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Existence theorems for trapped modes

Published online by Cambridge University Press:  26 April 2006

D. V. Evans ,
M. Levitin  and
D. Vassiliev
D. V. Evans
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
M. Levitin
Affiliation:
Department of Mathematics, Heriot–Watt University, Riccarton, Edinburgh EH14 4AS, UK
D. Vassiliev
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, UK
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Abstract

A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 261 , 25 February 1994 , pp. 21 - 31
Copyright
© 1994 Cambridge University Press

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