Brode [3] obtained detailed results of pressure, particle velocity and density as
functions of position and time outside the very strong shock region, so only a brief
account of some numerical result that correspond to the later stages of the spherical
expansion into the surrounding atmosphere are presented here.
6.6 Initial Conditions Using the Strong-Shock, Point-Source
Solution
The point-source solution for the pressure, particle velocity and density as outlined
in Chap. 5 is taken as the initial conditions for the numerical integration of the
difference equations in Lagrangian form. The point-source solution starting at 1000
atmospheres pressure at the shock front is used as the initial condition. This pressure
is large enough to ensure that the similarity solution can be used as the initial
condition for the numerical procedure.
6.6.1 The Pressure
From Eq. (5.25) we have the following relation for the strong shock wave pressure;
p ¼
1
γB γ
ð Þ
E 0 f
R
3
:
Dividing across by p 0 , the ambient air pressure, we have
p
p 0
¼
1
γB γ
ð Þ
E 0
p 0
f
R
3
,
and in terms of energy-reduced dimensionless units the latter equation becomes
p
p 0
¼
1
γB γ
ð Þ
ε
R
3
f ,
where ε
3
¼ E 0 /p 0 , hence, the normalized pressure (in Atmospheres) is
e p ¼
1
γB γ
ð Þ
R
ε
À3
f η
ð Þ,
and with η ¼ r/R ¼ (r/ε)/(R/ε), so that λ s ¼ R/ε and λ ¼ r/ε, hence, η ¼ λ/λ s , then
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6 Numerical Treatment of Spherical Shock Waves