Babenko's equation for periodic gravity waves on water of finite depth: derivation and numerical solution

TL;DR

This paper derives and numerically solves Babenko's equation for steady periodic gravity waves on finite depth water, extending the deep water case and analyzing wave profiles and bifurcations.

Contribution

It introduces a modified Babenko's equation for finite depth water waves and develops a numerical method for solving it, including analysis of wave profiles and bifurcation behavior.

Findings

Numerical bifurcation curves for finite depth waves are obtained.

Wave profiles of the extreme form are computed.

The equation generalizes the deep water case with a depth-dependent operator.

Abstract

The nonlinear two-dimensional problem, describing periodic steady waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko's equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the $2 \pi$-periodic Hilbert transform $\mathcal{C}$ used in the equation for deep water, the equation obtained here contains a certain operator $\mathcal{B}_r$, which is the sum of $\mathcal{C}$ and a compact operator whose dependence on the parameter involves on the depth of water. Numerical computations are based on an equivalent form of Babenko's equation derived by virtue of the spectral decomposition of the operator $\mathcal{B}_r \D / \D t$. Bifurcation curves and wave profiles of the extreme form are obtained numerically.