Effect of molecular relaxation on nonlinear evolution of N-waves

Abstract

The propagation of an initially antisymmetric disturbance through a relaxing medium in one-dimension is considered. If dissipation and dispersion effects are small compared with the effect of nonlinearity, the disturbance approaches the classic N-wave profile with narrow shocks controlled by relaxation processes. As the N-wave propagates it spreads and decays in amplitude, affecting key balances between competing physical processes. In this paper we analyse the change in the shock structure as the outer solution evolves, using asymptotic analysis supplemented by numerical results. Two numerical schemes are described - an implicit scheme with variable spatial mesh which allows good resolution of the shock structure, and a pseudospectral scheme which is used when multiple relaxation modes are considered. Experimental measurements (Pawlowski et al 2005 and Yuldashev et al 2008) reveal the appearance of a slowly decaying shock tail previously unexplained by analysis of the augmented Burgers equation. In this paper we demonstrate that this phenomenon occurs when one of the relaxation timescales is comparable to the time of pulse duration.