On the non-existence of trapped modes in acoustic wave-guides

It is well known that trapped modes exist in certain types of acoustic waveguides. These modes correspond to localized fluid oscillations and occur at frequencies at which propagating modes down the guide are not able to exist, below a so-called ‘cut-off frequency’. For example, antisymmetric trapped-mode motions are known to occur in two-dimensional, parallel-plate waveguides containing bodies, at wavenumbers which are less than π/2d, where 2d is the width of the guide. So far, however, these modes have only been found in waveguides that have acoustically hard walls and either contain acoustically hard bodies or have variable cross-section. The purpose of this work is to investigate the existence or otherwise of trapped modes when one or more of the boundaries is replaced by an acoustically soft boundary. We prove here that trapped modes do not exist below the cut-off frequency for a large class of sound-soft guides containing both sound-soft and sound-hard bodies. In addition, we show that antisymmetric trapped modes do not exist below the cut-off frequency in many two-dimensional, sound-hard guides containing sound-soft bodies. This second result is also generalized to certain types of trapped-mode motion in axisymmetric waveguides. The method of proof relies on finding a strictly positive function w which satisfies a certain field inequality within the guide and boundary inequalities on the guide walls and body surfaces. A vector identity is established which relates w to the possible trapped-mode potential φ in such a way that it may be deduced that φ must be identically equal to zero throughout the guide.