Circular islands as resonators of long-wave energy

Abstract
A study is made of the long gravity waves trapped around isolated, cylindrically symmetrical island-continental shelf topographies. Numerical evaluation of the discrete complex spectra of the trapped wave modes reveals that the oscillations are of two, essentially different kinds. The ‘ trapped-leaky ’ wave-modes are the analogue of the trapped (edge) wave-modes along straight shorelines and many of their properties, including the Coriolis split in frequency, are deducible from the simpler geometry. On the other hand, the ‘shelf-island’ modes have no counterpart in the motions trapped along extended shorelines; they are virtually generated in the ocean round a vertical-walled circular island of radius equal to that of the island-shelf system at the sea floor. It is shown that the trapped wave-modes do not necessarily have the ‘inner critical circle’ property elucidated by Longuet-Higgins (1967) for the similar modes of oscillations of the waters over a circular seamount. On the other hand, the modes do have ‘wave’ domains adjacent to the coast whenever the undisturbed depth of water at the island’s shoreline is zero; there may still be critical circles in the shallow water region over the continental shelf. For those islands where the water has non-zero depth at the shoreline, the computation verifies Longuet-Higgins’s hypothesis (Longuet-Higgins 1967, § 13) concerning the affect on the trapped wave-modes of the presence of an island in the middle of the sea-mount. It is also shown that the fundamental ‘ trapped-leaky’ modes dominate the disturbance observed at the coast when plane wave radiation from the ocean interacts with the island-shelf system. For the particular example where the excitation has the form of a rectilinear pulse, it is shown that power spectra of the resulting oscillations exhibit some of the features of the spectra of real wave records made at islands following the passage of tsunamis.

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