where u 2 has been substituted for u p in the latter equation. Let us now apply
Eq. (2.17) across the expansion wave:
du ¼ À
2
γ À 1
dc
so that
Z
u 4 ¼0
u 3
du ¼ À
2
γ À 1
Z c 4
c 3
dc,
hence
u 3 ¼
2c 4
γ À 1
1 À
c 3
c 4
,
ð4:55Þ
where c 3 and c 4 are the local sound speeds in regions 3 and 4. However, the flow is
isentropic between regions 3 and 4 and, as a result, we have
Fig. 4.30 Shock tube showing the air motion shortly after the diaphragm is ruptured
4.8 Numerical Examples of Plane Shocks
175
Eq. (2.17) across the expansion wave:
du ¼ À
2
γ À 1
dc
so that
Z
u 4 ¼0
u 3
du ¼ À
2
γ À 1
Z c 4
c 3
dc,
hence
u 3 ¼
2c 4
γ À 1
1 À
c 3
c 4
,
ð4:55Þ
where c 3 and c 4 are the local sound speeds in regions 3 and 4. However, the flow is
isentropic between regions 3 and 4 and, as a result, we have
Fig. 4.30 Shock tube showing the air motion shortly after the diaphragm is ruptured
4.8 Numerical Examples of Plane Shocks
175