Leakage and response of waves trapped by round islands

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Abstract
A study of surface wave trapping by typical islands of small seabed slope is reported. A new formulation is proposed which includes all the established approximations, but also permits treatment of trapping with energy leakage and complex eigenfrequencies. On this basis, we prove existence of eigenvalues with very small leakage and correspondingly large response to excitation by incident waves. Indeed, certain types of trapped wave modes have e x p o n e n t i a l l y s m a l l leakage and therefore, a violently resonant character. Two physical effects are found to promote such resonance: It is typical of waves that bear a close local resemblance to coastal edge waves. It also occurs for waves of large radial wavenumber component. Both effects can combine to produce especially strong resonance. The strongly resonant wave modes have a structure more complicated than is commonly considered important, but still clear‐cut enough to permit characterization by an eigenvalue problem for an ordinary differential equation. It has three relevant transition points, and extant theory for it is in effect only of first order. However, careful attention to energy conservation makes it possible to extract exponential precision for the imaginary part of the eigenvalues.

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