On stability of generalized short-crested water waves

In this paper, we take up the question of dynamic stability of genuinely two-dimensional “generalized” hexagonal traveling wave patterns on the surface of a three-dimensional ideal fluid (i.e., stability of Generalized Short-Crested Wave (GSCW) solutions of the water wave problem). We restrict ourselves to a study of spectral stability which considers the linearization of the water wave operator about one of these traveling generalized hexagonal waves, and draws conclusions about stability from the spectral data of the resulting linear operator. Within the class of perturbations we are allowed to study, for a wide range of geometrical parameters, we find stable traveling waveforms which eventually destabilize with features that depend strongly on the problem configuration. In particular, we find “Zones of Instability” for patterns shaped as symmetric diamonds, while such zones are absent for asymmetric configurations. Furthermore, we note that within a geometrical configuration, as a “generalized SCW” ratio is varied (essentially the character of the linear solution), these waves become more unstable as the waves become more asymmetric.