Air Absorption

This section provides some further details regarding acoustic air absorption [321]. For a plane wave, the decline of acoustic intensity as a function of propagation distance $x$ is given by

$$I(x) = I_0 e^{-x/\xi},$$

where

$$\begin{eqnarray*} I(x) &=& \hbox{intensity $x$\ meters from the source}\\ & & \hbox{(\sref {intensity} defines acoustic intensity), }\\ I_0 &=& \hbox{intensity at the plane source (\textit{e.g.}, a vibrating wall),}\\ \xi &=& \hbox{intensity decay constant (meters)}\\ & & \hbox{(depends on frequency, temperature, humidity, and pressure).} \end{eqnarray*}$$

Tables B.1 and B.2 (adapted from [317]) give some typical values for air.

Table B.1: Attenuation constant $m = 1/\xi$ (in inverse meters) at 20 degrees Celsius and standard atmospheric pressure

Relative	Frequency in Hz
Humidity	1000	2000	3000	4000
40	0.0013	0.0037	0.0069	0.0242
50	0.0013	0.0027	0.0060	0.0207
60	0.0013	0.0027	0.0055	0.0169
70	0.0013	0.0027	0.0050	0.0145

Table B.2: Attenuation in dB per kilometer at 20 degrees Celsius and standard atmospheric pressure.

Relative	Frequency in Hz
Humidity	1000	2000	3000	4000
40	5.6	16	30	105
50	5.6	12	26	90
60	5.6	12	24	73
70	5.6	12	22	63

There is also a (weaker) dependence of air absorption on temperature [184].

Theoretical models of energy loss in a gas are developed in Morse and Ingard [321, pp. 270-285]. Energy loss is caused by viscosity, thermal diffusion, rotational relaxation, vibration relaxation, and boundary losses (losses due to heat conduction and viscosity at a wall or other acoustic boundary). Boundary losses normally dominate by several orders of magnitude, but in resonant modes, which have nodes along the boundaries, interior losses dominate, especially for polyatomic gases such as air.^B.34 For air having moderate amounts of water vapor ($H_2O$ ) and/or carbon dioxide ($CO_2$ ), the loss and dispersion due to $N_2$ and $O_2$ vibration relaxation hysteresis becomes the largest factor [321, p. 300]. The vibration here is that of the molecule itself, accumulated over the course of many collisions with other molecules. In this context, a diatomic molecule may be modeled as two masses connected by an ideal spring. Energy stored in molecular vibration typically dominates over that stored in molecular rotation, for polyatomic gas molecules [321, p. 300]. Thus, vibration relaxation hysteresis is a loss mechanism that converts wave energy into heat.

In a resonant mode, the attenuation per wavelength due to vibration relaxation is greatest when the sinusoidal period (of the resonance) is equal to $2\pi$ times the time-constant for vibration-relaxation. The relaxation time-constant for oxygen is on the order of one millisecond. The presence of water vapor (or other impurities) decreases the vibration relaxation time, yielding loss maxima at frequencies above 1000 rad/sec. The energy loss approaches zero as the frequency goes to infinity (wavelength to zero).

Under these conditions, the speed of sound is approximately that of dry air below the maximum-loss frequency, and somewhat higher above. Thus, the humidity level changes the dispersion cross-over frequency of the air in a resonant mode.