Pattern Selection in Faraday Waves

TLDR

The study develops a systematic nonlinear theory of pattern selection for Faraday waves applicable beyond low‑viscosity fluids. A standing‑wave amplitude equation derived from Navier–Stokes equations, in gradient form, is used to compute a Lyapunov function that predicts the pattern selected near threshold. For high‑viscosity fluids the theory predicts parallel stripe patterns, while at lower viscosities it yields square symmetry in the capillary regime and a sequence of hexagonal, octagonal, and higher‑order patterns in the mixed gravity–capillary regime, with stability regions matching recent large‑aspect‑ratio experiments.

Abstract

We present a systematic nonlinear theory of pattern selection for parametric surface waves (Faraday waves), not restricted to fluids of low viscosity. A standing wave amplitude equation is derived from the Navier-Stokes equations that is of gradient form. The associated Lyapunov function is calculated for different regular patterns to determine the selected pattern near threshold. For fluids of large viscosity, the selected wave pattern consists of parallel stripes. At lower viscosity, patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of six-fold (hexagonal), eight-fold, ... patterns are predicted. The regions of stability of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.