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 V1). The velocity V1 undergoes a discontinuity across (Ω) to take a value V2 and we write:
The quantity w = V2 – V1 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. V1 + 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):