3-D kinematical conservation laws (KCL): Evolution of a surface in [formula omitted] – in particular propagation of a nonlinear wavefront

Abstract

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Ω t in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL. These are the most general equations in conservation form, governing the evolution of Ω t with singularities which we call kinks and which are curves across which the normal n to Ω t and amplitude w on Ω t are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of Ω t . The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Ω t ) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m > 1 and pure imaginary when m < 1 . Finally we give some examples of evolution of weakly nonlinear wavefronts.