∂u
∂t
¼ À
ε
2
λ
2
r 2
0 ρ λ 0 , 0
ð
Þ
∂p
∂χ
∂χ
∂r 0
¼ À
λ
2
ρ λ 0 , 0
ð
Þε
∂p
∂χ
:
ð6:7Þ
Let us now measure the particle velocity u in terms of the ambient sound speed c 0 ,
the pressure p in terms of atmospheric pressure p 0 , and the density ρ normalized to
ambient air density ρ 0 , then the following dimensionless quantities are obtained;
e u u=c 0 , e p p=p 0 , e ρ λ 0 , 0
ð
Þ¼ ρ λ 0 , 0
ð
Þ=ρ 0 , hence, Eq. (6.7) becomes
c 0
∂e u
∂t
¼ À
λ
2 p 0
ρ 0 e ρ λ 0 , 0
ð
Þε
∂e p
∂χ
so that
∂e u
∂t
¼ À
λ
2 p 0
ρ 0 e ρ λ 0 , 0
ð
Þc 0 ε
∂e p
∂χ
:
However, the speed of sound c 0 in ambient air is given by, c
2
0 ¼ γp 0 =ρ 0 , so that
the latter equation becomes
∂e u
∂t
¼ À
λ
2 c 0
γεe ρ λ 0 , 0
ð
Þ
∂e p
∂χ
:
ð6:8Þ
Clearly, the ratio, ε/c 0 , has dimensions of time, so defining a dimensionless time
according to
τ ¼ c 0 t=ε,
ð6:9Þ
then Eq. (6.8) can be written in the form;
∂e u
∂τ
¼ À
λ
2
γe ρ λ 0 , 0
ð
Þ
∂e p
∂χ
,
ð6:10Þ
or writing the latter equation in terms of the normalized specific volume e υ λ 0 , 0
ð
Þ
rather than the normalized density e ρ λ 0 , 0
ð
Þ, where e υ λ 0 , 0
ð
Þ ¼1=e ρ λ 0 , 0
ð
Þ, then
Eq. (6.10) becomes,
∂e u
∂τ
¼ À
λ
2
e υ λ 0 , 0
ð
Þ
γ
∂e p
∂χ
ð6:11Þ
When artificial viscosity is included, however, the previous equation becomes
284
6 Numerical Treatment of Spherical Shock Waves