On singular and highly oscillatory properties of the Green function for ship motions

The Green function used for analysing ship motions in waves is the velocity potential due to a point source pulsating and advancing at a uniform forwa rd speed. The behaviour of this function is investigated, in particular for the case when the source is located at or close to the free surface. In th e far field, the Green function is represented by a single integral along o ne closed dispersion curve and two open dispersion curves. The single integ ral along the open dispersion curves is analysed based on the asymptotic ex pansion of a complex error function. The singular and highly oscillatory be haviour of the Green function is captured, which shows that the Green funct ion oscillates with indefinitely increasing amplitude and indefinitely decr easing wavelength, when a field point approaches the track of the source po int at the free surface. This sheds some light on the nature of the difficu lties in the numerical methods used for predicting the motion of a ship adv ancing in waves.