Absorption
One should distinguish between attenuation, which is a dimunition in intensity for any cause whatever, and absorption, in which energy in the sound wave is transformed into some other form, usually heat. No simple mechanism for the absorption of sound energy by a perfect gas immediately suggests itself, and, in fact, sound does propagate with remarkably little absorption. It is attenuated mainly by spreading, scattering, and absorption by surfaces. The finite viscosity, heat conduction, and molecular mean free path of the medium do give rise to an absorption that increases at higher frequencies. One interesting effect is that the temperature changes in the adiabatic processes may equilibrate with molecular internal degrees of freedom at low frequencies, while at higher frequencies equilibration does not occur. The result is an effective change in γ, which not only changes the phase velocity, but also introduces an absorption because of the phase lag that is effective in the band of frequencies where the phase velocity is changing. The main other contribution to absorption is called viscothermal because it involves these transport properties of the gas, and increases as the square of the frequency. It is also called classical absorption. The distance at which the amplitude of a sound wave is diminished by a factor of 1/e by viscous absorption is (3c/8π2ν)λ2, where ν is the kinematic viscosity, 0.132 cm2/s for air. At 1000 Hz, this distance is more than 10 km. The classical intensity absorption coefficient for air is given in tables as α = 1.61 x 10-10f2 dB/m, where ln(I0/Id) = 2αd. At low pressures, absorption occurs when the wavelength becomes comparable to the molecular mean free path (about 66nm in air at STP). Most of these effects are considerable only at frequencies well above the audible range or at very low pressures. If the mean free path is taken as inversely proportional to the pressure, then the mean free path becomes 1/10 of the wavelength for 1000 Hz at a pressure of about 1.3 μHg, a low vacuum.
The effect of humidity on sound propagation is small, but rather complicated. At 68°F, the phase velocity increases from about 1127 ft/s for dry air to about 1131 ft/s at 100% humidity, mainly due to the decrease in density. The absorption at 100 Hz is 1.67 dB/km for dry air, 0.38 dB/km at 50% humidity, and 0.22 dB/km at 100% humidity. At 2000 Hz, the figures are 4.14, 7.14 and 6.29 dB/km. These figures are much larger than the classical absorption calculated from the equation in the preceding paragraph, and are probably due to vibrational and rotational relaxation in water vapour, oxygen, and carbon dioxide.
Although absorption in gases is well accounted for by viscothermal and relaxation effects (observed absorption only slightly higher than predicted), absorption in some liquids, such as water or alcohols, is much higher than would be expected on these grounds. The exess absorption can be explained as due to a structural relaxation, a change in the molecular arrangement, during the passage of the wave.
Fine wires and threads offer little resistance to the passage of sound. Tyndall found that a piece of felt half an inch thick stopped sound less well than a wet pocket handkerchief. In the latter case, the water closed the pores in the cloth so that it acted like a solid sheet, while the felt did not. This is the reason a hedge is a poor sound barrier, but a tight fence is a much more satisfactory one. Rain and fogs similarly have little effect on sound. In fact, the calmness often found in fogs may actually improve the transmission of sound. However, the presence of moisture catalyzes the absorption by vibrational relaxation in oxygen.
Sound Outdoors
In spite of small absorption, sound cannot be heard for any great distance outdoors. One reason for this is that the temperature of the atmosphere decreases rapidly with altitude (roughly 6 °C for each 1000 m). The lower phase velocity at altitude means that an initially vertical wave front will be tilted backwards, so the rays of sound are bent upwards, creating a shadow at ground level. A strong temperature inversion, or a wind blowing from the source of sound towards the observer, will have the opposite effect, and sound may be heard at a considerable distance. What is important in the case of the wind is that the wind speed generally increases with altitude, a wind shear, not simply a uniform wind. We will consider the effect of the wind in detail below. The unpredictability of the range of foghorns is well known, and can be largely ascribed to such effects. Fog and rain have little effect, since the scale of the disturbances is much smaller than a wavelength. Raising the source of sound has the effect of increasing its range; bells in church towers take advantage of this.
A strange effect was long noticed in the audibility of very large explosions and similar noises. The sound is observed in a region surrounding the source, perhaps extending 50 km or more. Then there is a zone of silence, but at several hundred kilometres, the sound, or at least its lower-frequency components, is sometimes again heard, and with extra delay. Guns on the continent were heard in England, for example. In the 1930's, this anomalous propagation was finally recognized as the effect of temperatures comparable to those on the ground, far above the stratosphere. The sound was reflected by these hot layers and again bent downward towards the earth. Sound from sources like jet aircraft or rockets can also be trapped as in a waveguide in the stratosphere, with higher temperatures both above and below, and can be detected thousands of miles from its source by a receiver in the stratosphere.
A plane wave propagating in a thermally stratified atmosphere can be represented by a ray normal to the wave front whose inclination to the horizontal changes so that the velocity with which the line of intersection of the wave front with a horizontal plane moves, called the trace velocity, is constant. If θ is the inclination, and c(h) the phase velocity as a function of altitude, then c(h) sec θ = c(0) = 340 m/s, if the wave started horizontally at the surface. As c(h) decreases, θ increases, and the wave climbs. At about 11.5 km, the stratosphere is reached, where c = 295 m/s. The inclination of the ray remains at about 30° through the stratosphere. At 20 km, the temperature begins to climb again (due to absorption of solar radiation by ozone), and reaches a maximum at about 49 km, where c = 330 m/s. The ray bends over, and at the maximum is inclined only 14° with the horizontal. Should the temperature up here be a little hotter than normal, and the temperature on the ground a little colder than normal, or a strong wind shear in the direction of propagation exist on high, or else a strong inversion exist on the ground, the ray will become horizontal, and then follow the mirror of its previous path down to the ground again, about 200km from the source. At 50 km, the mean free path of the air molecules is still no more than 0.1 mm, so the wave will not be strongly absorbed, especially at lower frequencies. The effect is analogous to the reflection of radio waves by the ionosphere, which also show 'skip' phenomena. Wind directions favour east to west propagation in the summer in northern temperate latitudes, and make it extremely unlikely in the winter.