We overview the recent result [3, Theorem 1.1] about the high-frequency instability of pure gravity Stokes waves subject to longitudinal perturbations. The spectral bands of unstable eigenvalues away from the origin form a sequence of isolas parameterized by an integer \( \texttt{p}\ge 2 \) for any value of the depth \( \texttt{h}> 0 \) such that an explicit coefficient \(\beta _1^{(\texttt{p})}(\texttt{h}) \) is not zero. In [3] it is proved that the map \( \texttt{h}\mapsto \beta _1^{(\texttt{p})}(\texttt{h}) \) is analytic and it is not identically zero for any \( \texttt{p}\ge 2 \) , by showing that \( \lim _{\texttt{h}\rightarrow 0^+}\beta _1^{(\texttt{p})}(\texttt{h}) = - \infty \) . In this manuscript we compute the asymptotic expansion of \(\beta _1^{(\texttt{p})}(\texttt{h}) \) in the deep-water limit \( \texttt{h}\rightarrow + \infty \) –it vanishes exponentially fast to zero– for \(\texttt{p}=2\) , 3, 4.
