Acoustic trapped modes in a three-dimensional waveguide of slowly varying cross section

Abstract

In this paper, we develop an asymptotic scheme to approximate the trapped mode solutions to the time harmonic wave equation in a three-dimensional waveguide with a smooth but otherwise arbitrarily shaped cross section and a single, slowly varying 'bulge', symmetric in the longitudinal direction. Extending previous research carried out in the twodimensional case, we first use a WKBJ-type ansatz to identify the possible quasi-mode solutions that propagate only in the thicker region, and hence find a finite cut-on region of oscillatory behaviour and asymptotic decay elsewhere. The WKBJ expansions are used to identify a turning point between the cuton and cut-off regions. We note that the expansions are non-uniform in an interior layer centred on this point, and we use the method of matched asymptotic expansions to connect the cut-on and cutoff regions within this layer. The behaviour of the expansions within the interior layer then motivates the construction of a uniformly valid asymptotic expansion. Finally, we use this expansion and the symmetry of the waveguide around the longitudinal centre, x=0, to extract trapped mode wavenumbers, which are compared with those found using a numerical scheme and seen to be extremely accurate, even to relatively large values of the small parameter. © 2012 The Author(s) Published by the Royal Society. All rights reserved.