pressure and U s is the velocity of the shock front as previously noted. The velocity
u in Eq. (5.72) is the air velocity immediately behind the shock which we have
previously seen to be given by the Eq. (3.38b), that is, u ¼ 2U s /(γ + 1), hence,
Eq. (5.72) becomes,
d
dt
4π
3
ρ 0 R
3 2
γ þ 1
U s
¼ 4πR
2
α
2
γ þ 1
ρ 0 U
2
s ,
ð5:73Þ
after cancelling common terms in the latter equation we can write
1
3
d
dt
R
3 U s
À
Á ¼ αR
2 U
2
s ,
and, furthermore, writing this as
1
3
dR
dt
d
dR
R
3 U s
À
Á ¼ αR
2 U
2
s
so that
1
3
d
dR
R
3 U s
À
Á ¼ αR
2 U s
ð5:74Þ
since U s ¼ dR/dt. Writing Eq. (5.74) in the form;
1
3
d R
3 U s
À
Á
R
3 U s
¼ α
dR
R
and after integrating we obtain
1
3
ln R
3 U s ¼ α ln R þ constant
or
U s ¼ aR
À3 1Àα
ð
Þ ,
ð5:75Þ
where a is a constant of integration.
Let us now determine the constants, a and α in these equations by taking into
account the energy generated in the explosion. The total energy as we have previously seen is made up of kinetic and internal energy according to
E 0 ¼
1
2
mu
2
þ mc V T:
ð5:76Þ
254
5 Spherical Shock Waves: The Self-similar Solution