On the stability of large-amplitude gravity-capillary surface waves

TL;DR

This paper analyzes the linear stability of finite-amplitude gravity-capillary waves at small surface tension, revealing how surface tension influences wave stability and the emergence of instabilities.

Contribution

It provides a comprehensive eigenvalue spectrum analysis for these waves, including the effects of small surface tension on stability and solution branches.

Findings

Superharmonic instability occurs at smaller amplitudes with surface tension.

Finite-amplitude solutions are stabilized against modulational instability by surface tension.

Stabilization effects occur at lower surface tension values than previously known.

Abstract

We consider the stability of periodic gravity-capillary waves of finite amplitude for small values of the surface tension. Linear stability with respect to both superharmonic and subharmonic perturbations is calculated for each solution, and our methodology obtains the full eigenvalue spectrum consisting of growth rates and temporal frequencies. For small surface tension, the gravity-capillary wave solution space consists of a countably-infinite number of solution branches that coalesce in the small-surface-tension limit, which forms one of the main complications of our study. When the energy is fixed as an amplitude constraint, we find that the superharmonic instability associated with near-limiting gravity waves emerges at smaller amplitudes in the presence of surface tension. Further, the modulational (long-wave) instability is seen to be stabilised for finite-amplitude solutions in the presence of surface tension. This occurs at surface tension values well below that previously obtained via weakly-nonlinear theory, and the stabilisation is nonmonotonic as very small fluctuations in the surface tension of solutions produce large changes in their stability properties.