Computation of long lived resonant modes and the poles of the S-matrix in water wave scattering

Water wave scattering by variable bathymetry and fixed objects in two-dimensions with particular interest in cases where long-lived resonant, or near-trapping, modes arise is studied. The S-matrix (or scattering matrix), which is derived from the frequency domain solution, is introduced and a numerical scheme to compute the elements for complex frequencies by the analytic extension is given. Various examples of the S-matrix are computed and the importance of the singularities or poles of the S-matrix are highlighted. The time-domain problem is then considered, in particular the fluid motion excited by the scattering of an incident wave packet. The singularity expansion method approximation for the time-dependent solution as a sum over resonant modes is obtained using the poles of the S-matrix. The method is illustrated with some numerical examples.