The paper fully answers a long standing open question concerning the stability/instability of pure gravity periodic traveling water waves—called Stokes waves—at
the critical Whitham–Benjamin depth hWB = 1.363... and nearby values. We prove that
Stokes waves of small amplitude O() are, at the critical depth hWB, linearly unstable
under long wave perturbations. The same holds true for slightly smaller values of the
depth h > hWB − c2, c > 0, depending on the amplitude of the wave. This problem
was not rigorously solved in previous literature because the expansions degenerate at the
critical depth. To solve this degenerate case, and describe in a mathematically exhaustive
way how the eigenvalues change their stable-to-unstable nature along this shallow-todeep water transient, we Taylor-expand the computations of Berti et al. (Arch Ration
Mech Anal 247:91, 2023) at a higher degree of accuracy, starting from the fourth order
expansion of the Stokes waves. We prove that also in this transient regime a pair of
unstable eigenvalues depict a closed figure “8”, of smaller size than for h > hWB, as the
Floquet exponent varies
