One observes immediately that the terms of the order of u/c 1 drop out, hence,
collecting terms of the order of u
2
=c
2
1 , we have
γ γ þ 1
ð
Þ
4
À
γ
2
2
À
γ 3 À γ
ð
Þ
4
þ
γ
2
¼
1
4
γ
2
þ γ À 2γ
2
À 3γ þ γ
2
þ 2γ
À
Á
¼0
and, therefore, terms of the order of u
2
=c
2
1 also drop out. Collecting terms of the order
of u
3
=c
3
1 , we have the following terms,
γ γ þ 1
ð
Þ
2
32
À
γ
2
γ þ 1
ð
Þ
4
þ
γ
3
3
À
γ γ
2
À 14γ þ 17
ð
Þ
32
þ
γ 3 À γ
ð
Þ
4
À
γ
3
:
By simplifying this latter expression it is easy to show that it reduces to
γ γ
2
À 1
ð
Þ
12
hence, the entropy change in the case of a weak shock can be written as
s 2 À s 1
c V
¼
γ γ
2
À 1
ð
Þ
12
u
3
c
3
1
,
ð3:62Þ
which is of third order when expanded in powers of the shock strength. Consequently, the increase in the entropy of the fluid as it crosses the shock is zero for
terms up to second order. This implies that the isentropic equations can be used for
determining the flow properties in a disturbance up to second order in the shock
strength.
3.12.7 Change in the Riemann Invariant R 2 for Weak
Shocks
We already know that the Riemann invariant is constant across a simple wave,
whereas the Riemann invariant ahead of the shock can be written as
R
ahead
À
¼ À
2c 1
γ À 1
,
while the Riemann invariant behind the shock is
122
3 Conditions Across the Shock: The Rankine-Hugoniot Equations