Focusing of shock waves at smooth caustics

Abstract

Focusing of weak shock waves at smooth caustics is investigated both theoretically and numerically. The pioneering works of Buchal and Keller (1960) and Ludwig (1966) about geometrical theory of diffraction are extended to take into account 3D heterogeneity and fluid motion, along with nonlinear effects. We thus recover the nonlinear Tricomi equation derived by Guiraud (1965) and Hayes (1968). In most situations, nonlinear effects are shown to be negligible for incoming smooth signals. However, for incoming weak shock waves, the linear theory predicts an unphysical, infinite amplification of the signal discontinuity. To recover finite amplitudes, it is necessary to take into account nonlinearities. The resulting equation to be solved is the so-called nonlinear Tricomi equation, similar to a transonic equation. To solve it, an iterative algorithm is based on an unsteady version of the equation. The algorithm is a modification of the pseudospectral code used for solving the KZ equation of nonlinear acoustics. Convergence of the method is analyzed. The results are illustrated by simulations performed in various situations, and the application to sonic boom is discussed.