Spontaneous singularity formation in converging cylindrical shock waves

Abstract

We develop a nonlinear, Fourier-based analysis of the evolution of a perturbed, converging cylindrical strong shock using the approximate method of geometrical shock dynamics (GSD). This predicts that a singularity in the shock-shape geometry, corresponding to a change in Fourier-coefficient decay from exponential to algebraic, is guaranteed to form prior to the time of shock impact at the origin, for arbitrarily small, finite initial perturbation amplitude. Specifically for an azimuthally periodic Mach-number perturbation on an initially circular shock with integer mode number q and amplitude proportional to ε≪1, a singularity in the shock geometry forms at a mean shock radius R_(u,c)∼(q^2ε)^(−1/b1), where b_1(γ)<0 is a derived constant and γ the ratio of specific heats. This requires q^2ε≪1, q≫1. The constant of proportionality is obtained as a function of γ and is independent of the initial shock Mach number M0. Singularity formation corresponds to the transition from a smooth perturbation to a faceted polygonal form. Results are qualitatively verified by a numerical GSD comparison.

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