Question

A shock wave is the region of an elevated pressure propagating in the positive direction of the x-axis at a high velocity v. At the moment of arrival of the wave, the pressure abruptly increases. This dependence is plotted in Fig. 71.

Determine the velocity u acquired by a wedge immediately after the shock front passes through it. The mass of the wedge is m, and its size is shown in Fig. 72. 

Friction should be neglected, and the velocity acquired by the wedge should be assumed to be much lower than the velocity of the wave (u << v).

Answer 1

Let us choose the origin as shown in Fig. 205. Then the force acting on the wedge depends only

on the x-coordinate of the shock front. The horizontal component of this force is 

where x = vt is the wave front coordinate by the moment of time t from the beginning of propagation of the wave through the wedge. The acceleration imparted to the wedge at this moment of time is 

At the moment of time to w hen the wave front reaches the rear face of the wedge, i.e. when the wave front coordinate is b = vt₀, the acceleration of the wedge becomes 

Since the acceleration of the wedge linearly depends on time, for calculating the velocity u of the wedge by the moment of time t₀ we can use the mean value of acceleration

When the entire wedge is in the region of an elevated pressure, the resultant force acting on the wedge is zero. The answer to the problem implies that the condition u << v means that p₀ << 2mv²/(abc).