By using the binomial expansion; (1 + x)
À1
¼ 1 À x + x
2
À x
3 + .. for the
denominator term and by retaining terms of the order of u
3
=c
3
1 , we have
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
!
Â
1 À
γ À 3
4
u
c 1
À
1
2
γ þ 1
4
2 u
2
c 2
1
þ
γ À 3
4
2 u
2
c 2
1
þ
γ À 3
ð
Þ γ þ 1
ð
Þ
2
64
u
3
c
3
1
À
γ À 3
4
3 u
3
c
3
1
!
Multiplying the terms on the right-hand side and by retaining terms of the order of
u
3
=c
3
1 , we find that
ρ 2
ρ 1
¼ 1 þ
γ þ 1
4
À
γ À 3
4
n
o u
c 1
þ
1
32
γ
2 À 14γ þ 17
À
Á À
γ þ 1
ð
Þ γ À 3
ð
Þ
16
þ
1
2
γ þ 1
4
2
&
'
u
2
c 2
1
þ
γ À 3
ð
Þ γ À 1
ð
Þ
8
þ
γ þ 1
ð
Þ
4
γ
2
À 14γ þ 17
ð
Þ
32
À
1
2
γ þ 1
4
2 γ À 3
4
&
'
u
3
c 3
1
and after simplifying the expression on the right-hand side, we finally obtain the
following result for the density ratio for weak shocks,
ρ 2
ρ 1
¼ 1 þ
u
c 1
þ
3 À γ
4
u
2
c 2
1
þ
γ
2
À 14γ þ 17
ð
Þ
32
u
3
c
3
1
ð3:59Þ
Let us now compare this result with the density ratio in the case of simple waves.
According to Eq. (2.21) we have
c 2
c 1
¼
ρ 2
ρ 1
γÀ1
2
and by using Eq. (2.18) we can write the latter equation as
ρ 2
ρ 1
¼ 1 þ
γ À 1
2
u
c 1
! 2
γÀ1 :
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 have
3.12 Approximate Analytical Expressions for Weak Shock Waves
119