With ω and m defined, we have in relation to the momentum equation;
∂u
∂t
¼
du
dω
∂ω
∂t
¼ ÀU s
du
dω
and
∂
∂x
p þ q
ð
Þ¼
d
dω
p þ q
ð
Þ
∂ω
∂x
¼
d
dω
p þ q
ð
Þ,
hence, the momentum equation becomes
m
du
dω
¼
d
dω
p þ q
ð
Þ
ð4:16Þ
Similarly, the energy equation becomes,
de
dω
þ p þ q
ð
Þ
dυ
dω
¼ 0
ð4:17Þ
For the continuity equation, we have
∂υ
∂t
¼
dυ
dω
∂ω
∂t
¼ ÀU s
dυ
dω
and
Fig. 4.1 A steady-state plane shock wave moving down a tube
138
4 Numerical Treatment of Plane Shocks