A Nonlinear Ray Theory

  • A proof of the famous Huygens" method of wavefront construction is reviewed and it is shown that the method is embedded in the geometrical optics theory for the calculation of the intensity of the wave based on high frequency approximation. It is then shown that Huygens" method can be extended in a natural way to the construction of a weakly nonlinear wavefront. This is an elegant nonlinear ray theory based on an approximation published by the author in 1975 which was inspired by the work of Gubkin. In this theory, the wave amplitude correction is incorporated in the eikonal equation itself and this leads to a sytem of ray equations coupled to the transport equation. The theory shows that the nonlinear rays stretch due to the wave amplitude, as in the work of Choquet-Bruhat (1969), followed by Hunter, Majda, Keller and Rosales, but in addition the wavefront rotates due to a non-uniform distribution of the amplitude on the wavefront. Thus the amplitude of the wave modifies the rays and the wavefront geometry, which in turn affects the growth and decay of the amplitude. Our theory also shows that a compression nonlinear wavefront may develop a kink but an expansion one always remains smooth. In the end, an exact solution showing the resolution of a linear caustic due to nonlinearity has been presented. The theory incorporates all features of Whitham" s geometrical shock dynamics.

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Verfasserangaben:P. Prasad
URN (Permalink):urn:nbn:de:hbz:386-kluedo-4981
Schriftenreihe (Bandnummer):Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (101)
Sprache der Veröffentlichung:Englisch
Jahr der Fertigstellung:1994
Jahr der Veröffentlichung:1994
Veröffentlichende Institution:Technische Universität Kaiserslautern
Datum der Publikation (Server):01.01.1994
Fachbereiche / Organisatorische Einheiten:Fachbereich Mathematik
DDC-Sachgruppen:5 Naturwissenschaften und Mathematik / 510 Mathematik
Lizenz (Deutsch):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011