On the energy of nonlinear water waves.

David Henry

doi:10.1098/rspa.2021.0544

Abstract

This article presents results concerning the excess kinetic and potential energies for exact nonlinear water waves. In particular, it is proven, for periodic travelling irrotational water waves, that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, in the steady reference frame. A characterization of the total excess energy density as a weighted mean of the kinetic energy along the wave surface profile is also presented.

Highlights

This article establishes results concerning the energy generated by nonlinear water waves
As a by-product, we derive a succinct formulation for the total excess energy of a nonlinear water wave, which can be expressed in terms of the mean kinetic energy along the wave surface profile, weighted by the wave surface profile itself
We have considered the excess potential (Ep), excess kinetic (Ek) and total excess (Etot) energies for nonlinear periodic travelling waves

Summary

Introduction

This article establishes results concerning the energy generated by nonlinear water waves. Recent advances in mathematical analysis have enabled progress in tackling fundamental questions concerning nonlinear waves, and this article presents new results concerning the excess kinetic and potential energies for exact nonlinear periodic and travelling irrotational water waves. The fact that relations (1.1)–(1.3) hold within the confines of linear water wave theory has important implications for practical applications [2,3,4], since they provide a convenient means of estimating the total wave energy. We use an interplay between harmonic function theory and conformal mappings to establish the validity of relations (1.2) for exact periodic irrotational travelling wave solutions to the nonlinear governing equations for water waves. As a by-product, we derive a succinct formulation for the total excess energy of a nonlinear water wave, which can be expressed in terms of the mean kinetic energy along the wave surface profile, weighted by the wave surface profile itself

Preliminaries

Wavespeed determination

Wave energy

Findings

Discussion