Since the velocity of the disturbance is dependent on u according to Eq. (2.19);
dx
dt
¼ c 0 þ
γ þ 1
2
u,
then different parts of the wave travel at different speeds. Let us now consider two
neighbouring point in the positive slope of the profile, namely, x 1 and x 2 where the
velocities are u 1 and u 2 , respectively. After a time t the point x 1 has moved to
x
0
1 ¼ x 1 þ c 0 þ
γ þ 1
2
u 1
h
i
t
and the point x 2 has moved to
x
0
2 ¼ x 2 þ c 0 þ
γ þ 1
2
u 2
h
i
t:
The initial separation between x 1 and x 2 is x 2 À x 1 while the new separation is
x
0
2 À x
0
1 ¼ x 2 À x 1
ð
Þþ
γ þ 1
2
u 2 À u 1
ð
Þ t:
Since u 2 >u 1 , then x
0
2 À x
0
1 will increase with time and the disturbance will spread.
On the other hand, consider the points x 3 and x 4 on the negative slope as shown in
Fig. 2.1. After a time t the point x 3 has moved to
x
0
3 ¼ x 3 þ c 0 þ
γ þ 1
2
u 3
h
i
t
and x 4 has moved to
x
1
x
2
x
3
x
4
x
u
Fig. 2.1 Snapshot of a
wave profile
50
2 Waves of Finite Amplitude