Trapping of waves by a submerged elliptical torus.
M. McIver & R. Porter
(submitted)
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 plane 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 and 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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