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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on December 18, 2007
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbm025
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© The author 2007. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

LINEAR WAVES ON ROSEAU'S TWO-PARAMETER BEACH PROFILE: A GENERALIZED NEAR-SHORE WAVE AMPLIFICATION FORMULA

Ulf Ehrenmark{dagger}

( Department of Mathematics, University of Reading, Whiteknights, Reading, Berkshire )

{dagger} Hampstead Way, London NW11 7YA < ulfe{at}hotmail.co.uk>

Received 25 June 2007. Revise 10 October 2007.
  Abstract

Roseau's exact solution for infinitesimal waves on a two-parameter ({alpha}, a) convex beach profile is here revisited using an alternative solution approach. Unlike Roseau's original development, the revised method appears to yield a description of the solution which (in terms of integral structure hierarchy) is similar to that of the classical plane beach problem. Thus, computation can be undertaken at the same level of simplicity pointing the way for the model (and its three-parameter infinite channel extension) to become more widely used, for example, in validation and calibration studies of the various mild-slope equations that are currently the topic of much study. It is shown that the plane beach represents the limiting case as the parameter a->{infty} and the solution developed is amenable to exact examination of the shoreline wave amplification factor. A formula is established which is shown to reduce to the classical result developed for plane beaches and indicates that for each value of {alpha} (the beach angle at shore) the amplification has a peak for a certain finite value of far-field depth (a{alpha}) and that this peak is about 10% greater than that predicted in a plane beach theory with a similar slope. It is shown also that the restriction noted by Roseau on the acute nature of the angle {alpha} can be relaxed in the present solution to include the case of an overhanging cliff. Results for this case indicate some similarity to those for a plane beach but only for sufficiently large values of a{alpha}.


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