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• W2115808214 abstract "The problem of edge waves as an example within classical water-wave theory is described by presenting an overview of some of the theories that have been offered for this phenomenon. The appropriate governing equations and boundary conditions are formulated, and then the important discoveries of Stokes and Ursell, concerning the travelling edge wave, are presented. (We do not address the corresponding problem of standing waves.) Thus, the linear problem and its spectrum are constructed; in addition, we also present the linear long-wave approximation to the problem, as well as Whitham's weakly nonlinear extension to Stokes' original theory. All these discussions are based on the same formulation of the problem, allowing an immediate comparison of the results, whether this be in terms of different approximations or whether the theory be for an irrotational flow or not. Gerstner's exact solution of the water-wave problem is then briefly described, together with a transformation that produces an exact solution of the full equations for the edge wave. The form of this solution is then used as the basis for a multiple-scale description of the edge wave over a slowly varying depth; this leads to a version of the shallow-water equations which has an exact solution that corresponds to the edge wave. Some examples of the theoretical predictions for the run-up pattern are presented. We conclude with three variants of nonlinear model equations that may prove useful in the study of edge waves and, particularly, the interaction of different modes." @default.
• W2115808214 created "2016-06-24" @default.
• W2115808214 creator A5016815633 @default.
• W2115808214 date "2007-03-13" @default.
• W2115808214 modified "2023-09-30" @default.
• W2115808214 title "Edge waves: theories past and present" @default.
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• W2115808214 doi "https://doi.org/10.1098/rsta.2007.2013" @default.
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