New evolution equations for non-linear water waves in general bathymetry with application to steady travelling solutions in constant, but arbitrary, depth

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2007, 2007(Special): 75-84. doi: 10.3934/proc.2007.2007.75

New evolution equations for non-linear water waves in general bathymetry with application to steady travelling solutions in constant, but arbitrary, depth

G. A. Athanassoulis ^1, and K. A. Belibassakis ^2,

1.	School of Naval Architecture & Marine Engineering, National Technical University of Athens, Heroon Polytechniou 9, Athens 15773, Greece
2.	School of Technological Applications, Technological Educational Institute of Athens, Greece

Received September 2006 Revised February 2007 Published September 2007

Abstract
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A non-linear coupled-mode system of horizontal equations is derived with the aid of Luke’s [13] variational principle, which models the evolution of nonlinear water waves in intermediate depth and over a general bathymetry. The vertical structure of the wave field is exactly represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional modes, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The system fully accounts for the effects of non-linearity and dispersion. The main features of this approach are the following: (i) various standard models of water-wave propagation are recovered by appropriate simplifications of the coupled-mode system, and (ii) a small number of modes (up to 5) are enough for the precise numerical solution, provided that the two new modes (the free-surface and the sloping-bottom ones) are included in the local-mode series. In the present work, the consistent coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate to shallow-water wave conditions.

Keywords: coupled modes., nonlinear water waves.

Mathematics Subject Classification: 76B15, 76B0.

Citation: G. A. Athanassoulis, K. A. Belibassakis. New evolution equations for non-linear water waves in general bathymetry with application to steady travelling solutions in constant, but arbitrary, depth. Conference Publications, 2007, 2007 (Special) : 75-84. doi: 10.3934/proc.2007.2007.75

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