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

Arun, KR and Prasad, Phoolan (2009) 3-D kinematical conservation laws (KCL): Evolution of a surface in R-3 - in particular propagation of a nonlinear wavefront. In: Wave Motion, 46 (5). pp. 293-311.

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Abstract

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Omega(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 Omega(t) with singularities which we call kinks and which are curves across which the normal n to Omega(t) and amplitude won Omega(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 2, 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 Omega(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.

Item Type:	Journal Article
Publication:	Wave Motion
Publisher:	Elsevier Science
Additional Information:	Copyright of this article belongs to Elsevier Science.
Keywords:	Ray theory; Kinematical conservation laws;Nonlinear waves; Conservation laws;Shock propagation;Curved shock; Hyperbolic and elliptic systems;Fermat's principle.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	09 Jul 2009 08:10
Last Modified:	19 Sep 2010 05:36
URI:	http://eprints.iisc.ac.in/id/eprint/21322

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