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A Hamiltonian formulation for nonlinear wave-body interactions

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BEM-numerics and KdV-model analysis for solitary wave split-up E. F. G. van Daalen, E. van Groesen, S. R. Pudjaprasetya Abstract In this paper we consider travelling surface waves on a layer of water of decreasing depth. A numerical scheme based on the boundary element method is used to present calculations for the run-up of a solitary wave. The numerical results are compared with an analytical ap- proximation based on a modified Korteweg-de Vries equation. 1 Introduction In this paper we consider some aspects of travelling sur- face waves on a layer of water (considered as an ideal fluid) of decreasing depth. The very accurate numerical scheme based on the Boundary Element Method (BEM), as developed by van Daalen (1993, 1995), is used for two different aims: The first aim is to present BEM-calculations for the run- up of a solitary wave. The results will show that (for lim- ited changes in depth) the solitary wave splits into two clearly distinguishable waves. Such a splitting may be thought to be well-known in view of the vast amount of literature about this subject that is available since the se- venties, see e.g. Grimshaw (1970, 1971), Johnson (1972, 1973, 1994), Karpman and Maslow (1978), Knickerbocker and Newell (1980), and Newell (1985). However, all these references use some model equation, mostly a variant of the KdV-equation, with coefficients or perturbations that should take the changing depth into account. The BEM- calculation, in contrast, is based on the exact equations, that is without using some approximation like the Boussinesq conditions on the wave shape, and without a priori assumptions such as uni-directional wave pro- pagation. The second aim is to compare the numerical results (considered to be the ‘exact’ description of the phenom- enon) with a relatively simple analytical description. This analytical work uses a specific modification of the KdV- equation, called ‘KdV-top’, that was derived by van Groesen and Pudjaprasetya (1993) under the assumption of mild bo

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