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Whispering gallery waves near boundary inflection

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The Helmholtz equation typically admits two types of solutions of interest depending on the geometry of the domain: modal solutions or scattering solutions. However, in analogy with the Airy function facilitating the transition between sinusoidal and oscillatory asymptotic behaviours, there is not yet an equivalent object for wave solutions transitioning from having a discrete to a continuous spectrum. Based on work by Babich, V.M. and Popov, M.M., a boundary with an inflection point models this fundamental problem, where the concave part of the boundary exhibits whispering gallery modal solutions, and the convex part exhibiting scattered rays. The big question lies in a neighbourhood of the inflection point, where asymptotic analysis and Green’s functions methods are used in attempt to construct a uniformly valid expansion on the entire boundary. The boundary value problem in the inflection region is reduced to two Volterra integral equations with scope of solution in the form of a convergent Neumann series. A rigorous review of the whispering gallery asymptotics is presented as well as a plan for future work.

This talk is part of the Waves Group (DAMTP) series.

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