Primary Patterns in Faraday Surface Waves at High Aspect Ratio

Abstract

The pattern selection in Faraday surface water waves caused by the instability of the state of no waves is examined under the assumption of high aspect ratio. First, nonlinear evolution equations for the amplitudes of resonant capillary-gravity waves with different directions of wavenumber vector are derived. These equations include cubic nonlinearity and the effects of viscous damping and parametric forcing obtained from the energy equation. Then, using the method of a center manifold, quintic amplitude equations for unstable modes are derived in which cubic damping and forcing are included. From these equations, it is concluded that the squares are stable for sufficiently short waves, the hexagons and the 8-fold quasipatterns are stable when the wavelength is within two intermediate regions, and the stripes are always unstable.