Spectral study of the Laplace--Beltrami operator arising in the problem of acoustic wave scattering by a quarter-plane

RC Assier, C Poon, Nigel Peake

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The Laplace-Beltrami operator (LBO) on a sphere with a cut arises when considering the problem of wave scattering by a quarter-plane. Recent methods developed for sound-soft (Dirichlet) and sound-hard (Neumann) quarter-planes rely on an a priori knowledge of the spectrum of the LBO. In this article we consider this spectral problem for more general boundary conditions, including Dirichlet, Neumann, real and complex impedance, where the value of the impedance varies like α/r,r being the distance from the vertex of the quarter-plane and α being constant, and any combination of these. We analyse the corresponding eigenvalues of the LBO, both theoretically and numerically. We show in particular that when the operator stops being self-adjoint, its eigenvalues are complex and are contained within a sector of the complex plane, for which we provide analytical bounds. Moreover, for impedance of small enough modulus |α| , the complex eigenvalues approach the real eigenvalues of the Neumann case.
Original languageEnglish
Pages (from-to)281-317
Number of pages37
JournalQuarterly Journal of Mechanics and Applied Mathematics
Volume69
Issue number3
DOIs
Publication statusPublished - 18 Aug 2016

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