High-frequency instabilities of Stokes waves

Euler's equations govern the behaviour 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. 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, with excellent agreement.

Automated summary

This study investigates the spectral stability of small-amplitude, one-dimensional Stokes waves on the surface of an incompressible, inviscid and irrotational fluid. By using a perturbation method, the first few high-frequency instabilities of the small-amplitude Stokes waves are accurately described and compared through asymptotic and numerical computations.