The profile of a steady plane shock wave

Abstract

Introduction The physical problem of the propagation of small disturbances (sound waves) corresponds to linearization of the equations in the appropriate mathematical formulation. In an inviscid fluid this leads to plane waves which can be either compressive or rarefactive and which propagate at a constant speed (the sound speed) which is a property of the medium. Since differential equations inherently assume the differentiability of solutions, the propagation of discontinuous wave fronts requires special treatment. This is accomplished by the simple expedient of approximating an initially discontinuous wave front by a sequence of continuous initial functions. The corresponding sequence of continuous solutions of the differential equations is observed to converge to a discontinuous wave front propagating at the same sound speed. This serves to extend the manifold of solutions of the differential equations to include discontinuities which propagate unaltered whether they are compressive or rarefactive. If dissipation is introduced into the mathematical formulation, it is observed that initial discontinuities are wiped out, and all disturbances tend to broaden with time. On the other hand, it is a classical result that the propagation of non-dissipative waves of finite amplitude is no longer symmetric, compressive fronts tending to steepen and rarefactive ones to broaden. In particular, a compressive front will steepen until it attains a vertical slope, after which it becomes multiple-valued and can no longer represent a physical quantity. Furthermore, if it is attempted to approximate an initially discontinuous compressive wave by a sequence of smooth initial functions, the time of breakdown of each succeeding smooth wave will approach the initial instant, so that no limit exists which could be termed propagation of the initial discontinuity. By contrast, successive approximations of this sort do converge if the initial discontinuity is rarefactive, but it is found that the initial discontinuity is immediately wiped out and propagates as a broadening rarefaction wave. 1