Abstract
Using small-amplitude expansions, it is demonstrated that weakly nonlinear periodic edge waves, travelling along the shoreline of a beach, can be attenuated owing to radiation of oblique waves out to sea. A few beach profiles, for which edge-wave dispersion relations are known in closed form, are discussed, and necessary conditions are determined for such radiation to occur due to nonlinear self-interactions. In particular, it is shown that quadratic nonlinear interactions cause the second edge-wave mode on a uniformly sloping beach of slope α to radiate when 1/18π < α < ⅙π; a detailed derivation to find the amplitude of the radiated wave and the attendant decay rate of the edge wave is presented, using the full water-wave theory. Also, it is pointed out that a concomitant nonlinear mechanism can transfer energy from incoming oblique waves to subharmonic edge waves – a plausible mechanism for the generation of travelling edge waves in coastal waters – and the details of this process are discussed within the framework of a shallow-water model.
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Schedule a callMore From: Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
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