Multi-valued solutions of steady-state supersonic flow. Part 1. Linear analysis

The shock wave equations for a perfect gas often provide more than one solution to a problem. In an attempt to find out which solution appears in a given physical situation, we present a linearized analysis of the equations of motion of a flow field with a shock boundary. It is found that a solution will be stable when there is supersonic flow downstream of the shock, and asymptotically unstable when there is subsonic flow downstream of it. It is interesting that both flows are found to be stable against disturbances of the d'Alembert type which grow from point sources; it is only when larger-scale line sources are considered that one can discriminate between the stabilities of the two types of flow. The results are applicable to supersonic flow over flat plates at incidence, to wedges, and to some cases of regular reflexion, diffraction and refraction of shocks.