Evolution of perturbed planar shock waves

We consider the evolution of a planar gas-dynamic shock wave subject to smooth initial perturbations in both Mach number and shock-shape profile. A complex-variable formulation for the general shock motion is developed based on an expansion of the Euler equations proposed by Best [\textit{Shock Waves}, \textbf{1}, 4, (1991)], The zeroth-order truncation of Best's system corresponds to the equations of Whitham's geometrical shock dynamics (GSD) while higher-order corrections provide a hierarchical description that can be closed at any order, as detailed initial flow conditions for the flow immediately behind the shock are prescribed. Solutions to the first- and second-order closure of Best's system for the evolution of planar perturbations are explored numerically to investigate the development of a finite-time singularity in the shock shape profile. Results are compared to those obtained using GSD [Mostert \textit{et al.}, \textit{J. Fluid Mech.}, 846, (2018)].