 120, 1911.

8. Cm., напр мер, Eyring, J. Acous. Soc. Am. 1, 127, 1930; Fokker Physica 7, 198, 1927; Shutze u. Waetzman, Ann. Phys. 1, 671, 1929.

9. V. O. Knudsen, Architectural Acoustics, p. 132—141, J. Wiley a Sons, 1932.

10. V. O. Knudsen, J. Acous. Soc. Am. 3, 126, 1931.

11. V. O. Knudsen, J. Acous. Soc. Am. 5, 112, 1933.

12. H. O. Kneser, J. Acous. Soc. Am. 5, 122, 1933.

13. M. J. O. Strutt. Z. ang. Mech. 10, 260, 1930.

14. Schuster u. Wätzmann, Ann. Phys. 1, 671, 1929.

15. V. O. Knudsen, J Acous. Soc. Am. 4, 20, 1932.

16. F. V. Hunt, J. Acous. Soc. Am. 5, 127, 1933.

17. Meyer & Just, E. N. T. 5, 293, 1928; Barrow, J. Acous. Soc. Am. 3, 562, 1932.

1. F. R. Watson, Acoustics of Buildings, p. 102, J. Wiley & Sons, 1930.

2. Kühl und Meyar, Berl. Akad. Ber. Math. Phys. Kl. 26, 416, 1923; Gremer, E. N. T. 10, 242; 19, 302, 1933.

3. Grandell, Theory of Vibrating Systems and Sound, p. 100, Van Nostrand, 1926.

4. See E. T. Paris, On the Reflection of Sound from a Porous Surface, Proc. Roy. Soc. A. 115, 407, 1927.

5. Rayleigh, The Theory of Sound 11, 328—333, 1896.

6. Rayleigh, The Resonant Reflection of Sound from a Perforated Wall, Phil. Mag. 39, 225, 1920.

7. Grandall, Theora of Vibrating Systems and Sound, pp. 186—191, Van Nostrand, 1926.

8. C. Zwikker, Gelindabsorptie door Porenze Wanden, de Ingenieur, 20, 1932.

9. Penman and Richardson, Absorption of Sound by Porous Materials at Normal Incidence, J. Acoust. Soc. Am. 4, 322, 1933.

10. V. O. Knudsen, Architectural Acoustics, p. 377, J. Wiley & Sons, 1932.

11. H. Fletcher, Speech and Hearing, p. 272, Van Nostrand, 1929.

12. See, for example, Lifschiz, Phys. Rev. 27, 618, 1926; Watson, J. Am. Inst. Arch. 16, 259, 1928.

13. Bagenal and Wood, Planing for Good Acoustics, Methuen, 1931; G. v. Bekesy, Ann. Physik 8, 851, 1931.

14. Heinrich Benecke, Ann. Phys. 15, 259, 1932.

15. W. A. Mac Nair, Optimum Reverberation Time for Auditoriums, J. Acoust. Soc. Am. 1, 242, 1930.

16. Fletcher and Munson, J. Acous. Soc. Am. 5, 82, 1933.

17. V. O. Knudsen, Acoustics of Music Rooms, J. Acoust. Soc. Am. 2, 234, 1931.

18. H. Fletcher, Physical Characteristics of Speech and Music, Bell. Sys. Tech. J. 10, 349, 1931; Rev. Mod. Phys. 3, 258, 1931.

19. E. C. Wente, Am. Arch. August 20, 1928.

20. M. J. O. Strutt, Rev d’Acoustique 2, 1, 1933.

21. Meyer, Grundlegende Messungen zur Schallisolation von Einfachen Trennwänden, Sitzungsber. d. Preuss. Akad. d. Wiss. Phys.-Math. Klass, IX, 1931.

1. This method was proposed by R. Norris at a meeting of the American Acoustical Society on May 11, 1929. The assumption that the energy density decreases by the same fractional amount after each reflection—an assumption which led to the derivation of equation (6)—is a perfectly valid one only when the absorption is partially preserved in deriving equation (6), whereas this is not the case for equation (7). ↩

2. G. Bagenal gives valuable indications on this question, based on a study of European concert halls. See[^30]. He arrives at the optimum reverberation time for musical rooms on the basis of Präsenz-Zeit (the interval of time during which a changing sound phenomenon—for example, the decay of sound in a room—is perceived as a whole). He arrives at the conclusion that the reverberation time should be approximately 1.0 sec. in small musical rooms, increasing to 1.4 sec. in large concert halls. ↩