ρ
2 du
ð Þ
2 ¼ c
2 dρ
ð Þ
2 ,
hence,
du ¼ Æc
dρ
ρ
,
ð2:14Þ
which should be compared with Eq. (1.81) for ordinary sound waves in Chap. 1.
However, Eq. (2.12) gives
dx
dt
u
¼ u þ
1
ρ
dp
du
,
which can be written as
dx
dt
u
¼ u þ
1
ρ
dp
dρ
dρ
du
:
With dp/dρ ¼ c
2 and using Eq. (2.14), the latter equation becomes
dx
dt
u
¼ u Æ c u
ð Þ,
ð2:15Þ
where we have explicitly indicated that c is a function of u. This latter equation
implies that the speed of propagation of a disturbance is dependent on the local
particle velocity u and the local speed of sound c(u). Consequently, an observer
moving at the local particle velocity will observe that the speed of propagation of a
disturbance travels at the local speed of sound. The plus and minus signs in the latter
equation represent wave propagation in the positive and negative x-directions,
respectively. Returning again to Eq. (2.14) we note that
c
2
¼
γp
ρ
¼
γ
ρ
kρ
γ
¼ γkρ
γÀ1 ,
where k is a constant and taking logs of both sides we have
2 log c ¼ log γk þ γ À 1
ð
Þlog ρ,
hence,
2
dc
c
¼ γ À 1
ð
Þ
dρ
ρ
so that
2.2 Finite Amplitude Waves
47