On the Mach stem configuration with shallow angle

TL;DR

This paper explains the common weak stability criteria in steady Mach stem configurations and weakly nonlinear shock stability by analyzing normal modes and causality conditions in the compressible Euler equations.

Contribution

It reveals the link between stability criteria for steady Mach stems and the weakly nonlinear stability analysis through normal mode analysis and causality conditions.

Findings

The vanishing of the temporal frequency for specific tangential velocities.

The coincidence of this velocity with the one for steady Mach stem configurations.

Clarification of the relationship between causality conditions and the Lopatinskii determinant.

Abstract

The aim of this article is to explain why similar weak stability criteria appear in both the construction of steady Mach stem configurations bifurcating from a reference planar shock wave solution to the compressible Euler equations, as studied by Majda and Rosales [Stud. Appl. Math. 1984], and in the weakly nonlinear stability analysis of the same planar shock performed by the same authors [SIAM J. Appl. Math. 1983], when that shock is viewed as a solution to the evolutionary compressible Euler equations. By carefully studying the normal mode analysis of planar shocks in the evolutionary case, we show that for a uniquely defined tangential velocity with respect to the planar front, the temporal frequency which allows for the amplification of highly oscillating wave packets, when reflected on the shock front, vanishes. This specific tangential velocity is found to coincide with the expression given by Majda and Rosales [Stud. Appl. Math. 1984] when they determine the steady planar shocks that admit arbitrarily close steady Mach stem configurations. The links between the causality conditions for Mach stems and the so-called Lopatinskii determinant for shock waves are also clarified.