Speed of Sound

The so-called sound speed is the rate of propogation of a pressure pulse of infinitesimal strength through a still fluid. It is a thermodynamic property of a fluid.

A pressure pulse in an incompressible flow behaves like that in a rigid body. A displaced particle displaces all the particles in the medium. In a compressible fluid, on the other hand, displaced mass compresses and increases the density of neighbouring mass which in turn increases density of the adjoining mass and so on. Thus, a disturbance in the form of an elastic wave or a pressure wave travels through the medium. If the amplitude and theerfore the strength of the elastic wave is infinitesimal, it is termed as acoustic wave or sound wave.

Figure 39.1(a) shows an infinitesimal pressure pulse propagating at a speed " a " towards still fluid (V = 0) at the left. The fluid properties ahead of the wave are p,T and , while the properties behind the wave are p+dp, T+dT and . The fluid velocity dV is directed toward the left following wave but much slower.

In order to make the analysis steady, we superimpose a velocity " a " directed towards right, on the entire system (Fig. 39.1(b)). The wave is now stationary and the fluid appears to have velocity " a " on the left and (a - dV) on the right. The flow in Fig. 39.1 (b) is now steady and one dimensional across the wave. Consider an area A on the wave front. A mass balance gives

Fig 39.1: Propagation of a sound wave
(a) Wave Propagating into still Fluid (b) Stationary Wave

(39.1)

This shows that

(a)

if dρ is positive.

(b) A compression wave leaves behind a fluid moving in the direction of the wave (Fig. 39.1(a)).

(c) Equation (39.1) also signifies that the fluid velocity on the right is much smaller than the wave speed " a ". Within the framework of infinitesimal strength of the wave (sound wave), this " a " itself is very small.

Applying the momentum balance on the same control volume in Fig. 39.1 (b). It says that the net force in the x direction on the control volume equals the rate of outflow of x momentum minus the rate of inflow of x momentum. In symbolic form, this yields

In the above expression, Aρa is the mass flow rate. The first term on the right hand side represents the rate of outflow of x-momentum and the second term represents the rate of inflow of x momentum.

Simplifying the momentum equation, we get

(39.2)

If the wave strength is very small, the pressure change is small.
Combining Eqs (39.1) and (39.2), we get

(39.3a)

The larger the strength of the wave ,the faster the wave speed; i.e., powerful explosion waves move much faster than sound waves.In the limit of infinitesimally small strength, we can write

(39.3b)

Note that

(a) In the limit of infinitesimally strength of sound wave, there are no velocity gradients on either side of the wave. Therefore, the frictional effects (irreversible) are confined to the interior of the wave.

(b) Moreover, the entire process of sound wave propagation is adiabatic because there is no temperature gradient except inside the wave itself.

(c) So, for sound waves, we can see that the process is reversible adiabatic or isentropic.

So the correct expression for the sound speed is