time t. In addition, the wave profile lies on both sides of the vertical tangent, so that it
must also be a point of inflexion, hence, ∂
2 x/∂u
2
¼ 0. Hence, the time and place of
formation of the shock wave is determined by the solution of the following
equations;
∂x
∂u
t
¼ 0 and
∂
2 x
∂u 2
t
¼ 0:
ð2:23Þ
Fig. 2.5 The compression
front becomes steeper with
time with the eventual
formation of a shock wave
Fig. 2.4 Forward fronts of the disturbances and associated compression waves at two different
times (see text)
2.5 Time and Place of Formation of Discontinuity
53