The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on October 4, 2009
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbp021
Elastodynamic Equations: Characteristics, Wavefronts And Rays
( Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada )
( Department of Earth Sciences, Memorial University, St. John's, Newfoundland A1B 3X5, Canada )
lpbos{at}ucalgary.ca
Corresponding author mslawins{at}mun.ca
Received 3 January 2009. Revise 9 August 2009. Accepted 14 September 2009.
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Abstract |
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We derive the characteristic equations, and the so-called Christoffel equation, for the vector elastodynamic equations in terms of both hypersurfaces of nonuniqueness and as wavefronts based on a physical definition. We follow Courant and Hilbert in defining a wavefront as a surface for which a solution may be zero on one side but nonzero on the other. We show that wavefronts defined in this way must be characteristics. Moreover, we give a partial converse showing that in the case of analytic coefficients, characteristics must (locally) be wavefronts. Furthermore, we show how the equations defining wavefronts, which are characteristics, can be obtained by letting frequency tend to infinity in a certain trial solution expressed in the frequency domain. The last approach might suggest that ray theory, which results from the Christoffel equation, is an asymptotic theory. Taking the limit, however, is just a technicality and is unnecessary to obtain the Christoffel equation, which is contained in the elastodynamic equations and their characteristic equations.