Asymptotic and numerical analysis of pulse propagation in relaxation media

Abstract

We consider the case of disturbances propagating in one-dimension through a medium with multiple relaxation modes and thermoviscous diffusion. Each relaxation mode is characterized by two parameters and the evolution of the disturbance is governed by an augmented Burgers equation. We begin by considering travelling wave solutions for the propagation of a pressure step, of amplitude P, in the small viscosity limit. For a single relaxation mode, if the amplitude P is less than a certain critical value then the transition is controlled entirely by the relaxation mode whereas for larger P, a thin viscous sub-shock arises. We then consider the propagation of a rectangular pulse. We establish parameter ranges in which the waveform is described by an outer solution (obtained using characteristics) and a thin shock region. Analysis of the shock region then reveals the same richness of structure seen in the travelling wave case, with subtle changes in shock structure as the disturbance decays. This is illustrated by numerical results using a pseudospectral method. Finally, analysis of the case of two relaxation modes is presented demonstrating that in some parameter regimes the transition region consists of three separate sub-regions governed by the three different physical processes.