  Whitham and Benjamin predicted in 1967 that small-amplitude periodic
traveling Stokes waves of the 2d-gravity water waves equations are linearly
unstable with respect to long-wave perturbations, if the depth $\mathtt h$ is
larger than a critical threshold $\mathtt h_{WB} \approx 1.363$. In this paper
we completely describe, for any value of $\mathtt h > 0$, the four eigenvalues
close to zero of the linearized equations at the Stokes wave, as the Floquet
exponent $\mu$ is turned on. We prove in particular the existence of a unique
depth $\mathtt h_{WB}$, which coincides with the one predicted by Whitham and
Benjamin, such that, for any $0 < \mathtt h < \mathtt h_{WB}$, the eigenvalues
close to zero remain purely imaginary and, for any $\mathtt h > \mathtt
h_{WB}$, 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 proof
combines a symplectic version of Kato's perturbative theory to compute the
eigenvalues of a $4 \times 4$ Hamiltonian and reversible matrix, and KAM
inspired transformations to block-diagonalize it. The four eigenvalues have all
the same size $O(\mu)$ - unlike the infinitely deep water case in [6]- and the
correct Benjamin-Feir phenomenon appears only after one non-perturbative
block-diagonalization step. In addition one has to track, along the whole
proof, the explicit dependence of the entries of the $4 \times 4$ reduced
matrix with respect to the depth $\mathtt h$.
