¼
13γ
2
À 7γ þ 12
ð
Þ
5 3γ À 1
ð
Þ 2 À γ
ð
Þ
;
2α 2
5
þ 2 ¼
6
5
;
2α 3
5
þ β 2 À 1 ¼
2 γ À 1
ð
Þ
2γ þ 1
þ
3
2γ þ 1
À 1 ¼ 0;
and
β 3 þ 1 ¼ À
γ
2 À γ
:
Therefore, the terms γφ À 1 disappears and we finally have the following
equation for the pressure f;
f ¼
2γ
γ þ 1
ð
Þ
γ þ 1
7 À γ
5 À 3γ À 1
ð
Þφ
f
g
! 13γ 2 À7γþ12
5 3γÀ1
ð
Þ2Àγ ð
Þ
γ þ 1
γ À 1
1 À φ
ð
Þ
! Àγ
2Àγ
γ þ 1
2
φ
h
i 6
5
ð5:137Þ
It can be seen that analytical solutions have been obtained for all three quantities
but the final results are not very revealing as the expressions obtained are implicit.
Accordingly, it is clear that Taylor’s method in solving the coupled equations
numerically was perhaps the best approach to take.
Nonetheless, it is interesting to note that all three equations, namely, Eqs. (5.132),
(5.135) and (5.137) contain some of the following terms in square brackets that are
raised to various powers;
γ þ 1
7 À γ
5 À 3γ À 1
ð
Þ
ϕ
η
&
' !
,
γ þ 1
γ À 1
1 À
ϕ
η
!
,
γ þ 1
2
ϕ
η
!
and
γ þ 1
γ À 1
γ
ϕ
η
À 1
!
,
and all these terms are equal to 1 at the shock front (η ¼ 1) where ϕ ¼ 2/(γ + 1) as can
be verified by direct substitution. Consequently, Eqs. (5.135) and (5.137) for the
density and pressure reduce to
γ þ 1
γ À 1
and
2γ
γ þ 1
,
respectively, which are just the first multiplying factors that appear on the right-hand
side of each equation and these factors are consistent with those expected for the
density and pressure at the shock front.
276
5 Spherical Shock Waves: The Self-similar Solution