Consequently, the radius of the shock front has the following dependence;
R t
ð Þ /
E 0
ρ 0
1
5
t
2
5 :
Letting
R t
ð Þ ¼ η 0
E 0
ρ 0
1
5
t
2
5 ,
ð5:1Þ
where η 0 is a dimensionless constant. One can see that the dependence as expressed
in this latter equation is in accordance with the dimensionless combination referred
to above. The velocity of the shock front is
U t
ð Þ ¼
dR
dt
¼
2
5
η 0
E 0
ρ 0
1
5
t
À
3
5 ¼
2
5
η 0
E 0
ρ 0
1
5 t
2
5
t
¼
2
5
R
t
,
or alternatively,
U t
ð Þ ¼
2
5
η
5
2
0
E 0
ρ 0
1
2
R
À
3
2 AR
À
3
2 :
ð5:2Þ
where A is defined according to the equation,
A ¼
2
5
η
5
2
0
E 0
ρ 0
1
2 :
ð5:3Þ
Writing the dimensionless combination r(ρ 0 /E 0 t
2 )
1/5 as
η ¼
ρ 0
E 0 t 2
1
5
r
and noting that r ¼ R at the shock front, we have
η 0 ¼
ρ 0
E 0 t 2
1
5
R,
hence,
η ¼ η 0
r
R
:
220
5 Spherical Shock Waves: The Self-similar Solution