Ill-posedness of waterline integral of time domain free surface Green function for surface piercing body advancing at dynamic speed

In the linear time domain computation of a floating body advancing at a dynamic speed, the source formulation for the velocity potential of the hydrodynamic problem is commonly used so that the velocity potential is expressed as the integral of time domain free surface sources distributed on the two-dimensional wetted body surface and the one-dimensional waterline, which is the intersection of the wetted body surface and the mean free water surface. A time domain free surface source is corresponding to the time domain free surface Green function associated with a suitable source strength, which is to be solved from body boundary condition and normal velocity boundary integral equation of the source formulation. The normal velocity boundary integral equation contains an integral of the normal derivative of the time domain free surface Green function on the waterline. It is shown that the waterline integral is ill-posed. Thus the source strength of velocity potential is not obtainable.