Babenko’s Equation for Periodic Gravity Waves on Water of Finite Depth: Derivation and Numerical Solution

Abstract

The nonlinear two-dimensional problem describing periodic steady gravity 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 depending on a parameter related to 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 $${\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t$$ . Bifurcation curves and wave profiles of the extreme form are obtained numerically.