By substitution these values for α and n in the equation for ϕ(η) we obtain the
following approximate analytical formula for the velocity;
ϕ η
ð Þ ¼
η
γ
þ
γ À 1
ð
Þ
γ γ þ 1
ð
Þ
η
7γÀ1
γ 2 À1 ,
ð5:37Þ
which agrees with the exact values at η ¼ 0 and at η ¼ 1. Similarly, with this value of
ϕ(η) substituted in the Eq. (5.10), namely,
ψ
0
ψ
¼
ϕ
0
þ 2ϕ=η
η À ϕ
,
it is straightforward to show that (ψ
0 /ψ) η ¼ 1 also agrees with the exact value at η ¼ 1
according to Eq. (5.18).
5.11.2 The Pressure f
Using Eq. (5.31), let us write it in the form,
f
0
f
¼
γϕ
0
À 3 þ
2γϕ
η
η À ϕ
:
ð5:38Þ
We now substitute ϕ(η) and ϕ
0
(η) in this latter equation where ϕ(η) is given by,
ϕ η
ð Þ ¼
η
γ
þ αη
n
:
Hence, one obtains,
f
0
f
¼
n þ 2
ð
Þαγ
2
η
nÀ2
γ À 1
ð
ÞÀαγη nÀ1
½
Š
:
ð5:39Þ
Substituting the values for n and α, we obtain,
df
f
¼
γ 2γ
2
þ 7γ À 3
ð
Þ
γ þ 1
ð
Þ
η
nÀ2 dη
γ 2 À 1
ð
ÞÀ γ À 1
ð
Þη nÀ1 ,
and letting, x ¼ (γ
2
À 1) À (γ À 1)η
n À 1 , hence, dx ¼ À (n À 1)(γ À 1)η
n À 2 dη.
Consequently,
df
f
¼ À
2γ
2
þ 7γ À 3
ð
Þ
7 À γ
dx
x
:
238
5 Spherical Shock Waves: The Self-similar Solution