Initial-value-problem solution for isolated rippled shock fronts in arbitrary fluid media

Following the work of Roberts [Los Alamos Scientific Laboratory Report No. LA-299, 1945 (unpublished)], we investigate the effect of small two-dimensional perturbations on an isolated, planar shock front moving steadily through an inviscid fluid medium with an arbitrary equation of state (EOS). In the context of an initial-value problem, we derive explicit analytical expressions for the linearized, time-dependent Fourier coefficients associated with an initial corrugation of the front. The temporal evolution of these coefficients superficially resembles the attenuated “ringing” of a damped harmonic oscillator, but with the important distinctions that the frequency of oscillation is not constant, and that the damping factor is not simply an exponential function of time t. It is shown that at least two three-parameter families of stable solutions exist, one more strongly damped than the other. In both cases, we find that the envelope of oscillations decays asymptotically as t^-3/2, with shorter wavelengths dying out earlier than longer ones. For a particular perturbed-shock system, the strength of the front and the EOS properties of the material through which it propagates determine the applicable family of solutions. Theoretical predictions agree well with FAST2D numerical simulations for several examples derived from the CALEOS library.