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

For the nonlinear two-dimensional problem, describing periodic steady waves on water of finite depth in the absence of surface tension, a single pseudo-differential operator equation (Babenko's equation) is derived and 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 in 1987 and studied in detail by Buffoni, Dancer and Toland in 2000. Unlike the equation for deep water involving just the $2 \pi$-periodic Hilbert transform $\mathcal{C}$, the equation obtained in this paper contains an operator which is the sum of $\mathcal{C}$ and a compact operator depending, in particular, on the depth of water.