The speed of sound (acoustic speed)
The changing conditions imposed on individual particles of gas as the pressure pulse passes is now considered. As a first simple approach to defining the pulse and its speed of propagation, consider the stream tube to have a velocity such that the pulse is stationary, Fig. 6.6a. The flow upstream of the pulse has velocity u, density p and pressure p, while the exit flow has these quantities changed by infinitesimal amounts to и + би, p + 6p, p + 6p.
The flow situation now to be considered is quasi-steady, assumed inviscid and adiabatic (since the very small pressure changes take place too rapidly for heat transfer to be significant), takes place in the absence of external forces, and is one-dimensional, so that the differential equations of continuity and motion are respectively
Fig. 6.6 and
du 1 dp
11 dx p dx
Eliminating dujdx from these equations leaves
This implies the speed of flow in the stream tube that is required to maintain a stationary pulse of weak strength, is uniquely the speed given by fdpld~p (see Section 1.2.7, Eqn (1.6c)).
The problem is essentially unaltered if the pulse advances at speed w = J dp I dp through stationary gas and, since this is the (ideal) model of the propagation of weak pressure disturbances that are commonly sensed as sounds, the unique speed sjdpjdp is referred to as the acoustic speed a. When the pressure density relation is isentropic (as assumed above) this velocity becomes (see Eqn (1.6d))
It will be recalled that this is the speed the gas attains in the throat of a choked stream tube and it follows that weak pressure disturbances will not propagate upstream into a flow where the velocity is greater than a, i. e. ;/ > a or M > 1.