Complex Resonances in Acoustic Waveguides

We consider a two-dimensional innnitely 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 & Vassiliev (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 eigen-value 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. 1. Introduction The existence of trapped modes in an innnitely long acoustic waveg-uide containing an obstacle of a fairly general shape symmetric about the centreline of the waveguide was proved by Evans, Levitin & Vas-siliev (1994). The trapped modes under consideration are the eigen-functions of the Laplace operator in an unbounded domain (which is the strip without the obstacle) subject to Neumann boundary conditions. The corresponding eigenvalues are embedded in the continuous spectrum. The vibration modes studied in 1] are antisymmetric with respect to the centreline and describe acoustic waves decaying at innn-ity. Further rigorous mathematical results concerning the existence or non{existence of trapped modes were obtained by Davies & Parnovski (1998). There is also a substantial bibliography devoted to the numerical study of trapped modes. For example, Evans & Linton (1991) provided a technique for computing trapped modes in an acoustic waveguide