Spectral study of the Laplace-Beltrami operator arising in the problem of acoustic wave scattering by a quarter-plane
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Publication Date
2016Journal Title
Quarterly Journal of Mechanics and Applied Mathematics
ISSN
0033-5614
Publisher
Oxford University Press
Language
English
Type
Article
This Version
AM
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Assier, R., Poon, C., & Peake, N. (2016). Spectral study of the Laplace-Beltrami operator arising in the problem of acoustic wave scattering by a quarter-plane. Quarterly Journal of Mechanics and Applied Mathematics https://doi.org/10.17863/CAM.749
Abstract
The Laplace-Beltrami operator 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 Laplace-Beltrami operator. In this paper
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 $\textit{α/=r, r}$ 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 Laplace-Beltrami operator, 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.
Sponsorship
R.C. Assier would like to acknowledge the support by UK EPSRC (EP/N013719/1).
Identifiers
This record's DOI: https://doi.org/10.17863/CAM.749
This record's URL: https://www.repository.cam.ac.uk/handle/1810/256816
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