Approximations to Embedded Trapped Modes in Wave Guides

Abstract

This work is concerned with fluid motions in an infinitely-long, two-dimensional acoustic waveguide containing an obstacle. Cartesian coordinates (x,y) are chosen so that the x axis lies along the centre of the guide and the walls of the guide are at y = ±d. Under the usual assumptions of linear acoustics, time-harmonic motions of radian frequency ω may be described by a velocity potential Ø(x,y)e ^-iωt. Non-trivial solutions for (f) are sought within the guide (excluding the obstacle) that, on the guide walls, satisfy either a homogeneous Neumann condition (a ‘Neumann guide’) or a homogeneous Dirichlet condition (a ‘Dirichlet guide’). In addition, the solutions are required to satisfy a homogeneous Neumann condition on the obstacle and to decay to zero as |x| → ∞. Such a solution corresponds to a ‘trapped mode’, that is a free oscillation of the fluid with finite energy. The physical significance of the existence of a trapped mode at a particular frequency is that it corresponds to an ‘acoustic resonance’ in a forced problem.

This work is part of a collaboration with C. M. Linton, M. Mclver and J. Zhang that is funded by EPSRC grant GR/M30937.