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 is larger than a critical threshold
hWB ≈ 1.363. In this paper, we completely describe, for any finite value of h > 0, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent μ is turned on.
