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

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

“…In [1] we have developed a method to study the propagation of a surface Ω t in three-dimensional space R 3 . The paper contains two systems of equations for an isotropic evolution of Ω t .…”

“…As long as Ω t remains smooth, the systems (1.1)-(1.2) and (1.3)-(1.4) are equivalent, we refer the reader to [1] for a proof of this basic result.…”

“…Hence, suitable closure relations are to be supplied to both to get a completely determined system of equations. In [1] we have derived a closure relation by considering the energy propagation along the rays of a weakly nonlinear ray theory (WNLRT) under high frequency approximation. For a nonlinear wave front running into a polytropic gas at rest and in uniform state, the energy transport equation written in the form (7.4) of [1] is…”

“…A conservation form of the equation (1.5) is the equation (6.7) of [1] which is (1.6) (m − 1) 2 e 2(m−1) g 1 g 2 sin χ t = 0. We add the differential form the energy transport equation, namely (1.5) to the ray equations (1.1)-(1.2).…”

“…In our recent publication [1] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7×7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained in [1]. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues.…”