An investigation is made into the trapping of surface gravity waves by totally
submerged three-dimensional obstacles and strong numerical evidence of the existence
of trapped modes is presented. The specific geometry considered is a submerged
elliptical torus. The depth of submergence of the torus and the aspect ratio of its
cross-section are held fixed and a search for a trapped mode is made in the parameter
space formed by varying the radius of the torus and the frequency. A plane wave
approximation to the location of the mode in this space is derived and an integral
equation and a side condition for the exact trapped mode are obtained. Each of these
conditions is satisfied on a different line in the plane and the point at which the
lines cross corresponds to a trapped mode. Although it is not possible to locate this
point exactly, because of numerical error, existence of the mode may be inferred with
confidence as small changes in the numerical results do not alter the fact that the
lines cross.If the torus makes small vertical oscillations, it is customary to try to express
the fluid velocity as the gradient of the so-called heave potential, which is assumed
to have the same time dependence as the body oscillations. A necessary condition
for the existence of this potential at the trapped mode frequency is derived and
numerical evidence is cited which shows that this condition is not satisfied for an
elliptical torus. Calculations of the heave potential for such a torus are made over a
range of frequencies, and it is shown that the force coefficients behave in a singular
fashion in the vicinity of the trapped mode frequency. An analysis of the time domain
problem for a torus which is forced to make small vertical oscillations at the trapped
mode frequency shows that the potential contains a term which represents a growing
oscillation.
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