Ray theories for hyperbolic waves, Kinematical conservation laws (KCL) and applications

Abstract

Ray theory, for the construction of the successive positions of a wavefront governed by linear hyperbolic equations, is a method which had its origin from the work of Fermat (and is related to Huygen's method). However, for a nonlinear wavefront governed by a hyperbolic system of quasilinear equations, the ray equations are coupled to a transport equation for an amplitude of the intensity of the wave on the wavefront and some progress has been made by us in its derivation and use. We have also derived some purely differential geometric results on a moving curve in a plane (surface ∊IR³), these kinematical conservation laws are intimately related to the ray theory. In this article, we review these recent results, derive same new results and highlight their applications, specially to a challenging problem: sonic boom produced by a maneuvering aerofoil.