f
0
ψ
¼
3
2
γϕ þ γϕ
0
η À ϕ
ð
Þ:
Substitution this in Eq. (5.30) yields
f
0
f
η À ϕ
ð
Þ
2 ¼ γϕ
0
η À ϕ
ð
Þþ
3
2
γϕ À 3η þ ϕ 3 þ
γ
2
À
2γϕ
2
η
and by dividing across by η À ϕ we obtain
f
0
f
η À ϕ
ð
Þ¼γϕ
0
À 3 þ
2γϕ
η
:
ð5:31Þ
In the interior region f
0 /f can be neglected as the curve of pressure is relatively flat,
consequently,
γϕ
0
À 3 þ
2γϕ
η
ffi 0,
and the approximate solution of the latter equation for which ϕ vanishes as η ! 0 is
0
10
20
30
40
50
60
70
80
90
100
0
30
60
90
120
150
time (milliseconds)
Pressure (atm)
Fig. 5.6 Pressure in atmospheres at 100 meters as a function of time for a point source explosion
with energy equivalent to 20,000 tons of TNT (γ ¼ 1.4)
236
5 Spherical Shock Waves: The Self-similar Solution
0
ψ
¼
3
2
γϕ þ γϕ
0
η À ϕ
ð
Þ:
Substitution this in Eq. (5.30) yields
f
0
f
η À ϕ
ð
Þ
2 ¼ γϕ
0
η À ϕ
ð
Þþ
3
2
γϕ À 3η þ ϕ 3 þ
γ
2
À
2γϕ
2
η
and by dividing across by η À ϕ we obtain
f
0
f
η À ϕ
ð
Þ¼γϕ
0
À 3 þ
2γϕ
η
:
ð5:31Þ
In the interior region f
0 /f can be neglected as the curve of pressure is relatively flat,
consequently,
γϕ
0
À 3 þ
2γϕ
η
ffi 0,
and the approximate solution of the latter equation for which ϕ vanishes as η ! 0 is
0
10
20
30
40
50
60
70
80
90
100
0
30
60
90
120
150
time (milliseconds)
Pressure (atm)
Fig. 5.6 Pressure in atmospheres at 100 meters as a function of time for a point source explosion
with energy equivalent to 20,000 tons of TNT (γ ¼ 1.4)
236
5 Spherical Shock Waves: The Self-similar Solution