Journal of Fluid Mechanics

Existence theorems for trapped modes

D. V.  Evans a1, M.  Levitin a2 and D.  Vassiliev a3
a1 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
a2 Department of Mathematics, Heriot–Watt University, Riccarton, Edinburgh EH14 4AS, UK
a3 School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, UK

Article author query
evans dv   [Google Scholar] 
levitin m   [Google Scholar] 
vassiliev d   [Google Scholar] 


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.

(Published Online April 26 2006)
(Received February 17 1993)