Journal of Fluid Mechanics

An explicit Hamiltonian formulation of surface waves in water of finite depth

A. C.  Radder a1
a1 Rijkswaterstaat, Tidal Waters Division, P.O. Box 20907, 2500 EX The Hague, The Netherlands

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A variational formulation of water waves is developed, based on the Hamiltonian theory of surface waves. An exact and unified description of the two-dimensional problem in the vertical plane is obtained in the form of a Hamiltonian functional, expressed in terms of surface quantities as canonical variables. The stability of the corresponding canonical equations can be ensured by using positive definite approximate energy functionals. While preserving full linear dispersion, the method distinguishes between short-wave nonlinearity, allowing the description of Stokes waves in deep water, and long-wave nonlinearity, applying to long waves in shallow water. Both types of nonlinearity are found necessary to describe accurately large-amplitude solitary waves.

(Published Online April 26 2006)
(Received March 11 1991)