U s À u
ð
Þ
ρ 0
4
3
μ
du
dω
¼ u
1
2
γ þ 1
ð
ÞU s u þ c
2
0 À U
2
s
h
i
:
ð3:67Þ
One can observe that du/dω ¼ 0 when u ¼ 0 and also when
1
2
γ þ 1
ð
ÞU s u þ c
2
0 À U
2
s
h
i
¼ 0:
Solving this latter equation for u, one finds that
u ¼
2c 0
γ þ 1
U s
c 0
À
c 0
U s
¼
2c 0
γ þ 1
M À
1
M
,
ð3:68Þ
which is just Eq. (3.39b) where M is the Mach number and u is the particle or
material velocity behind the shock. Consequently, u ¼ 0 and u given by Eq. (3.68)
are the extreme values of the particle velocity downstream and upstream, respectively, of the propagating shock. In order to determine the width of the shock wave
region, one can integrate Eq. (3.67) to obtain the velocity variation across the shock.
Letting
u 1 ¼
2c 0
γ þ 1
ð
Þ
U s
c 0
À
c 0
U s
¼
2c 0
γ þ 1
ð
Þ
M À
1
M
and re-writing Eq. (3.67) in the form,
U s À u
ð
Þ
4μ
3ρ 0
du
dω
¼
γ þ 1
2
U s u u À u 1
ð
Þ:
ð3:69Þ
Since U s > u and 0 u u 1 for u lying between its values at x ¼ Æ 1, we can
see that the slope, (du/dω), is negative through the shock region.
Integrating this latter equation by using a partial fraction expansion for the
quantity, u(u À u 1 ), one can show that the following equation can be written as,
U s À u 1
u 1
ln
u 1 À u
U s
À
U s
u 1
ln
u
U s
¼
3 γ þ 1
ð
Þρ 0 U s
8μ
ω þ constant: ð3:70Þ
The constant of integration can be determined by choosing the origin (ω ¼ 0) at
u ¼ u 1 /2, which corresponds to the point of inflection (d
2 u/dω
2
¼ 0) of the velocity
profile [15], hence, the constant of integration is given by
constant ¼
U s À u 1
u 1
ln
u 1
2U s
À
U s
u 1
ln
u 1
2U s
and by substituting this value back in Eq. (3.70), we can write the equation in the
following form,
3.13 Thickness of the Shock Wave Region
127
ð
Þ
ρ 0
4
3
μ
du
dω
¼ u
1
2
γ þ 1
ð
ÞU s u þ c
2
0 À U
2
s
h
i
:
ð3:67Þ
One can observe that du/dω ¼ 0 when u ¼ 0 and also when
1
2
γ þ 1
ð
ÞU s u þ c
2
0 À U
2
s
h
i
¼ 0:
Solving this latter equation for u, one finds that
u ¼
2c 0
γ þ 1
U s
c 0
À
c 0
U s
¼
2c 0
γ þ 1
M À
1
M
,
ð3:68Þ
which is just Eq. (3.39b) where M is the Mach number and u is the particle or
material velocity behind the shock. Consequently, u ¼ 0 and u given by Eq. (3.68)
are the extreme values of the particle velocity downstream and upstream, respectively, of the propagating shock. In order to determine the width of the shock wave
region, one can integrate Eq. (3.67) to obtain the velocity variation across the shock.
Letting
u 1 ¼
2c 0
γ þ 1
ð
Þ
U s
c 0
À
c 0
U s
¼
2c 0
γ þ 1
ð
Þ
M À
1
M
and re-writing Eq. (3.67) in the form,
U s À u
ð
Þ
4μ
3ρ 0
du
dω
¼
γ þ 1
2
U s u u À u 1
ð
Þ:
ð3:69Þ
Since U s > u and 0 u u 1 for u lying between its values at x ¼ Æ 1, we can
see that the slope, (du/dω), is negative through the shock region.
Integrating this latter equation by using a partial fraction expansion for the
quantity, u(u À u 1 ), one can show that the following equation can be written as,
U s À u 1
u 1
ln
u 1 À u
U s
À
U s
u 1
ln
u
U s
¼
3 γ þ 1
ð
Þρ 0 U s
8μ
ω þ constant: ð3:70Þ
The constant of integration can be determined by choosing the origin (ω ¼ 0) at
u ¼ u 1 /2, which corresponds to the point of inflection (d
2 u/dω
2
¼ 0) of the velocity
profile [15], hence, the constant of integration is given by
constant ¼
U s À u 1
u 1
ln
u 1
2U s
À
U s
u 1
ln
u 1
2U s
and by substituting this value back in Eq. (3.70), we can write the equation in the
following form,
3.13 Thickness of the Shock Wave Region
127