# Chapter 22

# Unsteady Shock Wave, Contact Surface, and Wave Reflections

## 22.1. The shock wave equations and relations

We examine the problem of a shock wave (Ω) propagating parallel to the velocity of the upstream fluid in a fixed reference frame (*A*) (see Figure 22.1). We denote with *D* the propagation velocity of the shock wave *in relation to the upstream fluid* counted positive upstream: *D* is called the shock wave *celerity.*

The absolute velocity of the wave in the reference frame (*A*) is:

(since the flow is one-dimensional, we omit the vector sign above *V*_{1}). The velocity *V*_{1} undergoes a discontinuity across (Ω) to take a value *V*_{2} and we write:

The quantity *w* = *V*_{2} – *V*_{1} represents the *velocity difference* experienced by the fluid across the shock (Ω). To make the shock wave *steady*, we consider a reference frame (*R*) moving with the velocity of the shock; i.e. *V*_{1} + *D* (see Figure 22.2). In (*R*), the upstream flow (1) encounters the shock wave with a velocity:

and becomes a state (2) with the velocity in (*R*):

**Figure 22.1.** *Shock wave in the absolute reference frame*

**Figure 22.2.** *Shock wave in ...*