A Whitham–Boussinesq long-wave model for variable topography

Abstract

We study the propagation of water waves in a channel of variable depth using the long-wave asymptotic regime. We use the Hamiltonian formulation of the problem in which the non-local Dirichlet–Neumann operator appears explicitly in the Hamiltonian, and propose a Hamiltonian model for bidirectional wave propagation in shallow water that involves pseudo-differential operators that simplify the variable-depth Dirichlet–Neumann operator. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth. Analogous models were proposed by Whitham for unidirectional wave propagation. We first present results for the normal modes and eigenfrequencies of the linearized problem. We see that variable depth introduces effects such as a steepening of the normal modes with the increase in depth variation, and a modulation of the normal mode amplitude. Numerical integration also suggests that the constant depth nonlocal Boussinesq model can capture qualitative features of the evolution obtained with higher order approximations of the Dirichlet–Neumann operator. In the case of variable depth we observe that wave-crests have variable speeds that depend on the depth. We also study the evolutions of Stokes waves initial conditions and observe certain oscillations in width of the crest and also some interesting textures and details in the evolution of wave-crests during the passage over obstacles.