A two-dimensional 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 centreline, 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 Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.

We consider a two-dimensional infinitely long acoustic waveguide formed by two parallel lines containing an arbitrarily shaped obstacle. The existence of trapped modes that are the eigenfunctions of the Laplace operator in the corresponding domain subject to Neumann boundary conditions was proved by Evans, Levitin and Vassiliev (J. Fluid Mech. 2611994) for obstacles symmetric about the centreline of the waveguide. In our paper we deal with the situation when the obstacle is shifted with respect to the centreline and study the resulting complex resonances. We are particularly interested in those resonances which are perturbations of (real) eigenvalues. We study how an eigenvalue becomes a complex resonance moving from the real axis into the upper half-plane as the obstacle is shifted from its original position. The shift of the eigenvalue along the imaginary axis is predicted theoretically and the result is compared with numerical computations. The total number of resonances lying inside a sequence of expanding circles is also calculated numerically.

We study the elasticity operator in a semi-strip subject to free boundary conditions. In the case of zero Poisson ratio we prove the existence of a positive eigenvalue embedded in the essential spectrum. Physically, the eigenvalue corresponds to a 'trapped mode', that is, a harmonic oscillation of the semi-strip localized near the edge. This effect, known in mechanics as the 'edge resonance', has been extensively studied numerically and experimentally. Our result provides a mathematical justification.