On Stability of Generalized Short-Crested Water Waves

We take up the question of the dynamic stability of genuinely two-dimensional generalized hexagonal traveling wave patterns on the surface of a three-dimensional ideal fluid. That is, the stability of Generalized Short-Crested Wave (GSCW) solutions of the water wave problem. Our study is restricted to spectral stability, which considers the linearization of the water wave operator about one of these traveling generalized hexagonal patterns. We draw conclusions about stability from the spectral data of the resulting linear operator. Within the class of perturbations that we study, for a range of geometrical parameters, we find stable traveling waveforms which eventually destabilize, with features that depend strongly on the problem's configuration. We find Zones of Instability for patterns shaped as symmetric diamonds. Such zones are absent for asymmetric configurations; in these cases, once instability sets in, it remains. Within a given geometrical configuration, as a GSCW leading-order coefficient ratio is varied, these waves become more unstable as they become more asymmetric.