Speed of sound and bulk elasticity

The bulk elasticity is a measure of how much a fluid (or solid) will be compressed by the application of external pressure. If a certain small volume, V, of fluid is subjected to a rise in pressure, 6p, this reduces the volume by an amount —6V, i. e. it produces a volumetric strain of —6V/V. Accordingly, the bulk elasticity is defined as

The volumetric strain is the ratio of two volumes and evidently dimensionless, so the dimensions of К are the same as those for pressure, namely ML_1T~2. The SI units are Nm“2 (or Pa).

The propagation of sound waves involves alternating compression and expansion of the medium. Accordingly, the bulk elasticity is closely related to the speed of sound, a, as follows:

(1.6b)

Let the mass of the small volume of fluid be M, then by definition the density, p = М/ V. By differentiating this definition keeping M constant, we obtain

Therefore, combining this with Eqns (1.6a, b), it can be seen that

(1.6c)

The propagation of sound in a perfect gas is regarded as an isentropic process. Accordingly, (see the passage below on Entropy) the pressure and density are related by Eqn (1.24), so that for a perfect gas

(1.6d)

where 7 is the ratio of the specific heats. Equation (1.6d) is the formula usually used to determine the speed of sound in gases for applications in aerodynamics.