p 2
p 1
¼
c 2
c 1
2γ
γÀ1 ¼ 1 þ
γ À 1
2
u
c 1
! 2γ
γÀ1
By using the binomial expansion,
1 þ x
ð
Þ
n ¼ 1 þ nx þ
n n À 1
ð
Þ
2
x
2
þ
n n À 1
ð
Þ n À 2
ð
Þ
6
x
3
þ ::
we can write this pressure ratio as
p 2
p 1
¼ 1 þ
2γ
γ À 1
γ À 1
2
u
c 1
þ
1
2
2γ
γ À 1
2γ
γ À 1
À 1
γ À 1
2
2 u
2
c 2
1
þ ::
and on further simplifying this equation and retaining terms to third order we arrive
at the following result,
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
12
u
3
c
3
1
þ . . . ,
which shows that the pressure ratio for the so-called simple wave and the shock wave
agree up to terms of second order in the shock strength.
3.12.3 Density Ratio for Weak Shocks
The equation for the density ratio according to Eq. (3.36) can be written as
ρ 2
ρ 1
¼
U=c 1
U=c 1
ð
ÞÀ u=c 1
ð
Þ
and after substituting Eq. (3.57) in this latter equation we obtain
ρ 2
ρ 1
¼
1 þ
γ þ 1
4
u
c 1
þ
1
2
γ þ 1
4
2 u
2
c 2
1
1 þ
γ þ 1
4
u
c 1
þ
1
2
γ þ 1
4
2 u
2
c 2
1
À
u
c 1
¼
1 þ
γ þ 1
4
u
c 1
þ
1
2
γ þ 1
4
2 u
2
c 2
1
1 þ
γ À 3
4
u
c 1
þ
1
2
γ þ 1
4
2 u
2
c 2
1
:
118
3 Conditions Across the Shock: The Rankine-Hugoniot Equations