Hamiltonian model for water waves in a triangular domain

Abstract

We present a Hamiltonian model for free surface water waves in a 2-D symmetric triangular domain with sides that have a 45∘ slope. The construction uses a new amplitude expansion for the free surface, and a Lagrangian formulation of the water problem that is based on a formal mechanical/geometrical analogy of the fluid motion to geodesic motion in a potential force field. We define an approximate Lagrangian by choosing a physically plausible generalized coordinate and obtain the Hamiltonian by the Legendre transform. Hamilton’s equations are presented in spectral form using a known explicit construction of the linear normal modes for the domain considered. A key step is the approximation of a Dirichlet–Neumann operator appearing in the kinetic energy.