LIMITS OF APPLICABILITY OF SOME APPROXIMATE METHODS USED IN ARCHITECTURAL ACOUSTICS

Regarding the article by Morse and Bolt “Sound Waves in Rooms”*)

L. Brekhovskikh

The fundamental review by F. Morse and R. Bolt in one of its parts requires certain additions.

It does not clarify such a fundamental question as the limits of applicability of geometrical acoustics, although the authors speak about this a good deal. This question, apparently, has not yet been resolved by anyone at all. Here we shall consider it, basing ourselves on our works¹, ², ³. In addition, we have thought it useful to include here the discussion, contained in these same works, of the applicability of normal impedance in the solution of various kinds of boundary-value problems, since this concept is widely used in architectural acoustics and is often transferred, erroneously, into related fields.

A. LIMITS OF APPLICABILITY OF GEOMETRICAL ACOUSTICS

1. General considerations

The authors of the review consider two approaches to the solution of problems of architectural acoustics: a) the exact wave theory and b) the geometrical theory. By the latter they understand a theory in which, first, the concepts of geometrical rays figure and, second, the ergodicity condition is fulfilled, i.e., there are sufficient grounds for statistical averaging of the sound field. Here, without any basis, two factors of completely different character are mixed together, which not only obscures the essence of the matter but also leads, as we shall see below, to a number of valuable possibilities in the development of architectural acoustics being missed. It would be more correct to consider three variants of the theory: a) the wave theory**), b) the geomet-

) See Uspekhi fizicheskikh nauk, 32, issue 2, p. 185; issue 3, p. 333; issue 4, p. 417 (1947).
*) With statistical averaging or without it.

rical acoustics, operating with the concepts of geometrical rays, which is also equivalent to the picture of imaginary radiators; c) statistical geometrical acoustics, where an additional simplification occurs by virtue of statistical averaging. Morse and Bolt omit the second of these variants. As a consequence, they say that such phenomena as a nonuniform reverberation decay, the dependence of the absorptive capacity of a material on its position in a room, etc., can be understood only from the point of view of wave theory (§ 9). This assertion cannot be regarded as correct, since all these phenomena, in the case of not very small rooms (more precisely, see below), also receive an explanation under the second variant omitted by them, if one adds to it, when necessary, partial averaging of the results. Since, in comparison with wave theory, it already contains considerable simplifications, it is precisely with its aid that one may hope to obtain valuable results under conditions close to practical ones. Some work has already been done in this direction*), but the possibilities contained here have apparently been realized to only a very small extent. Here we investigate under what conditions geometrical acoustics [variant b] may be used instead of exact wave theory. We note that exact criteria for the applicability of statistical acoustics also do not yet exist, but we shall not dwell on this question.

2. Effective Zone

Let us consider a point sound radiator \(Q\) (Fig. 1) above a plane boundary separating two media and determine under what conditions the reflection of spherical waves incident on the boundary may be considered according to the laws of geometrical acoustics. § 53 of the survey is devoted to this, but the authors there confined themselves merely to certain transformations of the resulting integrals, without arriving at any result. We shall try here to present the solution of this problem in the most intuitive form, referring to the fact that an exact derivation leads to the same results, but is cumbersome and does not provide the necessary physical insight. We shall first consider a question which in architectural acoustics is of independent interest, but in the present case is auxiliary. Suppose we are interested in the value of the reflected wave at a point \(P\). One might think that not all points of the plane \(\Pi\) play an identical role in the formation of the wave reflected in the direction \(P\). A substantial role will be played by the point \(O\), from which the ray constructed according to the laws of geometrical acoustics is reflected, and also by a certain zone surrounding it (zone \(I\) in Fig. 1). We shall call the latter the effective zone and

*) Works 1–3, 5–8 in Sec. III and 6–9 in Sec. IX of the list of the principal literature on architectural acoustics in Russian (p. 476).

determine its shape and dimensions. First let us consider the simpler case of an absolutely reflecting boundary of separation.

It is known that in this case the acoustic potential at the point \(P\) will have the form

\[ \varphi_p=-\frac{e^{ikR_0}}{R_0}+\frac{e^{ikR}}{R}, \tag{1} \]

where the first term represents the direct spherical wave, and the second—the reflected one. The latter may also be regarded as emanating from the image source \(Q_1\), obtained by mirror reflection of \(Q\) in the boundary of separation (\(R=Q_1P\), see Fig. 1).

Fig. 1. \(\Gamma\)—the effective zone on the infinite boundary of separation \(\Pi\), \(R_0\)—the path traversed by the direct wave, \(R=QOP=Q_1OP=R' + R''\)—the same for the reflected wave, \(\chi_1\)—the grazing angle, \(z_0\) and \(z\)—the elevations of the source and receiver above the boundary of separation, \(O\)—the origin of the rectangular coordinate system, whose \(xy\)-plane coincides with the plane \(\Pi\).

Let us now remove all parts of the reflecting boundary except the zone \(\Gamma\). By the definition of the effective zone, the reflected wave at the point \(P\) must not be appreciably changed thereby. However, if we now use the representation of the image source, it is necessary to imagine that not the whole plane \(\Pi\) is transparent to its rays, but only the region \(\Gamma\), which now acts as an aperture in an infinite screen. Thus we arrive at a diffraction problem. It is required to determine sufficient dimensions and the shape of the aperture so that the wave which has passed through it should have the form

\[ \frac{e^{ikR}}{R}, \]

just as in the absence of any screen.

From the theory of diffraction it is known,^4 that in the presence of an opening the intensity of the wave at the point \(P\) is expressed by the formula (for the phase see below)

\[ \left|\varphi_P\right|^2=\frac{1}{2R^2}\left(C_1^2+S_1^2\right), \tag{2} \]

where \(C_1\) and \(S_1\) denote the integrals

\[ \begin{aligned} C_1&=\frac{1}{\sqrt{2}}\iint_{\Gamma_1}\cos\left[\frac{\pi}{2}\left(u^2+v^2\right)\right]\,du\,dv,\\ S_1&=\frac{1}{\sqrt{2}}\iint_{\Gamma_1}\sin\left[\frac{\pi}{2}\left(u^2+v^2\right)\right]\,du\,dv. \end{aligned} \tag{3} \]

The variables \(u\) and \(v\) are obtained from \(x,y\) (the original coordinates in the plane \(\Pi\)) by the substitution

\[ \begin{aligned} \pi u^2&=k\left(\frac{1}{R'}+\frac{1}{R''}\right)x^2\sin^2\chi_1,\\ \pi v^2&=k\left(\frac{1}{R'}+\frac{1}{R''}\right)y^2,\\ k&=\frac{2\pi}{\lambda}. \end{aligned} \tag{4} \]

The region of integration \(\Gamma_1\) in the \(u,v\) plane corresponds to the region \(\Gamma\) in the \(x,y\) plane. By \(\chi_1\) is denoted the “grazing angle” formed by the ray \(Q_1P\) with the horizontal. The values \(R'\) and \(R''\) are indicated in Fig. 1.

For simplicity let us consider the case when \(R'=R''=\dfrac{R}{2}\) (the source and the receiver have the same elevation above the boundary of separation). Then

\[ \frac{\pi u^2}{2}=\frac{2kx^2}{R}\sin^2\chi_1,\qquad \frac{\pi v^2}{2}=\frac{2k}{R}y^2. \tag{5} \]

With an infinite increase of the region \(\Gamma\), and hence also of \(\Gamma_1\), the integrals tend to their asymptotic values: \(S_1\to\sqrt{2}\), \(C_1\to0\), which, according to (2), gives the required value of the intensity at the point \(P\). It is easy to show that the phase in this case also proves to be correct. From the tables of Fresnel integrals, to which our integrals are reduced, one can obtain that the intensity and the phase will not differ from these limiting values by more than \(10\%\), if at all points of the boundary of the region \(\Gamma_1\) the condition

\[ \frac{\pi}{2}\left(u^2+v^2\right)\ge 50. \]

is satisfied. The minimum dimensions of the region are obtained by taking here the equality sign. As a result we see that in the \(u,v\) plane such a region will be—

gives a circle, while in the plane \(x, y\), according to (4), an ellipse

\[ x^2 \sin^2 \chi_1 + y^2 = \frac{25R}{k} \]

with semiaxes

\[ \frac{2}{\sin \chi_1}\sqrt{R\lambda} \quad \text{and} \quad 2\sqrt{R\lambda} \]

or, in general, with semiaxes

\[ \frac{2a}{\sin \chi_1}\sqrt{R\lambda} \quad \text{and} \quad 2\sqrt{R\lambda}, \tag{6} \]

where the coefficient \(a\) is equal to unity for an accuracy of \(10\%\) and increases as the admissible relative error is decreased.

Below we shall make use of these results. Let us note that the concept of the effective zone may be useful in architectural acoustics in all those cases where one has to deal with the reflection of spherical waves from regions of finite dimensions. In such cases this concept makes it possible to decide quickly how the finiteness of the dimensions affects the reflection. True, in real cases the material is not absolutely reflecting and is not placed in empty space, but is mounted on a wall, which is also reflecting. However, in all cases the result remains the same; only the magnitude of \(a\) changes somewhat. It may be thought that for an accuracy of \(10\%\), in all practical cases \(a\) does not exceed several units.

It is not difficult to give a prescription for constructing the effective zone for an arbitrary position of the source and receiver. For this purpose, on the reflecting boundary (the plane \(\Pi\) in Fig. 1) we determine the locus of points from the condition that the phase increment in the propagation of a ray from \(Q\) to each of these points and from it to \(P\) be greater by \(\frac{\lambda}{2}\) than in propagation along \(QOP\). This will be a closed curve enclosing the first Fresnel zone. In the same way one can construct the second, third, \(\ldots\), \(N\)-th zone, and in the latter case the phase difference in comparison with the ray \(QOP\) will be \(\frac{N\lambda}{2}\). It is known from optics that if one takes an aperture on which a sufficiently large number of Fresnel zones fit, then the wave will pass through it unhindered. Above we saw that such an aperture, in shape and dimensions, will coincide precisely with the effective zone. Thus, the latter must consist of a sufficiently large number of Fresnel zones, and this number must be the greater, the stricter the requirements on the coincidence of the reflected wave with the wave obtained upon reflection from an infinite boundary. The quantity \(a\) introduced above is approximately equal to

\[ \frac{\sqrt{N}}{4}, \]

where \(N\) is the number of included zones

Fresnel zones. Approximately, one may assume that the latter is related to the admissible relative error \(\varepsilon\) by the relation

\[ N \sim \frac{1}{6\varepsilon^2}. \tag{7} \]

With an accuracy of \(10\%\) in intensity it is necessary to include 16 Fresnel zones.

It is not difficult to verify that, with the radiator and receiver raised, the effective zone will be an ellipse situated as indicated in Fig. 1. In the case where the radiator (or receiver) is on the boundary of separation, such a zone will be an ellipse enclosing the radiator, whose center lies on the line joining the radiator to the projection of the receiver onto the boundary of separation. If the radiator and receiver are situated on the boundary of separation, then the effective zone will be an ellipse with foci at the points of their location.

3. Geometrical acoustics and corrections to it

Let us return again to reflection from an infinite, homogeneous boundary of separation. If the latter is absolutely rigid, then the reflected wave [the second term in (1)] may be regarded as emanating from the image source \(Q_1\). Since the intensity of this wave does not depend on \(\lambda\) and, consequently, remains unchanged as \(\lambda \to 0\), the same must also be obtained by means of geometrical acoustics. Indeed, if we take rays incident from \(Q\) on the boundary of separation and construct for each of them the reflected ray, then the extensions of all of them intersect at \(Q_1\). Thus geometrical acoustics is equivalent to the picture of image sources, and in the case of perfectly reflecting boundaries it is strictly valid.

Let us investigate its applicability in the general case. In § 53 Morse and Bolt try to analyze this question, but, not having obtained definite results, conclude that any conclusions based on this representation may lead to erroneous results. We shall show that in a whole series of cases, and in particular in large rooms, the picture of image sources will give correct results in a sufficiently good approximation.

In the case of an arbitrary boundary, as above, we may confine ourselves to reflections from the effective zone, whose shape and dimensions will be practically the same as for an absolutely reflecting boundary. However, now, owing to the dependence of the reflection coefficient on the angle of incidence, the waves reflected from different parts of the effective zone will have different amplitudes. If we again make use of the representation of an image source and an aperture in an infinite screen instead of the effective zone, then this aperture must be endowed with a “transparency,” different from unity, which is different at different places.

If the upper and lower media have, respectively, densities \(\rho\) and \(\rho_1\) and sound propagation velocities \(c\) and \(c_1\), then the coefficient

of reflection as a function of the ray grazing angle \(\chi\) will be, as is known,

\[ B(\chi)=\frac{m\sin\chi-\sqrt{\,n^{2}-\cos^{2}\chi\,}} {m\sin\chi+\sqrt{\,n^{2}-\cos^{2}\chi\,}}, \tag{8} \]

where \(\displaystyle m=\frac{\rho_{1}}{\rho}\), and \(\displaystyle n=\frac{c}{c_{1}}\) is the refractive index\(^*\).

Let us first consider the case of a reflection coefficient different from unity, but not depending on the angle, which is realized, for example, when \(c=c_{1}\) \((n=1)\), where we have:

\[ B=\frac{m-1}{m+1}=\frac{\rho_{1}-\rho}{\rho_{1}+\rho}. \]

In this case all the arguments are carried out in the same way as for an absolutely reflecting boundary. But only when passing through the aperture, by which we replace the effective zone in the diffraction formulation of the problem, is the amplitude of the ray multiplied by the constant quantity \(B\). As a result, at the point \(P\) we obtain a spherical wave with the constant factor \(B\), i.e.,

\[ B\frac{e^{ikR}}{R}, \tag{9} \]

which again can be represented as radiated by the fictitious source \(Q_{1}\). Thus, the representation in terms of a fictitious source is strictly valid in all cases when the reflection coefficient does not depend on the angle of incidence. Absolutely reflecting boundaries are included here as a special case \((B=1)\).

In the case of arbitrary boundaries, we note that the representation in terms of a fictitious source may be considered approximately valid if the reflection coefficient varies with the angle sufficiently slowly, so that within the limits of the maximum deviations from the angle \(\chi_{1}\), corresponding to the edges of the effective zone [in Fig. 2 they are denoted by \((\Delta\chi)_{\max}\)], it may be regarded as constant. To give this consideration a quantitative character, let us expand the reflection coefficient \(B(\chi)\) in a series in powers of the deviations of the angle \(\chi\) from the angle \(\chi_{1}\), corresponding to the ray \(Q_{1}P\) (Fig. 2). Restricting ourselves to second-order terms, we have:

\[ B(\chi)=B(\chi_{1}+\Delta\chi)=B(\chi_{1})+B'(\chi_{1})\Delta\chi+\frac{1}{2}B''(\chi_{1})(\Delta\chi)^{2}. \tag{10} \]

\(^*\) All our results can also be applied to the case of reflection of electromagnetic waves, if a vertical dipole is taken as the radiator and one sets \(\displaystyle n=\frac{k_{1}}{k}\), \(\displaystyle m=n^{2}\).

The field at the point \(P\) is formed from waves reflected from various portions of the boundary of separation within the effective zone*). The amplitude of these waves is obtained by multiplying the amplitude of the incident wave by the reflection coefficient, taken in the form (10). In computing the total action of all the waves, we note that, if only the first, constant term in (10) is taken into account, the field of the reflected wave at the point \(P\) will be

\[ B(\chi_1)\frac{e^{ikR}}{R}, \tag{11} \]

since this corresponds completely to the case of a constant reflection coefficient considered above. The effect of the second term in (10) on the field at the point \(P\) will be small because in one half of the zone \(\Delta\chi\) it is positive, and in the other negative. We shall estimate the correction to the field due to the third term in (10) only in order of magnitude. From comparison of the third term with the first it is seen that it, like (11), will contain the function

\[ \frac{e^{ikR}}{R}, \]

but multiplied not by \(B(\chi_1)\), but by a factor of the order of magnitude of \(B''(\chi_1)(\Delta\chi)^2_{\max}\). From our analysis of the effective zone given above, one can obtain that

\[ (\Delta\chi)_{\max}\sim \frac{1}{\sqrt{kR}}. \tag{12} \]

Thus, the addition to (11) will be

\[ \frac{B''(\chi_1)}{kR}\frac{e^{ikR}}{R}, \tag{13} \]

where \(kR=\dfrac{2\pi R}{\lambda}\) is everywhere assumed by us to be a large quantity. As \(\lambda\to0\) (\(k\to\infty\)) this addition tends to zero, and the reflected wave

Fig. 2. \(\chi\) is the grazing angle of an arbitrary ray within the effective zone; \(\Delta\chi_{\max}\) is its maximum deviation from the grazing angle \(\chi_1\) within the effective zone.

*) More precisely, in computing the field at the point \(P\) by means of Green’s theorem, we reduce the problem to computing an integral over the surface of separation, and only the integral over the effective zone will be significant. In doing this, in order to substitute the field at the boundary of separation into the integral, we proceed on the assumption of geometrical acoustics, which is analogous to Kirchhoff’s well-known assumption in the theory of diffraction.

will be determined by a single expression (11). Thus, the latter indeed gives the field obtained under the assumption of the validity of geometrical acoustics. The wave given by this term may also be regarded as emanating from an imaginary source \(Q_1\), which will now possess a definite directional characteristic.

An exact calculation\(^{2,3}\) gives, for the reflected wave with an arbitrary arrangement of the radiator and receiver,

\[ B(\chi_1)\frac{e^{ikR}}{R} -\frac{iN}{kR}\frac{e^{ikR}}{R}, \tag{14} \]

where

\[ N=\frac{1}{2}B''(\chi_1)\cos^2\chi_1-\sin\chi_1\,B'(\chi_1). \tag{15} \]

The second term in (14) is, in order of magnitude, the same as the correction term (13), which justifies the arguments given above, though in some points not sufficiently rigorous.

4. Criteria for the negligibility of the corrections

The total field in the upper medium will be given by the sum of the direct wave

\[ \frac{e^{ikR_0}}{R_0} \]

and the reflected wave (14). Let us find the conditions under which the correction to geometrical acoustics [the second term in (14)] may be neglected. We first consider the case when the source or receiver (for definiteness we shall assume it to be the former) is situated at the interface. Then \(R_0=R\), and the condition that the correction term be small in comparison with the sum of the direct and reflected waves is written as

\[ kR\left|1+B(\chi_1)\right|\gg |N| \]

or

\[ kR\gg \left|\frac{N}{1+B(\chi_1)}\right|. \tag{16} \]

Here, according to (8),

\[ 1+B(\chi_1)=\frac{2m\sin\chi_1}{m\sin\chi_1+\sqrt{\,n^2-\cos^2\chi_1\,}}. \tag{17} \]

Substituting (17) into (16) and taking into account that \(R\sin\chi_1=z\), we obtain:

\[ kz\gg \frac{1}{2m}\left|N\cdot\left(m\sin\chi_1+\sqrt{\,n^2-\cos^2\chi_1\,}\right)\right|. \tag{18} \]

An estimate of the right-hand side of this inequality can be made with the aid of (15) and (8). For \(n>1\) this gives

\[ kz\gg \left|\frac{w}{(w\sin\chi_1+1)^2}\right|, \tag{19} \]

where \(w=\dfrac{m}{n}=\dfrac{\rho_1 c_1}{\rho c}\) is the specific impedance of the boundary. In many cases,

preserving the correct order of magnitude, in the right-hand side of (19) one may neglect \(w \sin \chi_1\) in comparison with unity. Then the condition for applicability of geometrical acoustics is written in the form

\[ 2\pi \frac{z}{\lambda} \gg |w|. \tag{20} \]

Thus, the elevation of the receiver above the interface must be sufficiently large in comparison with the wavelength.

For large specific impedances of the boundary, such that

\[ |w| \sin \chi_1 > 1, \tag{21} \]

unity in the right-hand side of (19) may be neglected, as a result of which we obtain:

\[ 2\pi \frac{z}{\lambda} \gg \frac{1}{|w| \sin^2 \chi_1}. \tag{22} \]

The last condition admits the limiting transition to absolutely reflecting boundaries (\(|w| \to \infty\)). In this case the right-hand side in (22) tends to zero, as a consequence of which the condition will be satisfied for any \(z\).

Let us give several generalizations of the results obtained:

a) For \(n < 1\), conditions (20)—(22) remain in force, but instead of the impedance \(w\), which, as we shall see below, in this case has no meaning at all, there will appear \(m = \dfrac{\rho_1}{\rho}\).

b) With the radiator and receiver raised, instead of \(z\) in (20)—(22) one must substitute the total elevation of the radiator and receiver above the interface. In this case \(\sin \chi_1 = \dfrac{z + z_0}{R}\).

c) In the presence, instead of one, of two interfaces, when the sound wave undergoes multiple reflections, conditions (20)—(22) again remain in force, but instead of \(z\) one must substitute in them the distance between the boundaries (the thickness of the layer), and by \(\chi_1\) one must understand the inclinations of the rays with respect to the interfaces. The same will apply to a room where there are already not two boundaries but more; moreover, the role of \(z\) will be played by the smallest dimension of the room.

We note that generalizations b) and c) are in some cases accompanied by a decrease in the rigidity of the conditions written above.

From an analysis of conditions (20)—(22) for a room, taking into account generalization c), one can see that, if the minimum dimension of the room is large, and often even comparable with the wavelength, the use of geometrical acoustics (the image-source picture) is admissible. Thus, for \(|w| = 3\) we shall have \(2h \gg \lambda\), where \(h\) is the smallest dimension of the room. In this case one may expect that, with an accuracy of results of \(10\%\), the field can be calculated by means of geometrical acoustics if \(h \gg 5\lambda\). For \(|w| = 0.6\), this same accuracy is already attained when \(h \gg \lambda\).

B. On the Applicability of Normal Impedance

In architectural acoustics, the use of boundary conditions written with the aid of impedance is widespread. In this connection, the case in which the latter does not depend on the angle of incidence of the sound wave on the boundary is especially important; in this case it is often called the normal impedance. The boundary conditions are then formulated as follows: the ratio of the normal component of the velocity to the sound pressure at the boundary is equal to a constant quantity characteristic of the given interface. It is clear that this condition can be only approximate, since in the exact theory there appear not one but two boundary conditions—the continuity of the sound pressure in passing through the boundary and the same for the normal component of the velocity.

The question arises as to the limits of applicability of the concept of normal impedance.

Since what interests us to a considerable extent is only qualitative orientation in the question posed here, we shall consider the simplest case, namely the reflection of a sound wave from a plane interface between two media. At large distances from the radiator the wave process may be regarded as plane. Then for the sound potential in the lower medium \((z<0)\) we shall have the wave equation

\[ \frac{\partial^2 \varphi_1}{\partial x^2}+\frac{\partial^2 \varphi_1}{\partial z^2}+k_1^2\varphi_1=0. \tag{23} \]

We shall have the same equation also in the upper medium, but we shall not need it. At the interface the boundary conditions mentioned above must be satisfied; in terms of the sound potential they are written in the form

\[ z=0,\qquad \rho_1\varphi_1=\rho\varphi,\qquad \frac{\partial\varphi}{\partial z}=\frac{\partial\varphi_1}{\partial z}, \tag{24} \]

where \(\varphi\) is the potential in the upper medium.

Let us show that these two exact boundary conditions in certain cases pass over into one approximate condition.* To this end let us assume that in the lower medium the propagation occurs mainly along the \(z\)-axis, as a result of which the derivatives with respect to \(x\) are small, namely

\[ \left|\frac{\partial^2\varphi_1}{\partial x^2}\right|\ll |k_1^2\varphi_1|, \tag{25} \]

i.e. in the \(x\)-direction the function \(\varphi_1\) will be practically constant. Then the first term in (23) may be neglected, and the remaining

* In essence, these considerations are a transfer into acoustics of Greenberg’s ideas\(^5\) concerning electromagnetic waves, for which boundary conditions analogous to normal impedance were proposed by M. A. Leontovich.\(^6\)

the equation is immediately solved and gives:

\[ \varphi_1=Ae^{ikz}. \tag{26} \]

Substituting this solution into (24), eliminating \(A\), and taking into account that the sound pressure and the normal component of velocity are expressed through the sound potential by the relations \(p=\rho\,\dfrac{\partial\varphi}{\partial t}\), \(v_z=-\dfrac{\partial\varphi}{\partial z}\), we obtain a single boundary condition

\[ p/v_z=W, \tag{27} \]

where \(W=\dfrac{\rho_1\omega}{k_1}=\rho_1c_1\) is the normal impedance.

It remains only to decipher condition (25). A plane wave incident on the boundary is given by the expression

\[ \varphi=\varphi_0e^{i\omega t+i(k_xx+k_zz)}, \tag{28} \]

and the refracted one by

\[ \varphi_1=\varphi'_0e^{-i\omega t+(k_{1x}x+k_{1z}z)}, \tag{29} \]

where \(k_{1x}=k_x\). Substitution of (29) into 25 gives:

\[ |k_x|^2\ll |k_1|^2. \tag{30} \]

The modulus signs are necessary here because \(k_x\) and \(k_1\) may be complex. If they are real, then

\[ k_x=\frac{\omega}{c}\sin\vartheta,\qquad k_1=\frac{\omega}{c_1}, \]

and condition (30) is written as:

\[ c_1^2\sin^2\vartheta\ll c^2, \tag{31} \]

where \(\vartheta\) is the angle of incidence. Note that when the last condition is satisfied, the refracted ray will proceed practically along the normal to the surface.

Strengthening condition (30) somewhat, let us write it in the form (taking into account that \(k_x^2+k_z^2=k^2\))

\[ |k|^2\ll |k_1|^2. \tag{32} \]

In this form the condition is applicable for any type of wave. When it is fulfilled, the concept of normal impedance may be used, for example, in investigating the field of a point radiator located at the interface*).

\[ \rule{4em}{0.4pt} \]

*) Except for cases when a substantial part of the energy from the radiator is carried by a lateral wave\(^{2,3}\) propagating in the lower medium.

Condition (32) can be satisfied either on account of a large imaginary part of \(k_1\) (large absorption per wavelength), or on account of a large real part. In the latter case, for the use of the normal impedance it is necessary that the speed of sound in the lower medium be considerably less than the speed of sound in the upper medium. Some authors have used the concept of normal impedance in studying the reflection of sound from the sea bottom. However, since the speeds of sound propagation in soil and in water are of the same order, and, as a rule, the speed is greater in soil than in water, the normal impedance for oblique incidence of a wave on the bottom cannot have meaning.

CITED LITERATURE

1. L. Brekhovskikh, On the limits of applicability of certain approximate methods used in acoustics. To be published in DAN.
2. L. Brekhovskikh, Propagation of sound and radio waves in layers. Dissertation. FIAN.
3. L. Brekhovskikh, Izvestiya AN SSSR, physical series, 10, 491 (1946).
4. M. Born, Optics. ONTI (1937).
5. G. Grünberg, Journ. of Phys. USSR, 6, 185 (1942).
6. M. A. Leontovich, Izvestiya AN SSSR, physical series, 8, 16 (1944).

LIST OF BASIC LITERATURE ON ARCHITECTURAL ACOUSTICS IN RUSSIAN*)

I. Books

1. S. Lifshits, Course of Architectural Acoustics. VTU, Moscow (1927).
2. S. Lifshits, Acoustics of Buildings and Their Insulation from Noise and Shocks. GNTI, Moscow (1931).
3. S. Lifshits, Course of Architectural Acoustics. Moscow (1937).
4. A. Rabinovich and G. Goldberg, Radio Broadcasting. Svyazizdat, Moscow (1935).
5. A. Rabinovich and Yu. Sukharevskii, Broadcasting Studios and Microphones. Moscow (1939).
6. I. Dreizin, Course of Electroacoustics, vol. 1. Svyazradioizdat, Moscow (1939).
7. Works of the Acoustic Commission of the Academy of Sciences of the USSR (TAK), 3. Sound-absorbing materials. Moscow (1939).
8. Acoustic Materials and Their Application. Collection of the Central Scientific-Research Institute of Industrial Structures. Stroiizdat, Moscow (1940).

II. Questions of the Optimum of Reverberation

1. S. Lifshits, Duration of sound and the musical optimum of reverberation. ZhTF, 4 (1934).

*) Compiled by L. D. Rozenberg.

1. S. Lifshits, Optimal frequency characteristic of sound-absorbing material. ZhTF, 6, 2127 (1936).
2. S. Lifshits, Experimental investigations of the frequency optimum of reverberation. DAN, 15, 317 (1937).

III. Statistical-geometrical consideration of problems in room acoustics

1. G. Chigrinsky, The picture of reflections and its application in architectural acoustics. DAN, 23, 631 (1939).
2. G. Chigrinsky, The picture of reflections and the reverberation of unclosed spaces. ZhTF, 9, 1484 (1939).
3. G. Chigrinsky, The picture of reflections and elements of the acoustics of prismatic polyhedra. ZhTF, 9, 2920 (1939).
4. L. Rosenberg, On the character of the sound field obtained when music is reproduced by a distributed system of radiators. ZhTF, 12, 211 (1942).
5. L. Rosenberg, A method for calculating sound fields formed by distributed systems of radiators. ZhTF, 12, 102 (1942).
6. L. Rosenberg, A method for calculating sound fields formed by distributed systems of radiators operating in enclosed rooms. ZhTF, 12, 220 (1942).
7. L. Rosenberg, Nonuniformity of the field produced by an infinite chain of omnidirectional radiators. ZhTF, 12, 573 (1942).
8. L. Rosenberg, On the placement of sound-absorbing material in an enclosed room. DAN, 51, 599 (1946).
9. L. Rosenberg, On the influence of the average sound-absorption coefficient on the level of sound intensity. ZhTF, 10, 1634 (1940).
10. I. Dreizen, On the calculation of sound pressure in the field of radiation of an ensemble. ZhTF, 4, 649 (1934).

IV. Acoustic processes in coupled rooms

1. L. Rosenberg, Total reverberation in the recording and reproduction of sound. ZhTF, 2, 139 (1932).
2. M. Sapozhkov, Remarks on room acoustics. ZhTF, 2, 395 (1932).
3. M. Sapozhkov, On the question of coupled rooms. ZhTF, 4, 822 (1934).
4. L. Rosenberg, Some considerations on total reverberation. ZhTF, 7, 2167 (1937).
5. M. Sapozhkov, On the question of determining the optimal reverberation in coupled rooms. ZhTF, 4, 1588 (1934).

V. Wave theory of room acoustics

1. I. Dreizen, Distribution of sound-absorbing material in a radio studio. ZhTF, 6, 2131 (1936).
2. Dreizen, ZhTF, 7 (1937).
3. V. Tsikunov, On oscillations inside a niche whose open surface is excited in a prescribed manner. TAK, 3, 37 (1939).
4. A. Rimsky-Korsakov and K. Struve, On the reflection of sound waves from a surface whose dimensions are comparable with the wavelength. TAK, 2, 69 (1939).
5. M. Sapozhkov, Remarks on the theory of oscillations of a one-dimensional space. ZhTF, 4, 1169 (1934).

VI. Theory of sound absorption

1. N. Andreev and E. Lysenko, Sound absorption of a porous material allowing for porosity and an air interlayer. TAK, 2, 7 (1939).
2. N. Andreev and E. Lysenko, On the sound absorption of perforated materials. TAK, 2, 17 (1939).
3. I. Pustovoitenko, Calculation of the sound-absorption coefficient of a material penetrated by tubes with absorbing walls. TAK, 2, 25 (1939).
4. M. Sapozhkov, Effectiveness of sound absorption by niches of various forms. TAK, 2, 49 (1939).
5. G. Malozhinets, Continuous sound-absorbing structures. Information Bulletin on the Construction of the Palace of Soviets, No. 5–6 (1941).

VII. Resonant sound absorption

For a review and exhaustive bibliography see S. N. Rzhevkin, UFN, 30, 40 (1946).

VIII. Architectural-acoustical measurements

1. Yu. Shneider, Measurements of certain sound-absorbing materials by the reverberation method. ZhTF, 6, 2147 (1936).
2. L. Ipatov, Measurements of absorption coefficients on an angular installation. ZhTF, 6, 2151 (1936).
3. M. Mishcherin and N. Mikheeva, An apparatus for reverberometric determination of sound-absorption coefficients. Proceedings of NIKFI, issue 6, 164 (1937).
4. N. Mikheeva, Measurement of the absorption coefficients of materials used for damping in a cinema hall. Proceedings of NIKFI, issue 6, 173 (1937).
5. Yu. Shneider, Combating noise in ventilation ducts. ZhTF, 8 (1938).
6. A. Belov and M. Fainshtein, Experimental study of sound damping in ventilation ducts. ZhTF, 9, 1499 (1939).
7. G. Gol’dberg, Modern methods for measuring reverberation. TAK, 1, 43 (1939).
8. A. Kharkevich, Acoustic measurements in enclosed rooms. TAK, 1, 65 (1939).
9. G. Gol’dberg, Measurements of sound-absorbing materials. TAK, 3, 33 (1939).
10. G. Gol’dberg, On the question of the dependence of the sound-absorption coefficient on the dimensions of the sample. TAK, 3, 37 (1939).

IX. Miscellaneous questions

1. I. Verkhovskaya, Vowel sounds and their role in the acoustics of rooms. Acoustical Collection of the Moscow State Conservatory, issue 1, p. 46 (1936).
2. A. Rabinovich, “Distance effect” in radio studios. ZhTF, 4 (1934).
3. I. Goron, Studios of the National Broadcasting Company in New York. “Elektrosvyaz’,” No. 1 (1938).

1. A. Rabinovich, “Acoustics of the Moscow Television Center,” Elektrosvyaz’, No. 4 (1940).

2. V. Grossman, “The Sound-Recording House in Moscow,” Architecture of the USSR, No. 10 (1939).

3. L. Rozenberg, “Acoustics of the Great Hall of the Palace of Soviets,” Architecture of the USSR, No. 4 (1939).

4. L. Rozenberg and B. Tartakovskii, “Acoustics in the Palace of Soviets,” Construction Industry, Nos. 11–12 (1939).

5. G. Gol’dberg and B. Tartakovskii, “Designing the Sound Absorption of the Dome of a Large Hall,” Information-Technical Bulletin of the Construction of the Palace of Soviets, Nos. 5–6 (1941).

6. A. Rabinovich, “On the Perceptibility of Echo and Its Influence on the Intelligibility of Speech,” Journal of Technical Physics, 10, 605 (1940).