1
2
u
2
a
¼ c 0 þ
γ þ 1
2
u
h
i u
a
þ x 0 u
ð Þ
which determines the constant of integration x 0 (u) as
x 0 u
ð Þ ¼
1
2
u
2
a
À c 0 þ
γ þ 1
2
u
h
i u
a
:
Substituting this back in the equation for x gives
x u, t
ð Þ ¼
1
2
u
2
a
À
c 0
a
u À
γ þ 1
2
u
2
a
þ c 0 þ
γ þ 1
2
u
h
i
t,
and simplifying yields,
x u, t
ð Þ ¼ À
c 0
a
u À
γ
2
u
2
a
þ c 0 þ
γ þ 1
2
u
h
i
t:
ð2:25Þ
Differentiating the latter equation we have
∂x
∂u
t
¼ À
c 0
a
À
γu
a
þ
γ þ 1
2
t:
The time t shock of formation of the shock wave is given by,
∂x
∂u
t
¼ 0 for u ¼ 0,
hence, t shock is given by the equation;
À
c 0
a
þ
γ þ 1
2
t ¼ 0:
so that,
t shock ¼
2c 0
γ þ 1
ð
Þa
,
ð2:26Þ
and the place of formation of the shock is at the forward front of the wave, namely, at
u
x
Fig. 2.7 Piston moving
with uniform accelerated
velocity in a tube
2.5 Time and Place of Formation of Discontinuity
55