  Euler's equations govern the behavior of gravity waves on the surface of an
incompressible, inviscid, and irrotational fluid of arbitrary depth. We
investigate the spectral stability of sufficiently small-amplitude,
one-dimensional Stokes waves, i.e., periodic gravity waves of permanent form
and constant velocity, in both finite and infinite depth. Using a nonlocal
formulation of Euler's equations developed by Ablowitz et al. (2006), we
develop a perturbation method to describe the first few high-frequency
instabilities away from the origin, present in the spectrum of the
linearization about the small-amplitude Stokes waves. Asymptotic and numerical
computations of these instabilities are compared for the first time to
excellent agreement.
