∂ρ
∂t
þ
dx
dt
ρ
∂ρ
∂x
¼ 0
Comparing this latter equation with Eq. (2.9), it is clear that
dx
dt
ρ
¼ u þ ρ
du
dρ
:
ð2:11Þ
Similarly, u ¼ u(x, t), so that
∂u
∂t
þ
dx
dt
u
∂u
∂x
¼ 0
and, similarly, comparing this latter equation with Eq. (2.10) we have
dx
dt
u
¼ u þ
1
ρ
dp
du
:
ð2:12Þ
However, if u is constant so is ρ since u ¼ u(ρ), then (dx/dt) ρ ¼ (dx/dt) u and this
implies from Eqs. (2.11) and (2.12) that
u þ ρ
du
dρ
¼ u þ
1
ρ
dp
du
or
ρ
du
dρ
¼
1
ρ
dp
du
:
The latter equation can be written as
ρ
du
dρ
¼
1
ρ
dp
dρ
dρ
du
:
ð2:13Þ
However, p/ρ
γ
¼ constant and therefore, dp/dρ ¼ γp/ρ which we identify as the
square of the local sound velocity; that is, c
2
¼ γp/ρ, and with this substitution in
Eq. (2.13), we have,
ρ
du
dρ
¼
c
2
ρ
dρ
du
,
which yields
46
2 Waves of Finite Amplitude