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 \( \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}= 1.363... \) and nearby values. We prove that Stokes waves of small amplitude \( \mathcal {O}( \epsilon ) \) are, at the critical depth \( \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}\), linearly unstable under long wave perturbations. The same holds true for slightly smaller values of the depth \( \texttt{h}> \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}- c \epsilon ^2 \), \( c > 0 \), 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-to-deep 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 \( \texttt{h}> \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}\), as the Floquet exponent varies.

本文完全回答了一个长期悬而未决的问题，涉及纯重力周期性行进水波（称为斯托克斯波）在临界惠瑟姆-本杰明深度的稳定性/不稳定性\( \texttt{h}_{\scriptscriptstyle {\textsc {WB} }}= 1.363... \)和附近的值。我们证明小振幅斯托克斯波\( \mathcal {O}( \epsilon ) \)在临界深度\( \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}\)呈线性在长波扰动下不稳定。对于深度稍小的值来说也是如此\( \texttt{h}> \texttt{h}_{\scriptscriptstyle {\textsc {WB}}}- c \epsilon ^2 \) , \( c > 0 \)，取决于波的振幅。这个问题在以前的文献中没有得到严格解决，因为展开在临界深度处会退化。为了解决这种退化情况，并以数学上详尽的方式描述特征值如何沿着浅水到深水瞬态改变其稳定到不稳定的性质，我们对 Berti 等人的计算进行了泰勒展开。（Arch Ration Mech Anal 247:91, 2023）以更高的精度，从斯托克斯波的四阶展开开始。我们证明，在这种瞬态状态下，一对不稳定的特征值描绘了一个封闭的数字“8”，其尺寸小于\( \texttt{h}> \texttt{h}_{\scriptscriptstyle {\textsc {WB}} }\)，随着 Floquet 指数的变化。