The velocity of sound is a vector u whose magnitude |u| is the speed of sound u and whose direction is normal to the surface of constant phase. The speed of sound is a property of the medium through which the sound travels and is therefore usually of more interest than the velocity itself which depends upon both u and the manner in which the sound is generated.
In discussing the relation between u and the properties of the medium is it useful to distinguish between homogeneous fluids and solids. In the former, the speed of sound is the same in all directions whereas, in the latter, this is not necessarily the case.
In a perfectly elastic (nondissipative) fluid the speed of sound is given for small-amplitude sound waves by
where p is pressure, ρ density, cp specific heat at constant pressure, S entropy and κS is the isentropic compressibility. Consequently, u may be determined from the thermodynamic Equation of State of the fluid or, conversely, information about the equation of state may be deduced from the u2(T, p) [see Trusler (1991)]. For the special case of the Perfect Gas, for which ρ = Mp/RT, the speed of sound is given by
where M is the molar mass and γpg is the heat-capacity ratio for the perfect gas. Thus the speed of sound in a perfect gas is proportional to but independent of pressure. All real gases approach this behavior at sufficiently low pressures, but generally u(T, p)/ is a slowly varying function of both T and p. Some examples of u(T, p → 0) for gases at T = 300 K are: 4He, 1019 m/s; CH4, 451 m/s; N2, 352 m/s; air 348 m/s; and n-butane, 194 m/s [Kaye and Laby 1986].
For the saturated vapor of a pure substance, u at first increases with temperature and, after passing through a maximum, it then declines to zero at the gas-liquid critical point. The speed of sound in the pure saturated liquid is always greater than that in the coexisting vapor. For normal saturated liquids, u declines steadily from its value at the triple point to zero at the liquid-gas critical point. Since the speed of sound in liquids varies only slowly with pressure, u generally declines also with increasing temperature along an isobar. Liquid water is exceptional in this respect as (∂u/∂T)p < 0 at atmospheric pressure between 0 and 74 °C. Some examples of speeds of sound in liquids at atmospheric pressure and T = 300 K are: perfluorohexane, 505 m/s; CCl4, 915 m/s; methanol, 1097 m/s; ethanol, 1139 m/s; cyclohexane, 1244 m/s; toluene, 1298 m/s; mercury, 1448 m/s; water, 1500.7 m/s; and sea water (3.5 mass% salinity), 1537.9 m/s [Kaye and Laby (1986)].
All fluids exhibit some absorption of sound at all frequencies. For most fluids there exists a wide range of frequencies from zero upwards in which both the dissipation is slight and Equation (1) is obeyed with great accuracy. For monatomic fluids, this situation persists up to frequencies approaching the molecular collision frequency. However, various relaxation mechanisms exist in molecular fluids which, in certain frequencies ranges, give rise to greatly increased absorption and to dispersion (i.e., frequency dependence of u). For gases, these relaxation mechanisms are usually associated with the vibrational and rotational modes of the molecules. However, in liquids structural relaxation associated with the viscoelastic properties is possible. In all cases, Equation (1) is obeyed at sufficiently low frequencies.
When sound waves are channelled along a fluid-filled waveguide (e.g., a tube) the apparent speed of sound is not identical with that observed in free space. Three characteristic sound speeds may be defined. The phase speed up is the speed at which points of constant phase propagate along the axis in a monofrequency wave. This is the only characteristic speed of importance for continuous waves although it will be frequency dependent if the medium is dispersive. When pulses of sound are transmitted through a waveguide, it is useful to define also the signal speed us as the speed at which the leading edge of the pulse propagates and the group speeds ug as the speed at which the center of the pulse propagates. In the absence of dispersion, all three speeds are identical but when up is a function of the angular frequency ω > they differ slightly and up < ug < us.
The speed of longitudinal sound waves in an infinite isotropic solid specimen is given by
where B is the bulk modulus and G is the shear modulus [Kinsler and Frey 1982]. A shear wave is also possible with sound speed (G/ρ)1/2. In practice, the sample is not infinite and the observed longitudinal sound speed depends, though the boundary conditions, upon the lateral dimensions of the sample. For a bar with lateral dimensions small compared with the wavelength, free boundary conditions apply (the lateral stress components vanish) and
where E = 3(1 − 2σ)B is Young's modulus. As the lateral dimensions increase, the effective boundary condition for a volume element in the sample approaches one of zero lateral strain and u approaches the bulk value. Sound speed measurements are the primary means of determining the elastic moduli of solids. For anisotropic solids (e.g., all crystals except simple cubic structures) the elastic constants and hence also the speed of sound are different along different directions.
For complex materials, such as porous media, laminates and other composites, the speed of sound is not usually given by a simple expression but may nevertheless provide valuable information about the structure.
Kinsler, L. E., Frey, A. R., Coppens, A. B. and Sanders, J. V. (1982) Fundamentals of Acoustics, 3rd edn. Wiley, New York.
Trusler, J. P. M. (1991) Physical Acoustics and Metrology of Fluids, Adam Hilger, Bristol.
Kaye, G. W. C. and Laby, T. H. (1986) Tables of Physical and Chemical Constants, 15th edn. Longman, Harlow.