High frequency waves near cusp caustics

Kalligiannaki, E.; Katsaounis, Th.; Makrakis, G. N.

doi:10.1090/qam/1955226

Authors: E. Kalligiannaki, Th. Katsaounis and G. N. Makrakis
Journal: Quart. Appl. Math. 61 (2003), 111-129
MSC: Primary 78A05; Secondary 35Q60, 76Q05, 78A40
DOI: https://doi.org/10.1090/qam/1955226
MathSciNet review: MR1955226
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Abstract: It is well known that the usual harmonic ansatz of geometrical optics fails near caustics. However, uniform expansions exist which are valid near and on the caustics, and reduce asymptotically to the usual geometric field far enough from them. In this paper, we apply the Kravtsov-Ludwig technique for computing high-frequency fields near cusp caustics. We compare these fields, with those predicted by the geometrical optics, for a couple of model problems: first, the cusp generated by the evolution of a parabolic initial front in a homogeneous medium, a problem which arises in the high-frequency treatment of cylindrical aberrations, and second, the cusp formed by refraction of the rays emitted from a point source in a stratified medium with a weak interface. It turns out that inside and near the cusp, the geometrical optics solution is significantly different than the Kravtsov-Ludwig solution, but far enough from the caustic that the two solutions are, in fact, in very good agreement.

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