Page 422 - Introduction to Continuum Mechanics

406 Acoustic Wave

Thus, for a barotropic fluid

and

These equations are exactly analogous (for one- dimensional waves) to the elastic wave
equations of Chapter 5. Thus, we conclude that the pressure and density disturbances will
propagate with a speed c 0 = ^(dp/dp) p . We call c 0 the speed of sound at stagnation, the local
speed of sound is defined to be

When the isentropic relation ofpand/o is used, i.e.,

where y = c p/c v ( ratio of specific heats) and/? is a constant

so that the speed of sound is

(a) Write an expression for a harmonic plane acoustic wave propagating in the ej direction.
(b) Find the velocity disturbance vj.

(c) Compare dv/dt to the neglected vydv/ / dx;.
Solution. In the following ,p, p, vj denote the disturbances, that is, we will drop the primes,
(a) Referring to the section on elastic waves, we have

(b) Using Eq. (6.27.4), we have