Trapping structures in the three-dimensional water-wave problem

Abstract

Trapped modes in the linearized water-wave problem are free oscillations of the fluid which have finite energy. They are known to exist at isolated frequencies in the presence of certain special structures. The existence of a trapped mode implies the non-uniqueness, or non-existence, of the solution to physically relevant radiation and diffraction problems for such a structure.

Previous work on the three-dimensional problem has established the existence of vertically axisymmetric structures that support trapped modes with either a single interior free surface, or two concentric interior free surfaces. In the present work the existence of several new types of trapping structures is established. These include non-axisymmetric structures with a single interior free surface and various structures with multiple interior free surfaces. The method used is an indirect one in which flow fields without wave radiation are specified, and corresponding structures are found by constructing suitable stream surfaces. Computations of the added-mass coefficients for these structures provide independent support for the existence of a trapping mode and illustrate their hydrodynamic characteristics at other wavenumbers.