High-Frequency Instabilities of Stokes Waves

Abstract: 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.