Modified Babenko's equation for periodic gravity waves on water of finite depth

TL;DR

This paper introduces a modified operator equation for periodic gravity waves on finite depth water, providing a new approach to analyze wave profiles and bifurcation phenomena, extending Babenko's classical equation.

Contribution

It develops a new nonlinear operator equation for finite depth water waves that generalizes Babenko's equation, incorporating mean water depth as a parameter.

Findings

Derived bifurcation curves for wave solutions

Numerical comparison with existing results

Parametric representation of free surface profiles

Abstract

A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko's equation considered in \cite{KD}. Both operators in the proposed equation are nonlinear and depend on the parameter equal to the mean depth of water, whereas each solution defines a parametric representation for a symmetric free surface profile. The latter is a component of a solution of the two-dimensional, nonlinear problem describing steady waves propagating in the absence of surface tension. Bifurcation curves (including a branching one) are obtained numerically for solutions of the new equation; they are compared with known results.