Existence theorems for trapped modes
Abstract
A twodimensional 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 centerline, 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 Rayleigh quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.
 Publication:

Journal of Fluid Mechanics
 Pub Date:
 February 1994
 DOI:
 10.1017/S0022112094000236
 Bibcode:
 1994JFM...261...21E
 Keywords:

 Blocking;
 Eigenvalues;
 Existence Theorems;
 Operators (Mathematics);
 Sound Waves;
 Wall Flow;
 Waveguides;
 Acoustic Frequencies;
 Boundary Value Problems;
 Channel Flow;
 Differential Equations;
 Laplace Transformation;
 Oscillations;
 Acoustics