EXISTENCE THEOREMS FOR TRAPPED MODES

A two-dimensional acoustic waveguide of infinite extent described by t wo parallel lines contains an obstruction of fairly general shape whic h is symmetric about the centreline of the waveguide. It is proved tha t there exists at least one mode of oscillation, antisymmetric about t he centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance d own the waveguide away from the obstruction. Mathematically, this trap ped mode is related to an eigenvalue of the Laplace operator in the wa veguide. The proof makes use of an extension of the idea of the Raylei gh quotient to characterize the lowest eigenvalue of a differential op erator on an infinite domain.