The normal motion of a shock wave through a non-uniform one-dimensional medium

Abstract

The non-uniform medium is regarded as a succession of small-density discontinuities separated by uniform regions. Consideration of the interaction of a shock wave with a weak contact discontinuity gives a first-order relationship between change in shock strength and change in density across the discontinuity, which is integrated to give the shock strength as a function of the initial density of the non-uniform medium in closed form. Due to the passage of the shock, a wave is reflected back through the non-uniform medium, generating in turn a doubly reflected wave which eventually catches up the shock. A complete description of the flow as modified by the first reflected wave is obtained. The modifications to the flow caused by the doubly reflected wave are more difficult to formulate, and a complete description of the flow so modified is not given. The extra difficulty is partly due to the dependence of the doubly reflected wave on the initial density distribution, whereas the motion of the incident shock, and the flow behind it as modified only by the first reflected wave, are found to have the useful property that they are independent of the particular density-distance distribution being considered. Calculations of the total strength of the doubly reflected wave, and the strength of the incident shock when this wave has fully merged with it, have been made for a particular density distribution. A comparison of this calculated strength with the strength of the shock transmitted, after the interaction of a shock wave and a contact discontinuity, suggests that a description of the flow which takes account only of the single and double reflexions is satisfactory, even if the initial density distribution varies considerably.