p 1 / R
À3
0 f
½ Š η¼1 :
The pressure at R 0 at a later time t after the shock wave has passed and moved to
radius R is
p / R
À3 f η
ð Þ:
Forming the ratio,
p
p 1
¼
R 0
R
3 f η
ð Þ
f
½ Š η¼1
ð5:27Þ
and noting that η ¼ R 0 /R and R(t) / t
2/5
, hence, η ¼ (t 0 /t)
2/5 so that
p
p 1
¼ η
3
f η¼ t 0 =t
ð
Þ
2
5
f
½ Š η¼1
¼
t 0
t
6
5
f η¼ t 0 =t
ð
Þ
2
5
f
½ Š η¼1
:
ð5:28Þ
Defining a normalized time according to τ ¼ t/t 0 , hence, τ(η) ¼ η
À5/2 , and
therefore,
p
p 1
¼
1
τ η
ð Þ
6
5
f η
ð Þ
f
½ Š η¼1
:
ð5:29Þ
This pressure ratio p/p 1 is shown plotted in Fig. 5.5 as a function of τ.
For example, Fig. 5.6 shows the pressure (in atmospheres) at 100 meters from a
point source explosion with energy equivalent to 20,000 tons of TNT. One can
observe that the shock front arrives at approximately 11 ms following detonation
with a maximum pressure of about 130 atmospheres.
5.11 Taylor’s Analytical Approximations for Velocity,
Pressure and Density
Following Taylor’s analysis [4] and using his notation let us now proceed to obtain
approximate analytical expressions for f, ϕ and ψ.
234
5 Spherical Shock Waves: The Self-similar Solution