Complex resonances in acoustic waveguides

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 th e corresponding domain subject to Neumann boundary conditions was proved by Evans, Levitin and Vassiliev (J. Fluid Mech. 261 1994) for obstacles symme tric about the centreline of the waveguide. In our paper we deal with the s ituation when the obstacle is shifted with respect to the centreline and st udy the resulting complex resonances. We are particularly interested in tho se resonances which are perturbations of (real) eigenvalues. We study how a n eigenvalue becomes a complex resonance moving from the real axis into the upper half-plane as the obstacle is shifted from its original position. Th e shift of the eigenvalue along the imaginary axis is predicted theoretical ly and the result is compared with numerical computations. The total number of resonances lying inside a sequence of expanding circles is also calcula ted numerically.