SOUND WAVES IN ROOMS*

F. Morse and R. Bolt

CONTENTS

VII. Application of perturbation theory to the calculation of rooms of various shapes . . . 417
42. Influence of the shape of the room . . . 417
43. Perturbations in the boundary conditions . . . 420
44. Perturbations caused by the placement of absorbing material . . . 423
45. Cylindrical rooms . . . 426
46. Triangular rooms . . . 429
47. Perturbations of the second order . . . 431
48. Transition to an ergodic process . . . 433
49. Perturbations caused by a change in the shape of the walls . . . 436
50. Coefficient of disorder . . . 439

VIII. Method of plane free waves for disordered vibrations . . . 443
51. Reflection of a plane wave from a homogeneous wall . . . 443
52. Sabine coefficient and wall impedance . . . 446
53. Reflection of a spherical wave from a plane wall . . . 446
54. Diffraction edge corrections . . . 450
55. List of accepted notation . . . 455
56. Literature . . . 459

VII. APPLICATION OF PERTURBATION THEORY TO THE CALCULATION OF ROOMS OF VARIOUS SHAPES

42. Influence of the Shape of the Room

In the two preceding sections we studied the case of a rectangular room with a uniform distribution of absorbing material on the walls. This case corresponds to separation of variables. We found that the exact theory is rather complicated, but that in the great majority of cases (for higher frequencies, walls that are not too “yielding,” etc.) the normal components of the vibrations can be

* Conclusion. See Uspekhi Fizicheskikh Nauk, vol. XXXII, no. 2, p. 183; no. 3, p. 333.
Reviews of Modern Physics, 16, No. 2, 69 (1944).

broken down, according to their damping indices, into a small number of categories. Most vibrations, namely all or most oblique vibrations, interact with the wall material in the manner determined by the normal coefficient \(\alpha_n\). This coefficient is approximately equal to eight times the specific conductance of the wall material \(\gamma\). For vibrations parallel to a given wall, the effective coefficient will be the tangential coefficient or the supplementary coefficient, depending on which wall is more yielding—this one or the opposite one. In the limiting case of very rigid walls, both the tangential and the supplementary coefficients acquire a limiting value equal to one half of the normal coefficient.

Thus, in most cases of rectangular rooms with homogeneous walls, Sabine’s assumptions are nearly correct, but they are inaccurate by just enough to be practically unsuitable. A large part of the vibrations in a specified frequency range are oblique vibrations, having almost the same damping index, but the damping index of the other forms of vibration—tangential and axial—differs greatly. Therefore the resultant damping curve for a combination of all possible vibrations is not a straight line. It turns out that the normal coefficient is larger, and the sliding coefficient smaller, than the Sabine coefficient.

In the present section we investigate rooms of more complex outline, along the walls of which absorbing material is distributed nonuniformly. For most such cases an exact solution cannot be obtained, but an approximate perturbation method can be developed which, for sufficiently rigid walls, gives results that are for the most part satisfactory. These results show that many room shapes possess sufficient regularity that not all types of standing waves in them have the same damping index. They also show what degree of irregularity the outlines of a room must have for the Sabine assumptions to become valid.

Since Sabine’s time it has been firmly established that the shape of a room has a very great influence on its acoustic properties, and explanations of this empirical fact have been found, insofar as this could be done while remaining within the geometrical approximation. A good summary of general considerations on the selection of a proper shape for auditoria was given by Bagenal and Wood\(^2\), Knudsen\(^3\), and others. Two general principles underlie it: (a) to avoid such forms as produce evident acoustic defects, for example focusing of sound by concave surfaces, echoes, etc.; (b) to choose such a form of room as facilitates the propagation of the flow of sound energy to all listeners. The following methods are used to study these questions: construction of sound rays on sketches of room sections; spark photography of impulse waves on small models; observation of waves on the surface of a liquid, the boundaries of which—

which reproduce, on a small scale, the shape of the cross-section of the room under study^A. Three-dimensional models were also used, with light rays reflected from small mirrors arranged in the proper manner on the walls of the model. By all these methods^A one can trace the path of a sound ray arising at a definite place, corresponding to the actual location of the sound source in the room, and propagating through the room while undergoing a series of successive reflections. In this way one can investigate how the sound energy is distributed, as a result of multiple reflections, over the entire area occupied by listeners’ seats, and ascertain whether the sound focusing has been arranged correctly.

Attempts to express the influence of shape analytically, with the aid of the geometrical approximation, have proved not especially fruitful because of the very nature of the problem. Knudsen^K3 studied the influence of shape on the reverberation time by measuring the mean free path of sound with the aid of light rays. He was able to trace the path of a sound ray over as many as 25 successive reflections, for rays issuing from the source in certain typical directions. The mean free path of a ray enters into the derivation of the reverberation equations (Sections 7 and 8) and is taken equal to \(4V/S\), which is exactly equal to the asymptotic value for rectangular rooms. Knudsen measured it for several dozen different room shapes, including rectangular, fan-shaped, octagonal rooms with a dome, cruciform rooms, etc. The results of the measurements enter into equations (2.3)—(2.5) in the form of an effective value of the coefficient \(K\), the theoretical value corresponding to the mean free path \(4V/S\) being 0.049. Measurements of the indicated shapes gave values of \(K\) within the limits from 0.046 to 0.053.

Knudsen’s investigations also clarified another circumstance, of greater importance than a small correction to the reverberation time. He found that some surfaces have a greater probability of reflecting sound than other surfaces. This means that absorbing materials placed on these surfaces produce a greater effect because, on the average, a larger fraction of the energy of the incident sound falls upon them. This difference proved, in particular, to be very noticeable in a room with large horizontal dimensions and a low ceiling—the usual form of large offices. In a room of approximate dimensions \(15 \times 12 \times 3\) m, with absorbers placed exclusively on the ceiling, the reverberation time turned out to be 20% shorter than follows from equation (2.4). This result can now be computed from equations (6.19) or (6.20).

In this example the basic difference between geometrical and wave acoustics is clearly revealed.

With the aid of the geometrical method, proceeding from statistical assumptions, certain results were predicted [equations

reverberation [(2.3), (4.5)]. In particular cases the observed results proved to be different. Qualitative considerations show that the initial assumptions are not correct: the geometrical method is used to find the actual (not statistical) distribution of energy, and this result is used to change the numerical value of the multiplier in the reverberation equation while preserving Sabine’s absorption coefficient. The wave conception makes it possible to describe analytically the components of oscillation in a room and to calculate the effective damping of each kind of wave separately; for this it is necessary to take into account the shape of the room, the distribution in it of the absorbing material, and the absorbing properties of the latter. It is assumed here that the acoustic impedance is an invariant describing all the acoustic properties of the absorbing material. The components of the oscillations are combined according to their kinds and, by means of averaging processes, approximate formulae are found that are applicable to various particular cases of practical interest. In this way, instead of the single coefficient in the geometrical theory, new coefficients of different kinds arise. The final result of the wave conception is a modification of the concept of the absorption coefficient, whereas the geometrical method makes it possible to take account of the actual distribution of sound energy only by introducing empirical correction coefficients.

43. Perturbations in the boundary conditions

The method of perturbations most appropriate to the problem under consideration is based on Green’s theoremF1, F2, M2. We shall begin with a room \(R_0\) of simple shape with rigid walls, for which the wave equation has an exact solution. The eigenfunctions \(\psi_N(x)\) satisfy the usual equation

\[ \left. \begin{aligned} &\nabla_x^2 \psi_N(x) + (\omega_{0N}/c)^2 \psi_N(x) = 0;\qquad x=x,y,z;\\ &N=n_x,n_y,n_z;\qquad \omega_{0N}=2\pi\nu_N=2\pi c/\lambda;\\ &\iiint_{R_0} \psi_N(x)\psi_M(x)\,dV_x = V_0\,\varepsilon_N^0\,\delta_{NM}. \end{aligned} \right\} \tag{7.1} \]

Here \(V_0\) denotes the volume of the room \(R_0\), and \(\varepsilon_N^0\) represents the mean value of \(\psi_N^2\) over the whole volume \(R_0\).

We seek, for the oscillations produced by a source of strength equal to unity and frequency \(\nu=\omega/2\pi\), situated at the point \(X=(X,Y,Z)\) inside \(R_0\), the solution \(G\) satisfying the equations

\[ \left. \begin{aligned} &\nabla_x^2 G_\omega^0(x,X) + (\omega/c)^2 G^0(x,X) = \delta(x-X),\\ &\nabla_x^2 G_\omega(x,X) + (\omega/c)^2 G_\omega(x,X) = \delta(x-X), \end{aligned} \right\} \tag{7.2} \]

SOUND WAVES IN ROOMS

where \(\delta(x-X)\) is the three-dimensional Dirac function. If this Green’s function satisfies the same boundary conditions as \(\psi_N\), namely \(\partial\psi/\partial n=0\), then it can be expanded in the usual series with respect to \(\psi_N\):

\[ G_\omega(x,X)=\sum_N \frac{c^2\psi_N(x)\psi_N(X)} {V_0\varepsilon_N^2(\omega^2-\omega_N^2)} . \tag{7.3} \]

Equations (5.20) and (5.21) take this form in the absence of absorption.

In studying the influence of small changes in the boundary conditions and in the shape of the boundary on the properties of the fundamental functions, we shall follow the work of Feshbach\(^2\). Let us imagine that the outlines of the room have changed slightly, so that they have assumed a new form \(R\), the walls of the new room being everywhere located inside the original room \(R_0\). Let us likewise assume that the new walls are yielding and have an impedance \(Z\), which may have values on the new walls varying from point to point. In order to ensure the convergence of the adopted approximation, we must suppose that \(R\) differs only slightly from \(R_0\) and that \(|Z|\) is everywhere large compared with \(\rho c\). The new boundary conditions for the walls of the room \(R\) will therefore be written as follows:

\[ \frac{\partial\varphi}{\partial n} = i\omega(\rho/Z)\varphi = i(\omega/c)\beta\varphi = (\omega/c)(\sigma+i\gamma)\varphi . \tag{7.4} \]

Here \(Z\), \(\beta\), \(\sigma\), and \(\gamma\) represent prescribed functions of a point on the surface of the wall.

Application of Green’s theorem together with the equation for \(G\) makes it possible to write the very general integral equation

\[ \varphi(x)= \int_S \!\!\int \left[ \varphi(X)\frac{\partial}{\partial n_x}G_\omega(x,X) - G_\omega(x,X)\frac{\partial}{\partial n_x}\varphi(X) \right]\,dS_X . \tag{7.5} \]

The integration is here extended over the surface \(S'\) of the new room \(R\). The solution \(\varphi\) of this integral equation represents, inside \(R\), a solution of the wave equation \(\nabla^2\varphi+(\omega/c)^2\varphi=0\). This solution is identically zero outside \(R\); in particular, it is zero in the interval between \(R_0\) and \(R\). The function \(\Psi\), defined by the equation

\[ \Psi_N(x)= \int_S \!\!\int \left[ \psi_N(X)\frac{\partial}{\partial n_X}G_{\omega_N}(x,X) - G_{\omega_N}(x,X)\frac{\partial}{\partial n_x}\psi_N(x) \right]\,dS_X , \tag{7.6} \]

is equal to \(\psi_N\) inside \(R\) and to zero outside \(R\).

Taking into account the new boundary conditions (7.4), one can show that the solution of the integral equation

\[ \varphi(x)= \int_S \!\!\int \varphi(X) \left[ \frac{\partial}{\partial n_X}G_\omega(x,X) - i\left(\frac{\omega\beta}{c}\right)G_\omega(x,X) \right]\,dS_X \tag{7.7} \]

will be equal to zero everywhere for almost all values of \(\omega\). But for a number of discrete values of \(\omega\), which we shall denote by \(\xi_N\), the solution \(\varphi\) inside \(R\) will not be equal to zero and will satisfy the wave equation under the new boundary conditions on the surface \(R\). Outside \(R\) it is equal to zero. Thus, the solutions of equation (7.7) are the desired eigenfunctions, and the quantities \(\xi_N\) represent the characteristic values for the new room \(R\) under the boundary conditions (7.4).

Equation (7.7), like the original differential equation, likewise has no exact solution if the surface \(S\) of the volume has a complicated shape and if the impedance of the wall varies in a complicated manner over the surface \(S\). However, the solution found has a form adapted for the calculation of perturbations, since the integration is extended precisely over the new surface \(S\) and provides for the new boundary conditions. It is sufficient in the integral to replace \(\varphi(X)\) by \(\psi_N(X)\) and to substitute, instead of \(G\), expression (7.3), and we obtain the first approximation for \(\varphi\). Further, one may again apply Green’s theorem \(F^2\) and obtain an equation for an approximate determination of the value \(\xi_n\):

\[ \xi_N^2=\omega_N^2+c^2 \left\{ \iint_S \varphi\left[ \frac{\partial}{\partial n}\psi_N - i\left(\frac{\omega\beta}{c}\right)\psi_N \right]dS \right\} \left\{ \iiint_R \varphi\psi_N\,dV \right\}^{-1}. \tag{7.8} \]

The difficulty arising when \(\psi_N\) is substituted instead of \(\varphi\) in the integral of expression (7.7) consists in the fact that the series then obtained converges very slowly, owing to the fact that \(\varphi\) undergoes a discontinuity on the surface \(S\). This difficulty can be avoided by subtracting the discontinuous value (or its principal part) by means of equation (7.6):

\[ \varphi(x)=\Psi_N(x)+ \iint_S \left\{ \varphi(X)\frac{\partial}{\partial n_X}G_\omega(x,X) - \psi_N(X)\frac{\partial}{\partial n_X}G_{\omega_N}(x,X) - G_\omega(x,X)\left(i\frac{\omega\beta}{c}\right)\varphi(X) + G_{\omega_N}(x,X)\frac{\partial}{\partial n_X}\psi_N(X) \right\}dS_X . \tag{7.9} \]

Here \(\Psi\) inside \(R\) is equal to \(\psi_N\) and is equal to zero outside \(R\).

Now one can obtain the first approximation for \(\varphi\), which converges quite satisfactorily. Substituting \(\psi_N\) instead of \(\varphi\) in the integral, we obtain:

\[ \varphi(x)\simeq \]

\[ \simeq \Psi_N- \iint_S G_{\omega_N}(x,X) \left[ \frac{i\omega\beta}{c}\psi_N(X) - \frac{\partial}{\partial n_x}\psi_N(x) \right]dS_X . \tag{7.10} \]

and, using equation (7.3), we finally find:

\[ \left. \begin{aligned} \varphi(x) &\simeq \Psi_N(x) + c^2 \sum_M' A_{MN}(\omega_N^2-\omega_M^2)^{-1}\psi_M(x),\\ A_{MN} &= (1/V\varepsilon_M)\iint_S \psi_M\left(\frac{\partial}{\partial n}\psi_N - i\frac{\omega_N\beta}{c}\psi_N\right)\,dS,\\ \iiint_R \psi_N^2\,dV &= V\varepsilon_N . \end{aligned} \right\} \tag{7.11} \]

In these expressions \(V\) denotes the volume of the room \(R\), \(\varepsilon_N\) is the mean square value of \(\psi_N\) inside \(R\) [see equation (5.17)], and the prime on the summation sign means that terms with identical indices \(M=N\) are excluded from the sum.

The corresponding equation for the characteristic values will be:

\[ \xi_N^2 \simeq \omega_N^2 + c^2 A_{NN} + c^4 \sum_M' (\varepsilon_M/\varepsilon_N) A_{MN}^2/(\omega_N^2-\omega_M^2). \tag{7.12} \]

Here expression (7.11) has been substituted into equation (7.8).

44. Perturbations caused by the placement of absorbing material

Let us first consider the case in which the outlines of the room are sufficiently regular and the boundary conditions (7.4) are the only perturbation. In this case \(R\) coincides with \(R_0\), and the integral \(A_{MN}\), which enters equations (7.11) and (7.12), takes the form

\[ A_{MN}=-i\omega_N(1/cV_0\varepsilon_M^0)\iint_S [\psi_M\beta\psi_N]\,dS. \]

This case was studied by Maa \(^{\mathrm{M2}}\) and by Feshbach and Klopston \(^{\mathrm{F1,F2}}\).

The first approximation for the frequency parameter \(\xi_N\) will in this case be the following expression, which should be compared with equation (5.9):

\[ \xi_N(-\omega) \simeq \omega_N -\frac{1}{2}\frac{c}{V_0\varepsilon_N^0} \iint (\sigma+i\gamma)\psi_N^2\,dS. \tag{7.13} \]

The function \(\varphi\) will have an exponential factor of the form

\[ \exp(-i\xi_N t) \simeq \exp\left\{ -i\left[ \omega_N-\frac{c}{2V_0\varepsilon_M^0}\iint \sigma\psi_N^2\,dS \right]t - \frac{ct}{2V_0\varepsilon_N^0}\iint \gamma\psi_N^2\,dS \right\}. \tag{7.14} \]

The mean-square value of $\varphi$ will therefore also have an exponential decay factor of the usual form $\exp[-(ca_N t/4V_0)]$, which it is useful to compare with equation (6.19). The effective wall coefficient for the walls of a room can be written in the form

\[ \left. \begin{gathered} \alpha_N=(a_N/S)\approx \frac{8}{2S\varepsilon_N^{0}}\iint \gamma \psi_N^2\,dS,\\ \gamma=\text{the real part of }[\rho c/Z]. \end{gathered} \right\} \tag{7.15} \]

It is a function of the wall area $S$, of $\varepsilon_N^{0}$—the mean square of the wave function $\psi_N$, taken over the volume $V$, and of the integral, taken over the surface of the walls, of the absorption coefficient $(8\gamma)$ multiplied by the weight factor $\psi_N^2$. Equation (7.15) is useful to compare with equations (5.26) and (5.29). The latter equations were derived for a homogeneous wall; equation (7.15), however, usually (but not always) proves valid for a wall covered with absorbing material not continuously, but in separate pieces.

We saw above that this approximation is valid if $(|Z|/\rho c\eta)>2$, where $\eta$ denotes the ratio of the “dimensions” of the room to the length of a half-wave.

From this first-order approximation one can derive several interesting conclusions. First of all, it turns out that a piece of absorbing material absorbs sound best if it is placed at such a location on the walls where most of the wave functions have maxima. Thus, for a rectangular room the most effective arrangement of the absorbing material is at the vertices of the solid angles; the next best arrangement is along the edges of the room. Further, if it is necessary to distribute the material over several walls, it is better to distribute it in an irregular manner than in regular figures. In the latter case it is almost impossible to avoid the result that, for some waves, the material will fall at minima of $\psi^2$, and these oscillations will be very weakly damped.

Equation (7.15) admits a further simplification if the absorbing material on each wall is distributed uniformly within regions large in comparison with the wavelength. In this case the integral $\iint \gamma\psi_N^2\,dS$ is approximately equal to the area $S$, multiplied by the constant number $\beta$ and by the mean-square value of $\psi_N$ on the given wall. Therefore, in the order of the first approximation, one may say that, for large samples of material (in comparison with the wavelength), the wall coefficient, for a given material on a given wall, is determined as follows:

\[ \alpha_N \approx (8\gamma)e_N, \tag{7.16} \]

where

\[ e_N=\frac{1}{2}\left[\frac{\text{mean value of }\psi_N^2\text{ within the limits of the given wall}}{\text{mean value of }\psi_N^2\text{ over the entire volume of the room}}\right] \]

(the formula is valid for \(|Z|/\lambda>4\rho cL\), where \(L\) is the largest dimension of the room).

The factor \(e_N\), through which in this first approximation only the type of wave enters, may be called the coefficient of the type of oscillation. It must be noted that \(a_N\) usually does not coincide with the Sabine coefficient. The rooms considered here have too regular a form to satisfy Sabine’s conditions.

Thus, even in the first approximation it follows that the absorption coefficient of a material depends not only on its acoustic impedance, but also on the position of the wall on which it is placed, and also on the particular form of oscillations. If we are interested in increasing the initial steepness of the decay curve, we must place our material so that the factor of the type of oscillation \(e_N\) is as large as possible for the majority of the modes of oscillation in the given range. If we wish to increase the final steepness of the decay curve, we must place our material so that for none of the natural oscillations does the entire absorbing material end up on a wall with small \(e_N\). This was established in Chapter VI for rectangular rooms with a uniform distribution of material. Here we see that this is true for any room of regular form and even for a nonuniform placement of material (at least in the first approximation).

To facilitate the practical use of these results we shall calculate the values of \(e_N\) for various frequently encountered forms of rooms. Thus, for example, for rectangular rooms, taking into account the known properties of the cosinusoidal terms in normal modes of oscillation, we obtain:

for all oblique oscillations:

\[ e_N=1 \quad \text{for all walls;} \]

for all tangential oscillations:

\[ e_N=\frac{1}{2} \quad \text{for two walls parallel to the direction of motion of the wave,} \]

\[ e_N=1 \quad \text{for the remaining four walls;} \]

for all axial oscillations:

\[ e_N=\frac{1}{2} \quad \text{for four walls parallel to the direction of propagation of the oscillations,} \]

\[ e_N=1 \quad \text{for the remaining two walls.} \]

It is recommended to compare this result with the considerations concerning equations (3.2) and (5.12).

One can go further and determine the values of all \(e\) for any room in which two opposite walls are plane and parallel, while all the remaining walls are perpendicular to the plane-parallel pair. An example of such a room is a room with vertical walls and with a plane horizontal floor and ceiling, but with some complicated plan. Let us direct the \(x\)-axis perpendicular to the plane-parallel walls. Although we cannot separate the wave equation in the two coordinates perpendicular to \(x\) (let us call them \(y\) and \(z\)), we can single out the \(x\)-th factor in the fundamental function for each normal mode:

\[ \psi_N=\cos(\pi n_x x/L_x)\,F_{n_y n_z}(y,z). \]

The mean value of \(\psi_N^2\) over the whole volume will be \(\left(K_{n_y n_z}/2e_{n_x}\right)\), where \(e_{n_x}\) is equal to unity (if \(n_x>0\)) or to one half (if \(n_x=0\)), and \(K\) denotes the mean value of \(F\) over all values of \(y,z\). Therefore, in the first approximation, for large pieces of absorbing material placed on any plane-parallel wall, we obtain:

\[ e_N=e_{n_x}= \begin{cases} 1 & \text{for waves incident on these walls,}\\[4pt] \dfrac{1}{2} & \text{for waves tangential to these walls }(n_x=0). \end{cases} \]

To obtain a more accurate answer it is necessary to use the methods discussed in Chapter V.

We shall now compute the value of \(e\) for cylindrical and triangular rooms. These rooms were considered by Po [8].

45. Cylindrical rooms

The fundamental functions for a cylindrical room of height \(L\) and radius \(R\) have the form

\[ \psi_N= \begin{matrix} \cos\\[-2pt] \sin \end{matrix} (n_\varphi\varphi)\, J_{n_\varphi}\!\left(\pi\tau_{n_\varphi n_r}\,r/R\right) \cos(\pi n_z z/L). \tag{7.19} \]

Here \(\tau\) denotes any number satisfying the equation

\[ \frac{d}{d\tau}J_{n_\varphi}(\pi\tau)=0; \]

we shall denote the smallest value by \(\tau_{n_\varphi 0}\), the next by \(\tau_{n_\varphi 1}\)

etc. For large \(n_r\) or \(n_\varphi\) the limiting values are:

\[ \tau_{n_\varphi,n_r} \simeq n_r+\frac{1}{2}n_\varphi+\frac{1}{4}; \qquad n_r \gg 1,\ (n_r/n_\varphi)\gg 1; \]

\[ \tau_{n_\varphi,n_r} \simeq (n_\varphi/\pi)+\frac{1}{2} \left[9\left(n_r+\frac{1}{4}\right)n_\varphi/\pi\right]^{1/3}; \]

\[ n_\varphi \gg 1,\ (n_\varphi/n_r)\gg 1; \]

\[ \tau_{n_\varphi,0} \simeq (n_\varphi/\pi)+0.2575\,n_\varphi^{1/3}; \qquad n_\varphi \gg 1,\ n_r=0. \]

The mean value of the square of the factor depending on \(\varphi\) is equal to unity for \(n_\varphi=0\), and is equal to one half for \(n_\varphi>0\). The same is also true for the factor depending on \(z\). The mean value of the square of the factor depending on \(r\) is equal to:

\[ (2/R^2)\int_0^R J_{n_\varphi}^2(\pi \tau r/R)\,r\,dr = J_{n_\varphi}^2(\pi\tau_{n_\varphi,n_r}) \left[1-\left(n_\varphi/\pi\tau_{n_\varphi,n_r}\right)^2\right]. \]

Therefore, on the basis of equation (7.16), the ratio of the wall coefficient for large samples of material to the normal coefficient \((8\gamma)\) is equal:

for material on both end walls:

\[ e_N \begin{cases} \dfrac{1}{2} & \text{for waves tangential to these walls }(n_z=0),\\[6pt] 1 & \text{for waves incident on these walls }(n_z>0); \end{cases} \tag{7.20} \]

for material on the cylindrical lateral surface:

\[ e_N=\frac{1}{2}\left[1-\left(n_\varphi/(\pi\tau_{n_\varphi,n_r})\right)^2\right]^{-1}. \]

The values of \(e_N\) for cylindrical surfaces and various \(n_r\) and \(n_\varphi\) are given in Table 1 and can be calculated by means of the following asymptotic formulas:

\[ e_N \simeq \frac{1}{2}\left[1+\left(n_\varphi/\pi n_r\right)^2\right], \qquad n_r\gg 1;\quad (n_r/n_\varphi)\gg 1; \]

\[ e_N \simeq 0.309\,n_\varphi^{2/3}+\frac{1}{8}, \qquad n_\varphi\gg 1;\quad n_r=0; \]

\[ e_N \simeq \frac{1}{2} \left[\frac{n_\varphi}{3\pi\left(n_r+\frac{1}{4}\right)}\right]^{2/3} +\frac{1}{8}; \qquad n_\varphi\gg 1;\quad (n_\varphi/n_r)\gg 1. \]

Table 1

Values of \(e_N\) for cylindrical rooms

\(n_r \backslash n_\varphi\)	0	1	2	3
0	0.50	0.70	0.86	1.00
1	0.50	0.52	0.55	0.57
2	0.50	0.51	0.52	0.54
3	0.50	0.51	0.52	0.53

The conditions for a cylindrical surface are quite different from those for flat end walls. Oscillations propagating around the circumference “parallel” to the curved wall \((n_r=0)\) have a coefficient \(\alpha_N=(8\gamma)e_N\), which (for \(n_\varphi \gg 1\)) is greater than the normal coefficient \((8\gamma)\). This is due to the circumstance that such oscillations slide along the surface of the outer walls, whereas oscillations tangential to flat walls have comparatively little energy near the walls, and therefore the corresponding tangential coefficients are equal to only one half of the normal ones. In cylindrical rooms that are large in comparison with the wavelength \((c/\nu)\), the greater part of the energy of the fundamental oscillations, for which both \(n_z\) and \(n_r\) are equal to zero and \(n_\varphi\) is large, is concentrated within half a wavelength near the walls, so that this energy is rapidly absorbed. Thus, for example, for \(n_z=n_r=0\) and \(n_\varphi=20\), the wall coefficient for material on the cylindrical surface exceeds the normal coefficient \((8\gamma)\) by a factor of 2.3. For \(n_\varphi=100\) it exceeds it by a factor of 6.7. For the end walls, however, it amounts to one half of the normal coefficient.

However, such circular oscillations represent a very special case. They are difficult to excite except by placing the sound source close up against the cylindrical walls, and they are difficult to measure except by placing the microphone likewise close up against these walls. It may be noted that the decay curve for cylindrical rooms depends very strongly on the position at which the microphone is installed. A curve taken close to the cylindrical wall has a greater initial steepness than a curve taken at the center of the room.

In contrast to oscillations tangential to the plane \((r,z)\), for waves propagating along the radius \(r\) (axial, \(n_\varphi=0\)), the wall coefficient for the cylindrical wall is equal to only one half of the normal coefficient. This occurs because the energy of such oscillations is focused on the axis of the room, and near the walls it is less than the volume average. For waves short in comparison with the dimensions of the room, most of the fundamental oscillations have \(e_N\) for cylindrical walls approximately equal to one half. Only those oscillations for which \(n_r\) is very small have \(e_N\) greater than unity. Therefore, for the central part of the room, the acoustic material,

placed on a cylindrical surface, is approximately half as effective as material placed on flat end surfaces. For sounds originating from the peripheral part of the room, on the contrary, the absorbing properties of the material on cylindrical walls are much higher.

Analogous investigations of rooms in the form of a half-cylinder with an added flat wall passing through the axis, as well as studies with a hemispherical wall, led to the following rules, valid within the limits of the first approximation:

for flat walls:

\[ e_N=\frac{1}{2} \quad \text{for tangential waves } (n_z \text{ or } n_\varphi \text{ is zero}), \]

\[ e_N=1 \quad \text{for waves incident on the wall } (n_z \text{ or } n_\varphi \text{ greater than zero}); \]

for curved walls:

\[ e_N>1 \quad \text{for waves tangential to the walls } (n_r\ll n_\varphi), \]

\[ e_N\approx \frac{1}{2} \quad \text{for waves incident on the wall } (n_r>n_\varphi). \]

Generally speaking, curved surfaces focusing sound energy in some region away from the walls reduce the action of the acoustic material. The greater the fraction of the walls occupied by concavity, the smaller on such walls are those portions on which the absorbing material can fully manifest its absorbing properties. This is true, at least, for the central part of the room. A similar analysis for a spherical chamber was carried out by Schuster \(^{11}\).

46. Triangular rooms

Let us imagine a room with vertical side walls, a flat horizontal floor and ceiling, whose plan has the form of a right isosceles triangle. For such a room one can obtain an exact solution for \(\psi_N\), although the wave equation is not separable. It can be proved that a function of the following form satisfies the boundary conditions both on the perpendicular walls—the legs—and on the diagonal wall—the hypotenuse:

\[ \psi_N=\left(\frac{1}{2}\right)\cos(\pi n_z z/L)\,T_{n_s n_d}(x,y). \]

Here the function \(T\) is equal to:

\[ \begin{aligned} T_{n_d n_s}(x,y) &=\frac{1}{2}\{\cos[\pi(n_d+n_s)x/L]\cos(\pi n_s y/L) \\ &\quad+(-1)^{n_d}\cos[\pi(n_d+n_s)y/L]\cos(\pi n_s x/L)\} \\ &=\frac{1}{2}\{\cos[\pi(n_d+2n_s)\xi/\Lambda]\cos(\pi n_d\eta/\Lambda) \\ &\quad+\cos[\pi(n_d+2n_s)\eta/\Lambda]\cos(\pi n_s\xi/\Lambda)\},\qquad (n_d\ \text{even}); \end{aligned} \tag{7.21} \]

\[ T_{n_d,n_s}(x,y)=\frac12\left\{\sin\left[\pi(n_d+2n_s)\xi/\Lambda\right]\sin(\pi n_d\eta/\Lambda)+\right. \]
\[ \left.+\sin\left[\pi(n_d+2n_s)\eta/\Lambda\right]\sin(\pi n_s\xi/\Lambda)\right\},\quad (n_d\ \text{odd}), \]

where

\[ \xi=\frac{1}{\sqrt2}(y+x),\qquad \eta=\frac{1}{\sqrt2}(y-x);\qquad \Lambda=\sqrt{2L}. \]

The number of natural frequencies smaller than \(\nu\) for such a room is approximately equal to one half of the same number for a square room with the same lengths of mutually perpendicular walls, since the diagonal wall eliminates the double degeneracy of most of the normal modes (the room has half the volume).

Integration of the above wave functions makes it possible to compute the factor of the type of oscillations \(e_N\) for different walls and different fundamental oscillations, with the normal coefficient \((8\gamma)\):

\[ \begin{aligned} &\text{for the cathetus walls:} \\ &\qquad e_N=\frac12\quad \text{when } n_s>0,\\ &\qquad e_N=\frac34\quad \text{when } n_s=0;\\[6pt] &\text{for the hypotenuse wall:}\\ &\qquad e_N=1\quad \text{when } n_d>0,\\ &\qquad e_N=\frac34\quad \text{when } n_d=0;\\[6pt] &\text{for the floor and ceiling:}\\ &\qquad e_N=1\quad \text{when } n_z>0,\\ &\qquad e_N=\frac12\quad \text{when } n_z=0. \end{aligned} \tag{7.22} \]

Thus, even in such a room there are oscillations “parallel” to a wall (one of the \(n\)’s is equal to zero), and the effective absorption of the material for such oscillations is reduced. The degree of reduction in effectiveness is not as great as in a rectangular room: \(e_N\) decreases only to \(3/4\) for the side walls. It may be considered that, in a triangular room, oscillations “parallel” to the walls are not so “parallel” as they could be in a rectangular room. This circumstance shows that a room with nonparallel walls is still not sufficiently irregular for Sabine’s assumptions to be valid for it. The decay curves for such rooms still differ appreciably from straight lines.

From consideration of Fig. 27 there follows a number of interesting circumstances. Nodal surfaces intersect the walls either at right angles

SOUND WAVES IN ROOMS

at an angle, or at an angle of \(45^\circ\). In the latter case the pressure will be in phase over the entire area of the wall; the corresponding value of \(e_N\) is equal to \(3/4\), and one may consider that the oscillation occurs “parallel” to this wall. If the nodal surface intersects the wall at a right angle, then the pressure on one half of the wall is opposite in phase to the pressure on the other half; \(e_N\) has the value unity; one may assume that the oscillation occurs nonparallel to this wall. Fig. 27 gives the arrangement of the nodal surfaces and the value of \(e_N\) for some of the lower natural frequencies, including the most typical cases.

47. Second-order perturbations

Up to now we have considered only first-order perturbations, assuming that the second-order term is negligibly small. Of course, this is not always true (this is true only if \(|Z|\lambda/4\rho c\) is greater than the largest dimension of the room), and therefore it makes sense to investigate also terms of higher order. For example, we shall see that if the second-order term in equation (7.12) is not sufficiently large and does not itself consist of a series of terms, then it is impossible to obtain an ergodic distribution of sound energy in the room. The relative magnitude of the second-order term in comparison with the first-order term may serve, at a given frequency, as an approximate criterion of whether the Sabine assumptions are valid for the given room or not.

Fig. 27. Arrangement of nodal figures of pressure in a triangular room and the values of the coefficient of type of oscillations \(e_N\). The integers—\(n_d\) and \(n_e\); the corresponding values of \(e_N\) are written beside each wall.

First let us examine the case when the outlines of the room \(R_0\) are simple, so that the only perturbation is the arrangement of absorbing material along the walls. The expression for the perturbed wave,

function will have the form

$$ \varphi(x)\simeq \Psi_N(x)- i\omega_N(c/V)\sum_M' \left[\iint \psi_M(X)\beta(X)\psi_N(X)\,dS_X\right] \frac{\psi_M(x)}{\varepsilon_M(\omega_N^2-\omega_M^2)}. \tag{7.23} $$

The prime on the summation sign means that, in summing, the term with identical indices \(M=N\) is omitted. Applying the methods described in Section 33, and also equation (7.3), one can establish that near a piece of absorbing material a rough approximation to equation (7.23) is

$$ \varphi(x)\simeq \Psi_N(x)+i\omega_N\rho \iint [\psi_N(X)/Z(X)]\cdot \frac{\exp(i\omega_ND/c)}{2\pi D}\,dS_X, \tag{7.24} $$

where the integration extends over the surface of the absorbing material near the point \(x\), and \(D\) denotes the distance between the point \(x\) \((x,y,z)\) in the room and the point \(X\) \((X,Y,Z)\) on the wall.

The last formula is of interest for the following reason: it shows that, in the first approximation, the pressure at the wall at the point \(X\) is equal to \(-i\omega_N\rho\psi_N(X)\); the normal component of the velocity at the wall, at the point with impedance \(Z(X)\), is equal to \(-i\omega_N\rho\psi_N(X)/Z(X)\). The motion of the surface element \(dS\) of the wall causes the radiation of sound into the room, corresponding to the strength of an elementary source:
\(- (i\omega_N\rho\psi_N/Z)dS(-\exp[i\omega_ND/c]/2\pi D)\). The result of integrating over all elementary sources located near \(x\) constitutes the second term. Thus, the first correction to the wave function \(\psi_N\), due to the presence of absorbing material on the wall, is the radiation produced by the motion of the absorbing material itself^VI or by the motion of the air in the pores of this material, caused by the standing wave \(\psi_N\) itself. The consequences of this simple derivation will be investigated in Chapter VIII.

Accurate to terms of second order with respect to \(\beta\), the equation for the frequency and for the damping coefficient in the case under consideration will have the form

$$ \xi_N^2\simeq \omega_N^2- \frac{i\omega_N c}{V\varepsilon_N}\iint \beta\psi_N^2\,dS+ \left(\frac{c\omega_N}{V\varepsilon_N}\right)^2 \sum_M'\frac{\varepsilon_N}{\varepsilon_M} \frac{\left[\iint \psi_M\beta\psi_N\,ds\right]^2} {(\omega_M^2-\omega_N^2)}. \tag{7.25} $$

This equation may also be approximated by taking into account only the coherent part of the perturbation:

$$ \xi_N^2\simeq \omega_N^2- \frac{i\omega_N c}{V\varepsilon_N}\iint \beta\psi_N^2\,d + \left(\frac{\omega_N^2}{V\varepsilon_N}\right) \iint dS_x \left\{ \psi_N(x)\beta(x)\iint \psi_N(X)\beta(X)\times \right. $$

$$ \left. {}\times [\exp(i\omega_ND/c)/2\pi D]\,dS_X \right\}. \tag{7.26} $$

Here the double integration need be extended only over small values of \(D\). A detailed investigation of this equation will be given in Chapter VIII.

48. Transition to an ergodic process

It should be assumed that, with sufficiently large quantities of absorbing material, distributed sufficiently irregularly over the walls of a rectangular room, the wave motion will be ergodic, and the attenuation curve will turn out to be rectilinear. This occurs when standing waves can no longer be regarded as purely tangential or axial waves and when all oscillations have the same attenuation exponent. From the point of view of perturbation theory this will happen when each standing wave \(\varphi\) represents a more or less random mixture of several different \(\psi_N\) in almost equal proportions, so that many oblique oscillations will participate in each \(\varphi\). As is evident from equations (7.23) and (7.25), this will occur when the last term in each expression consists of many terms nearly equal to one another, and their sum proves to be of the same order of magnitude as the first term.

In this case, of course, in order to obtain exact values of the wave function, still higher orders of approximation than the second are necessary. However, it seems possible to take the relative magnitude of the second-order correction, for example in equation (7.25) for \(\xi\), as a rough criterion of the ergodicity of the oscillations.

Thus, if the second-order term is equal in magnitude to the first term or greater than it, and if it consists of a large number of different wave functions having amplitudes of the same order, then the oscillations may be considered ergodic and Sabine’s assumptions valid. It is therefore of interest to obtain simple summary estimates of the second-order term in equation (7.25), in order to use them as a criterion of the disorderliness of the process.

In particular, in equation (7.25) the decisive importance in the second-order term is held by the exchange integral

\[ \beta_{MN} = \iint \psi_M^3 \psi_N \, dS . \]

It measures the ability of a given method of distributing the absorbing material to “scatter” oscillations, transforming waves \(M\) into waves \(N\). If this scattering is sufficiently great, then no oscillation will be strictly axial or tangential; all attenuation constants will tend toward a common mean value, and Sabine’s assumptions will come into force.

For this to be valid, of course, \(\beta_{MN}\) must differ from zero for the majority of values of \(M\) and \(N\). In reality most \(\beta_{MN}\) must have approximately the same magnitude. This ne-

directly shows that, under a uniform distribution of absorbing material over one or several walls in a room of regular shape, this is impossible: indeed, if \(\beta\) has one and the same value over the whole wall (for example, the walls \(x, y\)), then all \(\beta_{MN}\) will be zero, except those for which \(n_x=m_x\) and \(n_y=m_y\). As a result, most terms in the summation over \(M\) vanish, and the standing waves will not be random.

Similarly, in general, any sufficiently symmetric distribution of absorbing materials will lead to many integrals \(\beta_{MN}\) becoming zero. Therefore a “completely random” distribution of pieces of absorbing material is required. How this is to be achieved in practice is indicated in the works of Maxfield and Potwin \(^{M5}\), Boner \(^{B1}\), Meyer \(^{M6}\), and Follmann \(^{V1}\).

Let us consider a particular example. Suppose that \(m\) pieces of absorbing material with specific conductivity \(\beta=\rho c/z\), each of dimensions \(a\times b\), are “randomly” distributed over the walls of a rectangular room. The total area of the material will then be \(S_a=mab\). The first-order term in equation (7.25) then takes the form

\[ -2i\omega_N^c(\beta S_a/V). \]

Next it is necessary to determine the mean-square value of the integral

\[ \iint \psi_M \beta \psi_N\,ds, \]

where the integration is extended over the area of one piece of material, and the averaging is carried out over all \(M\) and \(N\). If a piece of material of dimensions \(a\times b\) is placed on the wall in a completely random way, then the nodal surface will intersect this piece at random, and the factor of the integrand depending on \(x\) will have the general form

\[ \beta\int_{-a/2}^{a/2} \sin\left[(\pi n_x x/L_x)+\Phi_x\right] \sin\left[(\pi m_x x/L_x)+\Phi'_x\right]\,dx. \]

This factor must be squared and averaged over \(\Phi_x,\Phi'_x,n_x\), and \(m_x\).

Averaging first over the phase angles \(\Phi_x\) and \(\Phi'_x\), we obtain:

\[ \frac{\sin^2\left[(\pi a/2L_x)(n_x-m_x)\right]} {\left[(\pi/L_x)(n_x-m_x)\right]^2} + \frac{\sin^2\left[(\pi a/2L_x)(n_x+m_x)\right]} {\left[(\pi/L_x)(n_x+m_x)\right]^2}. \]

Multiplying this expression by the corresponding factor depending on \(y\), and averaging over \(n_x,m_x,n_y\), and \(m_y\), one can obtain, for one piece, the following approximate expression:

\[ \frac14\beta^2\frac{a^2b^2}{1+(4ab/\lambda^2)}, \]

and for all \(m\) pieces,

\[ (\beta_{MN}^{2})_{\mathrm{cp}}\simeq \frac14\beta^2\frac{S_a^2}{1+(4S_a/m\lambda^2)}. \]

Here \(\lambda\) denotes the mean wavelength of the sound wave for the \(M\)-th and \(N\)-th fundamental oscillations. If in the sum of second-order terms the principal role is played by terms with \(\omega_M \simeq \omega_N\), then this approximation is acceptable.

A typical term of the second-order sum in equation (7.25) is approximately equal to:

\[ (c^2\omega_N/V^2)\beta^2 \frac{S_a^2}{1+(4S_a/m\lambda^2)} \cdot \frac{1}{\omega_M-\omega_N}. \]

As is seen from equation (3.4), the average difference between the eigenvalues \(\omega\) is equal to \((2\pi^2 c^3/V\omega^2)\), and therefore the magnitude of the largest terms in the second-order sum is approximately expressed as:

\[ (\omega^3/4\pi^2 cV)|\beta|^2 \frac{S_a^2}{1+(4S_a/m\lambda^2)}. \]

The ratio of these largest second-order terms to the first-order term is approximately:

\[ \Xi_a= \frac{\omega^2}{8\pi^2 c^2} \frac{|\beta|S_a}{1+(4S_a/m\lambda^2)} = \frac{(1/2)|\beta|S_a}{\lambda^2+(4S_a/m)}. \tag{7.27} \]

This number may be called the coefficient of disorder of oscillations in a rectangular room with irregularly placed pieces of absorbing material. Similar formulas can be derived for rooms of other regular shapes. If \(\Xi\) is less than unity, then the oscillatory process is non-ergodic; there is a noticeable difference between the attenuation indices of the various fundamental oscillations, and the attenuation curve is curvilinear. If \(\Xi\) is considerably greater than unity, then no mode of oscillation is purely axial, the attenuation indices tend toward their mean value, and one may expect that Sabine’s assumptions will prove valid. It remains only unclear how much \(\Xi\) must exceed unity for this.

It is easy to see that \(\Xi\) is much less than unity at low frequencies (\(\lambda\) large). Usually \(\beta\) has, for soft materials, a value of the order of \(0.2\), so that even for \(\lambda^2\) smaller than \(4S/m\), \(\Xi\) cannot greatly exceed unity unless \(m\) is small, i.e. unless the absorbing material is divided into many small pieces. The most effective size for each piece is approximately equal to the half-wavelength, i.e. \(\lambda^2 \simeq 4S_a/m\); then the resulting value of \(\Xi\) is \(m(|\beta|/8)\).

For example, for a cubic chamber with edge \(6\ \mathrm{m}\), fitted with square pieces of soft material having side equal to a half-wave, with \(|\beta|=0.2\), placed on average at a distance \(3\lambda/2\) from one another, there will be on average one piece for every \(4\lambda^2\) of wall area. The coefficient of disorder in this case will be \(\Xi=7.5/\lambda^2\). In order to obtain \(\Xi\) greater than 4 (which, presumably, is a sufficient criterion of disorder),

one must take a wave shorter by approximately \(0.45\ \mathrm{m}\), or a frequency higher than 800 cycles. At such a frequency approximately 250 pieces of material are required; each piece must account for approximately \(0.8\ \mathrm{m}^2\) of wall surface.

These results are very instructive; they may explain the success of the work of Maxfield and Potwin and other investigators \(^{B1,M5,M6,V1}\). However, pieces of absorbing material mounted on walls of regular shape are comparatively ineffective for proper scattering of sound. We shall now study the question of irregularities in the shape of the room and determine the influence of these irregularities on \(\Xi\).

49. Disturbances caused by a change in the shape of the walls

The second term in equation (7.10) is due to the difference in shape between \(S\) and \(S_0\). An example of such a difference is shown in Fig. 28, where \(S_0\) represents the plane \((x,y)\), and \(S\) is a protrusion of the wall into the room, bounded by the contour \(C\). Within this contour the equation of the surface will be

\[ z = B(x,y), \]

where the positive direction of \(z\) is directed into the room and nowhere has negative values, in accordance with our original assumptions. Let the unit vector along the normal to \(S_0\) be \(n_0\); it is directed toward negative \(z\). The unit vector normal to \(S\) will be denoted by \(n\); it forms an angle \(\vartheta(x,y)\) with the vector \(n_0\). In the case under consideration, in equation (7.11) each term \(A_{MN}\) will contain the expressions

Fig. 28. Irregularities on a plane wall.

\[ \frac{\partial}{\partial n}\psi_N = [n\cdot \operatorname{grad}\psi_N]_S = \left\{ -\left[\frac{d\psi_N}{dz}\right]_S + (\operatorname{grad}_{S_0} B)\cdot(\operatorname{grad}^{S}\psi_N) \right\}\cos\vartheta . \]

Here \(\operatorname{grad}_{S_0}\) represents the two-dimensional gradient in the plane \((x,y)\), and

\[ \cos^2\vartheta = 1 + \operatorname{grad}_{S_0}^{\,2}(B). \]

The area \(dS\) is equal to \(dx\,dy/\cos\vartheta\), and therefore the first term \(A_{MN}\) takes the form

\[ (1/V_{\varepsilon M})\left\{ \iint_S \psi_M \operatorname{grad} B\cdot \operatorname{grad}\psi_N\,dx\,dy - \iint_S \psi_M \frac{\partial}{\partial z}\psi_N\,dx\,dy \right\}. \tag{7.28} \]

If \(B\) is less than a quarter wavelength, \(\operatorname{grad}_S\psi_N\) may be replaced by \(\operatorname{grad}_{S_0}\psi_N\). Since we have assumed that \(S_0\) is a plane wall, the general form of the expression for \(\psi_N\) near this wall (\(z\) small) will be:

\[ \psi_N = F(x,y)\cos(\pi n_z z/L_z). \]

The approximate value of the expression \(-(\partial\psi_N/\partial z)\) at the point \(z=B\), provided that \(B\) is small in comparison with \((L_z/\pi n_z)\), will be equal to:

\[ (\pi n_z/L_z)^2 B[\psi_N]_{z=0}. \]

The final approximate expression for that part of \(A_{MN}\) which is due to the protrusion under consideration is equal to:

\[ A_{MN}\simeq \frac{1}{V_{\varepsilon M}} \iint_{S_0}\psi_M\left\{ \operatorname{grad}_{S_0}B\,\operatorname{grad}_{S_0}\psi_N - \frac{\omega}{c}(\sigma+i\gamma+\sigma_B)\psi_N \right\}dx\,dy . \tag{7.29} \]

Here \(\gamma-i\sigma\) represents the conductance on the protrusion; the term containing \(\partial\psi_N/\partial z\) is expressed in the form of an equivalent reactive compliance

\[ \sigma_B=-(\lambda/2\pi)(\pi n_z/L_z)^2B. \tag{7.30} \]

This compliance has an inertial character, since \(B\) is positive.

That part of the effect of the change in the shape of the room which can be expressed through \(\sigma_B\) is the result of a change in the total volume of the room. To determine this effect, let us consider the case when \(B_0\) is a rectangular room in which the only protrusion is located on the wall \((x,y)\). Let its dimensions be large in comparison with the wavelength. In this case the part of the quantity \(A_{MN}\) which is due to \(\sigma_B\) is approximately equal to:

\[ (2\overline{B}/L_z)(\pi n_z/L_z)^2 . \]

Here \(\overline{B}\) denotes the mean value of \(B\) over the entire wall. The last expression is valid for oblique oscillations \((n_z>0)\). The change in frequency caused by this term in the first-order approximation is:

\[ \xi_N^2 = c^2\left[ \left(\frac{\pi n_x}{L_x}\right)^2 + \left(\frac{\pi n_y}{L_y}\right)^2 + \left(\frac{\pi n_z}{L_z}\right)^2 + c^2\left(\frac{\pi n_z}{L_z}\right)^2 \right] (2\overline{B}/L_z). \]

As we see, in the first approximation this additional term is equivalent to a decrease of the length \(L_z\) by the segment \(\overline{B}\).

Thus, that part of the effect which is caused by the change of shape and which is expressed through \(\sigma_B\) can be interpreted in exactly the same way as the influence of an actual change of the impedance, which was discussed in the preceding section. In addition, there is another new term, containing \(\operatorname{grad}_{S_0}\psi_N\). In order to elucidate the role of this term, let us introduce a further simplification and consider the projection shown in Fig. 28 below. Its form is such that \(B\) changes from the value zero by a jump to the constant value \(B_0\) on the contour \(C\). In this case the expression for \(A_{MN}\) takes the form

\[ A_{MN}\approx \frac{1}{V_\varepsilon} \left\{ B_0\int_C \psi_M \mathbf n_S \operatorname{grad}_{S_0}\psi_N\,ds - \frac{\omega}{c}\int_{S_0}\int \psi_N(\sigma+i\gamma+\sigma_B)\psi_N\,dx\,dy \right\}. \tag{7.31} \]

Here \(\mathbf n_S\) denotes the unit vector in the plane \((x,y)\), normal to the contour \(C\). The first integral must be taken over the contour \(C\). This first term determines the influence of the boundaries of the projection. We shall see that such a projection is a very effective measure for scattering oscillations.

If the boundaries of the projection have the outline of a rectangle with sides \(a\) and \(b\), respectively parallel to the \(x\)- and \(y\)-axes, then one can calculate the mean-square value of the first term by using the same procedure that earlier led to equation (7.26). Assuming that neither \(a\) nor \(b\) reaches the corresponding dimensions of the chamber \(R_0\), we obtain:

\[ \frac{B_0}{2V}\, \frac{(a+b)\lambda}{\left[1+4(a+b)/\lambda\right]^{1/2}}. \]

If, however, we put \(b=L_y\), i.e. if the projection has become a strip of width \(a\), extending along the plane \((x,y)\) parallel to the \(y\)-axis from one wall \((x,z)\) to the opposite one, then the approximate mean-square value of the first term in the expression for \(A_{MN}\) in equation (7.29) will, for waves for which \(m_y=n_y\) and \(m_z=n_z\), have the form

\[ \frac{B_0}{2V}\, \frac{a/\lambda}{\left[(1+4a)/\lambda\right]^{1/2}}, \]

and for \(m_y\ne n_y\) or \(m_z\ne n_z\) it becomes zero.

Striving to make all \(A_{MN}\) differ from zero, we must arrange several long strips on different walls in such a way that at least one strip is perpendicular to each of the three coordinate axes. If the inhomogeneities are to occupy only part of the walls, it is necessary that they be distributed at least

measure, on three mutually perpendicular walls. If the projections are distributed sufficiently irregularly and if there are sufficiently many of them, then the approximate series in expression (7.11) will not be convergent, and an ergodic sound process will be obtained in the room.

Now we can determine the criterion for the transition to an ergodic process from equation (7.11). Let us consider the changes in the mean amplitude of the coefficient \(\psi_M c^2 A_{MN}/(\omega_N^2-\omega_M^2)\) for \(\omega_M\) close to \(\omega_N\). If this coefficient is small, then \(\varphi\) is determined mainly by the unperturbed oscillation \(\psi_N\), and if \(\psi_N\) belongs to a maximal oscillation, then \(\varphi\) will have a damping exponent sharply different from that for other oscillations. On the other hand, if several coefficients for different \(\psi_M\) are not small, then \(\varphi\) will represent a superposition of many unperturbed \(\psi_N\). Some of these oscillations \(\psi_N\) will be axial, some—oblique, so that each standing wave \(\varphi\) will be a mixture of axial, tangential, and oblique oscillations and will have almost identical damping exponents.

50. Coefficient of Irregularity

Suppose that we have \(m\) pieces of absorbing material with mean conductance \(\beta\) and mean area \(S_a/m\), distributed irregularly over the walls. Suppose, in addition, that we have \(n\) projections of mean height \(B\) and size \(a\), likewise distributed irregularly over the walls. If a projection occupies only part of a wall, then we shall take its semiperimeter as the size of the projection; if it extends over the whole length of the wall, then let its width be \(a\). Suppose that the mean sizes of the pieces of absorbing material and of the projections are of the order of the wavelength. Using the previous approximate results for pieces of material and for projections, one can obtain very approximate values of the mean amplitude of the coefficient \(c^2 A_{MN}/(\omega_N^2-\omega_M^2)\) in equation (7.11) for frequencies \(\psi_M\) close to \(\psi_N\):

\[ \Xi=(1/10\lambda)[nBa+|\beta_e|S_a] \tag{7.30} \]

(assuming that \(a^2\) or \(S_a/m\) is of the order of \(\lambda^2\)). Here

\[ \beta_e=\gamma-i\alpha-i\delta_B \simeq \gamma-i\alpha+(2\pi iB/\lambda). \]

If \(B\) is less than a half-wavelength, a somewhat more flexible, though not more exact, formula has the following form:

\[ \Xi=\frac{1}{2}\left\{\frac{(nBa/\lambda)}{\lambda+4a}+\frac{|\beta_e|S_a}{\lambda^2+(4S_a/m)}\right\}. \tag{7.31} \]

It is, however, valid for any ratios between \(\lambda\) and the dimensions

inhomogeneities, but is likewise valid only for a random distribution of these inhomogeneities; it is assumed in it that \(B\) is less than a half-wave.

If \(B\) is greater than a half-wave, then in these formulas one must substitute \(\lambda/2\) in place of \(B\).

The coefficient of disorder \(\Xi\), both for pieces of absorbing material and for projections, coincides with expression (7.27) for the second term. It may be used as a very rough measure of whether the sound phenomenon will be ergodic or not. If it is considerably less than unity, then the process cannot be ergodic, and the attenuation curve is not rectilinear. If it is considerably greater than unity, then the wave process has a random character, and it may be considered that Sabine’s assumptions are valid. Only in this case can we speak of the absorption coefficient of the given material, and not of the wall coefficient for the given wave, the given place on the wall, and the given material.

Several interesting consequences follow from this. First of all, the volume of the room does not explicitly enter into the expression for the coefficient of disorder. In this crude approximation this means that the same number of inhomogeneities on the walls of a room is sufficient, at a given frequency, to create a random distribution of oscillations both in a small room and in a large one. But since it is difficult, if not impossible, to place a given number of inhomogeneities in a small room in a random manner, at a given frequency it is harder to ensure the fulfillment of Sabine’s assumptions in a small room than in a large one. Secondly, it is easier to obtain ergodic oscillations in a given room for short waves than for long waves. This conclusion is the converse of the first conclusion.

As an example, let us consider the use of a material with acoustic conductivity \(|\beta| \approx 1.5\). Such a material is a very soft material. We shall take the projections in the form of rectangular strips on three or more walls, with some of the strips vertical and others horizontal. We shall take the thickness to be \(150\ \mathrm{mm}\). For a frequency of \(1000\) hertz \((\lambda = 0.3\ \mathrm{m})\), the strips should be taken only \(0.3\ \mathrm{m}\) wide, and the absorbing material in the form of separate pieces of size \(0.3\ \mathrm{m}\). In order to obtain an ergodic process \((\Xi\) greater than 2), more than a hundred pieces of absorbing material, scattered at random, are required, or more than 40 strips disturbing the smoothness of the walls.

For a frequency of \(250\) hertz \((\lambda = 1.2\ \mathrm{m})\), our material should be used in pieces of size \(1.2 \times 1.2\ \mathrm{m}\), and the strips should be \(1.2\ \mathrm{m}\) wide. (Both may be somewhat reduced, without noticeably diminishing the validity of the conclusions.) To obtain an ergodic process we shall again need a hundred such pieces of absorbing material or more than 160 strips. It is obvious that it is easy to obtain

ergodic process at 1000 cycles in a room of moderate dimensions ($3 \times 6 \times 9$ m), but it would be difficult to equip it with sufficiently large pieces or strips to achieve the same thing at 250 cycles. It is also clear that it is usually easier to obtain irregular oscillations by means of projections than by means of pieces of absorbing material. This is not surprising, since a projection scatters sound much better than a flat piece of absorbing material.

Experimental checks of this theory are fragmentary in character. To verify the general propositions of the theory of perturbations, some experiments were carried out with small models. In one series of experiments B⁸ the chamber had dimensions of the order of 0.3 m, so that even the lowest frequencies caused no difficulties in measurement. In this chamber, natural oscillations were excited and the distribution of nodes and antinodes of pressure was investigated. The shape of the room was varied from rectangular to trapezoidal, and rectangular projections and recesses were made on various walls. The pressure distribution for the first 10–12 natural oscillations and the corresponding resonance frequencies were compared with the conclusions of the perturbation theory set forth in this chapter. B¹⁰

In general, good agreement between theory and experiment was obtained, although to obtain a satisfactory coincidence it was necessary to take into account the term of second order. Fig. 29 shows one case. At the top are given the shape of the chamber and the lines of equal pressures for one of the natural oscillations. The other three diagrams illustrate the agreement between the experimental and theoretical values of the acoustic pressure. For this type of perturbation (a trapezoidal chamber) the first-order term is zero, since the volume of the chamber remained constant. For other types of change in the shape of the room, the first-order terms become important, and the calculations are somewhat simplified.

Much work still remains to be done in order to obtain confidence in the degree of applicability of perturbation theory. At present we are compelled to use it as the only means for studying the properties of rooms of complex outline, but it is still unknown how reliable these results are.

The general conclusions obtained in this section and theoretically substantiated here with the aid of the coefficient of disorder have already been applied in calculations in architectural acoustics by many authorities in this field B¹¹, M⁵, V¹. Experimental study of the scattering action of cylinders was carried out by Sabine S⁶. In this case the applications appeared before the theory. The calculations were based empirically on a large quantity of experimental data obtained in the construction of auditoriums and in the study of their properties. It may be noted with satisfaction that the results of the theory presented here more or less coincide with the practical conclusions made earlier. It is possible that the theory will be able to explain these practical rules and

Fig. 29. Pressure distribution in a standing wave according to measurements in a trapezoidal room. The lower curves compare experimental results (dotted lines) with the results of calculation of second-order disturbances (solid lines, calculated theoretically)^[10].

to give them a more quantitative character.

It has long been found that two rooms with the same reverberation time do not always have the same acoustic qualities. An additional characteristic has received the somewhat indefinite name of the “liveliness” of a room^[2]. This property is connected with the ability of a room to “hold” sound and at the same time distribute it uniformly.

At the present time there is a tendency to increase the reverberation time in comparison with the value that was formerly considered necessary, but to make it, as far as possible, independent of frequency (see Figs. 3, 6). At the same time efforts are made to obtain a uniform distribution of sound. Large flat surfaces of hard materials are not allowed. The walls are covered with irregularly distributed projections or pieces of absorbing material. The irregularities on the walls are placed as nonuniformly as possible. All these tendencies are in agreement with the conclusions of this chapter. It may be hoped that further investigations will increase this agreement still more.

VIII. THE METHOD OF FREE PLANE WAVES IN DISORDERED OSCILLATIONS

Rooms sufficiently irregular in shape, or with acoustical material irregularly distributed, at frequencies above a certain limit have an ergodic distribution of sound energy. In this case each standing wave consists of a disordered combination of plane waves propagating in various directions, and the system of nodal figures and antinodes is arranged in an irregular manner. The damping factor for each oscillation is approximately equal to the factor for any other oscillation of a resonant frequency close to it.

Only in this case can we be sure that sound from all possible directions is incident on each section of the wall. And only then can we speak of an “absorption coefficient” that depends neither on the room nor on the form of the oscillations, but only on the material itself.

When the conditions of ergodicity are satisfied and one may deal with plane waves propagating in all possible directions, we obtain a new simplification of the problem. One may consider the absorption or reflection of a plane wave from any section of the wall and then determine the properties for the case of ergodicity by averaging over all possible directions of the incident wave.

51. Reflection of a Plane Wave from a Homogeneous Wall

To illustrate this method and at the same time derive the formulas needed for comparing the Sabine absorption coefficient with the impedance of a wall, we shall first consider reflection of a plane wave from a plane wall characterized by the constant impedance \(Z=R-iX=\rho \xi=\rho c\beta=|Z|e^{-i\varphi}\). Let the angle of incidence be \(\vartheta\), and let the velocity potential be written in the form

\[ \psi=e^{i(\omega/c)(x\sin\vartheta-z\cos\vartheta-ct)} +Ae^{i(\omega/c)(x\sin\vartheta+z\cos\vartheta-ct)} . \]

In this expression \(z\) denotes the distance of the point from the wall along the normal to it, \(x\) the distance along the wall parallel to the plane of incidence, and \(A\) the complex amplitude of the reflected wave.

The amplitude \(A\) is determined from the consideration that, on the surface of the wall, the ratio of the pressure

\[ p_s=-i\omega\rho(1+A)e^{i(\omega/c)(x\sin\vartheta-ct)} \]

to the negative value of the normal component of velocity

\[ u_s=-(i\omega\cos\vartheta/c)(1-A)e^{i(\omega/c)(x\sin\vartheta-ct)} \]

must be equal to the impedance \(Z\) of the wall. The final expression for \(A\) is obtained as follows:

\[ A=\frac{\zeta \cos \vartheta-1}{\zeta \cos \vartheta+1} =\frac{\cos \vartheta-\beta}{\cos \vartheta+\beta} =e^{-2\pi \Gamma(\vartheta)} =e^{-2\pi \tau+2\pi i\nu}. \tag{8.1} \]

Here it has been set that

\[ \operatorname{cth}[\pi \Gamma(\vartheta)] =\operatorname{cth}[\pi(\tau-i\nu)] =\zeta\cos\vartheta =(|Z|/\rho c)e^{-i\varphi}\cos\vartheta; \]

\[ \operatorname{th}[\pi \Gamma(\vartheta)]=\beta\sec\vartheta. \]

The number \(e^{-2\pi\tau}\) represents the diminution of the amplitude of the reflected wave; \(2\pi\nu\) is equal to the change of phase upon reflection.

If the sound intensity in the incident wave is equal to unity, \(|p_i|^2/2\rho c=1\), then the amplitude of the pressure at the wall has the magnitude

\[ (2\rho c)^{\frac12}(1+A) =(2\rho c)^{\frac12}\frac{2\zeta\cos\vartheta}{\zeta\cos\vartheta+1} \]

\[ =(2\rho c)^{\frac12}\frac{2\cos\vartheta}{\cos\vartheta+\beta} =(8\rho c)^{\frac12}e^{-\pi\tau}\operatorname{ch}(\pi\Gamma), \tag{8.2} \]

the normal component of the velocity has the amplitude

\[ (2/\rho c)^{\frac12}\cos\vartheta(1-A) =(2\rho c)^{-\frac12}\frac{2\cos\vartheta}{\zeta\cos\vartheta+1} \]

\[ =(2/\rho c)^{\frac12}\frac{2\beta\cos\vartheta}{\cos\vartheta+\beta} =(8/\rho c)^{\frac12}e^{-\pi\tau}\operatorname{sh}(\pi\Gamma). \tag{8.3} \]

If the wall is rigid \((\beta=0,\ \Gamma=0)\), then the pressure amplitude is equal to twice the amplitude in the incident wave, while the normal component of the velocity is, of course, zero. As the wall becomes more yielding \((|\beta|>0)\), the pressure amplitude at the wall, generally speaking, decreases, while the normal component of the velocity increases in amplitude.

The sound intensity in the reflected wave is

\[ |p_r|^2/2\rho c=|A|^2 =\frac{(\cos\vartheta-\gamma)^2+\sigma^2}{(\cos\vartheta+\gamma)^2+\sigma^2} =e^{-4\pi\tau}, \tag{8.4} \]

where \(\beta=\gamma-i\sigma\). For the absorbed sound energy,

\[ \alpha(\vartheta)=1-|A|^2 =1-e^{-4\pi\tau} =\frac{4\gamma\cos\vartheta}{(\cos\vartheta+\gamma)^2+\sigma^2}. \tag{8.5} \]

This expression represents the coefficient of the wall for plane waves at the angle of incidence \(\vartheta\). Its value may easily be determined from equation (8.5) as a function of \((|Z|/\rho c)\), \(\varphi\), and \(\vartheta\). It is necessary to note that its maximum lies at such an angle of incidence \(\vartheta\) that \((|Z|/\rho c)\cos\vartheta=1\).

The harder the wall, the closer the angle of maximum absorption approaches \(90^\circ\) (grazing incidence). With exactly grazing incidence, free sound waves, of course, are not absorbed at all*).

Equation (8.5) was verified experimentally by Cremer \(C^7\) and Willi \(W^{10}\) by measuring the intensity of the incident and reflected sounds in free propagation and using large pieces of material. A variant of this free-wave method, according to preliminary data, gave results that in their essential features coincided with the theory \(P^6\). This variant consisted in measurements of pressure at the maxima and minima of the interference pattern on the surface of the absorbing material, formed by the incident and reflected waves at various angles of incidence. Equation (8.5) was also used \(H^7\) to interpret the damping of various natural oscillations in a rectangular room, one of whose walls was entirely covered with absorbing material. In view of the fact that each natural oscillation, corresponding to a definite angle of incidence, is expressed by the numbers \(n\), it is possible to study absorption for all such angles by a suitable choice of the signal frequency in a room of given dimensions. The theory of free waves gives a good approximation to this case, with the essential exception corresponding to the “grazing” direction of the sound ray. However, for a rigorous solution it is necessary to use the theory of standing waves set forth in Chapter V.

Willi \(W^{10}\) developed a miniature pressure-gradient microphone in order to separate the incident waves from the reflected ones by directionality. The directionality proved sufficient for measuring angles of incidence between 15 and \(75^\circ\). He also determined the absorption at normal incidence by means of the tube method. Taking \(\sigma=0\), i.e. regarding the impedance as real, he selected such values of \(\gamma\) as would give the best agreement of the experimental points with the curve calculated from equation (8.5). The experimental errors proved minimal at \(\alpha=0.75\) and remained fairly large for \(\alpha\) less than 0.2 and greater than 0.9, since here it was necessary to determine small differences between large numbers. The results coincided with the calculated curves within the limits of experimental error and gave good confirmation of the theory.

*) This is incorrect. Owing to the viscosity of the boundary layer of air adjacent to the wall, in this case as well part of the sound energy is converted into heat. See B. P. Konstantinov, dissertation, LPhTI, “On the absorption of sound waves upon reflection from a rigid boundary,” ZhTF, vol. 9, No. 3, pp. 226–38, 1939; “On the damping of sound in a room with rigid walls and on the diffuse coefficient of sound absorption,” ZhTF, vol. 9, No. 5, pp. 424–432, 1939. (Editor’s note.)

52. Sabine coefficient and wall impedance

In order to obtain the Sabine absorption coefficient, it is only necessary to compute the mean value of the coefficient \(a(\vartheta)\), determined by equation (8.5), over all angles of incidence. In doing so, each direction must have a weight proportional to the relative amount of sound energy incident on a unit area from the given direction. The calculations give:

\[ \begin{aligned} a_{\text{stat}} &=\frac{1}{\pi}\int_{0}^{2\pi} d\varphi \int_{0}^{\pi} a(\vartheta)\cos\vartheta \sin\vartheta\, d\vartheta =8\gamma \int_{0}^{1}\frac{x^2 dx}{x^2+2\gamma x+\gamma^2+\sigma^2} \\ &=8\gamma\left\{1-\gamma\ln\left[1+\frac{2\gamma+1}{|\beta|^2}\right] +\frac{\gamma^2-\sigma^2}{\sigma}\arctg\left[\frac{\sigma}{|\beta|^2+\gamma}\right]\right\} \\ &=8\frac{\cos\varphi}{w}\left\{1-\frac{\cos\varphi}{w}\ln\left[1+2w\cos\varphi+w^2\right] +\frac{\cos 2\varphi}{w\sin\varphi}\arctg\left[\frac{w\tg\varphi}{w+\sec\varphi}\right]\right\} \\ &= \begin{cases} 8\frac{\cos\varphi}{w}\left[1-\dfrac{2\cos\varphi}{w}\ln(w)+\dfrac{\varphi}{w}\dfrac{\cos 2\varphi}{\sin\varphi}\right], & \text{as } w\to\infty, \\[1.2em] \dfrac{8}{3}w\cos\varphi\left[1-\dfrac{3}{2}w\cos\varphi\right], & \text{as } w\to 0. \end{cases} \tag{8.6} \end{aligned} \]

Here it has been put that \(w=|Z|/\rho c\); \(\cos\varphi/w=\gamma\); \(w\cos\varphi=R/\rho c\); \(w\sin\varphi=X/\rho c\).

In Fig. 30 the dependence of \(a_{\text{stat}}\) on the magnitude of the specific wall impedance \(w=|Z|/\rho c\) and on the phase angle \(\varphi\) is given.

Equation (8.6) and Fig. 30 determine the relation between the wall impedance and the Sabine absorption coefficient. The latter number is determined from reverberation measurements only when the sound motion in the room is completely random. In large halls at ordinary frequencies the sound motion for the most part has an ergodic character, so that the quantity \(a_{\text{stat}}\) can be used for most architectural calculations. On the other hand, there is much evidence that an ergodic process does not occur in the majority of acoustic measuring chambers at frequencies below 2000 hertz. At these, lower, frequencies the measurements usually give the normal coefficient \(\alpha_n=8\cos\varphi/w=8\gamma\). Therefore, in calculating the acoustic equipment of large auditoriums, it is better to use the quantity \(a_{\text{stat}}\), computed from equation (8.6) or found from Fig. 30 from the measured impedance of the material, than the coefficients measured in reverberation chambers (at least, at frequencies

Fig. 30. Sabine absorption coefficient \(\alpha_{\mathrm{stat}}\) as a function of the modulus and argument of the wall impedance. The diagram applies to cases in which an ergodic sound process takes place.

below 2000 hertz). Usually the normal coefficient \(a_p\) is numerically larger than \(\alpha_{\mathrm{stat}}\), so that the coefficients currently obtained from measurements in acoustic chambers turn out to be larger than the Sabine coefficients characterizing the absorbing capacity of a material in large rooms of irregular shape \(^{E5,S7,S14}\).

53. Reflection of a spherical wave from a plane wall

Some theoretical studies on reverberation were based on the application of the method of images, which makes it possible to follow the successive reflections of a spherical wave from the walls of a rectangular room. Unfortunately, this useful method gives exact results only for absolutely rigid walls. If the impedance of the wall is not infinitely large, then the reflected wave cannot be represented as the result of the action of a point image. For the proof we shall use the equation

Fig. 31. Notation for angles and distances adopted in calculating the reflection of a spherical wave from an absorbing wall.

\[ \psi_i=\frac{e^{ikR}}{4\pi R} =\frac{k}{4\pi i}\int_{1}^{i\infty} e^{ikRz}\,dz \]
\[ =\frac{k}{8\pi^2 i}\int_{0}^{2\pi} d\chi\cdot \int_{i\infty-\pi/2}^{0} e^{ikR\cos u}\sin\theta\,d\theta =\frac{k}{8\pi^2 i}\iint_s e^{ikR}\,d\Omega . \tag{8.7} \]

Here \(k=\omega/c=2\pi/\lambda\), the angles and distances are shown in Fig. 31, \(d\Omega\) denotes the elementary solid angle for the direction \(k\), the sign \(s\) under

double integral means that one must integrate with respect to the azimuthal angle in the range from \(0\) to \(2\pi\) and with respect to the polar angle in the range from \(i\infty-(\pi/2)\) to \(0\).

Equation (8.7) represents the spherical velocity potential at the point \(P\), due to a unit point source of sound at the point \(Q\), in the form of an integral of plane waves of the type \(e^{ikR}\) over all directions of the vector \(\mathbf{k}\). We can now make use of the results of Section 51 in order to satisfy the boundary conditions on the wall \(z=0\). Each elementary plane wave \(e^{ikR}\,d\Omega\) gives rise to a reflected wave
\[ \frac{\cos\vartheta-\beta}{\cos\vartheta+\beta}\,e^{ikR}\,d\Omega . \]
The amplitude of this reflected wave depends on the angle of incidence; therefore the radiation issuing from the image \(Q'\) does not possess spherical symmetry. This means that the sound strength in the reflected wave at the point \(P\) depends on the angle \(\vartheta'\). Therefore an analysis based on representing reverberant sound as the result of the superposition of waves issuing from imaginary sources can apparently lead to erroneous results. This critical remark applies to the works of Sabine\(^{S2}\), Norris\(^{N1}\), Eyring\(^{E3}\), Millington\(^{M7}\), and Sette\(^{S12}\).

In fact, the reflected wave is described by the expression
\[ \psi_r=\frac{k}{8\pi^2 i}\iint_S \frac{\cos\vartheta-\beta}{\cos\vartheta+\beta}\,e^{ikR'}\,d\Omega' = \frac{k}{8\pi^2 i} \int_0^{2\pi} d\chi \int_{i\infty-\pi/2}^{0} e^{-2x\Gamma(\vartheta)+ikR'\cos u'}\sin\vartheta\,d\vartheta . \tag{8.8} \]

This integral cannot be expressed by simple functions. If \(\beta\) is small \((|Z|\gg \rho c)\), then an approximation valid for \(R'\) large compared with the wavelength has the form
\[ \psi_r\simeq \frac{\cos\theta-\beta}{\cos\theta'+\beta} \frac{e^{ikR}}{4\pi R'} = e^{-2x\Gamma(\theta')+ikR'}\,/\,4\pi R' . \tag{8.9} \]

According to this approximation, the reflected wave at the point \(P\) is attenuated by reflection just as much as a plane wave incident at the angle \(\theta'\). If the point \(P\) is moved away from the line \(QQ'\), the angle \(\theta'\) changes, and with it the factor \(e^{-2x\Gamma(\theta')}\) changes as well. This circumstance again illustrates the fact that the reflected wave is devoid of spherical symmetry.

When the reactive part of the wall impedance is negative (\(X\) positive), the reflected wave can be represented as the result of the joint action of a simple point source \(Q'\) and a line source extending along the normal from \(Q'\) to minus infinity:
\[ \psi_r=\frac{e^{ikR'}}{4\pi R'} +2ik\beta\int_a^\infty e^{ik\beta(q-a)} \frac{\exp(ikR_q)}{4\pi R_q}\,dq, \tag{8.10} \]
where \(\beta=\gamma-i\sigma\); the meanings of \(q\), \(R_q\), and \(a\) are taken from Fig. 31.

If \(|\beta|\) is large \((|Z|<\rho c)\), then the reflected wave can be expanded in a series of spherical harmonics with respect to \(Q\):

\[ \psi_r=\left[1-\beta \ln \frac{\beta+1}{\beta-1}\right]\frac{e^{ikR'}}{4\pi R'} -\frac{2kR}{4\pi}\sum_{n=1}^{\infty}(-i)^n(2n+1)\times \]

\[ \times Q_n(\beta)P_n(\cos\theta')h_n(kR'), \tag{8.11} \]

where \(Q_n\) denotes the Legendre function of the second kind, and \(h_n\) the spherical Bessel function of the third kind. This expansion is valid for all values of \(\beta\), except real values lying between \(+1\) and \(-1\); however, for not very large \(|\beta|\) it converges slowly.

54. Diffraction edge corrections

Equation (8.6) establishes a relation between the Sabine absorption coefficient and the impedance of the wall. For large rooms, if the inhomogeneities ensure ergodicity of the sound motion and the impedance remains constant over sufficiently large portions of the walls, the attenuation coefficient can be calculated by means of equation (2.2). In this case the coefficient \(a_{\text{stat}}\) for each material is multiplied by the area occupied by this material—in exact agreement with Sabine’s conclusion. In reality, however, the value of \(a_{\text{stat}}\) is computed from the impedance on the assumption that the wall is infinite and has constant impedance. Near the boundaries of each piece of material diffraction phenomena arise. Therefore the absorption here will differ from the absorption far from the boundaries.

In the present section we shall compute an approximate edge correction which must be introduced into Sabine’s formula in order to obtain a more accurate expression for the duration of reverberation in rooms with an irregular oscillatory process. These diffraction corrections are not applicable to rooms in which the oscillations are not disordered; in that case it is necessary to use the methods of Chapter VII. The calculations show that the edge corrections can be interpreted as an addition to the area of a piece of material, proportional to the perimeter of the piece or to the length of its boundary. In this case those boundaries of the piece which coincide with the boundaries of the wall (or are no farther than \(\lambda/2\) from them) do not require an edge correction. These parts of the boundaries of the pieces should not be included in the “perimeter” when computing edge corrections. If the wall is entirely covered with acoustic material, no corrections are required.

In order to show how diffraction changes absorption, let us consider the simplest case. Let a plane wave, corresponding to sound intensity unity, be incident normally on a plane wall (the plane \(x,y\)). Let the positive half-plane of it have impedance \(Z_+\), and the negative half-plane \((x<0)\) have impe—

impedance \(Z\). If the wall impedance were everywhere infinitely large, then the pressure would be equal to:

\[ p_0=(2\rho c)^{\frac12}\left[e^{-i(\omega/c)z}+e^{+i(\omega/c)z}\right]e^{-i\omega t}= \]

\[ =2\cdot(2\rho c)^{\frac12}\cos(\omega z/c)e^{-i\omega t}. \tag{8.12} \]

The normal component of the velocity in the direction toward the wall

\[ u_0=(i/\rho\omega)(\partial p_0/\partial r)_{z=0} \]

would be equal to zero.

Since the wall is not rigid, \(u_0\) is not equal to zero. The wall vibrates, or at least the air in the pores vibrates. This motion gives rise to a pressure wave \(p_1\), which is added to \(p_0\), so that their sum satisfies the boundary conditions. The total pressure at any point of the wall is equal to \(p_0+p_1\), and, according to the definition of the concept of impedance, the velocity

\[ u_0=(\beta/\rho c)(p_0+p_1)_{z=0}. \]

The pressure wave \(p_1\) at the point \((x,y,z)\) is expressed as follows:

\[ p_1(x,y,z)= \]

\[ =i(\omega/c)\int_{-\infty}^{+\infty}dy'\int_{-\infty}^{+\infty}dx'\, [\beta(x',y')e^{i\omega R/c}/2\pi R]\,[p_0(x',y',z',0)+ \]

\[ +p_1(x',y',z',0)], \tag{8.13} \]

where

\[ R^2=(x-x')^2+(y-y')^2+z^2. \]

\(p_1\) is composed of elementary waves excited by each surface element \(dx'\,dy'\). Equation (8.13) is an integral equation with respect to \(p_1\). Its exact solution gives an exhaustive answer. Usually, however, only an approximate solution can be found.

In the simple case under consideration, \(\beta(x,y)\) depends only on \(x\) (\(\beta=\beta_+\) for \(x>0\); \(\beta=\beta_-\) for \(x<0\)). Therefore the integration with respect to \(y\) can be carried out directly, and for the pressure on the surface of the wall we obtain:

\[ p_1(x)=-(\omega/2c)\int_{-\infty}^{\infty}\beta(x')\,[p_0(x')+p_1(x')]\times \]

\[ \times\{J_0[\omega(x-x')/c]+iN_0[\omega|x-x'|/c]\}\,dx'. \tag{8.14} \]

Here \(J_0\) and \(N_0\) denote the Bessel and Neumann functions of zero order.

In view of the fact that \(p_0\) for \(z=0\) is equal to \(2\cdot(2\rho c)^{\frac12}e^{-i\omega t}\) and does not depend on \(x\), while \(\beta\) changes its value only at \(x=0\), one may

make use of the formula

\[ (\omega/c)\int_{0}^{\infty}\{J_0[\omega(x-x')/c]+iN_0[\omega|x-x'|/c]\}\,dx' = \]

\[ = (\omega/c)\int_{-\infty}^{+\infty}\{J_0[\omega\tau/c]+iN_0[\omega|\tau|/c]\}\,d\tau = \]

\[ = \left\{ \begin{array}{ll} 1-Ji_0(-\omega x/c)-iNi_0(-\omega x/c), & (x<0),\\[3pt] 1+Ji_0(\omega x/c)+iNi_0(\omega x/c), & (x>0) \end{array} \right\} = \]

\[ = Fr(\omega x/c)+iFi(\omega x/c), \tag{8.15} \]

where

\[ Ji_0(\mu)=\int_{0}^{\mu}J_0(u)\,du;\qquad Ji_0(0)=0;\qquad Ji_0(\infty)=1, \]

\[ Ni_0(\mu)=\int_{0}^{\mu}N_0(u)\,du;\qquad Ni_0(0)=0;\qquad Ni_0(\infty)=0. \]

The graph of the functions \(Fr(z)\) and \(Fi(z)\) is given in Fig. 32. Let us note the following relations, which will be needed later:

Fig. 32. Diffraction functions entering into the calculation of edge corrections.

\[ \int_{-\infty}^{0} Fr(z)\,dz=0;\qquad \int_{0}^{\infty} Fi(z)\,dz=-\frac{\pi}{2}. \]

Therefore the integral equation for \(p_1(x)\) at \(z=0\) assumes the following final form:

\[ \begin{aligned} p_1(x)=&-(2\rho c)^{\frac12}\{\beta_+ [Fr(\omega x/c)+iFi(\omega x/c)]+\\ &\quad+\beta_- [Fr(-\omega x/c)+iFi(-\omega x/c)]\}e^{-i\omega t}-\\ &\quad-(\omega/2c)\int_{-\infty}^{+\infty}\beta(x')p_1(x')\{J_0[\omega(x-x')/c]+\\ &\quad+iN_0[\omega|x-x'|/c]\}\,dx'. \end{aligned} \tag{8.16} \]

Here the relation

\[ [Fr(-z)+iFi(-z)]=2-[Fr(z)+iFi(z)] \]

has been taken into account.

If \(\omega x/c\) is large compared with unity \((x\gg \lambda/2\pi)\), then \(Fr(\omega x/c)\simeq 2,\ Fr(-\omega x/c)\simeq 0,\ Fi(\pm\omega x/c)\simeq 0\). The approximate solution of equation (8.16) will have the form

\[ \left. \begin{aligned} p_1 &\simeq -2\cdot(2\rho c)^{\frac12}\beta_+ e^{-i\omega t}-\beta_+p_1 \end{aligned} \right\} \]

or

\[ \left. \begin{aligned} p_1 &\simeq -(2\rho c)^{\frac12}[2\beta_+/(1+\beta_+)]e^{-i\omega t}. \end{aligned} \right\} \tag{8.17} \]

Therefore the total pressure on the surface of the wall sufficiently far to the right of the boundary line \(x=0\) is equal to:

\[ p_0+p_1\simeq (2\rho c)^{\frac12}\left[2/(1+\beta_+)\right]e^{-i\omega t}. \]

This expression should be compared with expression (8.2), bearing in mind that in the present case \(\vartheta=0\). For \(-x\gg 2\pi/\lambda\) analogous expressions are valid if \(\beta_+\) is replaced by \(\beta_-\).

In the first approximation \(p_1\) is equal to these expressions, and along the line \(x=0\) there exists a discontinuity:

\[ p_1\simeq \begin{cases} -(2\rho c)^{\frac12}[2\beta_+/(1+\beta_+)]e^{-i\omega t}, & \text{for } x>0,\\ -(2\rho c)^{\frac12}[2\beta_-/(1+\beta_-)]e^{-i\omega t}, & \text{for } x<0. \end{cases} \tag{8.18} \]

In this approximation diffraction phenomena are not taken into account at all, and the edge corrections have not entered into the final expressions for the absorption.

A more exact expression, taking account of diffraction phenomena in the first approximation, will be obtained if the expressions (8.18) are substituted into the integral in the right-hand side of equation (8.16). After a number of transforma-

this approximation takes the form

\[ p_1(x)\approx -(2\rho c)^{\frac12}\left\{ \frac{\beta_+}{1+\beta_+}\,[Fr(\omega x/c)+iFi(\omega x/c)]+ \frac{\beta_-}{1+\beta_-}\,[Fr(-\omega x/c)+iFi(-\omega x/c)] \right\}e^{-i\omega t}, \tag{8.19} \]

\[ p_0+p_1\approx \begin{cases} (2\rho c)^{\frac12}\left\{ \dfrac{2}{1+\beta_-}+ \left[ \dfrac{\beta_-}{1+\beta_-}-\dfrac{\beta_+}{1+\beta_+} \right]\,[Fr(\omega x/c)+iFi(\omega x/c)] \right\}e^{-i\omega t}, & \text{for } x<0,\\[1.2em] (2\rho c)^{\frac12}\left\{ \dfrac{2}{1+\beta_+}+ \left[ \dfrac{\beta_+}{1+\beta_+}-\dfrac{\beta_-}{1+\beta_-} \right]\,[Fr(-\omega x/c)+iFi(-x/c)] \right\}e^{-i\omega t}, & \text{for } x>0. \end{cases} \tag{8.20} \]

The acoustic power scattered per unit area of the wall and per unit strength of the incident sound at a distance \(x\) from the boundary line is equal to \(\gamma |p_0+p_1|^2/2\rho c=\alpha(0)\). When both values \(|\beta_+|\) and \(|\beta_-|\) are less than unity \((|Z|>\rho c)\), a good approximation for \(\alpha(0)\) is:

\[ \alpha(0)\approx \begin{cases} \dfrac{4\gamma_-}{(1+\gamma_-)^2+\sigma_-^2}\, \left[1+(\gamma_- -\gamma_+)Fr(\omega x/c)+ (\sigma_- -\sigma_+)Fi(\omega x/c)\right], & \text{for } x<0,\\[1.2em] \dfrac{4\gamma_+}{(1+\gamma_+)^2+\sigma_+^2}\, \left[1+(\gamma_+ -\gamma_-)Fr(-\omega x/c)+ (\sigma_+ -\sigma_-)Fi(-\omega x/c)\right], & \text{for } x>0. \end{cases} \tag{8.21} \]

Here the first factor is exactly the expression for the wall coefficient for normal incidence \((\vartheta=0)\) at a large distance to the right or to the left of the boundary line \(x=0\). If this expression is averaged over all angles of incidence, then Sabine’s coefficient is obtained. The second and third terms in the square brackets represent corrections caused by diffraction phenomena near the boundary. The values of the functions \(Fr\) and \(Fi\) become small at distances from the boundary greater than approximately one wavelength. In order to determine the total absorption of the entire wall, this expression must be integrated. Then the first term will give the usual coefficient multiplied by the total area of the given material. The second and third terms will give edge corrections, which must be

add to the area of each material having a “free” boundary. By a “free” boundary is meant one which, over the greater part of its extent, is removed from the boundaries of the room. In particular, if a piece of material covers half of a wall, flush with the edges of the room, then the only free boundary will be the boundary passing through the middle of the wall.

Taking into account the expressions for the integrals \(Fr\) and \(Fi\), we obtain in the end the following approximate rules for applying edge corrections in rooms with disordered sound motion. The ordinary expression for the total absorption remains valid, but the Sabine coefficient, determined from equation (8.6) or from the nomogram in Fig. 31 for each piece of absorbing material, must be multiplied by the effective area of the piece. The effective area is equal to the actual area of the piece plus the length of its free boundary multiplied by

\[ (\lambda/4)(\sigma_+ - \sigma_-) = \frac{\lambda pc}{4} \left[ -\frac{X_+}{R_+^2+X_+^2} + \frac{X_-}{R_-^2+X_-^2} \right], \tag{8.22} \]

where \(\lambda\) denotes the wavelength of sound, \(\gamma_+ - i\sigma_+\) the acoustic conductance of the given piece, \(R_+\) and \(X_+\) the corresponding acoustic resistances, active and reactive, \(\gamma_- - i\sigma_-\) the conductance of the adjoining piece of material. It is assumed that \(|Z| \gg pc\).

It should be noted that for each boundary separating two kinds of material on one and the same wall, two edge corrections are obtained, equal to one another but opposite in sign. One of them refers to the material on one side of the boundary, the other to that on the other side of the boundary. It is assumed that the boundary does not pass too close to an edge of the room. The correction for the material with the greater conductance (\(\sigma\) larger) is positive, and conversely. The sum of the effective areas is exactly equal to the total area of the wall. Since \(\sigma=-pcX/(R^2+x^2)\) measures the compliance of the wall (the reciprocal of stiffness), one may say that the effective area of the more compliant material is obtained somewhat increased, and conversely. Equation (8.22) was derived for normal incidence. The corrections for oblique oscillations are somewhat different.

As an example, let us take a piece of material with active resistance \(R=1.5(pc)\) and reactive resistance \(-2(pc)\). The Sabine absorption coefficient is approximately equal to \(0.75\), and the specific reactive conductance is \(-0.33\). Suppose that a piece of this material of size \(3 \times 3\,m^2\) is mounted on a wall with impedance \((14+20i)(pc)\), \(\alpha_{\text{stat}}=0.15\), \(\sigma=0.03\), and that its edges lie far from the vertices of the room corners. At 500 hertz the effective area of the piece will be \(9.54\,m^2\). The diffraction effect has appeared in the fact that 4.5 sabins must be added to the absorption of the piece and 1 sabin subtracted from the absorption of the unequipped part of the wall. At lower frequencies, provided only that the impedances remain the same (which in fact does not occur), the relative magnitude of the correction increases.

55. List of adopted notation

Here are listed the most commonly used symbols, with a reference to the section or sections where their definitions are given. Included here are only those symbols that occur in more than one section.

Symbol	Meaning	Unit	Definition given in section
$a$	room absorption coefficient	$\mathrm{cm}^2$	32
$a_p$	normal absorption coefficient	$\mathrm{cm}^2$	33
$A$	amplitude $\psi$	$\mathrm{cm}^2/\mathrm{sec}$	14, 25
$A$	conditional conductance	$\mathrm{cm}^2/\mathrm{sec}$	37
$B$	coefficient in the expansion of $Q$	$\mathrm{sec}^{-1}$	29
$c$	speed of sound in air	$\mathrm{cm}/\mathrm{sec}$	10, 17
$e$	wave-type coefficient		27, 44
$G$	normalizing factor for $\psi$		27, 28
$i$	imaginary unit	$i=\sqrt{-1}$	10
$I$	sound intensity	$\mathrm{erg}/\mathrm{sec}\cdot\mathrm{cm}^2$	3.1
$j$	negative imaginary unit	$j=-i$	10
$J$	Bessel function
$k$	attenuation constant		6, 16, 27, 32
$k_{ON}$	damping constant of the natural oscillations		3, 7
$K$	constant in the reverberation formula		3.1, 4, 6
$K$	effective stiffness of the panel		21
$L$	dimensions of the room	$\mathrm{cm}$	14
$L$	thickness of the wall material	$\mathrm{cm}$	17
$m$	density coefficient of the material		17
$M$	effective mass of the panel		21
$n$	integer		14
$n$	real part of the refractive index		17
$N$	triple of whole numbers	$(n_x, n_y, n_z)$	27
$P$	porosity of the material (percent of volume occupied by pores)		17
$q$	imaginary part of the refractive index		17
$q$	distribution function of sound sources	$\mathrm{sec}^{-1}$	29
$Q$	amplitude of the source		29
$Q_0$	strength of a point source	$\mathrm{cm}^3/\mathrm{sec}$	29

Continuation

Symbol	Meaning	Unit	Definition given in section
$r$	coefficient of resistance of the material	g/cm·sec	17, 24
$R$	acoustic resistance	pressure/velocity	10
$S$	area of the wall surface	cm$^2$
$t$	time	sec
$T$	reverberation time	sec	3·1, 4
$\mathbf{u}$	acoustic velocity vector
$u$	$x$-component of acoustic velocity	cm/sec	14, 25
$U$	unit function		37
$v$	$y$-component of acoustic velocity	cm/sec	14
$V$	volume of the room	cm$^3$
$w$	$z$-component of acoustic velocity	cm/sec	14
$W$	density of sound energy	erg/cm$^3$	6
$x,\ y,\ z$	rectangular coordinates	cm
$X$	reactive acoustic resistance	pressure/velocity	10
$Z$	acoustic impedance	pressure/velocity	10, 27
$\alpha$	absorption coefficient or wall coefficient		4, 10
$\alpha_p$	normal coefficient of the wall		10, 32
$\alpha_t$	tangential coefficient of the wall		32
$\alpha_s$	additional coefficient of the wall		32
$\alpha_{\mathrm{stat}}$	Sabine or statistical absorption coefficient		4, 10, 52
$\beta$	specific acoustic conductance $\dfrac{\rho c}{Z}=\gamma-i\delta$		10, 27
$\gamma$	specific active acoustic conductance		10, 27
$\gamma$	parameter of resistance to blowing-through of the material		17

Continuation

Symbol	Meaning	Unit	Definition given in section
$\Gamma$	parameter of the material impedance		19
$\Gamma$	phase angle of the quantity $\xi$		37
$\Delta$	$=4\pi\mu x$		27
$\varepsilon$	normalizing factor	$\left(\dfrac{1}{8},\ \dfrac{1}{4},\ \dfrac{1}{2},\ 1\right)$	14, 28
$\zeta$	specific acoustic impedance $(Z/\rho c)$		10
$\eta$	frequency parameter $(2L/\lambda)$		25
$\vartheta$	angle of incidence		51
$\vartheta$	normalizing phase angle for $\psi$		27
$x$	attenuation parameter		25
$\lambda$	wavelength	cm
$\Lambda$	normalizing constant		28
$\mu$	wave-number parameter		25
$\nu$	frequency	hertz	14
$\xi$	characteristic number $(\omega+ik)$		27, 37
$\Xi$	coefficient of disorder		47
$\pi$	3.1416
$\Pi$	power of the sound source	erg/sec	6
$\rho$	mean density of air	g/cm$^3$	10
$\sigma$	specific reactive acoustic conductance		10, 27
$\sigma$	frequency parameter for the material		19
$\Sigma$	summation sign
$\tau$	parameters of the reflected wave		51
$\upsilon$	parameters of the reflected wave
$\Upsilon$	$=\tau-i\upsilon$		51
$\varphi$	phase angle for $Z$		10, 27
$\varphi_i$	angle of incidence		17
$\Phi$	phase angle of the velocity potential		25, 37
$\chi$	characteristic value of $\psi$		25
$\psi$	velocity potential	cm$^2$/sec	14, 25
$\Psi$	total velocity potential	cm$^2$/sec	29, 37
$\omega$	angular frequency		10, 27
$\omega_{ON}$	natural frequency		37
$\Omega$	$=\mu^2-x^2$		27

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