Sound Absorption*

V. O. Knudsen

Recent investigations in acoustics have substantially changed our ideas about the absorption of sound in rooms, in the open air, and also in pure and mixed gases. Until the last three years all theories of sound absorption in rooms were based on two assumptions: 1) that, after the source is stopped, the sound energy is conserved in rays of sound energy having the same frequency as the initial sound produced in the room, and that the sound energy is completely dissipated during the entire time of decay; and 2) that the absorption of sound in the medium (air) is insignificant and may be neglected. It is obvious, however, that a simple rectangular room is a bounded three-dimensional space capable of executing free vibrations, whose frequency \(f\) is given by the formula:

\[ f=\left(\frac{c}{2}\right)\left(\frac{n^{2}}{l_{1}^{2}}+\frac{p^{2}}{l_{2}^{2}}+\frac{q^{2}}{l_{3}^{2}}\right)^{1/2}, \]

where \(c\) is the speed of sound in air, \(l_1, l_2, l_3\) are the dimensions of the room, and \(n, p\), and \(q\) are ordinal numbers taking the values \(0, 1, 2,\ldots\). The author has shown that these characteristic frequencies are very sharply expressed in small rectangular rooms and that the reverberation of sound is made up of these free vibrations. In general, with the exception of very low frequencies, a number of adjacent free vibrations participate in the decay, and in general the listener identifies the dying sound with sound of the same frequency as that imparted to the room. However, when a small rectangular room is excited by a sound whose frequency differs only by a barely perceptible amount from the frequency of one of the principal modes of vibration of the room, the ear can easily detect that the decaying vibrations have the frequency of the room’s free vibrations and not that of the imposed (forced) frequency. It is evident, therefore, that an exact theory of reverberation must take into account the rate of damping of free vibrations in a room. Some work in this direction has already been done by Stratton and Schuster and by Wezmann².

* Review of Scientific Instruments, No. 12, 4, 637–639, 1933; translated by Ya. Kopilovich.

Sound Absorption

Even more, however, remains to be done before an exact formula can be found for practical calculations of sound absorption in rooms.

The assumption that the absorption of audible sound in air can be neglected is no longer acceptable. All previous theories of reverberation were based on the assumption that all absorption occurs at the boundaries, i.e. that the absorption of the medium, in view of its insignificance, can be neglected; and many investigators made an analogous assumption when calculating the propagation of audible sound through air. According to the classical theories of Stokes, Kirchhoff, and Rayleigh on the absorption of sound in gases, attenuation in the medium would be negligibly small (at least for all the principal calculations of architectural acoustics) for frequencies up to 8000 cycles. These earlier theories showed that attenuation becomes significant at very high frequencies—the attenuation constant is proportional to the square of the frequency—but for frequencies below 8000 cycles the attenuation is not so great that it would have to be taken into account in ordinary calculations of reverberation or in many other problems of sound signaling. However, as has been shown by recent experiments, this does not correspond to reality. It has been shown, for example, that the absorption of audible sound in air is about 10–100 times greater than was assumed by the classical theory, and that attenuation is characteristically dependent on temperature and on the presence of other gases, such as water vapor; moreover, this dependence could by no means have been expected from the classical theory.^3 The attenuation of a plane sound wave is represented by the equation \(I_x = I_0 e^{-mx}\), where \(I_0\) is the intensity of the sound wave at the position \(x = 0\), \(I_x\) is its intensity after it has traveled the distance \(x\), and \(m\) is the attenuation constant, i.e. \(1/m\) is the distance that the wave must travel in order for its intensity to decrease to \(1/e\) of its initial intensity.

Measurements of the rate of decay in two experimental rooms with identical bounding materials but different mean free paths make it possible to determine this attenuation constant \(m\), as well as the coefficient of surface absorption of the bounding surfaces. Thus, at 10,000 cycles, \(m = 0.00003\ \mathrm{cm}^{-1}\) for dry air at a temperature of \(20^\circ\mathrm{C}\); when water vapor is added to the air, the value of \(m\) increases to a maximum of \(0.00065\ \mathrm{cm}^{-1}\) at a relative humidity of 18%; with a further increase in humidity the value of \(m\) again decreases, falling to \(0.00022\ \mathrm{cm}^{-1}\) at a relative humidity of 80%. At 6000 cycles the maximum absorption occurs at a relative humidity of 13%, and the value of \(m\) at this maximum is \(0.00039\ \mathrm{cm}^{-1}\), which is just 60% of the magnitude of the maximum at 10,000 cycles. Analogous data at these and other

...their frequencies show that the maximum value of \(m\) at any frequency is proportional to the first, and not the second, power of the frequency, as classical theory requires. For comparison, the values of \(m\) given by the classical theory would be \(0.000027\ \mathrm{cm}^{-1}\) for \(10\,000\) hertz and \(0.000010\ \mathrm{cm}^{-1}\) for \(6000\) hertz, and there should have been practically no change in \(m\) under the above-mentioned changes of humidity and temperature.

Analogous measurements of the absorption of sound in oxygen and water vapor show that the coefficient \(m\) reaches a maximum value approximately five times greater than for air, but the maximum occurs at a higher concentration of water vapor. Further, the absorption in nitrogen differs little from the results obtained according to classical theory, and is not altered by the presence of water molecules. These results are satisfactorily explained by Kneser\(^{4}\) on the assumption that, in collisions between oxygen and certain other molecules, such as water molecules, the entire translational energy of the colliding molecules is transformed into the vibrational energy of the oxygen particles. When the gas is compressed adiabatically, during the compression phase of the sound oscillations, the number of oscillating molecules increases, while when the gas expands the number of oscillating molecules decreases. If the compression and expansion take place sufficiently slowly, the gas will always remain in a state of thermal equilibrium, and the transformation of translational energy into vibrational energy during compression will be exactly equal to the transformation of vibrational energy into translational energy during expansion. In general, however, a finite time (equal to the “mean lifetime of the vibrating molecule”) is required for the establishment of thermal equilibrium between normal and excited molecules. If the cycle of compression and expansion takes place in a period of time comparable with the “mean lifetime,” the process is no longer reversible and, consequently, sound energy is transformed into heat. On the other hand, if the cycle of the sound wave takes place in a time interval very short in comparison with this “mean lifetime,” only a very small part of the energy is transformed and, consequently, the sound energy will not be absorbed. Kneser’s theory is based on one of Einstein’s early papers\(^{5}\), which shows that measurement of the scattering of sound in a partially dissociated gas must provide a means for determining the rate of dissociation of that gas. In applying this to the problem of sound absorption, the reaction takes place not between atoms and molecules, as in Einstein, but between normal and excited oxygen molecules, and water vapor acts as a catalyst, exerting a very large influence on the “mean lifetime” of the vibrational quantum. From these theoretical considerations and the known value of the vibrational energy for

O$_2$ (4420 cal/mole) Kneser can provide an explanation of the author’s experimental results on the absorption of sound in air and oxygen.

The new results concerning the absorption of sound in gases are of great importance for problems of sound signaling in air, architectural acoustics, and the reproduction of sounds both in enclosed rooms and in the open air. In addition, these new results show that sound absorption must be explained in terms of quantum, not classical, mechanics, and they place in the hands of the investigator a new technique for studying molecular reactions in gases. It is possible that measurements of sound absorption in gases will reveal still more about molecular structure and molecular reactions than can be discovered by measurements of the velocity of sound, which have long been applied in connection with problems of the molecular physics of gases. Absorption is a direct and large effect of energy transformation in molecular collisions, whereas scattering is an indirect and almost always immeasurably small effect. The results of measurements of absorption in gases, which demonstrate the possibilities of this new technique for investigating the properties of molecules, will be published soon.

Literature

1. M. I. O. Strutt, Zs. ang. Math. Mech. 10, 360, 1930.
2. K. Schuster and F. Waezmann, Ann. d. Phys. 1, 671, 1929.
3. V. O. Knudsen, J. Acous. Soc. Am. 5, 112, 1933.
4. H. O. Kneser, J. Acous. Soc. Am. 5, 112, 1933.
5. A. Einstein, Ber. d. Berl. Akad. 5, 380, 1920.