∂p
∂ρ
s,ρ 0
¼
γp 0
ρ 0
so that
c 0 ¼
γp 0
ρ 0
1=2
:
ð1:73Þ
Substituting these values of u(x, t), ρ(x, t) and c
2 in Eqs. (1.68) and (1.72) and
neglecting terms of the order of Δu
2 and product terms like ΔuΔρ etc., we have the
resulting acoustic equations,
1
ρ 0
∂Δρ
∂t
þ
∂Δu
∂x
¼ 0
ð1:74Þ
and
∂Δu
∂t
þ
c
2
0
ρ 0
∂Δρ
∂x
¼ 0
ð1:75Þ
where only quantities to first order have been retained. Differentiating each of the
equations above we have
1
ρ 0
∂
2 Δρ
∂x∂t
þ
∂
2 Δu
∂x 2 ¼ 0 and
∂
2 Δu
∂t 2 þ
c
2
0
ρ 0
∂
2 Δρ
∂t∂x
¼ 0,
so that
∂
2 Δu
∂x 2 ¼
1
c 2
0
∂
2 Δu
∂t 2 :
ð1:76Þ
Similarly, eliminating Δu from Eqs. (1.74) and (1.75) we have
∂
2 Δρ
∂x 2 ¼
1
c 2
0
∂
2 Δρ
∂t 2 ,
ð1:77Þ
and by using the isentropic conditions it is straightforward to show that similar
equations are obtained for the pressure and temperature perturbations.
We recognise these latter equations as the one-dimensional wave equations where
c 0 has the dimensions of speed which we identify as the speed of sound or the speed
of propagation of small amplitude disturbances and where the disturbance
1.8 Small Amplitude Disturbances: Sound Waves
35