Arun, K. R. ; Prasad, Phoolan (2009) 3D kinematical conservation laws (KCL): evolution of a surface in R^{3}in particular propagation of a nonlinear wavefront Wave Motion, 46 (5). pp. 293311. ISSN 01652125

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S01652...
Related URL: http://dx.doi.org/10.1016/j.wavemoti.2009.03.004
Abstract
3D KCL are equations of evolution of a propagating surface (or a wavefront) Ω_{t} in 3space 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 3D 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.
Item Type:  Article 

Source:  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 
ID Code:  37665 
Deposited On:  25 Apr 2011 10:20 
Last Modified:  17 May 2016 20:33 
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