On Head Seas Travelling Along a Horizontal Cylinder

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Abstract
Consider a wave train of arbitrary wavelength travelling without change 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 longitudinal mid-plane, and that the wave motion is also symmetrical about this plane. At a distance from the cylinder the motion is supposed to approximate to the incident wave train. This wave motion is 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 the usual assumptions of linearized wave theory there can be no such wave motion. In other words, according to the linearized theory a head sea must deform as it travels along a horizontal cylinder. (The proof fails for those wavelengths, if any, for which the Fredholm determinant of a certain integral equation vanishes. There is as yet no general uniqueness theory without such a limitation.) To illustrate this conclusion a particular problem is treated, corresponding to a head sea travelling along a wall which is slightly inclined to the vertical along part of its length and is exactly vertical elsewhere. For this case the progressive deformation can be calculated.

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