Trapped Modes

Introduction

These notes describe some mathematical results on the linearised problem of the interaction of water waves with a freely-floating structure in open water. Conventionally, this problem is solved by first taking a Fourier transform in time, and then decomposing the resulting frequency-domain problem into the so-called scattering and radiation problems. In the scattering problem the structure is held fixed in incident waves of a prescribed frequency. In the radiation problem the incident waves are removed and the structure is forced to oscillate at a prescribed frequency. The scattering and radiation problems are closely related and differ only by the "forcing" boundary condition imposed on the surface of the structure. The two problems are linked through the equations of motion for the freely-floating structure.

The uniqueness of solutions in the frequency-domain to the scattering and radiation problems has been a subject of research since at least the early 1950's. In a seminal paper, John (1950) established uniqueness for a particular class of geometries and since that time many other partial results have been obtained (see, for example, Simon & Ursell 1984). Many (probably most) researchers in the field believed that it was only a matter of time before a general uniqueness proof would be obtained that was valid for any structural geometry and for all frequencies.

For a specified geometry, uniqueness of the solution to a forcing problem at a particular frequency is equivalent to the non-existence of a trapped mode at that frequency. A trapped mode is a solution of the corresponding homogeneous problem and represents a free oscillation with finite energy of the fluid surrounding the fixed structure. For a given structure, trapped modes may exist only at discrete frequencies. Mathematically, a trapped mode corresponds to an eigenvalue embedded in the continuous spectrum of the relevant operator.

The reason for the absence of a general uniqueness proof became clear when M. McIver (1996) showed how to construct fixed structures that support a trapped mode at a particular frequency. Subsequently, quite a variety of trapping structures have been discovered. All of those found so far are characterised by one of more holes that essentially isolate a portion of the free surface. The fluid motion consists of a sloshing supported by the hole, and a decay to zero on paths directed away from the structure. Trapped modes supported by a fixed structure are termed here "sloshing modes".

P. McIver (2005) has pointed out that the study of the individual scattering and radiation problems does not provide direct information about the uniqueness of a solution to the problem for a freely-floating structure. In general, there is a unique solution for the problem of a freely-floating structure even at frequencies corresponding to the existence of sloshing trapped modes supported by the fixed structure. Conversely, it has been shown by P. McIver & M. McIver (2005a,b) that the freely-floating structure can support trapped modes at frequencies for which the scattering and radiation problems have a unique solution. This latter type of trapped mode is a coupled motion of the fluid and structure that possesses finite energy; such modes will be termed here "motion modes".