Standing wave description of nearly conservative, parametrically excited waves in extended systems

Abstract

We consider the standing wavetrains that appear near threshold in a nearly conservative, parametrically excited, extended system that is invariant under space translations and reflection. Sufficiently close to threshold, the relevant equation is a Ginzburg-Landau equation whose cubic coefficient is extremely sensitive to wavenumber shifts, which can only be understood in the context of a more general quintic equation that also includes two cubic terms involving the spatial derivative. This latter equation is derived from the standard system of amplitude equations for counterpropagating waves, whose validity is well established today. The coefficients of the amplitude equation for standing waves are obtained for 1D Faraday waves in a deep container, to correct several gaps in former analyses in the literature. This application requires to also consider the effect of the viscous mean flow produced by the surface waves, which couples the dynamics of the surface waves themselves with the free surface deformation induced by the mean flow.