On Head Seas Travelling Along a Horizontal Cylinder

Consider a wave train of arbitrary wavelength travelling withoutchange of form along a partially immersed fixed horizontal cylinder,the wave crests being normal to the generators of the cylinder.It is supposed that the cylinder is symmetrical about its longitudinalmid-plane, and that the wave motion is also symmetrical aboutthis plane. At a distance from the cylinder the motion is supposedto approximate to the incident wave train. This wave motionis a limiting form of the motion near a long ship in head seas.It is the purpose of the present work to show that under theusual assumptions of linearized wave theory there can be nosuch wave motion. In other words, according to the linearizedtheory a head sea must deform as it travels along a horizontalcylinder. (The proof fails for those wavelengths, if any, forwhich the Fredholm determinant of a certain integral equationvanishes. There is as yet no general uniqueness theory withoutsuch a limitation.) To illustrate this conclusion a particular problem is treated,corresponding to a head sea travelling along a wall which isslightly inclined to the vertical along part of its length andis exactly vertical elsewhere. For this case the progressivedeformation can be calculated.