Aerospace Engineering e-Mega Reference

u

v

r

vx

þ

r

vu

vx

¼ 0 (4.8.31)

and

u

vu

vx

þ

1vp

r

vx

(4.8.32)

Eliminating vu/vx from these equations leaves

u

2

¼

vp

v

r

(4.8.33)

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

ffiffiffiffiffiffiffiffiffiffiffiffi

vp=v

r

p

.

The problem is essentially unaltered if the pulse advances at speed u ¼

ffiffiffiffiffiffiffiffiffiffiffiffi

vp=v

r

p

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

ffiffiffiffiffiffiffiffiffiffiffiffi

vp=v

r

p

is referred to as the acoustic speed a. When the pressure density relation is isentropic (as

assumed above) this