Complex resonances in the water-wave problem for a floating structure

This work is concerned with the linearized theory of water waves applied to the motion of a floating structure that restricts in some way the motion of a portion of the free surface (an example of such a structure is a floating torus). When a structure of this type is held fixed in incident monochromatic waves, or forced to move time harmonically with a prescribed velocity, the amplitude of the fluid motion will have local maxima at certain frequencies of the forcing. These resonances correspond to poles of the scattering and radiation potentials when extended to the complex frequency domain. It is shown in this work that, in general, the positions of these poles in the scattering and radiation potentials will not coincide with the positions of the poles that appear in the velocity potential for the coupled problem obtained when the structure is free to move. The poles of the potential for the coupled problem are associated with the solution for the structural velocities of the equation of motion. When physical quantities such as the amplitude of the fluid motion are examined as a function of (real) frequency, there will in general be a shift in the resonant frequencies in going from the radiation and scattering problems to the coupled problem. The magnitude of this shift depends on the geometry of the structure and how it is moored.