Dispersion of Sound

the dependence of the phase velocity of monochromatic sound waves on frequency. Dispersion of sound is the reason for the change in shape of a sound wave (sound pulse) upon its propagation in a medium. A distinction is made among dispersion of sound caused by the physical properties of the medium, dispersion caused by the presence of boundaries of the body in which the sound wave is propagating, and dispersion that is independent of the properties of the body.

The first type of dispersion of sound is produced by various causes. The instances of dispersion of sound associated with relaxation processes that occur during the passage of a sound wave are the most important. The initiating mechanism of relaxation dispersion of sound may be understood by using an example of multiatomic gas. Upon propagation of sound in the gas, its molecules perform translational motion. If the gas is monatomic, its atoms are not able to perform any motion other than translational. However, if the gas is multiatomic, then rotational motion of the molecules, as well as vibration of the atoms that make up the molecules, may arise. Part of the energy of the sound wave is expended on the excitation of these vibrational and rotational motions. The transfer of energy from the sound wave (that is, from translational motion) to inner degrees of freedom (vibrational and rotational motion) occurs not instantaneously but over a certain time interval, which is called the relaxation time τ. This time is determined by the number of collisions between molecules that is necessary for redistribution of energy among all degrees of freedom. If the period of the sound wave is small compared to τ (high frequencies), then during the period of the wave there will not be time for the excitation of inner degrees of freedom or the redistribution of energy. In this case the gas will behave as if no inner degrees of freedom whatever exist. However, if the period of the sound wave is considerably greater than τ (low frequencies), then during the period of the wave, there will be time for the redistribution of the energy of translational motion to the inner degrees of freedom. In this case, the energy of translational motion will be less than in the case when there are no inner degrees of freedom. Since the elasticity of a gas is determined by the energy necessary for translational motion of the molecules, then the elasticity of gas, as well as the speed of sound, will also be less than in the case of high frequencies. In other words, in a certain frequency range close to the relaxation frequency, which is equal to ωp = 1/τ, the speed of sound increases with increasing frequency—that is, so-called positive dispersion takes place. If the speed of sound is c₀ for low frequencies (ωτ<c_∞ for very high frequencies, then the speed of sound for a random frequency is described by the formula

Increased absorption of sound is observed as a result of the irreversibility of processes of redistribution of energy in the frequency range in which dispersion of sound occurs.

Relaxation dispersion of sound occurs not only in gases but also in liquids, where it is associated with various intermolecular processes; in electrolyte solutions; in mixtures, in which chemical reactions between components are possible under the influence of sound; in emulsions; and also in certain solids.

The magnitude of dispersion of sound may be very different in different substances. For example, the dispersion of sound in carbon dioxide is about 4 percent; in benzene, ~10 percent; and in seawater, less than 0.01 percent. In highly viscous liquids and high-polymer compounds the speed of sound may change by 50 percent. However, in the majority of materials, the diffusion of sound is an extremely small value, and measurements are rather complex. The frequency range in which dispersion of sound occurs is also different for different substances. Thus, in carbon dioxide at normal pressure and a temperature of 18°C the relaxation frequency is equal to 28 kilohertz (kHz); in seawater it is 120 kHz. In compounds such as carbon tetrachloride, benzene, and chloroform, the area of relaxation lies in the frequency range of the order of 10⁹-10¹⁰ Hz, where ordinary ultrasonic methods of measurements are not applicable and dispersion of sound can be measured only through the use of optical methods.

Heat conduction and viscosity of the medium are classified as diffusion of sound of the first type; however, they are not of a relaxational nature. These types of diffusion of sound result from the exchange of energy between areas of contraction and dilation in a sound wave and are especially important for microscopically nonuniform mediums. Diffusion of sound may also be manifested in a medium with disseminated nonuniformities (resonators)—for example, in water containing bubbles of gas. In this case, at a sound frequency close to the resonance frequency of the bubbles, part of the energy of the sound wave is expended on excitation of the vibrations of the bubbles; this leads to dispersion of sound and to an increase in the absorption of sound.

“Geometric” dispersion, which is caused by the existence of limits of a body or medium in which sound is propagating, is the second type of dispersion of sound. It occurs during the propagation of sound in rods and plates and in any type of acoustic wave guide. Dispersion of sound is observed for flexural waves in thin plates and rods (the thickness of the plate or rod must be much less than the length of the wave). Upon bending of a thin rod, the transverse elasticity at the point of deflection increases as the section being bent decreases. Upon propagation of a flexural wave, the length of the section being bent is determined by the length of the wave. Therefore, upon reduction of the length of the wave (with an increase in frequency), the elasticity and, correspondingly, the rate of propagation of the wave increases. The phase velocity of such a wave is proportional to the square root of the frequency—that is, positive dispersion occurs.

Upon propagation of sound in wave guides, the sound field may be represented as a superposition of normal waves whose phase speeds for a rectangular wave guide with rigid walls have the form

where n is the number of the standard wave (n = 1, 2, 3, . . .), c is the speed of sound in free space, and d is the width of the wave guide. The phase velocity of the normal wave is always greater than the speed of sound in a free medium and decreases as frequency increases (“negative” dispersion).

Dispersion of sound of both types leads to blurring of the shape of the pulse during its propagation. This is especially significant for hydroacoustics, atmospheric acoustics, and geoacoustics, which deal with the propagation of sound over large distances.