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 h\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathtt h} $$\end{document} is larger than a critical threshold hWB≈1.363\documentclass[12pt]{minimal}
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\begin{document}$$\texttt{h}_{\scriptscriptstyle {\textsc {WB}}}\approx 1.363 $$\end{document}. In this paper, we completely describe, for any finite value of h>0\documentclass[12pt]{minimal}
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\begin{document}$$ \mathtt h >0 $$\end{document}, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} is turned on. We prove, in particular, the existence of a unique depth hWB\documentclass[12pt]{minimal}
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\begin{document}$$ \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$\end{document}, which coincides with the one predicted by Whitham and Benjamin, such that, for any 0<h<hWB\documentclass[12pt]{minimal}
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\begin{document}$$ 0< \mathtt h < \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$\end{document}, the eigenvalues close to zero are purely imaginary and, for any h>hWB\documentclass[12pt]{minimal}
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\begin{document}$$ \mathtt h > \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}$$\end{document}, a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As h→hWB+\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathtt h} \rightarrow \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}^{\, +} $$\end{document} the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits h→+∞\documentclass[12pt]{minimal}
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\begin{document}$$ \texttt{h}\rightarrow +\infty $$\end{document} (deep water) and μ→0\documentclass[12pt]{minimal}
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\begin{document}$$ \mu \rightarrow 0 $$\end{document} (long waves) do not commute. In finite depth, the four eigenvalues have all the same size O(μ)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}(\mu )$$\end{document}, unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth h\documentclass[12pt]{minimal}
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\begin{document}$$\texttt{h}$$\end{document} is carefully tracked.
