Generalized eigenfunction method for floating bodies

We consider the time domain problem of a floating body in two dimensions, constrained to move in heave and pitch only, subject to the linear equations of water waves. We show that using the acceleration potential, we can write the equations of motion as an abstract wave equation. From this we derive a generalized eigenfunction solution in which the time domain problem is solved using the frequency-domain solutions. We present numerical results for two simple cases and compare our results with an alternative time domain method.