Kinematical Conservation Laws in Inhomogeneous Media

The system of kinematical conservation laws (KCLs) in two dimensions involves a pair of first-order partial differential equations in a ray coordinate system written in the conservation form. The KCL system governs the evolution of a propagating front (a wavefront or a shock front) in 2D media, which involves four unknown variables, and therefore, we need additional equations to close the system. Such additional relation(s) can be obtained by a weakly nonlinear ray theory (WNLRT) for wavefront propagation and a shock ray theory (SRT) in the case of shock front propagation. The WNLRT and the SRT are well-studied for front propagation in homogeneous media and are successfully applied for an uniform medium filled with a polytropic gas. As these theories are shown to be applicable in the study of sonic boom propagation, it is important to develop these theories in the case of inhomogeneous media. This article summarizes the derivation and a basic numerical test of these two theories in an inhomogeneous medium. We also show that the derived systems are hyperbolic under the condition that the wave speed is greater than the sound speed in the unperturbed medium ahead of these waves.