Faraday Waves — Parametric Resonance

Standing waves on a vertically vibrated fluid layer; tongue diagram of instability

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ω_d/2ω₀ = - Growth: - Pattern: - t = 0
Faraday waves arise in a fluid layer vibrated vertically with acceleration g + A cos(ωdt). The effective gravity oscillates, creating parametric resonance for surface waves near half-integer multiples of the drive: ωwave ≈ ωd/2n. The Mathieu equation d²η/dt² + (δ+ε cos t)η = 0 describes onset; instability tongues open at δ = (n/2)² (Arnold tongues). Above the critical amplitude Ac(γ) (set by damping), subharmonic standing waves lock in at ωd/2. Top: 2D surface height map with superimposed mode pattern. Bottom: mode amplitude time series.