On the short-wave asymptotic motion due to a cylinder heaving on water of finite depth. III

Abstract

AbstractThis paper concludes the investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we now present a rigorous method for an arbitrary smooth cylinder which intersects the free surface normally. The aim is to justify those assumptions made in a previous paper concerning the asymptotic values on the cylinder of the two related auxiliary potentials–only one of which need be considered–which were introduced to enable an evaluation of the descriptive wave-making and virtual-mass coefficients. This is done by constructing an integral equation of the second kind whose kernel is small and deducing the leading term in the iteration solution. The method is an extension of the method for infinite depth which is treated first and it is found that there is a simple relationship between the kernels. The argument depends to some extent on results established in another previous paper–which is generalized herein–on the rigorous method for the special case of a half-immersed circular cylinder.