Drift instability of standing Faraday waves

Abstract

We consider the weakly nonlinear evolution of the Faraday waves produced in a vertically vibrated two-dimensional liquid layer, at small viscosity. It is seen that the surface wave evolves to a drifting standing wave, namely a wave that is standing in a moving reference frame. This wave is determined up to a spatial phase, whose calculation requires consideration of the associated mean flow. This is just the streaming flow generated in the boundary layer attached to the lower plate supporting the liquid. A system of equations is derived for the coupled slow evolution of the spatial phase and the streaming flow. These equations are numerically integrated to show that the simplest reflection symmetric steady state (the usual array of counter-rotating eddies below the surface wave) becomes unstable for realistic values of the parameters. The new states include limit cycles (the array of eddies oscillating laterally), drifted standing waves (patterns that are standing in a uniformly propagating reference frame) and some more complex attractors.