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Whitham and Benjamin predicted in 1967 that small-amplitude periodictraveling Stokes waves of the 2d-gravity water waves equations are linearlyunstable with respect to long-wave perturbations, if the depth $\mathtt h$ islarger than a critical threshold $\mathtt h_{WB} \approx 1.363$. In this paperwe completely describe, for any value of $\mathtt h > 0$, the four eigenvaluesclose to zero of the linearized equations at the Stokes wave, as the Floquetexponent $\mu$ is turned on. We prove in particular the existence of a uniquedepth $\mathtt h_{WB}$, which coincides with the one predicted by Whitham andBenjamin, such that, for any $0 < \mathtt h < \mathtt h_{WB}$, the eigenvaluesclose to zero remain purely imaginary and, for any $\mathtt h > \mathtth_{WB}$, a pair of non-purely imaginary eigenvalues depicts a closed figure"8", parameterized by the Floquet exponent. As $\mathtt h \to \mathtt h_{WB}^+$this figure "8" collapses to the origin of the complex plane. The proofcombines a symplectic version of Kato's perturbative theory to compute theeigenvalues of a $4 \times 4$ Hamiltonian and reversible matrix, and KAMinspired transformations to block-diagonalize it. The four eigenvalues have allthe same size $O(\mu)$ - unlike the infinitely deep water case in [6]- and thecorrect Benjamin-Feir phenomenon appears only after one non-perturbativeblock-diagonalization step. In addition one has to track, along the wholeproof, the explicit dependence of the entries of the $4 \times 4$ reducedmatrix with respect to the depth $\mathtt h$.

Benjamin-Feir instability of Stokes waves in finite depth