Here, we will take η ¼ r/R as the dimensionless similarity parameter; η 0 will be
determined in due course. The pressure, p s , the gas velocity, u s and the density, ρ s
just behind the shock front satisfy the following Rankine-Hugoniot equations (see
Chap. 3; Eqs. (3.26a), (3.38a) and (3.17a));
U
2
c 2
0
¼
1
2γ
γ À 1
ð
Þþ γ þ 1
ð
Þ
p s
p 0
!
,
ð5:4aÞ
u ¼
2
p s
p 0
À 1
U
γ À 1
ð
Þþ γ þ 1
ð
Þ
p s
p 0
,
ð5:4bÞ
and
ρ s
ρ 0
¼
γ À 1
ð
Þþ γ þ 1
ð
Þ p s =p 0
ð
Þ
γ þ 1
ð
Þþ γ À 1
ð
Þ p s =p 0
ð
Þ
,
ð5:4cÞ
(Taylor uses the symbol y 1 for the pressure ratio, p s /p 0 )
In the case of very strong shocks, namely, ( p s /p 0 ) >> 1, we have previously noted
in Chap. 3 that the latter equations become,
U
2
c 2
0
¼
γ þ 1
2γ
p s
p 0
ð5:5aÞ
u s
U
¼
2
γ þ 1
ð5:5bÞ
ρ s
ρ 0
¼
γ þ 1
γ À 1
ð5:5cÞ
where U ¼ dR/dt is the velocity of the shock front and c 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
γp 0 =ρ 0
p
is the speed of
sound in the undisturbed atmosphere. Hence, the pressure, particle velocity and
density immediately behind the shock front are
p s ¼
2γp 0
γ þ 1
ð
Þc 2
0
U
2
¼
2ρ 0
γ þ 1
ð
Þ
U
2 ,
u s ¼
2
γ þ 1
U,
and
ρ s ¼ ρ 0
γ þ 1
γ À 1
,
respectively. Substituting Eq. (5.2) in these latter equations yields;
5.4 Taylor’s Analysis of Very Intense Shocks
221