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 and 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 McIver and McIver 2006 and McIver and McIver 2006b 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".
Sloshing modes
The first examples of trapping structures were constructed by M. McIver 1996 using an inverse procedure. The structures are two-dimensional and have two surface-piercing elements. The corresponding sloshing trapped mode consists of a persistent oscillation of the fluid that is essentially confined to the region between the two elements of the structure.
Subsequent work by McIver and McIver 1997 established the existence of axisymmetric trapped modes in three-dimensional problems. The supporting structures are surface-piercing toroids and the fluid motion is essentially confined to the central hole. It is now clear that trapping structures can be constructed almost at will. For example, subsequent work on the three-dimensional problem has established the existence of non-axisymmetric sloshing trapped modes supported by both axisymmetric and non-axisymmetric structures (Kuznetsov and P. McIver 1997, P. McIver and Newman 2003). Further, convincing numerical evidence has been obtained for the existence of submerged trapping structures in both the two- and three-dimensional problems (M. McIver 2000 , M. McIver and Porter 2002).
McIver, McIver and Zhang 2003 have shown how sloshing trapped modes may be excited in the time domain by the forced motion of a trapping structure, that is when the velocity of the structure is specified. When the structure is forced to oscillate at the trapped-mode frequency the energy in the trapped mode cannot escape to infinity and resonant growth of the fluid motion is observed. When the structure is forced to oscillate at a frequency that differs from the trapped-mode frequency, the initial motion of the structure excites a trapped mode so that the long-time behaviour is a combination of oscillations at the forcing frequency and the trapped-mode frequency.
Subsequently, P. McIver 2005 has shown that sloshing trapped modes cannot be excited by any free motion of a sloshing trapping structure. Thus, if such a trapping structure is displaced and released from rest, or is subject to incident waves, the trapped mode is not observed.
Motion modes
For a structure that supports a sloshing trapped mode at a particular frequency it may be shown that the scattering potential exists at that frequency, but that the radiation potential does not. However, when these potentials are combined through the equation of motion to obtain the potential for a freely-floating structure there is no manifestation of the sloshing trapped mode. In other words the existence of a sloshing trapped mode does not preclude a unique solution to the problem of a freely-floating structure at the trapped-mode frequency. Conversely, the uniqueness of the solution to the scattering and radiation problems does not establish uniqueness for the corresponding freely-floating structure.
A trapped-mode solution of the problem for a freely-floating structure corresponds to a coupled motion of the fluid and structure that persists for all time. To achieve this the hydrodynamic properties of the structure must satisfy two conditions at a particular frequency. The first requires that there be no radiation of waves to infinity when the structure is forced to oscillate, and the second requires the added mass coefficient to satisfy a resonance condition obtained from the equation of motion. McIver and McIver 2006 and McIver and McIver 2006b have established that there are indeed structures that satisfy these conditions in both two and three dimensions. These structures have been termed motion trapping structures. Superficially, motion trapping structures have a very similar appearance to the sloshing trapping structures discussed earlier, although their hydrodynamic properties at the trapped-mode frequency are very different.
It was noted earlier that within the linearised problem there is no mechanism for the excitation of sloshing trapped modes. The same is not true of motion modes which can be excited if a motion trapping structure is displaced from equilibrium and released from rest. However, an incident wave cannot excite a motion trapped mode.
