Trapped modes of the Helmholtz equation in infinite waveguides with wall indentations and circular obstacles

Trapped modes of the Helmholtz equation are investigated in infinite, 2D acoustic waveguides with Neumann or Dirichlet walls. A robust boundary element scheme is used to study modes both inside and outside the continuous spectrum of propagating modes. An effective method for distinguishing between genuine trapped modes and spurious solutions induced by the domain truncation is presented. The method is also suitable for the detection and study of nearly trapped modes. These are of great practical importance as they display many features of trapped modes but do not require perfect geometry. An infinite, 2D channel is considered with one or two discs on its centreline. The walls may have rectangular, triangular or smooth cavities. The combination of a circular obstacle and a rectangular cavity, in both Neumann and Dirichlet guides is studied, illustrating the possible use of a movable disc to detect wall irregularities. The numerical method is validated against known results and many new modes are identified, both inside and outside the continuous spectrum. Results obtained suggest that at least one symmetry line is an important condition for the formation of trapped mode-type resonances. The addition of a symmetry-preserving geometric parameter to a problem which has a discrete embedded trapped mode solution for a specific geometry, tends to lead to a continuous set of trapped modes.