Complex resonances in acoustic waveguides

Abstract

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. 261 1994) 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.