On uniqueness and trapped modes in the water-wave problem for a surface-piercing axisymmetric body

Abstract

The linear water-wave problem is considered for the case of an axisymmetric structure floating in water of infinite depth. The structure is toroidal so that the free surface is in two distinct parts. The paper has two main sections. The first considers structures satisfying the so-called 'John' condition which, for vertically axisymmetric structures, requires that the submerged part of the structure is contained within concentric cylinders extending downwards from the circles of intersection between the surface of the structure and the free surface of the water. For each azimuthal mode, it is shown that there is an infinite sequence of frequency intervals for which the solution is unique. In the second part of the paper, trapped mode solutions, that is examples of non-uniqueness, are constructed from ring-source potentials in the free surface. The radius of the ring is chosen to eliminate outgoing waves and then stream surfaces are traced numerically to obtain possible structural surfaces. Axisymmetric solutions found in a similar way were considered in another paper; here non-axisymmetric solutions are also investigated. Although the structures constructed in this way do not satisfy the John condition, the solutions obtained are entirely consistent with the uniqueness results established in the paper.