# Harmonic Stability of Standing Water Waves

## TL;DR

This paper introduces a numerical Floquet-based method to analyze the stability of standing water waves, revealing stability thresholds, new instability phenomena, and stability of large-amplitude waves in various water depths.

## Contribution

It develops a novel numerical approach using Floquet theory and Fourier truncation to study water wave stability, including a method to track eigenvalues across wave amplitudes.

## Key findings

- Most deep water standing waves with crest acceleration below 0.889 are stable.
- Identifies previously unreported instability bubbles at lower crest accelerations.
- Some large-amplitude gravity-capillary waves are stable, others are unstable.

## Abstract

A numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory. A Fourier truncation of the monodromy operator is computed by solving the linearized Euler equations about the standing wave with initial conditions ranging over all Fourier modes up to a given wave number. The eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions, which we introduce as a method of retaining only accurately computed Floquet multipliers. The mean wave number matches up with analytical results for the zero-amplitude standing wave and is helpful in identifying which Floquet multipliers collide and leave the unit circle to form unstable eigenmodes or rejoin the unit circle to regain stability. For standing waves in deep water, most waves with crest acceleration below $A_c=0.889$ are found to be linearly stable to harmonic perturbations; however, we find several bubbles of instability at lower values of $A_c$ that have not been reported previously in the literature. We also study the stability of several new or recently discovered time-periodic gravity-capillary or gravity waves in deep or shallow water, finding several examples of large-amplitude waves that are stable to harmonic perturbations and others that are not. A new method of matching the Floquet multipliers of two nearby standing waves by solving a linear assignment problem is also proposed to track individual eigenvalues via homotopy from the zero-amplitude state to large-amplitude standing waves.