Canonical variables for steep planar water waves over nonuniform bed

TL;DR

This paper derives explicit canonical variables and Hamiltonian formulations for fully nonlinear 2D water wave dynamics over arbitrary nonuniform beds, extending previous conformal mapping methods.

Contribution

It provides a new explicit canonical variable formulation for nonlinear water waves over uneven beds, building on and extending prior conformal mapping approaches.

Findings

Explicit canonical variables for nonlinear water waves over nonuniform beds.

Hamiltonian functional expressed in terms of these variables.

Discussion of weakly nonlinear Hamiltonian models.

Abstract

An explicit expression in terms of canonical variables is obtained for the Hamiltonian functional determining the fully nonlinear dynamics of two-dimensional potential flows of an ideal fluid with a free surface over an arbitrary nonuniform depth. The canonically conjugate variables are derived from the previously developed non-canonical conformal description of water waves over a strongly undulating bottom [V. P. Ruban, Phys. Rev. E {\bf 70}, 066302 (2004)]. Also an alternative approach to the problem is discussed, which gives weakly nonlinear Hamiltonian models of different orders.