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The equations that describe the classical problem of water waves - inviscid, no surface tension and constant pressure at the surface - are non-dimensionalised and scaled appropriately, and the two examples: traditional gravity waves and edge waves, are introduced. In addition each type of wave is allowed to propagate over an existing flow field that is rotational and also admits a shoreline; some examples of such background flows are presented. Then, for each problem, a suitable asymptotic solution is constructed; for gravity waves, this is chosen to be that which gives a balance between nonlinearity and dispersion far from the shore (so that a soliton-type problem is recovered there), and then the behaviour of this solution is examined as the shoreline is approached. Sufficiently close to the shore, the asymptotic expansion is not valid, resulting in the formulation of a new, scaled problem. It is then shown - not surprisingly - that the wave, close inshore, is dominated by nonlineaity, with the amplitude of the wave growing according to Green's law. The problem of edge waves is formulated in a similar fashion, although the relevant scales are different; in particular, the background flow must be roughly of the same size as the edge wave itself, for a self-consistent asymptotic theory of the type presented here. The development follows closely that used in the absence of a background flow, but with the background flow now appearing in the solution to leading order. This has the effect of distorting, for example, the run-up pattern of the edge waves at the shoreline, to the extent that, under certain conditions, the two solutions of the earlier theory can now be replaced by one (unique) solution.
Author(s): Johnson RS
Publication type: Article
Publication status: Published
Journal: Journal of Nonlinear Mathematical Physics
Year: 2008
Volume: 15
Issue: 2
Pages: 133-156
Print publication date: 01/08/2008
ISSN (print): 1402-9251
ISSN (electronic): 1776-0852
Publisher: World Scientific Publishing Co. Pte. Ltd.
URL: http://dx.doi.org/10.2991/jnmp.2008.15.s2.10
DOI: 10.2991/jnmp.2008.15.s2.10
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