Review of Studies on Resonant Sound Absorbers

S. N. Rzhevkin.

§ 1. The Present State of the Question of Sound Absorbers

The foundations of architectural acoustics were laid by the works of the American physicist-acoustician Wallace Sabine, carried out at the beginning of our century[^1]. These works showed that, in order to obtain good acoustic conditions in a room, it is necessary to use special materials—sound-absorbing materials. In subsequent years, in a number of countries, especially in America, a large number of new types of porous sound-absorbing materials were developed. Usually this was done by the purely empirical manufacture of new materials, obtaining, by gradual adjustment of the composition of the porous mass and variation of the construction, one or another desired frequency characteristic of absorption. In this way there arose dozens of different types of materials now produced by numerous firms.*)

The empirical method of development made it almost impossible to foresee what changes would occur in the frequency characteristic of sound absorption with a given change in the composition of the mass or in the construction. To design a new material with a modified sound-absorption characteristic—for example, with a very large coefficient of sound absorption in one or another frequency region or over the entire sound range—was impossible because of the absence of guiding theoretical considerations.

In 1937, intensive preparation began for the acoustic design of the large hall of the Palace of Soviets, the height of whose dome was planned to be about 100 m (Fig. 1). In connection with this, the need arose to create new sound-absorbing materials that would give practically complete absorption of sound in a very broad range from 100 to 4000 hertz. Even a negligible reflection of sound from the dome of so large a hall threatened the danger of a strong echo, which would distort the transmission of speech and music. The task of creating a material with a sufficiently high coefficient of sound absorption,

*) In the well-known book by V. Knudsen on architectural acoustics, published in 1932, about 14 pages are devoted to the description and enumeration of various kinds of absorbing materials.

close to unity seemed extremely difficult, and the ways of solving it in practice were not known. However, thanks to the work of a number of Soviet acousticians, grouped in the Academy of Sciences of the USSR and in the construction of the Palace of Soviets, significant successes were achieved in the development of the theory of sound-absorbing materials, and the full possibility of solving the above-mentioned problem was demonstrated.

Fig. 1.

In this article I would like to present the basic information about one of the methods for solving this problem, developed by a collective of acousticians who worked together with me. These works led to the creation of sound-absorbing materials which we called resonance sound absorbers. In the course of work on sound absorbers, ways were found to design new types of materials with a prescribed frequency characteristic of absorption. These materials—or, more precisely, structures—proved suitable for use in construction practice: for theaters, halls, studios, cinemas, and other rooms.

At the present time, the complete possibility is being established of abandoning the empiricism that previously prevailed in the development of sound absorbers and of proceeding to their rational design on the basis of the theory

In this review I shall try to illuminate the history of the development of resonant sound absorbers, the basic principles of their construction, and questions of their application.

§ 2. HISTORY OF THE USE OF RESONATORS IN ARCHITECTURAL ACOUSTICS.

Already in ancient Greece and Rome, in open theaters, resonators were used in the form of metal vessels placed in the backs of the stone benches of amphitheaters and opening, by their apertures, toward the listeners seated on the benches. These vessel-resonators (echeia—Greek: ἠχεῖα) were tuned to various tones of the scale and, by their resonance, apparently produced a certain impression of boominess characteristic of enclosed rooms and now known by the name of reverberation. The details of the construction of the echeia, their form, dimensions, number, and placement are known to us only in general outline from the data of the Roman architect Vitruvius.

Fig. 2.

The architectural-acoustic heritage of the ancients passed to Byzantium, chiefly along the line of church construction. In Byzantine churches, and later in the churches of Europe, in particular in ancient Russian churches, metal or clay vessel-resonators were embedded in the walls and vaults; they served, in all probability, to strengthen the resonance of the voices of the choir or of music, i.e., to increase reverberation. Figure 2 shows the placement of resonators (golosniki) in ancient churches of Novgorod and Pskov. Similar vessels were often installed in theaters as well. Thus, in the Maly Theater in Moscow there formerly existed vessel-resonators in the walls, which were destroyed later during repairs to the building.

Recently, resonators have been applied to the acoustics of rooms in an entirely different, diametrically opposite direction. In Sabine’s works it was found, for example, that the sound absorption of a felt layer of a certain thickness has a maximum in a certain frequency region. A layer two, three, etc. times as thick has an absorption maximum at increasingly lower frequencies. This points to the resonant character of the absorption phenomenon.

In 1932 Erwin Meyer² showed that a resonant absorption maximum is observed in closed cavities covered by a sheet of metal, plywood, or another material. The resonance frequency corresponds to the natural frequency of the membrane supported by the elastic air cushion of the cavity lying beneath it. In Glover’s well-known book Practical Acoustics³ there is a reference to the finishing of certain American theaters with peculiar resonators in the form of round metal disks fastened at some distance from the wall on a mesh or fabric. In this case the resonant cavities have a cylindrical form, with the bases of the cylinder closed and the lateral surface open.

Dimensions in mm

Fig. 3.

In 1933 Wintergerst⁴ published a study of sound absorption by a perforated sheet made of a rigid porous mass (of the “Celotex” type), installed at some distance from the wall (Fig. 3). The cavity behind the sheet was either empty or filled with loose porous material. Wintergerst noted that this absorber may be regarded as a system of individual resonators. Materials of the “Sanacoustic” type and similar ones, consisting of a rigid perforated facing sheet with a loose porous mass beneath it, long known in practice, could likewise be regarded as resonant sound-absorbing systems.

In the examples mentioned, the resonators were sound absorbers and, consequently, their action led to a decrease in the reverberation of the room.

§ 3. THE INFLUENCE OF RESONATORS ON THE REVERBERATION OF A ROOM

Analyzing the data given above on the use of resonators, it was natural to pose the question: how do resonators act on the acoustics of a room? When do they increase and when do they decrease the reverberation of a room?

As a result of work in 1934–1936, I succeeded in illuminating this question from both the theoretical and the experimental points of view⁵. It was shown that a resonator, accumulating sound energy

in its cavity, is equivalent, as it were, to some added volume of the room. This volume is the greater, the smaller the damping of the resonator. On the other hand, a resonator possessing certain frictional losses continuously absorbs sound energy, drawing it from the store of energy of the room. Sabine’s well-known formula for the reverberation time of sound in a room was presented in a modified form:

\[ T = 0.162 \cdot 10^{-2} \frac{V + V'}{A + A'}, \tag{1} \]

where \(V\) is the volume of the room, and \(A\) is the total absorption. Thus an added volume \(V'\) and an added absorption \(A'\) were introduced into Sabine’s formula. The quantity \(V'\) is proportional to the number of resonators and inversely proportional to the square of the total damping coefficient. The added absorption \(A\) is proportional to the number of resonators and to the coefficient of friction in the neck of the resonators. Analysis of this formula shows that resonant systems with slightly damped resonators will give a considerable added volume \(V'\) and a small added absorption \(A'\), which, according to the formula given, will cause an increase in reverberation. Resonant systems with large damping, caused by placing porous material in the neck of the resonator or in its cavity, will give a large value of \(A'\) with small \(V'\). This will lead to a decrease in the reverberation time \(T\).

Fig. 4.

To verify these theoretical considerations, M. S. Antsiferov\(^{6}\) carried out the following experiment. Eighty resonators with small damping were introduced into a reverberation chamber. The resonators were ordinary wide-necked milk bottles. Calculation showed that, at a resonant frequency of 230 hertz, each such resonator was equivalent to an added volume of \(2\ \text{m}^3\), and all together—to a volume of \(160\ \text{m}^3\). The damping of all the resonators gave an additional 14.5 units of absorption. The reverberation time of the empty chamber at 230 Hz was found to be \(0.48\) sec. It increased to \(0.96\) sec in the presence of 80 resonators. This experiment represented a good

...a good example of the action of lightly damped resonators. “Echo chambers” and “voice cavities” must have produced a similar effect.

The opposite effect was observed by Antsiferov on cylindrical resonators with a free lateral surface, resembling in their construction mushrooms with a large cap. A system of such resonators (diameter 80 cm, height 3 cm) gave an average absorption coefficient

Fig. 5.

of \((a)\) per \(1\ \mathrm{m}^2\) of surface, as shown in Fig. 4 (below); a series of maxima corresponded to the calculated natural frequencies of the resonators. When a thin strip of fabric was placed along the lateral surface of the resonators, \(a\) increased sharply (Fig. 4, upper curve). Similar observations made by us later at the Physics Institute of the Academy of Sciences on resonators in the form of caissons, niches, and wooden panels installed against a wall, etc., showed that in all resonant systems a small amount of absorbent placed at the entrance to the resonator, where the highest velocities of air oscillation prevail, gives a strong increase in sound absorption. Thus, in the example shown in Fig. 4, the same amount of fabric placed directly on the floor gave an absorption 10 times smaller than it did in combination with the resonators.

These observations showed that the use of resonant systems gives a very effective utilization of porous absorbing material when it is placed in the throat of the resonator and allows,

thus to achieve a saving of costly absorbent, and also makes it possible to use less dense and cheaper materials (thin fabric, mesh) instead of large masses of absorbent. The principle of economy of absorbent formulated here was subsequently used in the development of more complex and more advanced systems. The possibility of conveniently solving the problem of sound absorption in the low-frequency region, where ordinary absorbents are only slightly effective, was used in the development of a resonant sound absorber for the ceiling of one of the studios of the House of Sound Recording in Moscow (Fig. 5). The resonators are located in the center of the rosettes; in the recesses there is a porous absorber covered with a perforated layer.

§ 4. POSSIBILITY OF COMPLETE ABSORPTION OF SOUND.

Ordinary absorbents used in construction practice rarely give an absorption coefficient greater than 0.6–0.7 (in energy), and in the low-frequency region it falls very rapidly, and at 100 hertz most absorbents are completely ineffective. The above-mentioned acoustical problem of the Palace of Soviets rested on the realization of a sound absorbent with an absorption coefficient greater than 0.9 in the low-frequency region and reaching 0.98 in the high-frequency region. A solution of this problem could be envisaged on the basis of the works of Berger8, Békésy9, Vysotsky10, and others. It consisted in the use of a large number of layers of fabric or a very thick layer of finely fluffed cotton wool. Structurally, however, these solutions proved unsatisfactory, and therefore the idea arose of seeking new ways of solving the problem. The idea of using resonators was very attractive, but it was not yet clear how, by means of resonators, one could obtain very strong absorption.

G. D. Malozhinets11 indicated, proceeding from theoretical considerations, the fundamental possibility of complete absorption of sound incident at a certain angle upon a perforated rigid sheet, behind which there is a packing of porous material (a material of the “Sanacoustic” type). For this, a certain relation must be maintained between the dimensions of the perforations and the friction of the material; the paths of practical realization of this relation (for the given construction) remained unclear because of the difficulty of taking account of the friction parameter.

In 1937 I considered12 a very simple scheme of a resonant absorber (Fig. 6, a and b), consisting of a row of resonators with holes arranged on a square lattice, with frictional resistance \(R\) concentrated in the holes, and showed that complete absorption of sound at normal incidence can be obtained at the resonance frequency provided the condition

\[ R = \frac{\sigma^{2}}{\Sigma}\rho c, \tag{2} \]

where $\sigma$ is the area of one opening; $\Sigma$ is the area of the cell per one resonator; $\rho$ is the density of air and $c$ is the speed of sound in air. The quantity $R$ represents the frictional resistance in the opening, the value of which could easily be calculated and, by selecting $\sigma$ and $\Sigma$, condition (2) could be satisfied. Thus, the condition of complete absorption for such a system could practically be realized. It should be noted that the question of the exact calculation of sound absorption by resonators or by a system of resonators still raised a number of doubts at that time. The problem had been highly idealized, and the practical calculations were too schematic. Therefore acousticians working in this field were by no means certain of the reliability of theoretical predictions regarding the possibility of obtaining high absorption coefficients. Experimental verification of the theory was necessary.

Fig. 6.

Experiments on measuring sound absorption by resonators, carried out in the Acoustics Laboratory of the Palace of Soviets by S. T. Ter-Osipyan in 1937*), showed that a gradual increase of $R$ (by placing a number of layers of gauze over the opening $\sigma$) leads, in the end, to a value corresponding to formula (2), and gives a sound absorption coefficient $\alpha$ equal to unity, i.e. complete absorption. These experiments gave complete confidence in the correctness of the theoretical scheme I had proposed and served as an impetus for the further development of the theory and practice of resonant sound absorbers. Complete absorption of sound in these experiments was achieved with the aid of a very small quantity of a rare frictional material, such as gauze or metal mesh. These experiments suggested the possibility of using such a strong and durable material as metal meshes for creating effective absorber structures. The idea of using meshes as a frictional material was subsequently used by G. D. Malozhinets in designing multilayer frictional sound absorbers of high efficiency for the Palace of Soviets[^13].

*) The works remained unpublished.

S. N. RZHEVKIN

The importance of placing the frictional material in the zone of maximum velocities, i.e., in the neck of the resonator, was not duly taken into account by most authors who investigated this question in subsequent years (Zwikker^14, Jordan^15, Djigle^16). Since they proceeded from the Wintergerst scheme (Fig. 6), in which the resonators lay behind a perforated layer of porous structure (Celotex, etc.) (that is, the frictional elements had a very complex configuration), they were unable to take quantitative account of the magnitude of the damping and, consequently, of the magnitude of the sound-absorption coefficient. The work carried out by me jointly with Ter-Osipyan^17, and the work of Nesterov^18, brought complete clarity to the question of allowing for the damping of resonant systems with frictional material in the neck of the resonator, and made it possible to indicate methods for calculating the active resistance of resonators and the sound-absorption coefficient of a system of resonators. This made it possible to proceed with confidence to the design of more complex systems.

High absorption, close to unity, can be obtained in a simple resonant system with a single layer of resonators only near the resonant frequency. For producing high absorption over a wide range of frequencies, such a system proved unsuitable, and in solving this problem we turned to more complex systems. However, the single-layer absorber proved very convenient for solving a whole series of practical problems in architectural acoustics, as will be discussed somewhat below.

As a more rigorous solution^12 shows, a single-layer resonant sound absorber has not only one main absorption maximum. The expression for the absorption coefficient \(a\) under normal incidence of sound has the form:

\[ a = \frac{ 4R \frac{\sigma^2}{\Sigma}\rho c }{ \left( R + \frac{\sigma^2}{\Sigma}\rho c \right)^2 + \left( 2\pi f M - \frac{\sigma^2}{\Sigma}\rho c \operatorname{ctg}\frac{2\pi f L}{c} \right)^2 }, \tag{2a} \]

where \(f\) is the sound frequency, \(M\) is the mass of air in the opening of the resonator, and \(L\) is the depth of the layer of resonators. As is easy to see, at the frequency

\[ f_1 = \frac{c}{2L}, \]

and also at the frequencies \(2f_1\), \(3f_1\), etc., the absorption must be equal to zero (Fig. 7). Between these frequencies the absorption will attain certain maximum values, lying the closer to unity the more accurately the value \(R\) satisfies the condition

\[ R = \frac{\sigma^2}{\Sigma}\rho c. \]

Here it is appropriate to mention two initial attempts to calculate broadband sound absorbers, dating from 1937.^26 One of them belongs to G. D. Malyuzhinets, who proposed using a system of a large number of perforated screens,

installed parallel to one another at the wall, with sufficiently high sound absorption over a broad band being created by selecting the magnitude of the air friction in the openings; this determined the choice of the size of the openings. The magnitude of the friction was chosen on the basis of Wintergerst’s data on the passage of sound through openings[^27]. Calculations of an absorber giving \(\alpha > 0.9\) from 100 cps to 4,000 cps led to a complex structure of 22 parallel perforated sheets with a total thickness of about 1 m. Wintergerst’s data concerning the passage of sound through openings, from which Maluzhinets proceeded, give, when recalculated as a coefficient of friction, a value that differs greatly from other experimental data[^28], and therefore the calculation that was made raised doubts. The considerable bulkiness of the structure and the dubious nature of the data underlying its calculation were the reason why this interesting idea was abandoned and not even tested experimentally.

Fig. 7.

Another attempt to create an effective broadband absorber was made by me and consisted in arranging systems of resonators, tuned to different regions in a wide frequency range, not in series but parallel to one another. In this case the openings into the large resonators, absorbing the lower frequencies, would have to pass through one or two layers of resonator-absorbers tuned to higher frequencies. It was immediately found that such a construction was practically difficult to realize; an exact calculation of a system of resonators arranged in parallel was also very difficult. All this led to the idea of a parallel system of absorbers likewise being abandoned, although there is no doubt that within certain limits it may fully justify itself.

After this, intensive development was begun of methods for calculating and designing resonant multilayer absorbers, which promised quite real prospects for creating highly effective absorption over a broad frequency range.

§ 5. Multilayer Resonant Sound Absorber

As I have succeeded in showing,^19 a system consisting of several resonant cells connected in series (Fig. 8), with suitable values of the frictional resistance \(R_i\) in the openings, can give high absorption over a wider frequency range than the simple resonant system described above. The absorber schematically depicted in Fig. 8, under the assumption that the wavelength is large, may be regarded as a lumped system with several degrees of freedom. Its absorption coefficient will have a number of maxima equal to the number of layers. A number of detailed works by V. S. Nesterov^20 were devoted to the study and calculation of complex resonant absorbers. The absorption coefficient \(a\) as a function of frequency, for given dimensions and values of the resistances \(R_i\) (the direct problem), is found, for normal incidence of sound, by the well-known formula

Fig. 8.

\[ a = 1 - \left|\frac{\dfrac{Z}{\Sigma} - \rho c}{\dfrac{Z}{\Sigma} + \rho c}\right|^2, \tag{3} \]

where \(Z\) is the input impedance over the area of one resonator cell. The quantity \(Z\) is determined for any number of layers through their impedances by an elegant formula, in the form of a continued fraction, found by Nesterov.^21 In practice, however, what is of interest is the solution not of the direct problem, but of the inverse problem—the problem of finding the parameters (dimensions) of the system and the values of the resistances \(R_i\), from the value of the absorption coefficient \((a)\) in a given frequency range. This difficult problem was solved by V. S. Nesterov^20 for two- and three-layer resonant systems. He showed that, with the aid of a two-layer system, it is possible to obtain easily \(a > 0.9\) over a range of about \(2 \tfrac{1}{2}\) octaves and \(a > 0.95\) over a range of one octave. With the aid of a three-layer system it is possible to obtain \(a > 0.9\) over a range of about 3–4 octaves.

An example of the frequency characteristic \(a\) for a two-layer system at normal incidence of sound is given in Fig. 9, and for a three-layer system in Fig. 10. All the conclusions of the theory were carefully verified experimentally by Nesterov in a large number of experiments (by the standing-

waves in a tube), and exceptionally good agreement between theory and experiment was found^18.

It should be noted that a great step forward in the field of the theory of sound was made in the work of Academician V. A. Fock^22, who, having become interested in our work on sound absorption, solved the very difficult problem of finding the acoustic conductance \((K)\) of an opening of diameter \(d\) in a partition standing across a tube of diameter \(D\). Fock showed that the conductance is

\[ K=\frac{D}{1-\varphi\left(\frac{d}{D}\right)}, \tag{4} \]

where \(\varphi\left(\frac{d}{D}\right)\) is expressed in the form of an infinite series in powers of \(\frac{d}{D}\).

Fig. 9.

For \(D=\infty\), i.e., for an opening in an infinite screen, the function \(\varphi\) becomes zero, and we obtain the well-known Rayleigh expression \((K=D)\). As \(d\) approaches the diameter of the tube, \(\varphi\) tends to unity, and the conductance \(K\) to infinity, which means the absence of added kinetic energy (over and above the energy of the plane wave) and of added mass in such an opening. For \(\frac{d}{D}<0,2\) Fock’s correction is small, and it can often be neglected in practice. The conductance \(K\) makes it possible to determine the additive or

Fig. 10.

the added mass

\[ \left(m=\frac{\pi d^2}{4}\frac{\rho}{K}\right), \]

which determines the inertial properties of the air in an opening of diameter \(d\). The introduction of the correction into Fock’s formula made it possible to carry out reliable calculations of resonant absorbers at large ratios \(d/D\), since the problem of calculating a resonant absorber under normal incidence of sound is identical with the problem of calculating the propagation of sound in a tube resting on an area occupied by one resonator.

Fig. 11.

V. S. Nesterov has given a method for the rapid calculation of the absorption of a multilayer system by means of a special nomogram[^21]. With the aid of this method it is possible to simplify very considerably the solution of the inverse problem for complex systems by the gradual fitting of the parameters through a series of trials. Thus, for example, the parameters of a four-layer system with absorption \(a > 0.9\) in the range—

Fig. 12.

zone from 100 Hz to 4000 Hz. The frequency characteristic \(\alpha\) is given in Fig. 11.

As a result of the entire body of work mentioned above, the full possibility was established of constructing a resonant absorber that provides a sufficiently small reflection of sound and eliminates echo.

The beginning of the war in 1941 delayed the broad development of work on resonant sound absorbers and their introduction into practice. A two-layer sound absorber was installed on the ceiling of the large studio of the House of Sound Recording in Moscow (Fig. 12). Measurements carried out in 1945 by S. T. Ter-Osip’yants showed that the absorption proved to be close to that expected according to the calculation.

§ 6. RESONANT SOUND ABSORBERS IN THE DIFFUSE SOUND FIELD OF A ROOM.

The cases of sound absorption by a resonant absorber considered above refer to normal incidence of sound. For oblique incidence of sound at an angle \(\vartheta\) on an absorber with separate resonant cells, for calculating \(a_\vartheta\) one may use the formula following from the work of Rayleigh and Paris \({}^{23}\)

\[ a_\vartheta = 1- \left| \frac{\dfrac{Z}{\Sigma}\cos\vartheta-\rho c} {\dfrac{Z}{\Sigma}\cos\vartheta+\rho c} \right|^{2} = \]

\[ = \frac{4R_1\cos\vartheta} {(R_1\cos\vartheta+1)^2+\cos^2\vartheta\,(kM_1-\operatorname{ctg}kL)^2}, \tag{5} \]

where \(Z=R+jY\) is the impedance over the area of the cell of one resonator for normal incidence of sound; \(k=\dfrac{2\pi}{\lambda}\); \(\lambda\) is the wavelength; \(R_1=\dfrac{R}{\Sigma\rho c}=\dfrac{r}{\rho c}\cdot\dfrac{\Sigma}{\sigma}\) is the mean dimensionless active resistance per unit area of the absorber, expressed in units of \(\rho c\), where \(\rho\) is the density of air and \(c\) is the speed of sound in air, and \(r\) is the coefficient of friction of the porous material. The quantity \(M_1\) represents the mean dimensionless mass per unit surface, and \(L\) the depth of the resonant layer. The resonant maximum of absorption will be obtained at all angles of incidence at one and the same frequency, but its magnitude will be different; it will be the greater, the closer the quantity

\[ \frac{R}{\Sigma\rho c}\cos\vartheta=R_1\cos\vartheta \]

is to unity. If, as must be done in broadband absorbers, the quantity \(R_1\) is of the order of 3–4, the absorption maximum will be obtained at a very oblique angle of incidence.

The mean absorption coefficient is determined by Paris’s formula

\[ \bar a = 2\int_{0}^{\frac{\pi}{2}} a_\vartheta \cos\vartheta \sin\vartheta\, d\vartheta . \tag{6} \]

Calculation shows that if \(a_\vartheta\) reaches a maximum at angles on the order of \(70^\circ\)—\(80^\circ\), then \(\bar a\) will be somewhat greater than \(a_0\) at normal incidence \((\vartheta = 0)\).

If the partitions between the individual resonators are absent, then another variety of resonance absorber is obtained, which may be called a “layer-resonance” absorber. At normal incidence of sound the properties of an absorber with partitions (compartments) and without them are the same, but at oblique incidence they differ substantially. As shown by K. A. Vital’[^24], the expression for \(a_\vartheta\) of a layer-resonance absorber is obtained from an expression of type (5), but with the layer depth \(L\) replaced by \(L\cos\vartheta\).

The expression for \(a_\vartheta\) has the form:

\[ a_\vartheta = \frac{4R_1\cos\vartheta} {(R_1\cos\vartheta + 1)^2 + [kM_1\cos\vartheta - \operatorname{ctg}(kL\cos\vartheta)]^2}. \tag{7} \]

The resonance maximum of absorption, according to formula (7), is obtained at a different frequency for each angle of incidence.

The calculation of \(a_\vartheta\) and \(\bar a\) by the formulas given above is not a very complicated, but a very laborious computational problem. These calculations are of particular interest for the simplest systems with one layer of resonators, which are very convenient for building practice. Below are given the results of such calculations for a number of absorbers.

§ 7. RESONANCE SOUND ABSORBERS FOR BUILDING PRACTICE

For the purposes of building practice, in halls, theaters, studios, cinemas, etc., sound absorbers of very high efficiency are usually not required; however, requirements are imposed for good architectural and construction characteristics and for the possibility of widely varying the acoustic characteristics in order to obtain one effect or another. The simplest resonance sound absorbers, consisting of one layer of resonators arranged along a wall or ceiling, present in this respect a number of advantages. They can be realized in the form of a very strong construction of light weight; especially important is the strength of the outer layer, consisting of a rigid sheet with small openings. The constructions, examples of which we give below, are quite simple to implement in the...

and can be made on the construction site. The construction of the absorber can easily be made fire-resistant and is not subject to damage from dampness, rodents, or moths. The acoustic indicators can be varied at the designer’s discretion within very broad limits. It is possible to design an absorber for a band of low, medium, or high frequencies. By combining these types, any variants of the frequency characteristic can be obtained. It is also possible to calculate and construct an absorber with a narrow absorption band, intended to correct acoustic defects observed in a room that is already finished or is at the stage of finishing and acoustic adjustment.

Fig. 13.

The solution of the inverse problem for a diffuse sound field, i.e., finding the parameters of the absorber from a prescribed form of the frequency characteristic, is a problem of extraordinary complexity, and, in general, it has not yet been possible to indicate ways of solving it. Therefore one has to resort to the method of trial calculations according to the formulas given above, as a result of which it is possible, in the end, to find the structural parameters (the depth of the resonators \(L\), the distance between them \(a\), the diameter of the holes \(d\), and the coefficient of friction \(r\)) that satisfy the specified requirements for the frequency characteristic of absorption. This path is extremely cumbersome and unacceptable for practitioners engaged in the construction of studios, theaters, etc. Therefore we have outlined\(^ {25}\) a fundamentally different way of solving the problem, which greatly simplifies it and makes it possible to provide methods of engineering calculation accessible to non-specialists in acoustics.

It is possible to choose in advance several types of absorption frequency characteristics suitable for solving the main architectural-acoustic problems and then to indicate ways of their constructive implementation (by means of resonant absorbers), ensuring the production of such characteristics. Figure 13 shows, for example, characteristics selected in this way for a broadband absorber for low frequencies (ЗП—3б) or for high frequencies (ЗП—4а). Both of these characteristics correspond to the value of the specific active resistance \(R_1 = 4\) and prove convenient for solving a number of practical problems. If \(R_1 < 4\), then the absorption characteristic is obtained with a sharper and higher

maximum; if, however, \(R > 4\) is taken, then the characteristic is smoothed out and lowered. Both are undesirable, in view of which it was decided to settle on the standard characteristic with \(R_1 = 4\) and to use it in different frequency ranges, for which it was necessary to specify the corresponding design parameters (\(L\), \(a\), \(d\), and \(r\)).

Fig. 14.

To realize the first type of characteristic ZP—3b, the construction shown in Fig. 14 may be used. It has a wooden frame forming compartments every 250 mm, which prevent the propagation of sound along the wall and ensure the operation of the entire system in that frequency region where \(\lambda > 250\) mm practically as if all the resonant cells were separated from one another by partitions. In the perforations of the thin covering sheet (\(l = 0.5\) mm), fabric with a specific resistance of 5 mechanical ohm/cm\(^2\) (coarse calico, cambric) is installed. This construction is not the only one solving the problem of obtaining the characteristic shown in Fig. 13. Any number of such solutions may be given, depending on the resistance of the fabric used and the thickness of the covering sheet in which the holes are made. All other possible variants have the same depth of the resonator layer \(L = 160\) mm, while the values of \(d\) and \(a\) may be calculated from

based on simple formulas or graphs, if the friction coefficient \(r\) and the thickness of the front wall are specified \([25]\). Thus, for example, the same characteristic can be obtained with a facing sheet of plywood \(5\ \mathrm{mm}\) thick (instead of \(0.5\ \mathrm{mm}\) in Fig. 14) and with the aid of a fabric having a resistance of about \(20\ \mathrm{mech.\ ohm/cm^2}\), and with holes \(11\ \mathrm{mm}\) in diameter located \(28\ \mathrm{mm}\) apart.

If the partitions in the air cavity between the facing sheet and the wall are removed, as is shown in Fig. 14 at the top, then the absorption characteristic will shift into the higher-frequency region, as is shown in Fig. 13 (ZP—3a). In this case, to increase rigidity, the facing sheet should be fastened to the wooden frame in the form of crosspieces.

The design of an absorber with the high-frequency characteristic ZP—4a can be realized according to the scheme shown in Fig. 14 (bottom), i.e., without partitions. In this case the distance of the facing sheet (of thickness \(0.5\ \mathrm{mm}\)) from the wall is taken as \(80\ \mathrm{mm}\); the hole diameter may be taken, for example, as \(5\ \mathrm{mm}\), and their mutual spacing as \(17\ \mathrm{mm}\); the fabric in the holes must in this case have about \(12\ \mathrm{mech.\ ohm/cm^2}\). When using fabric with a resistance of about \(30\ \mathrm{mech.\ ohm/cm^2}\), in order to obtain the same characteristic it will be necessary to take holes \(22\ \mathrm{mm}\) in diameter at spacings of \(45\ \mathrm{mm}\), with the same sheet thickness and the same distance from the wall.

Fig. 15.

To obtain selective absorption with a maximum in a narrow band of 1–2 octaves, it is convenient to use the design of a resonant absorber with compartments. Fig. 15 shows an approximate design of a narrow-band absorber, for which \(\alpha > 0.5\) in the range from 55 to 220 cycles. The depth of the resonators is \(160\ \mathrm{mm}\); the holes may be taken as \(30\ \mathrm{mm}\) in diameter at spacings of \(200\ \mathrm{mm}\); the fabric has a resistance of \(3\ \mathrm{mech.\ ohm/cm^2}\). Absorbents of this type make it possible to obtain high absorption easily with comparatively small dimensions, which is especially important in the low-frequency region, where ordinary absorbents are completely ineffective.

Approximate data on the dimensions of such an absorber may be obtained by the method of an approximate solution of the inverse problem for

of normal incidence of sound[^19]. The depth of the layer \((L)\), the diameter of the resonator openings \((d)\), and the distances \((a)\) between the openings may be calculated, on the basis of the specified absorption \(a \geq a_1\), in the frequency interval from \(f_1\) to \(f_2\), by the following formulas:

\[ L_{\text{cm}}=\frac{c}{4\pi}\frac{a_1}{\sqrt{1-a_1}}\cdot\frac{1-f_1/f_2}{f_1} \simeq 2700\frac{a_1}{\sqrt{1-a_1}}\frac{1-f_1/f_2}{f_1}, \tag{8} \]

\[ d_{\text{cm}}=\frac{4}{\pi^2\varphi}\,r\, \frac{\sqrt{1-a_1}}{2-a_1}\frac{1/f_1}{f_2/f_1-1} -\frac{4}{\pi}l \simeq 330r\frac{\sqrt{1-a_1}}{2-a_1}\frac{1/f_1}{f_2/f_1-1} -\frac{4}{\pi}l, \tag{9} \]

\[ a_{\text{cm}}= \sqrt{\frac{\pi\rho c}{4}\,d_{\text{cm}}} \sqrt{\frac{2-a_1}{r a_1}} \simeq 5.67d_{\text{cm}}\sqrt{\frac{2-a_1}{r a_1}}. \tag{10} \]

The maximum absorption, equal to \(a_m=a_1(2-a_1)\), is obtained at the frequency \(f_m=\sqrt{f_1f_2}\). In these formulas \(l\) is the thickness of the covering sheet in cm, and \(r\) is the specific friction of the fabric placed in the openings, in mechanical ohms/cm\(^2\). The specific value of the active resistance is then specified as \(R_1=\frac{2-a_1}{a_1}\), i.e., it is always greater than unity. Observance of this condition ensures, for a specified absorption at the boundaries, the widest range \(f_2-f_1\) in which \(a\geq a_1\); moreover, the nonuniformity of the frequency characteristic, determined in our method of calculation by the quantity \(\frac{a_m}{a_1}=2-a_1\), will in practice be quite small \((1.2—1.6)\).

To obtain \(a\geq 0.7\) in the range from 50 to 70 cycles, with a covering-sheet thickness of 6 mm, we find from the formulas \(L=210\) mm, \(d=30\) mm, and \(a=350\) mm; the fabric in the openings is taken with a resistance of only 0.5 mechanical ohms/cm\(^2\) (gauze or metal mesh). To increase the rigidity of the front wall of resonators of such large size (in order to avoid its oscillations), ribs of rigidity may be made under it. Such an absorber can successfully attenuate, for example, fan noises, which usually lie in a very low-frequency region. Narrow-band absorbers are also extremely valuable for correcting the frequency characteristic of studios and theaters, since they make it possible to eliminate excessive reverberation in one or another frequency region.

An interesting modification of the resonant sound absorber is obtained if the resistance is produced only by the friction of air in the opening. Calculation shows that this kind of absorber gives a rather convenient solution of the problem for low absorption coefficients. It is possible, for example, to obtain \(a\geq 0.3\) \((a_m=0.51)\) in the range from 50 to 100 cps by means of resonators formed behind a sheet of thin tin \((l=0.3\) mm), placed at a distance of 10 cm from the wall, with small openings of only 1 mm diameter, spaced 7 cm apart. The presence of fairly considerable absorbing properties in such a system is, at first sight, somewhat unexpected.

The absorption coefficient in a diffuse sound field for narrow-band absorbers will always be somewhat greater than is assumed in formulas (8), (9), and (10) for the case of normal incidence. The point is that, in deriving these formulas, as indicated above, the condition

\[ R_1=\frac{2-a_1}{a_1}. \]

is imposed. When \(a_1\) is chosen within the limits \(0.5\)—\(0.8\), \(R\) will lie between 3 and 1.5. Consequently, according to the condition indicated above, \(R_1\cos\vartheta=1\), maximum absorption will occur not at \(\vartheta=0\), but at oblique angles of incidence, and the mean diffuse coefficient will be greater than \(a_0\).

The openings of the resonators need not be made round, but may be cut in the form of slots, rectangles, stars, etc. (Fig. 16), and may be arranged not on a square lattice but according to some pattern. If the same total area of openings \(N\sigma\) per unit area of absorber is preserved, approximately the same coefficient \(a\) will be obtained, independently of the shape of the openings.

Fig. 16.

In the practical construction of resonant sound absorbers, the co-oscillation of the front wall covering the resonators may be of considerable importance if it is not sufficiently rigidly fastened. At first glance it seems that the wall is a heavy system, and that the velocity of its oscillations under the action of sound cannot reach significant values; however, the volume velocity produced by the wall, i.e. the product of the velocity by the area \((\Sigma)\), may in some cases (a light wall, very small and sparse openings) prove to be of the same order as the volume velocity through an opening of area \((\sigma)\). As a result, oscillations of the wall will have a substantial influence on the behavior of the entire system.

The possibility that wall oscillations influence absorption could already have been supposed on the basis of Wente’s theoretical considerations\(^{30}\), concerning the influence on sound absorption of the rigid frame of a porous material, and of the experimental works of E. Meyer\(^{2}\) and Lauffer\(^{29}\). In 1941 Malozhinets\(^{31}\) calculated the impedance of a freely suspended perforated sheet and showed that the inertial properties of such a system, which determine the passage of sound through it, depend not only on the mass \((m_1)\) of the air in the openings (mainly the added mass), but also on the mass \((m_2)\) of the wall (screen) attributable to one opening. The total acoustic mass that determines the inertial properties of the system is found from the expression:

\[ M=\frac{M_1M_2}{M_1+M_2}, \tag{11} \]

where \(M_1=\dfrac{m_1}{\sigma^2}\) and \(M_2=\dfrac{m_2}{\Sigma^2}\) are the acoustic mass of the air in an opening of area \(\sigma\) and the acoustic mass of a piece of wall of area \(\Sigma\), falling to one opening. Thus, the mass of the system consists, as it were, of a parallel connection of the masses \(M_1\) and \(M_2\) (“sound mushroom”) and turns out to be smaller than one mass in the opening. In cases where \(M_1\) and \(M_2\) are of the same order, this should lead to a considerable increase in the resonant frequency of the system.

The front wall of resonators used in resonant sound absorbers is not a freely suspended screen; it is rigidly fixed along its edges to a certain frame. Nevertheless, under the action of sound pressure the wall can be set into vibration as an elastic membrane. The influence of such vibrations of the front wall of the resonator on sound absorption was analyzed by me in a recent theoretical work\(^{32}\). The problem is idealized by the assumption that the entire front wall of the resonator vibrates as a flat rigid plate-piston, without bending; the elastic element is assumed to be concentrated only along the edges of the plate (Fig. 17).

Fig. 17.

Fig. 18.

The problem is solved by setting up the Lagrange equations of the system; an interesting feature of it is the circumstance that the kinetic energy of the added mass of the opening is a function of the relative velocity of the wall and of the air in the opening, and not of the absolute velocity of the air; the same also applies to the energy dissipated by friction in the opening. The resulting expression for the impedance of a resonant sound absorber with a compliant wall is rather complicated and therefore is not given here.

As a special case, for a very large stiffness of the plate mounting, one obtains the case of a resonant absorber already examined in detail above\(^{12}\). With stiffness equal to zero, one obtains the case analyzed by Malyuzhinets\(^{31}\).

The formulas I obtained also made it possible to analyze the case where the resonance of the wall lies in the region close to the natural frequency of the resonant sound absorber. Fig. 18 presents the resul—

coefficient of the sound absorption of a system calculated for absorption \(a > 0.6\) from 100 to 400 Hz. Such a system can be realized by means of resonators with a front wall of iron \(0.5\) mm thick, spaced \(33\) cm from the wall, with cells of area \(a^2 = 2.6 \times 2.6\ \mathrm{cm}^2\) and holes of diameter \(d = 0.4\) cm, covered with fabric having a friction coefficient \(r = 3\) mech. ohm/\(\mathrm{cm}^2\).

Assuming that the resonant frequency of the wall lies at 214 Hz, i.e. exactly in the center of the absorption range, one can further calculate the impedance and absorption of the system. The parameter characterizing the friction in the plate can be determined only experimentally from the decrement of damping, as was done on the model. The calculation illustrated in the drawing of Fig. 18 was carried out for three different friction parameters (expressed in acoustic ohms): \(R_2 = 0\); \(R_2 = 0.1 R_1\), and \(R_2 = R_1\), where \(R_1\) is the friction parameter in the aperture of the resonant sound absorber. The value \(R_2 = 0.1 R_1\) is closest to the value found experimentally. From Fig. 18 it is seen that, for \(R_2 = 0\), the absorption has a sharp dip to zero near the resonant frequency of the plate; for \(R_2 = 0.1 R_1\), the curve \(\alpha\) runs below that for a rigidly fixed plate up to the resonance of the plate, and above it after the resonance; for \(R_2 = R_1\), the whole curve runs higher. Since the individual cells of a resonant sound absorber are in practice not constructed strictly identically (especially with respect to the conditions of fastening of the front wall), one may expect substantial differences in the resonant frequencies and friction parameters of the individual cells. On the average, for a large number of cells, one may assume that the co-oscillations of the wall will not cause significant deviations from the calculations made under the assumption of an absolutely rigid fastening, as is done in the formulas given above for practical calculations.

In order that the co-oscillations of the wall should have practically no effect on the absorption of a resonant system, it is sufficient to make the natural frequency of the wall approximately an octave higher than the frequency of the resonator; for this the wall must be sufficiently rigidly fastened.

The data set forth in this review are the result of many years of work (1936–1945) by a large collective, and I take this opportunity to express my sincere gratitude to all its participants.

LITERATURE

1. W. Sabine. Collected Papers on Acoustics, Harvard Univ. Press, Cambridge USA, 1927.
2. E. Meyer. El. Nachr. Techn., 13, issue 3, 1936.
3. C. W. Glover. Practical Acoustics for the Constructor, London, 1933.
4. E. Wintergerst. ZS. d. VDI, 77, 91, 1933.
5. S. N. Rzhevkin. ZhTF, 6, 2103, 1936; Techn. Phys., 3, 1, 1936.
6. M. S. Antsyferov. ZhTF, 6, 2118, 1936.
7. S. N. Rzhevkin. Proceedings of the Acoustical Commission of the Department of Technical Sciences of the Academy of Sciences of the USSR, collection 3, 17, 1939.

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