SOUND WAVES IN ROOMS

F. Morse and R. Bolt*)

CONTENTS

V. Steady sound regime in rectangular rooms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
25. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
26. Characteristic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
27. Resonance frequencies and damping coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 336
28. Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
29. Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30. Phenomena at low frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
31. Phenomena at high frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
32. Wall coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
33. Coherent and incoherent waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
34. Mean-square pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
35. Approximate formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
36. Measurements in the steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

VI. Transient regimes in rectangular rooms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
37. Operational calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
38. Pulse wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
39. Reverberation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
40. Details of the decay curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
41. Approximate formula for the decay curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

V. STEADY SOUND REGIME IN RECTANGULAR ROOMS

In rooms of rectangular shape with smooth walls, sound is not statistically diffuse, and in order to study it one must use wave acoustics. For this purpose the wave equations must be separated, and it is necessary to find solutions satisfying the boundary conditions. Thus, theoretical analysis is limited to forms of rooms corresponding to the eleven coordinate systems that permit separation of the wave equation E2. Fortunately, the most commonly encountered form of room—the rectangular parallelepiped—is the easiest for

) Continuation. See Usp. Fiz. Nauk, vol. XXXII, no. 2, p. 185. Rev. Mod. Phys. 16* No. 2 April (1944).

calculations. But even for this case the analysis is sufficiently complicated that the general physical picture is often obscured by complicating details. Nevertheless, the general picture cannot be considered complete without a study of these details.

25. Boundary conditions

Let us consider the acoustic properties of a rectangular room with dimensions \(L_x, L_z, L_y\), assuming that its walls*) are uniformly covered with acoustic material. First we shall examine the steady-state regime, assuming that the sound source has frequency \(\nu=\omega/2\pi\). Subsequently, methods of operational calculus will be applied, making it possible to describe the processes that become established on the basis of the results of the analysis of steady-state regimes. We shall soon see that the form of the solution for the steady-state regime depends on the sign of the frequency in the exponential factor \(e^{i\omega t}\) that is constantly present. As was mentioned earlier (Section 10), in analyzing steady-state regimes we choose the negative sign before \(\omega\). In applying operational calculus, however, both signs are needed. It can be shown that when the sign before \(\omega\) is changed, the fundamental functions and characteristic numbers are replaced by the complex quantities conjugate to them. For simplicity, let us agree to regard the fundamental functions, characteristic numbers, and other complex quantities as functions of the variable \(\omega\), which may take both positive and negative values, whereas the real and imaginary parts of these complex numbers are functions only of positive values of \(\omega\). For example, the complex impedance of a wall \(Z(\omega)\) will be different for positive and negative values of \(\omega\), while \(R(\omega)\) and \(X(\omega)\)—the active and reactive components of it—are defined only for positive values of \(\omega\). Therefore \(Z(\omega)=R+iX\), while \(Z(-\omega)=R-iX\). Thus the inertial resistance is always positive, and the elastic resistance is always negative.

Returning to the rectangular room, let us denote the specific acoustic impedance \(Z/\rho c\) for the wall \(x=0\) by \(\xi_{x1}\), for the wall \(x=L_x\) by \(\xi_{x2}\), etc. The velocity potential for standing waves in the room will have the general form

\[ \psi(\omega; x, y, z)=D(x)\cdot E(y)\cdot F(z)e^{i\omega t}, \]

where

\[ D(x)=\operatorname{ch}\left[(\pi i x/L_x)\chi_x(\omega)-\Phi_x\right] \tag{5.1} \]

and there are similar expressions for \(E\) and \(F\). The characteristic number \(\chi_x\) has a real part, which we shall call the wave number and denote by \(\mu_x\), and an imaginary part, which we shall call the parameter

*) The floor and ceiling of the room are also denoted here by the term “wall.”

attenuation and denote it by \(\chi_r\). In accordance with the foregoing one may write \(\chi(\omega)=\mu+i\chi\), \(\chi(-\omega)=\mu-i\chi\). The acoustic pressure \(p\) and the acoustic velocity \(\mathbf u\) are related to the velocity potential by the relations:

\[ p=\rho'(\partial\psi/\partial x),\quad \mathbf u=-\operatorname{grad}(\psi). \tag{5.2} \]

The boundary conditions determining the values of the constants \(\chi\), \(\mu\), and \(\Phi\) are the requirements that the ratio of the pressure to the normal velocity at each surface be equal to the acoustic impedance of this surface. Thus, for the exciting frequency \((\omega/2\pi)\) (\(\omega\) positive) at \(x=0\):

\[ i\omega\rho\psi=\rho c\zeta_{x1}(\partial\psi/\partial x)\quad\text{or}\quad \operatorname{cth}(\Phi_x)=-\,(\zeta_{x1}/\eta_x)\chi_x. \]

We shall call the parameters \(\eta_x=\omega L_x/\pi c=2L_x/\lambda\) and the corresponding \(\eta_y\) and \(\eta_z\) the frequency parameters. They determine the dimensions of the room in half-wavelengths. Introducing the boundary conditions for \(x=L_x\), we obtain the characteristic equation for \(\chi_x\):

\[ \pi i\chi_x+\operatorname{Arcth}\left[(\zeta_{x1}/\eta_x)\chi_x\right] +\operatorname{Arcth}\left[(\zeta_{x2}/\eta_x)\chi_x\right]=0 \tag{5.3} \]

and two similar equations for \(\chi_y\) and \(\chi_z\).

26. Characteristic numbers

The simplest case is that in which none of the impedances \(\zeta\) depends on the angle of incidence of the sound ray. Then the three characteristic equations separate, and \(\chi_x\) depends only on \((\zeta_{x1}/\eta_x)\) and \((\zeta_{x2}/\eta_x)\) and does not depend on \((\zeta_{y2}/\eta_y)\) and the other constants for the remaining two pairs of walls. On the basis of equation (3.1) (and recalling that here \(\omega\) is positive) we write the expression for the specific acoustic admittance of the walls as \(\beta=1/\zeta=(\rho c/|Z|)e^{-i\varphi}\). Then the characteristic equation may be expressed in the form:

\[ \pi i\chi+\operatorname{Arcth}(\chi/\beta_1\eta)+\operatorname{Arcth}(\chi/\beta_2\eta)=0. \tag{5.4} \]

Appending the signs \(x\), \(y\), and \(z\), we obtain a system of three characteristic equations.

This system is transcendental; it has an infinite number of roots. When \(\beta_1\) and \(\beta_2\) have positive real parts, there is at least one (and not more than two) such value of the expression \(\chi=\mu+i\chi\) for which \(\mu\) lies between zero and unity. There is one root between 1 and 2, one between 2 and 3, and so on. The different roots and the corresponding fundamental functions may, if necessary, be distinguished from one another by indices \(n\), where \(n=0\) for the value of \(\chi\) with the smallest value of \(\mu\), \(n=1\) for the next smallest value of \(\mu\), and so on.

In many cases the numbers \(\beta_1\eta\) and \(\beta_2\eta\) are much less than unity, so that it is possible to confine oneself to the first term of the expansion of the expression

(5.4) in the series:

\[ \left. \begin{aligned} \chi_0 &=(i\eta/\pi)^{1/2}(\beta_1+\beta_2)^{1/2} \left[ 1-\frac{i\pi}{6}\eta\,\frac{\beta_1^3+\beta_2^3}{(\beta_1+\beta_2)^2}+\cdots \right] \\ &\qquad (n=0;\ \beta_1\eta,\ \beta_2\eta<1);\\ \chi_n &= n+(i\eta/\pi n)(\beta_1+\beta_2)+(\eta^2/\pi^2 n^3)(\beta_1+\beta_2)^2+\cdots \\ &\qquad (n>0;\ \beta_1\eta,\ \beta_2\eta<n). \end{aligned} \right\} \tag{5.5} \]

These formulas cease to correspond to reality when one or both walls are made yielding.

However, in reverberation chambers where acoustic materials are investigated, the latter are placed on one of the walls (or on the floor). It is therefore important also to obtain the solution of equation (5.4) which corresponds to one of the \(\beta\)'s (for example \(\beta_1\)) being small, while the other \(\beta\)'s are not small. In this case \(\chi\) may be split into two terms:

\[ \chi \simeq \chi_2+(i\beta_1\eta/\pi\chi_2) \left[ 1+\frac{i\beta_2\eta/\pi}{\chi_2^2-\beta_2^2\eta^2} \right]^{-1}, \quad [\,(\beta_1\eta<\chi_2)\,]. \tag{5.6} \]

Here \(\chi_2\) is one of the roots of the transcendental equation

\[ (1/\chi_2)\operatorname{cth}(-\pi i\chi_2)=1/\beta_2\eta=2Z_2L/\rho c\lambda, \tag{5.7} \]

where \(Z_2\) denotes the impedance of the yielding wall. The value of \(\chi\) for the remaining four rigid walls of the reverberation chamber can be determined from equations (5.5).

In the literature \(^{\mathrm{H9,M13}}\) one may find graphs representing the real and imaginary parts of equation (5.7) as functions of the modulus and magnitude \(\beta_2\). These graphs correspond to the conformal transformation from \(\ln(\beta_2\eta)\) to \(\chi_2\), determined by equation (5.7). This transformation is multivalued; it has an innumerable number of sheets, corresponding to the infinitely large number of roots \(\chi_2\). The latter is evident from Fig. 15, which represents the transformation from \(\ln(\beta\eta)\) to \(\chi^2\). The transformations of \(\chi_2^2\) into \(\ln\left(\dfrac{1}{\beta_2\eta}\right)\) will be discussed in detail later. Graphs of several sheets are given in Figs. 16–19.

27. Resonance Frequencies and Damping Coefficients

The solution of the characteristic equations (5.4) makes it possible to compute the values of the characteristic numbers and to determine the fundamental functions of our boundary-value problem. For positive values of \(\omega\), these functions have the form:

\[ \psi_N(\omega;\ x,\ y,\ z)=D(x)E(y)F(z). \]

Here the notation is

\[ D_{n_x}(x)=\operatorname{ch}\{(\pi i x/L_x)\chi_{x n_x}+\operatorname{Arcth}(\chi_{x n_x}/\beta_{x1}\eta_x)\}; \tag{5.8*} \]

\(\chi_{x n_x}(\omega)\) denotes the \(n_x\)-th root of equation (5.4) for the wall \(x\); the letter \(N\) is put in place of the triple of numbers \(n_x, n_y, n_z\). These fundamental functions satisfy the differential equation:

\[ \nabla^2 \psi_N(\omega) + +(1/c^2)\,[\omega_N(\omega)+ + i k_N(\omega)]^2 \psi_N(\omega)=0, \]

corresponding to the characteristic value:

\[ [\xi_N(\omega)]^2 = [\omega_N+i k_N]^2 = \]
\[ = (\pi c)^2\{[\chi_{x n_x}(\omega)/L_x]^2+ \]
\[ +[\chi_{y n_y}(\omega)/L_y]^2+ \]
\[ +[\chi_{z n_z}(\omega)/L_z]^2\}. \tag{5.9} \]

We shall call the quantity \(\omega_N\) the resonance frequency of the standing wave \(\psi_N\), and \(k_N\) the attenuation constant.

In accordance with our assumptions, the resonance frequency and the attenuation coefficient are defined only for positive values of \(\omega\).

In most cases of practical interest, the attenuation coefficient \(k_N\) is much smaller than the resonance frequency \(\omega_N\). Therefore the following approximate equations are valid:

Fig. 15. Conformal transformation of the wall impedance parameter \(r e^{i\varphi}\) to the squares of characteristic values \(\chi^2=(\mu+i\kappa)^2\). Branch points are shown by black circles; branch-cut lines by hatched bands.

\[ \omega_N(\omega) \simeq \pi c \left[ \frac{\mu_x^2-\chi_x^2}{L_x^2} + \frac{\mu_y^2-\chi_y^2}{L_y^2} + \frac{\mu_z^2-\chi_z^2}{L_z^2} \right]^{\frac12}, \]
\[ k_N(\omega) \simeq (c/4) \left[ \frac{4\pi\mu_x\kappa_x}{\eta_x L_x} + \frac{4\pi\mu_y\kappa_y}{\eta_y L_y} + \frac{4\pi\mu_z\kappa_z}{\eta_z L_z} \right]. \tag{5.10} \]

It will be shown below that experimentally \(\omega\) and \(k\) are usually determined more easily than the wave parameters \(\mu\) and \(\chi\). Therefore, for our calculations it is more convenient to determine \(\mu^2-\chi^2\) and \(2\mu\chi\), rather than the values of \(\mu\) and \(\chi\) themselves. These values represent the real and imaginary parts of the quantity \(\chi^2\); therefore the greatest benefit may be obtained from the conformal transformation relating \(\chi^2\) and \(\ln(1/\beta\eta)\), according to equations (5.4) or (5.7).

The expansion (5.5) makes it possible for us to compute \(\omega_N\) and \(k_N\), if all the walls are rigid. The result is most easily expressed in terms of the real and imaginary parts of the specific conductivity \(\beta^2\). Let us denote the specific acoustic active conductivity by \(\gamma\), and the specific acoustic reactive conductivity by \(\sigma\), so that

\[ \beta(\omega)=\gamma+i\sigma, \]

\[ \beta(-\omega)=\gamma-i\sigma. \]

Fig. 16. Conformal transformation
\(\chi^2=\Omega+i\Delta/2\pi\) into \(\ln \zeta/\eta\) for the first sheet shown in Fig. 15. The branch point has coordinates \(|\zeta/\eta|=1.19;\ \psi=-38^\circ.7\).

From equations (5.5) we obtain:

\[ \begin{aligned} \mu_n^2-\chi_n^2 &\simeq \begin{cases} -(\eta/\pi)(\sigma_1+\sigma_2), & n=0,\\ n^2-(2n/\pi)(\sigma_1+\sigma_2), & n>0, \end{cases} \\[4pt] 4\pi\mu_n\chi_n/\eta &\simeq \begin{cases} 2(\gamma_1+\gamma_2), & n=0,\\ 4(\gamma_1+\gamma_2), & n>0, \end{cases} \end{aligned} \tag{5.11} \]

\[ \gamma=(\rho c/|Z|)\cos\varphi,\qquad \sigma=-(\rho c/|Z|)\sin\varphi \quad \text{for } |\zeta|\gg \eta/(n+1). \]

Therefore the approximate expression for the attenuation coefficient for very rigid walls is as follows:

\[ k_N \simeq (c/8V)\,[8e_{nx}(\gamma_{x1}+\gamma_{x2})S_x -8e_{ny}(\gamma_{y1}+\gamma_{y2})S_y+ 8e_{nz}(\gamma_{z1}+\gamma_{z2})S_z]. \tag{5.12} \]

Here \(V\) denotes the volume of the room, \(S_x\) the area of the walls with coordinates \(x=0\) and \(x=L_x\), etc.; the number \(e_n\) is equal to unity for \(n>0\) and to \(1/2\) for \(n=0\). Since \(e_n\) depends on the form of the standing wave, and not on the walls, we shall call it a coefficient of wave type.

In the case considered, the effects from each wall are additive, and the formula for the attenuation coefficient becomes similar to Sabine’s formula (2.2) for an ergodic sound process. This similarity is nevertheless not complete, since Sabine’s formula assumes that all waves have the same coefficient \(k\), whereas equation (5.12) distinguishes oblique, tangential, and axial waves by means of the multiplier \(e_n\). Axial oscillations, to which correspond values of \(e_n\) equal to \(1/2\), decay much more slowly than oblique ones, to which correspond all three \(e_n\)’s equal to unity.

Fig. 17. Conformal transformation \(\chi_2^2\) into \(\ln(\zeta/\eta)\) for the second sheet. The additional branch point has coordinates \((|\xi|/\eta)=0.56;\ \varphi=21^\circ.1\).

For the case of a room in which one wall is yielding, we may use equation (5.6). First we determine the quantities \(G\) and \(\vartheta\) by means of the equation

\[ G_2 \exp(i\vartheta_2)= \left\{ 1+\frac{i\beta_2\eta/\pi}{\chi_2^2-\beta_2^2\eta^2} \right\}^{-1}, \tag{5.13} \]

where \(\chi_2\) is determined by equation (5.7). For small values of \(\beta_2\), \(\chi_2\) will also be small, and \(G\) is equal to unity if \(n>0\), and equal to \(1/2\) if \(n=0\). In other words, \(G\) is approximately expressed by the value of \(e_n\) in equation (5.12). The values of \(G\) and \(\vartheta\) for large values of \(\beta/\eta\) may be determined from the diagrams of Figs. 20 and 21, where the first two sheets of the transformation from \(\beta_2\eta\) to \(\chi_2\) are shown. Then the term ve-

quantities \(\omega\) and \(k\), due to the yielding wall and the wall opposite to it, are equal to:

\[ \begin{aligned} \mu^2-\chi^2&=\Omega_2+\left(2\rho c\eta G_2/\pi |Z_1|\right)\sin(\psi_1-\vartheta_2),\\ 4\pi\mu\chi/\eta&=\Delta_2/\eta+\left(4\rho c G_2/|Z_1|\right)\cos(\psi_1-\vartheta_2),\\ \chi_2^2&=\Omega_2+\left(i\Delta_2/2\pi\right). \end{aligned} \tag{5.14} \]

The values of \(\Omega\) and \(\Delta\) can be read from the diagrams in Figs. 16–19 as functions of the impedance and the phase angle of the yielding wall, and

Fig. 18. Conformal transformation of \(\gamma^2\) into \(\ln(\zeta/\eta)\) for the third sheet. The additional branch point has coordinates \((|\zeta|/\eta)=3.55,\ \varphi=-15^\circ.0\).

the values of \(G\) and \(\vartheta\) can be read from the diagrams in Figs. 20 and 21. For the subsequent sheets of the transformation and for values of \(Z_2/\rho c\eta\) larger than those shown in the diagrams, one may, with sufficient accuracy, set \(G=1,\ \vartheta=0\). For the first sheet the limiting value will be \(G=1/2\).

The quantities \(\Omega\) and \(\Delta/2\pi\) are the real and imaginary parts of the complex number \(\chi^2\), related to \(\beta\eta\) by the relation

\[ (1/\chi)\operatorname{cth}(-\pi i\chi)=1/\beta\eta . \]

Therefore, according to equation (5.7), the transformation from the real and imaginary parts of the expression \(\ln(1/\beta\eta)\) to \(\Omega\) or \((\Delta/2\pi)\) is a conformal mapping; it is multivalued: to a prescribed value of \(\beta\eta\) there corresponds an infinite set of values of \(\Omega\) and \(\Delta\). Each sheet

Fig. 19. Approximate transformation of \(\chi^2\) into \(\ln(\zeta/\eta)\) for the \((n+1)\)-th sheet. The regions adjacent to the branch points are not shown, since the adopted approximation is inapplicable there.

has its own branch point and lines of cut connecting this sheet with the adjacent sheets above and below. In the diagrams of Figs. 16–19 the branch points are shown by heavy dots, and the lines of cut are indicated by white gaps. Fig. 15 represents the inverse transformation, with the lines of cut marked by double hatched bands. Another representation of these transformations was given earlier in H9, M13.

Equations (5.14) show that the absorption of the wall opposite the compliant wall is non-additive. The term ve-

values of \(\omega\) and \(k\), due to this wall (wall number 1 in equations 5.14), has a factor \(G_2\) and a phase angle \(\vartheta_2\), depending on the impedance of the compliant wall. This is due to the fact that the presence of a compliant wall substantially distorts the form of the standing wave in the room. In some cases (\(n = 0\), or positive \(\varphi_2\) when \(n > 0\)) the pressure at the compliant wall has a larger amplitude,

Fig. 20. Conformal transformation of the normalizing function \(G e^{i\vartheta}\) into the wall-impedance function \(\ln(\zeta/\eta)\) for the first sheet.

than at the opposite rigid wall. Thus, the relative significance of the rigid wall is reduced. In other cases (when \(n > 0\) and for negative values of \(\varphi\) for the compliant wall) the pressure amplitude at the compliant wall is reduced in comparison with the pressure amplitude at the rigid wall, so that \(G_2\) becomes greater than unity. In the most extreme case—when \(n = 0\), \(|\zeta_1|/\eta\) is less than unity, and \(\varphi_2\) is negative—\(G_2\) becomes very small, so that the rigid wall practically does not absorb sound. In these cases the pressure amplitude decreases according to an exponential law as one moves away from the compliant wall; the amplitude at the rigid wall is practically negligible, so that the energy capable of being absorbed by wall number 1 is quite insignificant.

28. Eigenfunctions

The preceding considerations cover the properties of the characteristic numbers determined by equation (5.9). We must now complete the consideration of the eigenfunctions \(\psi_N\). These functions are orthogonal, i.e. the integral of the product \(\psi_N(\omega)\psi_{N'}(\omega)\), taken over the entire volume of the room, is zero if and only if the triple of numbers \(N\) does not coincide with the triple \(N'\).

Fig. 21. Conformal mapping from \(Ge^{i\vartheta}\) to \(\ln(\zeta/\eta)\) for the second sheet.

It should be noted that we multiply \(\psi_N\) by \(\psi_{N'}\), and not by the function complex-conjugate to \(\psi_{N'}\), since the latter would correspond to the eigenfunction for \(-\omega\). The normalizing factors are determined as follows:

\[ \left. \begin{aligned} \Lambda_N(\omega) &= \iiint \psi_N^2(\omega)\,dV = \Lambda_{n_x}\Lambda_{n_y}\Lambda_{n_z}, \\[6pt] \Lambda_n(\omega) &= \frac{L}{2} \left\{ 1+ \frac{i\beta_1\eta/\pi}{\chi_n(\omega)-\beta_1^2\eta^2} + \frac{i\beta_2\eta/\pi}{\chi_n^2(\omega)-\beta_2^2\eta^2} \right\}. \end{aligned} \right\} \tag{5.15} \]

where in the second equation the signs \(x, y\), and \(z\) must be appended so as to form the three factors for \(\Lambda_N\). If both opposite walls are rigid, then expansion in a series gives:

\[ \Lambda_n \simeq \begin{cases} L\left[1-\left(\pi i \eta/3\right)\left(\beta_1^3+\beta_2^3\right)(\beta_1+\beta_2)^{-2}\right], & n=0;\ \beta_1\eta,\ \beta_2\eta<1;\\[6pt] L/2\left[1+\left(i\eta/\pi n^2\right)(\beta_1+\beta_2)\right], & n>0;\ \beta_1\eta,\ \beta_2\eta<n. \end{cases} \tag{5.16} \]

If all the walls are rigid, then the normalizing factor has the more usual value:

\[ \Lambda_N \to (V\varepsilon_N), \]

\[ \varepsilon_N= \frac{\frac{1}{8}}{e_{n_x}e_{n_y}e_{n_z}} = \begin{cases} 1, & n_x=n_y=n_z=0,\\[4pt] \dfrac{1}{2}, & \text{only two } n \text{ are equal to zero},\\[6pt] \dfrac{1}{4}, & \text{only one } n \text{ is equal to zero},\\[6pt] \dfrac{1}{8}, & \text{not one } n \text{ is equal to zero}. \end{cases} \tag{5.17} \]

Here \(V=L_xL_yL_z\) is the volume of the room.

In the case when one of the walls (for example, wall number 2) of a parallel pair is yielding, while the opposite one is rigid, in determining the corresponding factor in (5.15) we may use equations (5.13) and (5.14):

\[ \Lambda_n \simeq (L/2)\left[(1/G_2)\exp(-i\vartheta_2)+\left(i\beta_1\eta/\pi\chi_2^2\right)\right]. \]

In exactly the same way, for the eigenfunctions themselves, the \(x\)-th factor in the expression for \(\psi_N\) takes a simpler form in comparison with equation (5.8) in the case when the wall with coordinate \(x=0\) is rigid (\(\beta_{x1}\eta_x\) small):

\[ D_{nx}\simeq \operatorname{ch}\left[(\pi i x/L_x)\chi_x+\left(\eta_x\beta_{x1}/\chi_x\right)\right]. \tag{5.18} \]

If the opposite wall is also rigid, then the following approximate formulas are valid:

\[ D_{xn}\simeq \begin{cases} \operatorname{ch}\left\{ \left[\dfrac{\pi\eta_x}{i(\beta_{x1}+\beta_{x2})}\right]^{1/2} \cdot \dfrac{1}{L_x} \left[\beta_{x1}(x-L_x)+\beta_{x2}x\right] \right\}, & (n_x=0),\\[10pt] \operatorname{ch}\left\{ (\pi i n_x/L_x) -\dfrac{\eta_x}{nL_x} \left[\beta_{x1}(x-L_x)+\beta_{x2}x\right] \right\}, & (n_x>0). \end{cases} \tag{5.19} \]

This completes the analysis of standing waves in a rectangular room with a uniform covering of the walls by such a material whose impedance does not depend on the angle of incidence \(\varphi_i\). The exact solution for the case in which the impedance depends on \(\varphi_i\) lies beyond the scope of the present survey. When all the walls are rigid, the angle of incidence of the sound ray, for example on the \(x\)-wall, is approximately equal to:

\[ \arccos\left\{(n_x/L_x)\left[(n_x/L_x)^2+(n_y/L_y)^2+(n_z/L_z)^2\right]^{-1/2}\right\}. \]

For this approximate value of the angle, the impedance of the \(x\)-walls may be determined, as well as the values of \(\mu_x\), \(x_x\), etc. If the accuracy of this result is insufficient, a more exact value of the angle of incidence can be obtained by substituting the obtained values of \(\mu_x\), \(x_x\) into the equation

\[ \cos\varphi_i \simeq (\pi c/\omega_N L_x)(\mu_x^2-x_x^2)^{1/2} \]

and then recalculating the impedance values and the corresponding resonance frequency and damping coefficient.

29. Steady-state regime

Let us apply the results obtained to the study of the steady-state regime in rectangular rooms. Let some distribution of sound sources be specified, i.e. suppose that from one cubic centimeter about the point \(x,y,z\) at time \(t\) there flows out a volume of air \(q(x,y,z,t)\ \mathrm{cm}^3\) per second. In this section we shall consider only point sources, although a field of sources of arbitrary form can be investigated by superposition of point sources of different strengths*). The equation for the velocity potential in the presence of a given distribution of sources is

\[ \nabla^2\Psi-(1/c^2)(\partial^2\Psi/\partial t^2)=-q(x,y,z,t). \]

If the source is simply harmonic,

\[ q=Q(x,y,z)e^{-i\omega t}, \]

then \(\Psi\) and \(Q\) can be expanded in series in the eigenfunctions considered above. These series must satisfy the boundary conditions for the frequency \((-\omega/2\pi)\). Substituting the series into the equation for \(\Psi\) and using the relation connecting the fundamental

) And phases. (Editor’s note.*)

functions \(\Psi_N(-\omega)\) with characteristic values \(\omega_N-ik_N\), we obtain the following series:

\[ \left. \begin{aligned} \Psi &=-c^2 \sum_N \frac{B_N \Psi_N(-\omega; x,y,z)} {\omega^2-(\omega_N-ik_N)^2}\, e^{-i\omega t},\\[6pt] B_N&=\frac{1}{\Lambda_N(-\omega)} \iiint Q\,\Psi_N(-\omega)\,dv . \end{aligned} \right\} \tag{5.20} \]

The pressure at the point \((x,y,z)\) in the steady-state regime is expressed as: \(-i\rho\omega\Psi\).

If the reproducer is regarded as a simple point source of strength \(Q_0\) at the point \((x_0,y_0,z_0)\), then

\[ B_N=Q_0\Psi_N(-\omega;x_0,y_0,z_0)/\Lambda_N(-\omega). \tag{5.21} \]

This means that the relative value of the \(N\)-th wave in the series of oscillations excited by a point source is proportional to the amplitude of the wave at the location of the source. In order to eliminate some proper oscillation in the steady-state regime, the sound source should be placed at the point where the amplitude of the undesired wave is zero*).

Of greater importance, however, is the resonant denominator in each term of the series. It is small when the exciting frequency is close to the resonant frequency \((\omega_N/2\pi)\). Then the corresponding term becomes large. The resonance peaks for each standing wave are sufficiently narrow. The half-width of the resonance curve (i.e., the amount by which the excitation frequency must differ from the resonant frequency in order that the mean-square value of the pressure be reduced to one half of its resonant value) is equal to \(k_N/2\pi\). For low frequencies, i.e., when the wavelength is not small in comparison with the dimensions of the room, the distance between neighboring resonant frequencies is on average greater than this half-width. The strength of the sound at these frequencies varies strongly with frequency. It is small when the exciting frequency is not equal to any of the resonant frequencies, and very large at resonance. In this case we have the possibility of studying each standing wave separately by exciting oscillations of the corresponding resonant frequency.

The limiting frequency below which individual standing waves can be excited independently of the others is determined from equation (3.4), which gives the mean value of the number of resonant frequencies less than \(\nu\). Differentiating equation (3.4) and putting \(dn=1\), one can find the corresponding value \(d\nu\). It determines

*) If such a place exists at all. It should have been stated thus: “in order to weaken as much as possible some oscillation..., where the amplitude is minimal.”

(Editor’s note.)

average frequency interval in the range between resonances. Retaining only the first term in the expansions, we shall see that the mean value of this number is equal to \(c^3/4\pi V\nu^2\). In order that individual resonance peaks can be separated, this number must be greater than \(k_N/2\pi\). Thus, individual standing waves can be excited in a room at such frequencies \(\nu\) as satisfy the inequality

\[ \nu < \left(c^3/2Vk_N\right)^{1/2}. \tag{5.22} \]

Here \(V\) denotes the volume of the room, and \(k_N\) is determined by equation (5.10).

30. Phenomena at low frequencies

Up to now we have set ourselves the task of calculating the acoustic properties of a room when the impedances of the walls are given. But there also exists the inverse problem, of no less importance: to find the impedance of the material covering one wall of a room by measuring the acoustic properties of this room. The considerations just set forth give two different methods for solving this problem.

Both methods require small chambers so that the resonance frequencies are separated from one another in the required range, and the corresponding standing waves can be excited separately. Chambers with a greatest dimension of 3 feet (0.9 m) and the remaining dimensions in the ratios \(1.5:2.5:3.5\) have proved most suitable for such measurements \(^{\mathrm{B5},\mathrm{B9},\mathrm{H7}}\). The sound source may be a tube passed through a channel in one of the walls and acting as a point source. For this case the coefficients \(B_N\) in equation (5.20) are determined by equation (5.21). With a proper placement of the end of the tube on the surface of the wall, it is sometimes possible to determine separately the intensity even of such standing waves whose resonance peaks partially overlap. The chamber must be built of a very heavy, rigid, impermeable material, so that the value of \(k_N\) due to all the walls not covered by the absorber is negligibly small.

The first of the methods mentioned assumes measurement of the pressure in the chamber as a function of frequency \(^{\mathrm{B5},\mathrm{H7}}\). By measuring the pressure near the resonance frequencies, one can calculate the values of \(\omega\) and \(k_N\). Comparing the values of these quantities for a chamber with acoustically untreated walls and for a chamber with the material under investigation on one wall, one can determine from equation (5.14) the values of \(\Omega\) and \(\Delta\) corresponding to the absorbing wall. Using the graphs of Figs. 16–19, one can determine the specific impedance of the absorbing material at various resonance frequencies.

In practice it is much more difficult to determine \(\Omega\) than \(\Delta\). In the first case it is necessary to determine the change in frequency

of the resonance caused by the introduction of the material under investigation*), whereas for determining $\Delta$ only a comparison of the half-widths of the resonance curves in the absence and in the presence of the material under investigation is needed. If only the value of $\Delta$ has been determined, while the value of $\Omega$ is unknown, then the quantity $\zeta$ cannot be determined. However, if the values of $\Delta$ are known for two or several different standing waves, close in frequency but having different values of $n_x$, then it is sometimes possible to determine also the values of $\zeta$.

Fig. 22. Mean-square pressure in decibels as a function of one coordinate.

those having different values of $n_x$, then it is sometimes possible to determine also the values of $\zeta$. In particular, if the values of $\Delta$ are known for a grazing sound ray ($n_x = 0$) and for a ray with almost normal incidence ($n_n \gg 1$), then, by comparing Figs. 16 and 17, one can find which value of $\zeta$ gives this pair of values of $\Delta$. However, this method does not always give an unambiguous result B5, H7, and small errors in measurements of $k_N$ may lead to very large errors in the determined values of $\zeta$. In general, the method of frequency variation is more convenient when both quantities $\Omega$ and $\Delta$ can be measured.

The second method is based on measuring the pressure as a function of the spatial coordinates. To measure the pressure amplitude, a movable microphone is used, placed at various distances from the absorbing wall. If the absorbing material being investigated

* Shifts of the resonant frequency were experimentally observed B8, K5, but it was not possible to establish a quantitative relation with the impedance of the wall. (Editor’s note.)

material is located on the wall \(x=L_x\), and the wall \(x=0\) is absolutely rigid, then the dependence of the mean-square pressure value on \(x\) is determined by the factor

\[ (p^2)_{\mathrm{av}}=\frac{1}{2}\operatorname{ch}(2\pi \varkappa_x x/L_x)+\frac{1}{2}\cos(2\pi\mu_x x/L_x). \tag{5.23} \]

The family of curves representing ten times the logarithm of this factor (\(p^2\) in decibels) as a function of \(\mu x/L\) for various values of the parameter \(\varkappa/\mu\) is presented in Fig. 22. By comparing the scale along the abscissa axis of the experimental curves with the scale along the abscissa axis of this figure, one can determine \(\mu_x\), after which comparison of the shape of the experimental curve with the shape of one of the theoretical curves makes it possible to determine \(\varkappa_x\). When these two parameters have been determined, the values \(\Omega_x^2=\mu_x^2-\varkappa_x^2\) and \(\Delta x=4\pi\mu_x\varkappa_x\) can be calculated. Further, from the graphs of Figs. 16–19 one can find the values of the specific impedance \(\zeta_{x2}\). If the wall \(x=0\) is not perfectly rigid, then the zero point in equation (5.23) will be displaced, in comparison with its position for an absolutely rigid wall, by an amount determined from equation (5.18). In exactly the same way, according to equation (5.14), in order to find \(\Omega_x\) and \(\Delta_x\) for the subsequent calculation of \(\zeta_{x2}\), it is necessary to subtract the correction term from \(\mu_x^2-\varkappa_x^2\) and \(4\pi\mu_x\varkappa_x\). The magnitude of this correction can be determined by measuring the pressure distribution in the room before installing absorbing material on one of its walls.

The published data on the application of these methods are still not very numerous, but they have already made it possible to solve some questions and have shown that such an approximate method can provide information which would be difficult to obtain by other means. In particular, the study of individual natural oscillations in rectangular rooms can give impedance values as a function of the angle of incidence and of the conditions under which the specimens are fastened. For the result of the measurements to be complete, it is necessary that the specimens have sufficiently large dimensions.

Hunt \(^{7}\), using the first of the methods described, measured resonance peaks and determined the absorbing properties of acoustic materials for various types of sound oscillations; however, this work was carried out before Beranek \(^{83}\) and others showed by their measurements the important significance of the imaginary part of the impedance. Therefore, the reactive component was not taken into account in it: only \(\Delta\) was determined, while the quantity \(\Omega\) was ignored. The author used movable plates 5 cm thick, from which models of rooms of various sizes were made up. In this case the resonance peaks varied within the range from 250 to 1500 hertz. The sound source had a large impedance in order to reduce the feedback effect on it of the standing wave, since, according to equation (5.20), the impedance characterizing this feedback effect undergoes considerable changes near the resonance frequency. The width of the resonance peak was measured between the points at which \(p^2\) is equal to one half of the maximum resonance value. This width

(in hertz) is equal to twice the value of the damping coefficient \(k_N\) in equation (5.12).

By measuring the given standing wave in the presence and in the absence of an absorbing material, it was possible to determine the value of \(\gamma\) for the material, if all the other values of \(\gamma\) in equation (5.12) are equal to one another and have the value corresponding to rigid walls. At that time the wave acoustics of rooms had not been sufficiently developed; therefore the measurement results were interpreted by the method of free plane waves, set forth below in Chapter VIII. Hunt calculated \(\alpha(\vartheta)\) from an equation similar to equation (8.5) (under the assumption that the acoustic impedance does not depend on the angle of incidence of the sound ray), by comparing impedance measurements at one and the same point. It was found that the values of \(\alpha(\vartheta)\) measured by this method for different angles of incidence, generally speaking, reproduce this dependence well, except that for the grazing direction of the sound ray the absorption does not tend to zero, as is predicted by the theory of free waves. From the point of view of the analysis given in the present chapter, this is understandable. In a closed room, vibrations “grazing” along the wall (when one or two of \(n\) are equal to zero) are to a certain extent absorbed by this wall.

Batt\(^{85}\) applied a variant of the first of the above methods. He determined \(\Delta\) for two or several vibrations having almost identical frequencies but different angles of incidence. As has already been pointed out, this method does not always give an unambiguous answer. However, Batt showed that if one determines a series of values of \(\Delta\), constructs from such pairs of vibrations the dependence of these values on frequency, and smooths the resulting curve, then the true impedance curve can be obtained with sufficient accuracy. The values found in this way agree exactly with the values obtained by the hyperbolic-tangent method (Chapter IV). Instead of measuring the widths of the peaks, he used two other methods for determining \(\Delta\): (a) measurement of the damping index, and (b) pressure measurements for individual normal vibrations. The first method proved very accurate. It gave values of \(k_N\) with an error not exceeding a few percent. Measurements of the pressure peaks were much less accurate: unexplained deviations were found, possibly caused by changes in the conditions of fastening the specimens.

The second of the methods indicated—measurement of the pressure distribution in space—has also already been applied. A chamber with massive walls, similar to Batt’s chamber, having dimensions of about \(0.6 \times 0.9 \times 1.2\) m, was equipped with a miniature microphone capable of moving in all directions; this displacement was controlled from outside the chamber. Pressure curves were obtained both for bare walls and for walls covered in various ways with absorbing materials. These curves directly-

...are readily and easily compared with the curves of Fig. 22. The variation of impedance with the angle of incidence of the ray was investigated for porous felt absorbers. The impedance of materials of this type was first studied for normal incidence [^9]. Theoretically, this question was discussed above in Section 21. The difference expressed by equations (4.13) and (4.14) was clearly demonstrated, i.e. the dependence of the impedance on the angle of incidence in the case where sound vibrations can propagate in the space behind the absorber parallel to its surface, and the independence of the impedance from the angle when this space is divided by a lattice frame. This method will be a useful tool in the study of new sound-absorbing devices.

31. Phenomena at High Frequencies

Individual standing waves can be investigated independently of one another in small chambers at ordinary frequencies and in large rooms at very low frequencies. In rooms of ordinary size, in the “normal” range (from 200 to 5000 cycles), the resonance frequencies lie so close to one another that a single source can excite several standing waves with almost identical amplitudes. Therefore these oscillations cannot be studied separately. In this case it is very difficult to determine the value of the impedance of the walls from measurements of the field in the steady-state regime, although it is possible to predict the acoustic behavior of a room if the impedances of the walls are given.

Even if only a few standing waves are excited simultaneously, it is impossible to draw general conclusions about the acoustic pressure in a room. The distribution of sound pressure can be determined only by a detailed calculation using equation (5.20). The acoustic pressure in this range depends on frequency in a rather complicated manner, and the spatial pressure distribution is very far from uniform. However, when the natural frequencies are so closely spaced that within the half-width of the resonance curve there are one hundred or more different frequencies of standing waves, then statistical methods may be used for the calculation. By arguments similar to those used in deriving equation (5.22), one can verify that the statistical method may be applied in the frequency range determined by the inequality:

$$ \nu > \left(50c^3/Vk_N\right)^{\frac{1}{2}}. \tag{5.24} $$

In this case the number of standing waves is sufficient for the summation in expression (5.20) to be replaced by an integral. However, before applying the analysis given in Chapter III in order to carry out this transformation, we must introduce certain approximations, and also subdivide our standing waves into oblique, tangential, and axial ones, as was set forth in the named chapter.

First of all, at these higher frequencies the wave numbers $\mu$ are usually much larger than the attenuation parameter $\chi$, so that expression (5.17) may be used as the normalizing factor. This gives a good approximation for oblique vibrations, which constitute the majority; it is not valid for tangential and axial vibrations if one or more walls are yielding. Next we note that, although the introduction of absorption has changed the resonant frequencies in comparison with the simple expressions for them given in Chapter III, nevertheless the mean density of resonant frequencies in this range is the same as that given by equation (3.5). Therefore, in a first approximation, the mean number of vibrations of the three different types having resonant frequencies between $\omega$ and $\omega + d\omega$ is as follows:

\[ \begin{aligned} dn_p &\simeq (V\omega^2/2\pi^2 c^3)\,d\omega &&\text{for oblique vibrations;}\\ dn_t &\simeq (L_y L_z\,\omega/2\pi c^2)\,d\omega &&\text{for } yz\text{-tangential;}\\ dn_a &\simeq (L_x/\pi c)\,d\omega &&\text{for } x\text{-axial.} \end{aligned} \tag{5.25} \]

We may regard $\omega_N$ in equation (5.20) as the magnitude of a vector in frequency space. The summation is carried out over the entire first octant of this space, and the “density” of the vectors is determined by equations (5.25). This summation may be replaced by integration over all directions and over all lengths of the vector within the first octant. It is first necessary also to know the dependence of the attenuation coefficient $k_N$ on the frequency vector.

By virtue of equation (5.12), the attenuation coefficients for a room with sufficiently rigid walls depend on the magnitude of the vector $\omega_N$, but they do not depend on its direction (on the relative magnitudes $n_x$, $n_y$, and $n_z$), provided that oblique vibrations are meant, i.e. provided that this vector is parallel neither to a face nor to an edge of the room. All tangential $xy$-vibrations, whose vectors are parallel*) to the plane $xy$, have the same value of the attenuation coefficient, differing from that for oblique vibrations. Even in the case when one or more walls of the room are yielding, the majority of oblique vibrations have nearly equal attenuation coefficients. In particular, when the wall $x=L_x$ is moderately yielding, equations (5.14) and Figs. 16–19 show that the values $\Delta_2$ for $n_x=0$ or $n_x=1$ differ noticeably from the remaining roots, but when the value of $n_x$ is increased, the admissible values of $\Delta_2$ rapidly approach the limiting value determined by equation (5.11). In most practically interesting cases only the first two roots differ noticeably from the limiting value. Thus, even in this

*) In reality the vector cannot be exactly parallel to the wall, because in an absorbing room not one of the three components of the vector $\omega_N$, $(\pi c/L_x)(\mu_x^2-\chi_x^2)^{1/2}$, etc., is exactly equal to zero. However, for our statistical consideration we may regard the vector $\omega_N$ for tangential vibrations as parallel to the corresponding walls.

case for all oblique oscillations (except only those for which the vector \(\omega_N\) is almost parallel to the yielding wall) the same value \(k_N\) is obtained. These limiting cases may be called “almost tangential” oscillations.

32. Wall coefficients

We shall introduce one more modification in order, as far as possible, to bring our equations in form closer to Sabine’s equations. Let us define, in §2, H8, for the wall \(x\), the wall coefficient by means of the equation

\[ a_{n x1x}+a_{n x2x}=8\pi\mu_{n x}\chi_{n x}/\eta_x \]

and by two similar equations—the coefficients for the walls \(y\) and \(z\). These coefficients play in wave acoustics a role similar to that of absorption coefficients in geometrical acoustics. They are not equal to the ratio of the absorbed sound energy to the incident energy, but are approximately proportional to the ratio of the absorbed energy to the mean energy in the room. In some cases the energy is distributed far from uniformly over the whole room, and the sound intensity at some walls is greater than the mean sound intensity. Therefore the wall coefficients may in some cases be greater than unity.

Thus, to each wall of the room we can assign a series of coefficients \(\alpha\), depending, generally speaking, on the shape and dimensions of the room, on the standing wave under consideration, on the acoustic impedance of the wall, and in some cases even on the impedance of the opposite wall. If the wall is rigid (\(|Z|/\eta\) greater than approximately \(\rho c\)) and if the frequency vector \(\omega_N\) of the standing wave under consideration is not parallel to the wall \((n>0)\), then the wall coefficient has the value:

\[ \alpha_p=8\gamma=(8\rho c/|Z|)\cos\varphi . \tag{5.26} \]

Here \(Z\) and \(\varphi\) are the acoustic impedance of the wall under consideration and its phase angle. This number is called the normal coefficient. If the wall is rigid, but the frequency vector is parallel to the wall, then the wall coefficient has the value

\[ \alpha_s=(8\rho c\,G_2/|Z|)\cos(\varphi-\vartheta_2). \tag{5.27} \]

Here \(G_2\) and \(\vartheta_2\) depend on the impedance of the opposite wall. Their values can be obtained from the diagrams in Figs. 20 and 21. Such coefficients are called additional. If \(|Z|/\eta\) is greater than approximately \(4\rho c\), then both \(\alpha_s\) and \(\alpha_t\) are approximately equal to \(\frac{1}{2}\alpha_p\).

If the wall is yielding and if the frequency vector is parallel to it \((n=0)\), then the corresponding coefficient will be:

\[ \alpha_t=2\Delta/\eta . \tag{5.28} \]

Here the values of \(\Delta\) must be taken from Fig. 16 (the first transformation sheet). It is called the sliding or tangential coefficient. For \(n=1\), when the frequency vector is almost tangential, the coefficient is equal to \(2\Delta/\eta\), and the value of \(\Delta\) must be taken from Fig. 17 (the second transformation sheet). In most cases the coefficient for \(n=2\), read from Fig. 18, is almost equal to the normal coefficient, according to equation (5.26). For still larger values of \(n\), except in quite exceptional cases, the coefficients are determined quite accurately by equation (5.26) or by the graph in Fig. 13.

Thus, the damping coefficient (see equations 4.1 and 5.10) for some standing wave is determined by the formula

\[ k_N = ca_N/8V, \]

\[ a_N = L_y L_z(\alpha_{x1}+\alpha_{x2}) + L_x L_z(\alpha_{y1}+\alpha_{y2}) + L_x L_y(\alpha_{z1}+\alpha_{z2}). \tag{5.29} \]

Here \(a_N\) represents the room absorption factor for an oscillation determined by the triple of numbers \(N=(n_x,n_y,n_z)\). For a room with rigid walls and for oblique oscillations all values of \(\alpha\) are equal to the corresponding normal coefficients for each wall. For tangential or axial waves for walls perpendicular to the frequency vector, the coefficients \(\alpha\) are likewise equal to the normal ones, while for walls parallel to the frequency vector the coefficients \(\alpha\) are equal to the corresponding sliding coefficients, in this case equal to one half of the normal ones. If one of two parallel walls is yielding, and if the frequency vector is parallel to it, then the coefficient for the harder wall of this pair is equal to the additional coefficient, while for the more yielding wall it is equal to the sliding coefficient, whose value is determined from Fig. 16. In the case of oscillations whose frequency vector is almost parallel to the given pair of walls (\(n=1\)), for the harder wall one should take its normal coefficient; for the yielding wall one should take the coefficient \(\alpha_1\), obtained from Fig. 17. For oscillations with somewhat smaller angles of incidence (\(n>1\)), in most cases one may take, both for the yielding and for the rigid walls, their normal coefficients. If both walls of the given pair are yielding, it is necessary to find the values of \(\mu\) and \(\chi\) from equation (5.3), and from equation (5.26) to find the sum of the coefficients. In this case it cannot be split into two terms, one of which depends only on the properties of one wall and the other on the properties of the opposite one.

Equation (5.29) is similar in form to Sabine’s equation (4.1) for the ergodic distribution of sound. The difference is that in the present case the absorption coefficients depend not only on the properties of the walls, but also on the nature of the given standing wave, and sometimes also on the dimensions and shape of the room. For most oblique oscillations the absorption coefficients are the normal coefficients.

They do not depend on the dimensions of the room, and may be greater than unity. Tangential and axial vibrations always have unequal values of the coefficients, and in those cases when one or several walls are compliant, their sliding coefficients depend both on the dimensions of the room and on the impedances of these walls. If a wall is sufficiently compliant, then the coefficients for “almost tangential vibrations” have special values.

33. Coherent and incoherent waves

Returning to the question of sound energy in a room, we can now say that the attenuation coefficients for most standing waves in the expansion (5.20) are independent of the direction of the vector \(\omega_N\). When the source frequency satisfies the inequality (5.24), summation can sometimes be replaced by integration over all magnitudes of the vector \(\omega_N\) and over all its directions within the first octant of frequency space. In equation (5.20), each term of the series contains the product \(\psi_N(-\omega; x_0, y_0, z_0)\times \psi_N(-\omega; x, y, z)\). Expanding the hyperbolic cosines in terms of the exponential function and carrying out the multiplications, we obtain altogether 64 different exponential functions in the composition of each term of the expansion. Some of these functions will have the form \(\exp [(i/c)(\omega_N-i k_N)\cdot \mathbf R]\). The direction of the vector \((\omega_N-i k_N)\) is determined by the ratio of the numbers \(n_x, n_y, n_z\), while \(\mathbf R\) is the distance vector from the source \((x_0, y_0, z_0)\) to the point \(P(x, y, z)\). Along with these exponentials there will occur seven other exponentials of the same form, but instead of the vector \((\omega_N-i k_N)\) they will contain its mirror images, located in the seven other octants of frequency space. Therefore, instead of integrating eight exponentials within the first octant, one may integrate one exponential over all directions of frequency space. In exactly the same way there will occur exponentials in which the vector \(\mathbf R\) is replaced by the distance vector from the point \(P\) to the mirror image of the sound source in one or several octants (the mirror image of the source in the nearest walls).

When integrating the eight exponentials originally mentioned over all directions, a restriction has to be imposed on the magnitude of the vector \(\mathbf R\). The resonant denominator causes the integrand to be significant only for values of \(\omega_N\) lying within the limits between \(\omega+k_N\) and \(\omega-k_N\). When the vector \(\omega_N\) traverses values within this substantial spherical layer in frequency space, summation over certain discrete directions cannot be replaced by integration over all directions, unless \(\mathbf R\) is so small that \((1/c)\omega_N R\) changes in magnitude by less than \(\pi/2\) as \(\omega_N\) changes from one discrete direction to the next. On the basis of equations (5.25) and (5.29) this requirement can be reduced

to the following simple inequality:

\[ R < (a_p/64)^{1/2}. \tag{5.30} \]

Here the quantity \(a_p\) is equal to the quantity \(a_N\) determined by equation (5.29), if for each wall one may use the value of the normal coefficient \(a_p\). This means that the point \(P\) must be located very close to the sound source in comparison with the dimensions of the room; under these conditions such integration is admissible.

If \(R\) satisfies inequality (5.30), then the result of integrating the eight exponents gives the simple expression:

\[ \psi_c = (Q_0/4\pi R)e^{(i\omega/c)(R-ct)}. \tag{5.31} \]

Here equations (5.17) and (5.21), as well as (5.25), have been used to determine \(dn_p\). On the right-hand side of equation (5.31) stands the velocity potential of a point source in free space. This part of the velocity potential may be called the coherent part. If only the source is not located closer to any wall than the limit established by inequality (5.30), then in none of the 64 remaining exponents can summation be replaced by integration. Therefore these terms of the sum constitute the incoherent part. It does not correspond to any definite flux of sound energy or ordered wave motion. It is studied by the statistical methods set forth below in this same section.

If the sound source is located near a wall, then the distance from the point \(P\) to the mirror image of the source in that wall is sufficiently small that it satisfies inequality (5.30). Therefore, in this case, one more term is added to the coherent part, representing the reflection of sound waves from that wall. This term is multiplied by a coefficient arising from the absorption of energy upon reflection. This additional term is considered in Section 53. There the coherent part will be investigated by methods that are more direct, though also less rigorous.

If the point \(P\) recedes from the sound source, then the coherent part decreases in intensity, and when \(R\) reaches the limit established by inequality (5.30), it has the same mean value as the incoherent part. With a further increase of \(R\), the coherent part continues to exist, but it is rapidly lost against the background of the incoherent part of the velocity potential, and its experimental measurement becomes exceptionally difficult \(R^1\).

At high frequencies satisfying inequality (5.24), the incoherent part of the sound energy is distributed very uniformly throughout the entire volume of the room and possesses a high degree of isotropy, except for the space immediately adjacent to the walls.

This does not mean, however, that the incoherent part of the sound energy in a simple rectangular room has an ergodic distribution, since in it there remains the possibility of distinguishing oblique, tangential, and axial oscillations from one another. This distinction is analogous to the distinction between periodic motion with many degrees of freedom and ergodic motion in mechanics.

34. Mean square pressure

To investigate the incoherent part, let us determine its mean square amplitude. Taking into account expressions (5.17), (5.20), and (5.21), we find that the general expression for the mean square pressure at a point \(P\), due to the presence of a sound source at the point \((x_0, y_0, z_0)\), is equal to:

\[ \langle p^2\rangle_{\mathrm{cp}} = \frac{4c^4\rho^2\omega^2 Q_0^2}{V^2} \left| \sum_N \left(\frac{1}{\varepsilon_N}\right) \frac{\psi_N(-\omega; x_0,y_0,z_0)\psi_N(-\omega; x,y,z)} {\omega^2-(\omega_N-i k_N)^2} \right|^2 . \tag{5.32} \]

In measurements in reverberation chambers, two methods are customarily used for the placement of the microphone and the reproducer: either at the vertices of the trihedral angles of the room, or on a rotating arm, whereby spatial averaging is achieved. For the vertex of a chamber angle, \(\psi_N\) is approximately equal to \(\pm 1\), whereas the spatial mean \(\psi_N^2\) is approximately equal to \(\varepsilon_N\). Therefore equation (5.32) takes on three special forms, which it is useful to write out. One form corresponds to the case when both the microphone and the reproducer are in “averaging” positions in the room (far enough from one another that the coherent part may be neglected). The second form corresponds to the case when one instrument—either the microphone or the reproducer—is at the vertex of a chamber angle, while the other is moved for averaging. The third form corresponds to the case when one instrument is located in one vertex, the other in some other vertex of the room. In all three cases, terms containing different values of \(N\) drop out of the sum, either because of the orthogonality of the functions \(\psi\) under spatial averaging, or because of the equality of the number of terms with plus and minus signs in the case when the microphone and the source are located in different angles of the room. Thus, only quadratic terms are summed. The final result is:

\[ \langle p^2\rangle_{\mathrm{cp}} = \frac{c^4\rho^2\omega^2 Q_0^2}{2V^2} \sum_N \frac{E_N} {\left(\omega^2-\omega_N^2+k_N^2\right)^2+4\omega_N^2 k_N^2}. \tag{5.33} \]

Here

\[ E_N= \begin{cases} 1, & \text{spatial averaging for both instruments;}\\ (1/\varepsilon_N), & \text{averaging for one, the other in a corner;}\\ (1/\varepsilon_N)^2, & \text{both instruments in corners.} \end{cases} \]

The quantity \(\varepsilon_N\) is determined by equation (5.17).

Summation may be replaced by integration if the sound frequency is sufficiently high and satisfies inequality (5.24). As is clear from the preceding arguments and from equation (5.29), in most cases the quantities \(k\) for all oblique oscillations in a given frequency range are the same, since in computing them the normal coefficients are substituted into the expression for \(a_N\). We shall denote the corresponding values of \(a_N\) by \(a_p\). In the same way, the quantities \(k\) for \(yz\)-tangential oscillations can be obtained by taking the sliding coefficients for the \(x\)-walls and the normal coefficients for the remaining walls. We shall denote the corresponding values of \(a_N\) by \(a_{tyz}\). The damping coefficients for the remaining tangential and axial oscillations are determined in a similar way, and the values of \(a_N\) may be marked with the corresponding indices.

Recalling relations (3.5)—(3.7), one can see that, in those cases where summation is replaced by integration, one has to integrate the expressions \(\omega_N^2\,d\omega_N\), \(\omega_N\,d\omega_N\), or \(d\omega_N\), divided by the resonance denominator \(\left[(\omega^2-\omega_N^2+k_N^2)^2+4\omega_N^2 k_N^2\right]\), over the limits \(0<\omega_N<\infty\). In those cases where \(k_N\) is small in comparison with the source frequency \(\omega\), the approximate values of the integrals are, respectively, equal to:

\[ \pi/4k_N,\qquad \pi/4\omega k_N\quad \text{and}\quad \pi/4\omega^2 k_N . \]

After carrying out all integrations and substituting, in place of the values of \(k\), the expressions for them in terms of \(a\) given above, one obtains an approximate formula for the mean-square pressure of the incoherent part. In the case where both instruments—the source and the microphone—are moving for the purpose of averaging, this formula has the form:

\[ (p^2)_{\mathrm{av}}= \frac{\rho^2\omega^2 Q_0^2}{2\pi} \left[ \frac{1}{a_p} \left( 1-\frac{\pi Sc}{4V\omega} +\frac{\pi Lc^2}{8V\omega^2} \right) + \sum_{xy} \frac{\pi L_xL_yc}{\omega V d_{txy}} \left( 1-c\frac{l_x+l_y}{L_xL_y\omega} \right) + \sum_x \frac{2\pi L_xc^2}{\omega^2Vd_{aax}} \right]. \tag{5.34} \]

This formula is rather complicated, but in many cases only its first term is of substantial importance. This first term represents the value obtained from Sabine’s elementary theory for a point source in a room, if the walls of this room have an absorption coefficient equal to the normal coefficient determined by formula (5.26). Therefore most measurements in the steady state in rectangular rooms give values of the normal coefficients, and not of the Sabine absorption coefficients. In those exceptional cases where one wall is much more compliant than the others, one of the values of \(a\) for the tangential oscillations may turn out to be much smaller than \(a_0\), and therefore one of the terms of the sum \((xy)\) may turn out to be larger than the first term in expression (5.34). In these

cases the result of measurements in the steady-state regime corresponds more closely to the grazing coefficient of the compliant wall than to the normal coefficient. It is precisely this that accounts for the well-known fact that placing all the absorbing material on only one wall of a rectangular room does not permit its full utilization. Certain tangential and axial oscillations, propagating parallel to the material, are only weakly absorbed by it.

In the case when either the microphone or the sound source is installed at the vertex of a corner of the room, while the other instrument is moved for averaging, the first brackets of expression (5.34) must be multiplied by 8, the second brackets by 4, and the last term by 2. In the case when both the microphone and the source are installed in corners, the corresponding factors are 64, 16, and 4.

35. Approximate formula

If none of the walls has any special compliance, more precisely, if the smallest value of the quantity \(a\) for tangential oscillations is greater than \(a_p\) divided by the greatest dimension of the room in half-waves, then expression (5.34) may be simplified. On the basis of the definitions of the quantities \(S\) and \(L\) given by expression (3.4), and recalling that \(\eta_x=(2L_x/\lambda)\), etc., we may write the approximate expression, valid up to quantities of the second order relative to \((a_p/\eta_x a_{tyz})\), etc.,

\[ (p^2)_{\mathrm{cp}} \simeq \frac{\rho^2\omega^2 Q_0^2}{2\pi a_p} \left(1-\frac{1}{2\eta_x}+\frac{a_p}{\eta_x a_{tyz}}\right) \times \]

\[ \times \left(1-\frac{1}{2\eta_y}+\frac{a_p}{\eta_y a_{txz}}\right) \cdot \left(1-\frac{1}{2\eta_z}+\frac{a_p}{\eta_z a_{txy}}\right). \tag{5.35} \]

The expression \(\rho\omega^2 Q_0^2/8\pi c\) represents the total power radiated by a point source of strength \(Q_0\). Therefore the last equation can be generalized to the case of a weak source of sufficiently small dimensions, not substantially exceeding the wavelength. If the source radiates sound energy in the amount of \(\Pi\) ergs per second, then the approximate formula for determining the mean-square incoherent pressure in a rectangular room, with spatial averaging both for the source and for the microphone, is

\[ (p^2)_{\mathrm{cp}} \simeq \left(\frac{4\rho c}{a_p}\Pi\right) B_x B_y B_z, \]

where:

\[ B_x= \left[ 1-\frac{1}{2\eta_x}+\frac{a_p}{\eta_x a_{tyz}} \right] \quad \text{and so on.} \tag{5.36} \]

The first factor in parentheses is the number obtained from the Sabine theory formula by substituting into it

normal coefficients for each wall. The factors \(B\) represent correction coefficients arising from the deviation of the sound distribution in a rectangular room with homogeneous walls from the ergodic one.

When either the source or the microphone is located at the vertex of a room angle, and the other instrument is moved, the factor \(B_x\) takes the form \([2-(1/\eta_x)+(a_p/\eta_x a_{tyz})]\), etc. When both instruments are located in corners, the factor \(B_x\) takes the form: \([4-(2/\eta_x)+(a_p/\eta_x a_{tyz})]\), etc.

If one of the walls, for example \(x2\), is sufficiently yielding, so that the “almost tangential” vibrations are characterized by a coefficient \(\alpha\), which is not equal to \(\alpha_p\), then the expression for \(B_x\) takes the form:

\[ B_x = 1-\frac{3}{2\eta_x}+\frac{a_p}{\eta_x a_{1yz}}+\frac{a_p}{\eta_x a'_{1yz}} . \tag{5.37} \]

Here \(a_{1yz}\) denotes that value of \(a\) which is obtained if, instead of \(\chi_{x2}\), in expression (5.29) for \(K_{x2}\) one substitutes the “almost tangential” coefficient \(a_1\) of the yielding wall. The change of the formula for the case in which one or both instruments are placed at the vertices of angles consists in the fact that the first three terms are multiplied respectively by 2 or by 4, while the last term remains unchanged.

Equation (5.36) shows that the Sabine absorption coefficient cannot be measured in a steady state in a simple rectangular room. If the room is large (the quantities \(\eta\) are large), or if the frequency is high and if none of the walls differs in yielding, then the number obtained from the measurements is nearly equal to the normal coefficient. As we shall see below, it is not exactly equal to the Sabine coefficient and sometimes differs greatly from it. On the other hand, the interpretation of the results of such measurements encounters further difficulties connected with the fact that the factors \(B\) differ from unity. The number obtained as the result of such measurements is a weighted mean of the normal coefficient and the grazing coefficient, and the weights with which these coefficients enter depend on the dimensions of the room, measured in wavelengths.

36. Measurements in the Steady State

Experimental work with steady states developed much later than work with settling states. This was due in part to the fact that the sounds of music and speech have predominantly nonsteady character, and therefore it was natural to compare the acoustic qualities of a room with the decay of sound in it. On the other hand, Sabine and his contemporaries, who did not have modern electronic equipment, did not see in the duration of reverberation such a parameter as could be directly

observe with the ear. At the same time the ear is deprived of the possibility of judging the relative strength of the sound in the steady-state regime with accuracy sufficient for measuring absorption. With the advent of modern apparatus these considerations lost their force.

Some of the measurements that we mentioned above in this review will be described here in more detail. Wente^W8 in 1935 studied the transmission coefficient in rooms with the aid of a fast-acting self-recording instrument. The loudspeaker was supplied from a generator whose frequency changed very slowly. The sound signal was received at some other point in the room. The transmission coefficient thus determined in the steady-state regime characterized the properties of the room. This method is usually used in studying the transmission coefficient of electrical systems. In Fig. 23 two examples of Wente’s records are given. The upper one corresponds to a living room of volume about \(270\ \text{m}^3\); the lower—to the same room, additionally equipped with absorbing material. In the damped room the resonance peaks are greatly spread out, as follows from equation (5.20), and the whole curve is correspondingly smoothed.

Fig. 23. Acoustic characteristic of a room in the steady-state regime. Curve \(A\)—living room, curve \(B\)—room damped (W^8).

Wente took yet a further step, discovering a simple empirical relation between the “irregularities of the transmission coefficient” and the total amount of absorbing material in a room. This work of Wente’s gave rise to the studies by Hunt^H7 and Bätt^B5, which were mentioned in the preceding section. Measurements in the steady-state regime were used by Knudsen^K4 in the form of the “method of intensities” for determining absorption coefficients. This method requires averaging over a large number of standing waves and the adoption of special measures to eliminate the influence of individual resonance peaks. The equation playing the principal role in this method is obtained from Sabine’s approximate theory. It has the form (5.36):

\[ a \approx \frac{4pc\Pi}{(p^2)_{\mathrm{cp}}}, \]

Here \(a=\sum \alpha S\) represents the total absorption on the walls of the room; \(p\) denotes the acoustic pressure; \((p^2)_{\mathrm{av}}\) is proportional to the mean sound intensity in the room. According to this equation, the total absorption is inversely proportional to the sound intensity in the steady state. In applying this equation, it is first necessary to determine the total absorption of the chamber itself by equipping it with a “standard” absorbing material with a known absorption coefficient. After this, the coefficient of any material can be determined by direct comparison of two sound intensities in the steady state. This method has the advantage over reverberation methods that a moderate background of interference does not hinder the measurements, which can be made at higher powers. Conversely, determinations of decay can be carried out only in a very quiet environment, and it is not always easy to create one.

Unfortunately, at present we do not have sufficient numerical material to compare the experimental data of the “intensity method” with the conclusions of the wave theory set forth above. Only two papers have been published, and these likewise do not make it possible to calculate the impedances. However, under certain plausible assumptions it proves possible to calculate the correction factors in the simplified equation (5.36) for the case corresponding to Knudsen’s measurements\(^{K4}\). This calculation shows that the difference between the geometrical and wave theories in this case is experimentally insignificant, at least for relative values of absorption coefficients. For plaster with \(\alpha=0.083\), covering all the surfaces of a square room of dimensions \(5.4\times5.4\times4.8\ \mathrm{m}\), the product \(B_xB_yB_z\) of the correction factors in expression (5.36) is approximately equal to \(1.14\). This means that the actual value of the absorption is somewhat greater than the value derived from intensity measurements by means of the simplified formula. The correction factors, however, usually have the same value both for bare walls (\(\alpha=0.014\)) and for plastered ones. Therefore the ratio of the absorption in the presence and in the absence of plaster changes by less than \(0.5\%\) when the wave theory is used. Nevertheless, the wave theory reveals here one important point: the quantity that is measured is in fact not a statistical coefficient, but a normal one. The good agreement with reverberation measurements is due to the fact that the reverberation method below approximately 500 cycles also determines precisely the normal coefficient. The applicability of geometrical theory for obtaining correct relative values of absorption in the present case is due to the low values of \(\alpha\) and to the identical coating material of all surfaces. Most of the most important acoustic materials possess large

absorption than the plaster of which we have been speaking here, and usually only one surface, or even only part of it, is covered with them. In many such cases expression (5.36) would differ substantially from its simplest form.

V. TRANSIENT PROCESSES IN RECTANGULAR ROOMS

The characteristics of a room for transient regimes can be obtained from its characteristics for the steady-state regime by applying operational calculus. Equations (5.20) and (5.17) determine the velocity potential in the steady-state regime at the point \((x, y, z)\) for a room in which sound is excited by a point source of strength equal to unity and with frequency \((+\omega)\), placed at the point \((x_0, y_0, z_0)\):

\[ \Psi_s(\omega)= \]

\[ =-c^2\sum_N \frac{1/\Lambda_N}{\omega^2-\xi_N^2(\omega)}\, \psi_N(\omega,x,y,z)\psi_N(\omega,x_0,y_0,z_0)e^{i\omega t}. \tag{6.1} \]

Here \(\psi_N\) denotes the eigenfunction, and \(\xi_N\) the characteristic value (see 5.9) corresponding to the \(N\)-th standing wave for the source frequency \((+\omega)\). The properties of this potential were considered in the preceding section. According to our convention:

\[ \xi_N(\omega)=(\omega)_N+ik_N;\qquad \xi_N(-\omega)=\omega_N-ik_N. \]

37. Operational calculus

As we saw in the preceding section, in the immediate vicinity of the source the coherent part of the radiation predominates. It follows from equation (5.31) that, for the steady-state regime and for distances from the source \(R\) satisfying inequality (5.30), the approximate formula

\[ \Psi_s(\omega)\approx \Psi_c(\omega)=(1/4\omega R)e^{i\omega[t-(R/c)]}. \tag{6.2} \]

is applicable. At large distances the incoherent part of the radiation predominates, and it is necessary to take the complete series (6.1).

Operational calculus makes it possible to obtain expressions for transient processes in the system in general form through the conditional conductivity. The latter is a characteristic of the transient process in the system under the action of a unit impulse. It can be proved that the velocity potential in a room with a source at the point \((x_0, y_0, z_0)\), having strength zero for \(t<0\) and strength unity for \(t>0\), can be represented by an integra-

along the contour:

\[ A(t)=U(t)\Psi_s(0)+(1/2\pi i)\int_{-\infty}^{+\infty}[\Psi_s(\omega)-\Psi_s(0)](d\omega/\omega), \tag{6.3} \]

where the contour of integration is situated immediately above the real axis. Here

\[ U(t)= \begin{cases} 0 & \text{for } t<0,\\ 1 & \text{for } t>0 \end{cases} \tag{6.4} \]

represents Heaviside’s “unit function.” The function \(\Psi_s(0)\) represents the velocity potential under the indicated conditions for zero frequency. At this frequency most of the walls have infinitely large impedance, so that the corresponding characteristic numbers and eigenfunctions have a very simple form.

The presence under the integral in (6.3) of the potential \(\Psi_s(0)\) shifts the pole of the function \(A\) to the point \(\omega=0\). The remaining poles are due to the resonance denominators of the individual terms of the series. Each term has two poles: one at the point \(\omega=\xi_N(\omega_N+ik_N)\), the other at the point \(\omega=-\xi_N(-\omega_N+ik_N)\). These values of \(\omega\) are the self-consistent solutions of equation (5.9). In other words, the quantities \(\xi\) must have such a value that, if in equations (5.4), instead of \(\eta_x\), etc., the values \((\xi_N L_x/\pi c)\) are substituted in order to determine the quantity \(\chi\), then equation (5.9) will be satisfied by the same values \(\xi_N\). The real part (positive) of this self-consistent solution is called the \(N\)-th natural frequency of the room and is denoted by \(\omega_{0N}\). The imaginary part is called the natural damping index and is denoted by \(k_{0N}\). Let us note that these numbers for the transient regime of sound decay are not the same as the resonance frequency \(\omega_N(\omega_N)\) and the damping coefficient \(k_N(\omega_N)\) in the steady-state regime. In some special cases this difference becomes so large that it can be detected experimentally \(^{\text{k5}}\).

Taking into account that the two poles corresponding to the \(N\)-th term of the series are \(\omega_{0N}+ik_{0N}\) and \(-\omega_{0N}+ik_{0N}\), and computing the residues for each term of the sum, one can prove that the conditional conductivity for positive values of \(t\) is expressed as follows:

\[ A(t)=U(t)\Psi_s(0)-c^2\sum_N \frac{1}{\omega_{0N}^{2}+k_{0N}^{2}} \times \]

\[ \times W_N \exp(-k_{0N}t)\cos(\omega_{0N}t+\Gamma_N-2\Phi_N). \tag{6.6} \]

Here:

\[ \xi_N(\omega_N+ik_N)=\omega_{0N}+ik_{0N} =(\omega_{0N}^{2}+k_{0N}^{2})^{1/2}\exp(i\Phi_N); \]

\[ (1/\Delta_N)\psi_N(\omega_{0N}+ik_{0N};x_0,y_0,z_0)\phi_N(\omega_{0N}+ik_{0N};x,y,z) =W_N\exp(i\Gamma_N). \]

This general expression is useful in studying the incoherent part of the sound process. For the coherent part the conditional conductivity

takes the following simple form:

\[ A_c(t)=(1/4\pi R)\,U[t-(R/c)]. \tag{6.7} \]

Then the velocity potential in the room at time \(t\), due to a sound source of strength \(q(t)U(t)\), located at the point \((x_0,y_0,z_0)\), is expressed as:

\[ \Psi=A(t)q(0)+\int_0^t A(t-\lambda)q'(\lambda)\,d\lambda \tag{6.8} \]

(source \(q(t)U(t)\)).

Here \(q'(t)\) denotes \(dq(t)/dt\). This formula has a general significance: it may be used to determine the acoustic properties of a room both in steady-state and in transient regimes. The acoustic pressure is obtained from it by means of the formula \(p=\rho(\partial\Psi/\partial t)\).

For a simple harmonic sound source that was switched on indefinitely long ago, the corresponding acoustic process may be determined from expression (6.7), if the origin of time is shifted into the past. Then in the room, by definition, a steady-state regime exists and

\[ \psi_s(\omega)=\int_{-\infty}^t A(t-\lambda)q'(\lambda)\,d\lambda, \tag{6.9} \]

\[ (dq/dt=i\omega q). \]

If the sound source was switched on long ago and switched off at the moment \(t=0\), then the velocity potential after the source is switched off can be determined by subtracting (6.8) from (6.9):

\[ \psi=\int_0^\infty [A(t+\lambda)q'(-\lambda)]\,d\lambda-q(0)A(t) \tag{6.10} \]

(source: \(q(t)[1-U(t)]\)).

38. Impulse wave

Let us consider the velocity potential in the transient regime corresponding to a shock-like impulse:

\[ q_p=\lim_{\Delta t\to 0}\{(B/\Delta t)[U(t+\Delta t)-U(t)]\}, \tag{6.11} \]

in which a volume of air \(B\) is pushed out of the source during the time \(\Delta t\).

From equation (6.7) it is seen that the coherent part of the process consists in the propagation of a spherical impulse:

\[ \Psi_{\mathrm{ср}}=\lim_{\Delta t\to 0}\left\{(B/4\pi R\Delta t)\left[U\left(t+\Delta t-\frac{R}{c}\right)-U\left(t-\frac{R}{c}\right)\right]\right\}. \tag{6.12} \]

It reaches the point \(P\) after a time interval \((R/c)\) following the impulse produced by the source. In this equation that part of the coherent wave which is reflected from the walls is neglected. With such a sharp impulse, as is assumed here, the coherent part of the wave will still be observed after repeated reflections. From physical considerations it follows that, in this case, against the background of the incoherent part of the phenomenon one can detect several successive pulses of coherent waves.

For the one-dimensional case the sound impulse was investigated by Maa^[3]. In this case all sound waves are plane, the natural frequencies are harmonic (with rigid walls), and the wave retains coherence. Thus the exact solution consists of a sequence of impulses, similar in form to (6.12), arising as a result of a series of successive reflections of the source plane in two parallel walls. The solution in the form of a series precisely represents this sequence of impulses. Maa applied this solution to explain certain properties of flutter echo.

The agreement of the results with the actual phenomenon of flutter echo turned out to be only approximate. The point is that ordinary sources of sound impulses are closer to point sources than to plane sources, and that only in very rare cases is the sound reflected mainly from only two walls, so that the influence of the other four walls may be neglected. The formulas given in the present section are approximations. Their comparatively complicated form makes them of little use for explaining the properties of flutter echo.

The incoherent part is obtained from the complete series (6.6), which takes the following form:

\[ \Psi_p = c^2 B \sum_N \left(\omega_{0N}^{\,2}+k_{0N}^{\,2}\right)^{\frac12} W_N \exp(-k_{0N}t) \times \]

\[ {}\times \sin(\omega_{0N}t+\Gamma_N-\Phi_N). \tag{6.13} \]

The mean square pressure corresponding to this velocity potential is determined as follows:

\[ (p_p^2)_{\mathrm{av}} = \sum_N \frac{\rho^2 c^2 B^2}{2|A_N|^2} \left| \psi_N(\omega_{0N}+ik_{0N};\,x_0,y_0,z_0) \times \right. \]

\[ \left. {}\times \psi_N(\omega_{0N}+ik_{0N};\,x,y,z) \right|^2 \exp(-2k_{0N}t). \tag{6.14} \]

In the incoherent part of the process caused by an impulse-like excitation, all frequencies have equal weight. The decay curve—

...of the mean-square pressure*) is not a straight line, as in a simple oscillatory system—its steepness gradually decreases. Its initial steepness is rather large: it corresponds to strong damping of high-frequency oscillations. Its final steepness corresponds to the least-damped oscillations. Usually these are the lowest-frequency axial oscillations between the two most rigid walls.

The experimental damping curves do not have the smoothed form predicted by equation (6.14), owing to the impulse-like character of the coherent part of the establishing process. The sound impulse is very sharp and strong before it experiences its first reflection, and although in subsequent reflections it is smoothed and reduced in height (in particular, if the wall is yielding, as we shall see in Section 53), nevertheless the distortions it introduces into the damping curve are noticeable over a considerable interval after \(t=0\). As experience shows, at the point \(P\) nothing is heard until the instant \(t=R/c\), the arrival of the first impulse. The subsequent coherent impulses gradually diminish relatively in comparison with the smoothed incoherent damping.

39. Reverberation

Measurements of reverberation in rooms are not, however, made with such sharp sound impulses. Usually the sound source emits either one frequency or a band of frequencies (a howling tone) for so long a time that a steady state is attained, after which the source is switched off. We shall first consider the case of a simple harmonic signal; for a howling tone the result can be calculated by superposition of the results for the individual frequencies. Using expressions (6.10) and (6.6), one can carry out the integration and obtain, at the point \(P\), at the instant \(t\) (\(t>0\)), the following expression for the velocity potential due to a sound source
\(Q\sin(\omega t-\varphi)[1-U(t)]\), located at the point \((x_0,y_0,z_0)\):

\[ \Psi=-c^2 Q\sin\varphi\sum_N \frac{W_N\exp(-k_{ON}t)}{\omega_{ON}^{\,2}+k_{ON}^{\,2}} \cos(\omega_{ON}t+\Gamma_N-2\Phi_N)- \]

\[ -\frac{1}{2}c^2\omega Q\sum_N \frac{W_N\exp(-k_{ON}t)}{\omega_{ON}^{\,2}+k_{ON}^{\,2}} \left\{ \frac{\sin\omega_{ON}t+\Gamma_N-2\Phi_N-\varphi-\Omega_{N-}} {\left[(\omega_{ON}-\omega)^2+k_{ON}^{\,2}\right]^{1/2}} + \frac{\cos(\omega_{ON}t+\Gamma_N-\Phi_N+\varphi-\Omega_{N+})} {\left[(\omega_{ON}+\omega)^2+k_{ON}^{\,2}\right]^{1/2}} \right\}, \tag{6.15} \]

*) The term “damping curve” in this article always denotes the curve representing the logarithm of the mean-square pressure as a function of time (in seconds). It is often plotted on a decibel scale (the decimal logarithm of the pressure squared).

where the phase angles \(\Gamma_N\) and \(\Phi_N\) are determined by expressions (6.6), and we have put:

\[ (\omega_{ON}\pm\omega)+ik_{ON}=[(\omega_{ON}\pm\omega)^2+k_{ON}^2]^{1/2}\exp(i\Omega_{N\pm}). \]

The pressure corresponding to this potential is determined as follows:

\[ \begin{aligned} p={}&\rho c^2 Q\sin\varphi\sum_N W_N(\omega_{ON}^2+k_{ON}^2)^{-1/2}\exp(-k_{ON}t)\times\\ &\times \sin(\omega_{ON}t+\Gamma_N-\Phi_N)-\\ &-\frac{1}{2}\rho c^2\omega Q\sum_N \frac{W_N\exp(-k_{ON}t)}{(\omega_{ON}^2+k_{ON}^2)} \left\{ \frac{\cos(\omega_{ON}t+\Gamma_N-\Phi_N-\varphi-\Omega_{N-})} {[(\omega_{ON}-\omega)^2+k_{ON}^2]^{1/2}} +\right.\\ &\left.\qquad\qquad\qquad\qquad +\frac{\cos(\omega_{ON}t+\Gamma_N-\Phi_N+\varphi-\Omega_{N+})} {[(\omega_{ON}+\omega)^2+k_{ON}^2]^{1/2}} \right\}. \end{aligned} \tag{6.16} \]

In this expression the first series represents a pressure impulse caused by the sudden switching off of the source. We may call it a switch-off impulse. The magnitude of this impulse depends on the phase \(\varphi\) of the source at the moment of switch-off. The origin of the impulse is seen more clearly from the expression for the coherent part of the process. Using expressions (6.7) and (6.10), we find the velocity potential for the coherent part

\[ \Psi_c \simeq (Q/4\pi R)\sin\left[\omega\left(\frac{R}{c}-t\right)+\varphi\right]U\left(\frac{R}{c}-t\right). \tag{6.17} \]

The coherent part of the acoustic pressure is proportional to the time derivative of this potential. Switching off the source has no effect until the moment \(t=\frac{R}{c}\). At that moment the velocity potential begins to fall to zero. However, if \(\varphi\) is not equal to zero or \(\pi\), then a discontinuity of the potential is obtained, causing a sharp pressure impulse before the beginning of its decrease. This impulse is absent only in the case when the loudspeaker is switched off at the moment when the velocity of the membrane is zero, i.e. \(\varphi=0\) or \(\pi\). The nature of the impulse is indicated by the fact that the first series in expression (6.16) for the pressure is proportional to the series in expression (6.13) for the velocity potential under a point impulse. The pressure impulse in the case of an impulse contains two successive phases: compression is followed by rarefaction. The pressure impulse when the sound source is switched off contains either only compression or only rarefaction, depending on the phase \(\varphi\).

Of course, in reality the sound source is not switched off instantaneously. If \(\varphi\) is neither zero nor \(\pi\), then the loudspeaker membrane, before coming to rest, will perform its own settling-

Noted misprints

Page	Line	Printed	Should read	Whose fault
153	9 from bottom	\(\displaystyle \pm \frac{1h}{22\pi}\)	\(\displaystyle \pm \frac{1}{2}\,\frac{h}{2\pi}\)	printer
197	9 from top and 11 from top	\(\displaystyle \frac{W_c}{4}\,\alpha S\)	\(\displaystyle \frac{W_c}{4}\,\alpha S\)	»
289	1 from bottom	\(\displaystyle -x_0^2\left(\Phi_4^2-\Phi_s^2\right)\)	\(\displaystyle \frac{x_0}{8\pi}\left(\Phi_4^2-\Phi_s^2\right)\)	»
290	18 from top	\(\displaystyle S[\Gamma\Phi]\)	\(\displaystyle S=[\Gamma\Phi]\)	editor
299	14 from top	\(\displaystyle \int \frac{dk}{K}\)	\(\displaystyle \int \frac{dk}{K}\)	editor
309	1 from bottom	Laranjian	Lagrangian	printer
311	4 from top	\(\displaystyle M\ddot{x}=gF\)	\(\displaystyle M\ddot{x}=gF\)	»
328	table, column V, 7 from top	\(\displaystyle aD_{t_{1/2}}=\frac{1}{v}\)	\(\displaystyle R\gg D_{1/2}=\frac{1}{v}\)	»
343	1 from bottom	\(\displaystyle \frac{i\beta_{1\eta}/\pi}{\chi_n(\omega)-\beta_1^2\eta^2}\)	\(\displaystyle \frac{i\beta_{1\eta}/\pi}{\chi_n^2(\omega)-\xi_1^2\eta^2}\)	»
367	6 from bottom	\(\displaystyle \frac{\sin \omega_{ON}t+\Gamma_N-2\Phi_N-\varphi-\Omega_N}{\left[(\omega_{ON}-\omega)^2+k_{ON}^2\right]^{1/2}}\)	\(\displaystyle \frac{\sin\left(\omega_{ON}t+\Gamma_N-2\Phi_N-\varphi-\Omega_N\right)}{\left[(\omega_{ON}-\omega)^2+k_{ON}^2\right]^{1/2}}\)	editor
371	1 from bottom	\(\displaystyle \frac{1}{(\omega_{ON}+\omega)^2+\omega kO_N}\)	\(\displaystyle \frac{1}{(\omega_{ON}+\omega)^2+k_{ON}^2}\)	printer
375	caption to Fig. 26	line)\(^{9}\)	line) \(H^{9}\)	»

ing motion. If the loudspeaker has large damping, then the sound process will have almost the same character as if the membrane stopped instantaneously; the switch-off impulse will be one-sided, only it will be wider and lower than corresponds to the first term of equation (6.16). If the damping of the loudspeaker is small, then the transient regime of its membrane will contain a series of oscillations, and their natural frequency will in general not coincide with the excitation frequency. The mathematical series for the switch-off impulse in this case can be calculated as described above, provided only that the natural oscillation of the loudspeaker is known. The expressions obtained must be substituted into equation (6.16) in place of the first term. In this expression, those natural frequencies of the room which are close to the natural frequency of the loudspeaker will be especially singled out. The second term in equation (6.16) will single out those frequencies which are close to the exciting frequency.

40. Details of the decay curve

We can now make several general remarks on the properties of the transient regime observed at a point \(P\) after a harmonic sound source located at the point \((x_0, y_0, z_0)\) has been switched off at the moment \(t=0\). At the moment \(t=R/c\), a pressure impulse is observed—this is the first (direct) arrival of the switch-off impulse. The magnitude of the impulse depends on the phase of switching off, while its form depends on the transient process of the loudspeaker. After this moment, the sound pressure begins to decrease, with each standing wave having its own natural frequency and damping exponent. Reflections of the switch-off impulse from the walls of the room may also be noticeable. They produce characteristic irregularities on the decay curve. To obtain a smoothed decay curve it is necessary either to switch off the sound source at the moment when the velocity of its membrane is zero, or to average many curves corresponding to different values of the phase \(\varphi\). The smoothed curve corresponds to the second term in equation (6.16).

Several works by Hunt\(^{Н6}\) and other investigators are devoted to the question of switching off a sound source. Hunt’s automatic reverberation apparatus (see Section 8) makes it possible always to switch off the source at the same phase of the period of variation of the frequency of the beat tone. It would be possible to go a step further and arrange switching off at any prescribed phase of the fundamental frequency. However, apparently, it is experimentally much simpler to switch off at random moments but to average a set of repeated measurements. The dosing method used by Hunt itself smooths the small peaks of the decay curves, so that those peaks which are due to the switch-off impulse must also be smoothed.

On the other hand, the oscillatory character of sound decay appears quite distinctly on oscillograms and automatic

records of the loudness level. A detailed comparison of the fluctuations of these oscillations with the physical properties of the room is a very complex problem, on which work has only just begun. Jones J3 studied the effect of the mean damping on the magnitude of the fluctuations. Watson W3 experimentally investigated fluctuations by means of a recording device provided with compensation for exponential damping.

At low frequencies, corresponding to inequality (5.22), the natural frequencies \(\omega_{ON}\) are sufficiently far apart from one another, so that

Fig. 24. Oscillograms illustrating the occurrence of beats during the decay of sound for different excitation frequencies. The upper and lower curves were taken when sound was excited in the room at one of the resonant frequencies. In this case only one natural oscillation is strongly excited. The middle curve corresponds to the case in which the excitation frequency lies between the natural resonant frequencies. Then both natural oscillations are excited with equal strength. On the decay curve, beats are observed between the two natural oscillations K5.

usually one term of the second row (namely the one for which \(\omega_{ON}-\omega\) is close to zero) is much larger than all the others taken together. In this case only one standing wave is excited by the source; its decay curve is rectilinear, and its slope corresponds to the damping constant of this wave. We note that even in this case a switching-off transient is observed on the curves. In those cases where the frequency is somewhat higher than that corresponding to inequality (5.22), or where two eigenfunctions have close frequencies, two standing waves may be sharply expressed. While the sound source is operating, these oscillations have a frequency identical with the frequency of the source, as follows from the steady-state equation (5.20). After the source is switched off, both standing waves continue to exist, now with their own natural frequencies. If these frequencies differ somewhat, then irregularities arise on the decay curve—

... irregularities \(k_5\), caused by beats, which are added to those irregularities that are caused by the switching-off impulse. Their form depends on the relative amplitude of both oscillations and on the phase of the source at the moment of switching off. These circumstances follow from the analysis of equation (6.16).

Some of these phenomena were clearly demonstrated by Knudsen \(^{k_3,k_5}\). Typical curves are shown in Fig. 24. It depicts oscillograms of the decay of sound in a rectangular chamber of dimensions \(2.4 \times 2.4 \times 2.9\ \text{m}\). The curves were obtained at different excitation frequencies, indicated above each curve. The theoretical natural frequencies of the room in this range are: 70.3; 92.8; 99.8 and 115.9 cycles. In the oscillograms it is clearly seen that at frequencies 92.9 and 99.7 cycles the decay curves fall purely exponentially. At the same time, at the frequency 96.7 cycles, lying approximately midway, the decay is accompanied by sharply expressed beats of frequency about 7 cycles. At other exciting frequencies, both components have unequal amplitude, which gives rise to beats with different depths of modulation. However, at any excitation frequency in the range 92.8–99.8 cycles, beats of frequency about 7 cycles are obtained: this convincingly proves that only the two indicated natural oscillations take part in the phenomenon of sound decay. At a signal frequency of 118 cycles in this same chamber, Knudsen observed the coexistence of beats with frequencies 3.3 and 19.3 cycles, arising as a result of the simultaneous excitation of three natural oscillations with frequencies 99.8, 116 and 119.2 cycles. At somewhat higher frequencies, when many natural oscillations are excited simultaneously, the processes established in the room become much more complex.

41. Approximate formula for the decay curve

In those cases where many natural oscillations are excited, the reverberation process can be studied by statistical methods analogous to those discussed in the preceding section. This occurs either for frequencies so high that inequality (5.24) is satisfied, or for a warble tone with bandwidth \(\Delta \nu\) and mean frequency \(\nu\), satisfying the inequality \(\Delta \nu > 50c^3/V\nu^2\). Leaving aside the switching-off impulse and its reflections, to determine the mean square pressure it is sufficient to use the second term in expression (6.16). Applying methods analogous to those that led to the derivation of equation (5.33), we obtain:

\[ (p^2)_{\mathrm{cp}} = \frac{c^4 \rho_0^2 \omega^2 Q^2}{8V^2} \sum_N \frac{E_N \exp(-2k_{ON}t)}{\omega_{ON}^{\,2}+k_{ON}^{\,2}} \left\{ \frac{1}{(\omega_{ON}-\omega)^2+k_{ON}^{\,2}} + \frac{1}{(\omega_{ON}+\omega)^2+k_{ON}^{\,2}} \right\}. \tag{6.18} \]

This expression should be summed over all waves: oblique, tangential, and axial. Then one obtains the formula corresponding to equation (5.34):

\[ (p^2)_{\mathrm{av}}= \frac{\rho^2\omega^2Q^2}{2\pi} \left\{ [\exp(-a_p ct/4V)]\left(\frac{1}{a_p}\right) \left(1-\frac{\pi S c}{4V\omega}+\frac{\pi Lc^2}{8V\omega^2}\right) +\sum_{xy}[\exp(-a_{txy}ct/4V)] \left(\frac{\pi L_xL_yc}{\omega Va_{txy}}\right) \left(1-c\frac{L_x+L_y}{\omega L_xL_y}\right) +\sum_x[\exp(-a_{ax}ct/4V)] \left(\frac{2\pi L_xc^2}{\omega^2Va_{xa}}\right) \right\}. \tag{6.19} \]

This expression corresponds to spatial averaging by means of the motion of both instruments: the source and the microphone.

Finally, if none of the walls is distinguished by special compliance, and if all \(\eta\) are greater than, approximately, 3, then, analogously to (5.36), the following approximate expression may be obtained:

\[ (p^2)_{\mathrm{av}}\simeq(4\rho c\,\Pi/a_p)D_x(t)\cdot D_y(t)\cdot D_z(t), \]

\[ D_x(t)= \left\{ \left(1-\frac{1}{2\eta_x}\right) \exp[-L_yL_z(\alpha_{px1}+\alpha_{px2})ct/4V] + \left(\frac{\alpha_p\nu}{\eta_xa_{tyz}}\right) \exp[-L_yL_z(\alpha_{tx1}+\alpha_{tx2})ct/4V] \right\}. \tag{6.20} \]

In this, \(\alpha_{px1}\) denotes the normal coefficient for wall \(x1\), \(\alpha_{tx1}\) denotes the grazing coefficient for the same wall, etc. The number \(\nu\) is equal to unity in the case when the source and the microphone are moved; it is equal to \(1/2\) when one of the instruments is located at the vertex of a corner and when both instruments are at different vertices. The decay curves corresponding to this formula are given in Fig. 25.

The decay curve represents the dependence of \((p^2)_{\mathrm{av}}\) on time on a semi-logarithmic scale. We see that, in the approximation expressed by the last equation, the decay curve is the sum of three curves, each of which refers to one of the pairs of walls. Each of these curves begins with a slope corresponding to the normal coefficient of both walls of the pair, and ends with a slope corresponding to the grazing coefficient. The “break” of the curve, i.e. the transition from the initial slope to the final one, occurs at the moment when both terms in the expressions \(D\) become equal, i.e. at the moment:

\[ t_{bx}=(4L_x/c)(\alpha_{px1}+\alpha_{px2}-\alpha_{tx1}-\alpha_{tx2})^{-1} \times \ln\left[(a_{tyz}/\nu a_p)\left(\eta_x-\frac{1}{2}\right)\right]. \tag{6.21} \]

Generally speaking, this moment occurs later if the normal coefficients \(\alpha_p\) are small and do not differ greatly from the grazing coeffi-

ficients, and earlier, if one or both normal coefficients are large. Therefore, usually for the logarithmic curve \(D\), corresponding to the more yielding pair of walls, the “break” occurs earlier than for the other two. In addition, it has the greatest steepness of all three curves. Indeed, if only one wall of a rectangular room is yielding (as in a reverberation chamber), then that component in the decay curve which is due to the yielding wall lowers the ordinates of the decay curve so sharply that the “breaks” of the components from the harder walls are usually found below the limits of measurement. In such cases one can determine only the normal coefficients of the harder walls, and the single break noticeable on the experimental curves refers to the most yielding pair of walls.

\[ D_x(T)=\left(1-\frac{1}{2\eta_x}\right)\left|e^{-\tau_x}+A_x e^{-B_x\tau_x}\right| \]

\[ B_x=\frac{4(\alpha_{tx1}+\alpha_{tx2})}{(\alpha_{px1}+\alpha_{px2})} \]

\[ T_x=L_yL_z(\alpha_{px1}+\alpha_{px2})(ct/4V) \]

\[ \eta_x=2L_x/\lambda \]

\[ A_x=(\alpha_p/\alpha_{tyz})\,[2/(2\eta_x-1)]. \]

Fig. 25. Approximate decay indices for a rectangular room.

Suppose that the only yielding wall is perpendicular to the \(x\)-axis and that the frequency is sufficiently high, so that \(\eta_x=(2L_x/\lambda)\) is greater than ten. Substituting expression (6.21) into expression (6.20), one can verify that the first break in the decay curve will arise at a loudness level approximately

\[ (10a_p' L_yL_z)(\alpha_{px1}+\alpha_{px2}-\alpha_{tx1}-\alpha_{tx2}) -\frac{1}{2}\lg(\eta_x\alpha_{tyz}/\upsilon_x a_p) \quad \text{decibels} \]

below the initial loudness level for \(t=0\). This will occur at the instant determined by equality (6.21).

To summarize. Each decay curve for high frequencies \(\nu \ll (400c^2/a_p)^{1/2}\), taken in a reverberation chamber with one yielding wall, has a more or less irregular form because of the superposition of switching-off shocks and begins to fall only after the first shock from the sound source reaches the microphone. The averaged decay curve, obtained from a series of curves with different phases

at the moment of switching off, falls off very uniformly. Its initial slope depends on the normal coefficients of the walls. It usually has only one “break” over the length of the measurable portion. Its slope after the break corresponds to the sliding coefficient of the yielding wall, to the additional coefficient of the opposite wall, and to the normal coefficient for the other four walls. If the difference between the sliding and normal coefficients for the yielding wall is large, then the break may occur very close to the beginning of the curve. In this case it is difficult to determine the initial slope against the background of the switching-off shock. Such curves over most of their length depend on oscillations parallel to the yielding wall, which do not die out so rapidly and have a larger initial amplitude than the oblique oscillations.

On the other hand, as we shall see in the next chapter, the presence of any irregularities in the outlines of the walls, or in the uniformity of distribution of the absorbing material, or the use of rotating stirrers for scattering the sound—all this brings the numerical values of the normal and sliding coefficients closer together. It may turn out that, under the influence of these irregularities, the sliding coefficients will increase more rapidly than the normal coefficients will decrease. In this way the initial part of the decay curve can be so lengthened that even the first “break” occurs outside the measurable limit. At the same time the slope of this initial part of the curve will correspond very well to the values of the normal coefficients determined by equation (5.26) and pertaining to non-diffuse oscillations. Therefore, for rooms in which sound scattering is not so perfect as to ensure its ergodic distribution, the “absorption coefficient” determined from decay curves (which appear rectilinear over the first thirty or forty decibels) corresponds more closely to Sabine’s normal coefficient. Likewise, comparison with equation (5.36) shows that the coefficients obtained in steady-state measurements for such rooms usually turn out to be smaller than the normal coefficients determined from the initial slope of the decay curves. This is due to the fact that steady-state measurements give an average result over all excited oscillations, whereas the initial steepness of the decay curve is determined only by oblique oscillations.

Decay curves of the type shown in Fig. 25 are often obtained in practice, and it will be of interest to compare the wall impedances measured under these conditions with decay curves obtained in the usual way. Hunt, Beranek, and Maa9 used an equation similar to equation (6.20) for the analysis of decay curves. They obtained quite satisfactory agreement with theory, as is evident from Fig. 26. The measurements were made in a chamber measuring \(6 \times 4.2 \times 2.4\) m, with plastered walls and ceiling and with a floor, po-

covered with absorbing material. The experimental points are marked by circles; the solid lines represent the theoretical curves. The deviation of both from the straight line is conspicuous. A similar analysis by \(B^{12}\), in which the modified equation (6.20) and other experimental conditions were used, also gave satisfactory agreement of the measurement results with the theory.

Fig. 26. Decay curves. Agreement of experimental results (circles) with theoretical results (solid lines)\(^{9}\).

When using equation (6.20) it is necessary to remember that this is only an approximate expression; it becomes incorrect when any of the \(\eta\)’s becomes less than two. Likewise, when using the approximate expression \(\alpha_s=\alpha_t=4\gamma\), it is necessary to remember that it is valid only if \((|z|/\eta pc)\) is greater than approximately four.

(To be continued in the next issue.)