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The nonlinear two-dimensional problem, describing periodic steady waves onwater of finite depth is considered in the absence of surface tension. It isreduced to a single pseudo-differential operator equation (Babenko's equation),which is investigated analytically and numerically. This equation has the sameform as the equation for waves on infinitely deep water; the latter had beenproposed by Babenko and studied in detail by Buffoni, Dancer and Toland.Instead of the $2 \pi$-periodic Hilbert transform $\mathcal{C}$ used in theequation 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 whosedependence on the parameter involves on the depth of water. Numericalcomputations are based on an equivalent form of Babenko's equation derived byvirtue of the spectral decomposition of the operator $\mathcal{B}_r \D / \D t$.Bifurcation curves and wave profiles of the extreme form are obtainednumerically.

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