h ¼
γ
γ À 1
T 1 p 0
T 0 ρ 0
:
Before the passage of the disturbance the heat energy per unit mass of air is
h 0 ¼
γ
γ À 1
p 0
ρ 0
,
so that the increase in heat energy per unit mass is
Δh ¼
γ
γ À 1
T 1 p 0
T 0 ρ 0
À
γ
γ À 1
p 0
ρ 0
¼
γp 0
γ À 1
ð
Þρ 0
T 1
T 0
À 1
,
and the corresponding increase in the heat energy per unit volume is
Δh
V
¼
γp 0 ρ
γ À 1
ð
Þρ 0
T 1
T 0
À 1
¼
γp 0
γ À 1
ð
Þ
ψ
T 1
T 0
À 1
,
ð5:58Þ
where ψ ¼ ρ/ρ 0 according to Eq. (5.7c). Accordingly, the heat energy wasted when
the shock front has expanded to radius R is given by
E 1 ¼ 4πR
3
Z 1
0
γp 0 ψ
γ À 1
T 1
T 0
À 1
!
η
2 dη,
and substituting for the ratio T 1 /T 0 from Eq. (5.56) in the latter equation gives
E 1 ¼ 4πR
3 γp 0
γ À 1
Z 1
0
R
À3 E 0
γp 0 B γ
ð Þ
1=γ
f
1=γ
À ψ
"
#
η
2 dη:
ð5:59Þ
However, the pressure at the shock front is given by
p 1 ¼
R
À3 E 0
γB γ
ð Þ
f
½ Š η¼1 ,
ð5:60Þ
and dividing by p 0 on both sides gives,
p 1
p 0
¼
R
À3 E 0
γp 0 B γ
ð Þ
f
½ Š η¼1 y 1 ,
ð5:61Þ
so that y 1 is the pressure in atmospheres at the shock front. Using this in the integral
in Eq. (5.59) gives,
5.14 The Wasted Energy
245