and by retaining terms to second order we can show that the shock velocity can be
written in the form,
U ¼ c 1 1 þ
γ þ 1
4
u
c 1
þ
1
2
γ þ 1
4
2 u
2
c 2
1
!
:
ð3:57Þ
3.12.2 Pressure Ratio for Weak Shocks
Using Eq. (3.25) for the pressure ratio we have
p 2
p 1
¼ 1 þ
2γ
γ þ 1
U
2
c 2
1
À 1
:
Writing this in the form
p 2
p 1
¼ 1 þ
2γ
γ þ 1
U
c 1
À 1
U
c 1
þ 1
and after using Eq. (3.57) in this latter equation we obtain
p 2
p 1
¼ 1 þ
2γ
γ þ 1
γ þ 1
4
u
c 1
þ
1
2
γ þ 1
4
2 u
2
c 2
1
!
2 þ
γ þ 1
4
u
c 1
þ
1
2
γ þ 1
4
2 u
2
c 2
1
!
:
Performing the multiplication in this latter equation and after retaining terms of
the order of u
3
=c
3
1 , we have
p 2
p 1
¼ 1 þ
2γ
γ þ 1
2
γ þ 1
4
u
c 1
þ
γ þ 1
ð
Þ
2
8
u
2
c 2
1
þ
γ þ 1
ð
Þ
3
64
u
3
c 3
1
!
and, on further simplification, we finally arrive at the following expression for the
pressure ratio in the case of weak shocks,
p 2
p 1
¼ 1 þ γ
u
c 1
þ
γ γ þ 1
ð
Þ
4
u
2
c 2
1
þ
γ γ þ 1
ð
Þ
2
32
u
3
c
3
1
ð3:58Þ
The waves represented by Eqs. (2.14) to (2.19) of Chap. 2 are often described as
simple waves [8] and according to Eqs. (2.18) and (2.21) we have
3.12 Approximate Analytical Expressions for Weak Shock Waves
117