dρ
ρ
¼
2
γ À 1
dc
c
:
ð2:16Þ
Substituting this in Eq. (2.14) and integrating gives,
u ¼ Æ
2
γ À 1
Z
dc ¼ Æ
2
γ À 1
c À c 0
ð
Þ,
ð2:17Þ
where the constant of integration has been chosen to be c ¼ c 0 when u ¼ 0. Hence,
the local velocity of sound in terms of the local particle velocity is
c u
ð Þ ¼ c 0 Æ
γ À 1
ð
Þ
2
u,
ð2:18Þ
and substituting this in Eq. (2.15) yields,
dx
dt
u
¼ u Æ c 0 Æ
γ À 1
2
u
¼ Æc 0 þ
γ þ 1
2
u,
ð2:19Þ
which now gives the speed of propagation of the disturbance in terms of the particle
velocity and the ambient or undisturbed sound speed, c 0 . Let us assume that
propagation takes place in the positive x-direction, then from Eq. (2.19) we can
write the speed of propagation of the disturbance as,
c
0
¼ c 0 þ
γ þ 1
2
u:
Using Eq. (2.17) for u in the latter equation yields,
c
0
¼ c 0 þ
γ þ 1
2
2
γ À 1
c À c 0
ð
Þ:
ð2:20Þ
However, we note that,
c
2
¼
γp
ρ
and c
2
0 ¼
γp 0
ρ 0
and
48
2 Waves of Finite Amplitude