Focusing of weak acoustic shock waves at a caustic cusp

Abstract

This paper investigates theoretically the focusing of weak acoustic shock waves at a caustic cusp. Near the cusp, a diffraction boundary layer is introduced, the characteristic length-scales of which are determined with the help of Pearcey function which governs the field in linear acoustics. With the adequate variables, the nonlinear wave equation is then shown to reduce to the KZ equation. The proper matching to the geometrical acoustics approximation provides the complete boundary conditions, so that the problem is numerically tractable. A similitude is found for an incoming step wave, and the linear approximate solution is obtained in the general case.