A Problem in shock wave decay

Abstract

One of the elementary applications of the Rankine-Hugoniot shock relations which relate conditions on the two sides of a plane shock wave is that of determining the flow when a piston is pushed with constant velocity ū into a tube containing gas at rest. A shock wave races into the undisturbed gas at a constant speed Ū whose value can easily be found in terms of ū and the constants which specify the uniform condition of the gas at rest. If, however, the piston is suddenly brought to rest after a finite time the subsequent behaviour of the shock wave is very difficult to determine. A rarefaction wave is generated at the piston, and, as the velocity of the shock is subsonic relative to the gas behind it, this eventually overtakes the shock wave causing it to weaken. Since the energy supplied is finite the ultimate speed of the shock will tend to that of a sound wave. The analytical treatment of the flow behind the shock is made difficult by the entropy gradients which arise because of the variation in shock strength. It is further complicated by the disturbances which are reflected off the piston and give rise to a secondary interaction with the shock. Indeed, it seems safe to say that a complete description of the motion would certainly depend on some form of numerical integration.