The motion of a freely floating cylinder in the presence of a wall and the approximation of resonances

A linear theory, based on wide-spacing and high-frequency approximations, is developed to describe resonant behaviour in two-dimensional water-wave problems involving a freely-floating half-immersed cylinder in the presence of a vertical rigid wall. The theory is not able to describe the lowest-frequency resonance, but otherwise yields explicit approximations for the locations of resonances in the complex plane and for their corresponding residues. Two problems are investigated in detail: the time-domain motion following a vertical displacement of the cylinder from equilibrium, and the time-harmonic motion of the cylinder that is excited by an incident plane wave.