Focusing of acoustical 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; its characteristic length scales are determined based on the Pearcey function which governs the linear acoustic field. With these 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 for any transient signal, so that the problem is numerically tractable. A similitude is found for an incoming step wave, and is compared to focusing at a smooth caustic. The linear approximate solution is obtained in the general case. Application for an incoming N-wave to simulate sonic boom superfocusing is also discussed.