3f þ η f
0
þ γ
ψ
0
ψ
f Àη þ ϕ
ð
ÞÀϕ f
0
¼ 0:
ð5:12Þ
By substituting the right-hand side of Eq. (5.10) for ψ
0 /ψ in Eq. (5.12) and then
using Eq. (5.9) for ϕ
0 in the resulting equation, we can write f
0 as
f
0
¼ f
À3η þ ϕ 3 þ γ=2
ð
ÞÀ2γϕ
2
=η
Â
Ã
η À ϕ
ð
Þ
2 À f =ψ
h
i
:
ð5:13Þ
5.5 Derivatives at the Shock Front
On inspection of Eq. (5.13) above it can be seen that all quantities on the right-hand
side are known at the shock front (η ¼ 1), namely,
f 1
ð Þ ¼
2γ
γ þ 1
, ϕ 1
ð Þ ¼
2
γ þ 1
and ψ 1
ð Þ ¼
γ þ 1
γ À 1
,
hence, the value of f
0 (1) can be determined by directly substituting these values;
giving,
f
0
f
η¼1
¼
À3 þ
2
γ þ 1
3 þ
γ
2
À
8γ
γ þ 1
ð
Þ
2
1 À
2
γþ1
2 À
2γ γ À 1
ð
Þ
γ þ 1
ð
Þ
2
¼
2γ
2
þ 7γ À 3
γ 2 À 1
ð5:14Þ
and hence,
f
0 1
ð Þ ¼
2γ 2γ
2
þ 7γ À 3
ð
Þ
γ À 1
ð
Þ γ þ 1
ð
Þ
2
:
ð5:15Þ
Using this value for f
0 (1) in Eq. (5.9) above we obtain the following expression
for (ϕ
0 /ϕ) η ¼ 1 ;
ϕ
0
ϕ
η¼1
¼
γ þ 9
2 γ þ 1
ð
Þ
,
ð5:16Þ
so that
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5 Spherical Shock Waves: The Self-similar Solution