and applying the boundary conditions at x ¼ + 1 ahead of the shock, namely, u ¼ 0
and ρ ¼ ρ 0 , where ρ 0 is the ambient air density, we find that the constant of
integration is ÀU s ρ 0 , hence,
ÀU s ρ þ ρu ¼ ÀU s ρ 0
and, therefore,
ρ
ρ 0
¼
U s
U s À u
:
ð3:64Þ
The momentum equation yields,
ÀρU s
du
dω
þ ρu
du
dω
þ
d
dω
p À σ x
ð
Þ¼0,
that is,
ÀρU s þ ρu
ð
Þ
du
dω
þ
d
dω
p À σ x
ð
Þ¼0:
By using the continuity equation this becomes,
ÀU s ρ 0
du
dω
þ
d
dω
p À σ x
ð
Þ¼0,
and after integrating we obtain,
ÀU s ρ 0 u þ p À σ x
ð
Þ¼ constant:
Applying the boundary conditions at x ¼ + 1, namely, u ¼ 0, p ¼ p 0 and σ x ¼ 0,
where p 0 is the ambient air pressure, we find that the constant of integration is p 0 ,
hence,
ÀU s ρ 0 u þ p À σ x ¼ p 0
and, therefore,
p À σ x ¼ p 0 þ U s ρ 0 u
ð3:65Þ
In the same way, the energy equation becomes,
ÀU s
d
dω
1
2
ρu
2
þ ρe
h
i
þ
d
dω
u
1
2
ρu
2
þ ρe
þ u p À σ x
ð
Þ
h
i
¼ 0:
3.13 Thickness of the Shock Wave Region
125