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

The three-dimensional time domain free surface Green function is the sum of the potentials of a Rankine source and its image source and a wave integral. The wave integral is harmonic in the fluid domain upper bounded by the mean free surface plane $z=0$ and is highly oscillatory when field and source points are close to the mean free surface plane. It is obtained that the limit form of the wave integral in the dimensionless formulation is the elementary function $$\int^\infty_0 \sqrt{\lambda} J_0(\lambda)\sin(s\sqrt{\lambda})d \lambda=\frac{s}{\sqrt{2}}\sin\left(\frac{s^2}4\right) $$ on the mean free surface plane. The function $J_0$ is the zero order Bessel function of the first kind. The velocity potential of the fluid motion problem of a surface piercing body advancing in waves is defined by a normal velocity boundary integral equation, which contains waterline contour integral of the normal derivative of the time domain free surface Green function. This elementary function form implies the ill-posedness of the boundary integral equation due to the unboundedness of the waterline integral.