Session AN - Waves I.
ORAL session, Sunday morning, November 23
Preakness, SMH
The natural description of Faraday waves at low viscosity is usually made in terms of pairs of counterpropagating waves (CPWs), which reduce to standing waves (SWs) in a sufficiently small vicinity of threshold. The derivation of an amplitude equation describing the evolution of the SWs requires to consider higher order terms accounting for nonlinear damping and nonlinear forcing in the CPW-equations.Unfortunately, there are some additional higher order terms that are of the same order as those retained and have been systematically ignored in previous derivations of the SW-amplitude equation. These terms appear in a careful asymptotic derivation and are of two kinds.(i) Two of them result from dependence on wavenumber of the coefficients of the CPW-equations and lead to a O(1) correction in the scaled coefficient of the cubic term in the SW-amplitude equation. Thus this correction has a O(1) effect in the primary bifurcation at threshold. The corrected cubic coefficient is checked comparing with its exact counterpart in the viscous limit, as calculated using different methods by Chen amp; Viñals (1998) and Mancebo amp; Vega (2003). And (ii) the viscous mean flow provides additional terms that affect both the stability of the primary bifurcated branch and secondary bifurcations.