Why does the surface oscillate at half the drive frequency?

The Mathieu equation has a primary instability tongue centred at ωₐ = 2 ω₀: the parametric pump deposits energy most efficiently into modes whose natural frequency is half of the drive. That sub-harmonic response is the signature of Faraday waves.

What sets the pattern (stripes vs squares vs hexagons)?

The selection comes from weakly nonlinear interactions of competing modes — damping, fluid depth, and meniscus boundary conditions favour different symmetries. Our simulator only chooses a few Fourier modes, so the patterns we draw are illustrative cartoons rather than predictions.

How is this related to the swing-pumping demo?

Pumping a swing by squatting at twice the swing frequency is the same parametric resonance: the natural oscillator is excited by a periodic modulation of one of its parameters (effective length here, restoring force there).

Faraday Waves

A thin liquid layer sitting on a vertically vibrated plate develops standing surface waves at half the drive frequency — Faraday waves. Each Fourier mode of the surface obeys a damped Mathieu equation ä + 2γ ȧ + ω₀²(1 + ε cos ωₐ t) a = 0; when the parametric forcing ε exceeds a damping-dependent threshold, the sub-harmonic response a ∼ exp(σ t) cos(ωₐ t/2) blows up and saturates into stripes, squares, or hexagons depending on container geometry and dissipation. Our simulator integrates a small bank of such Mathieu oscillators and superposes their cosines to render the surface in real time.

Who it's for: Intro nonlinear dynamics, parametric resonance, and pattern-formation physics; pairs nicely with the Mathieu/Hill simulator.

Key terms

Faraday waves
parametric instability
Mathieu equation
sub-harmonic response
pattern formation
standing wave

How it works

Vertically shaking a thin layer of liquid drives the surface through a **Mathieu equation**. The fluid responds at half the drive frequency and forms regular **stripes, squares or hexagons** — a textbook example of pattern formation by parametric instability.

Key equations

A¨ + 2γ A˙ + ω₀² (1 + ε cos ω_d t) A = 0

sub-harmonic response: ω_response ≈ ω_d / 2

Frequently asked questions

Why does the surface oscillate at half the drive frequency?: The Mathieu equation has a primary instability tongue centred at ωₐ = 2 ω₀: the parametric pump deposits energy most efficiently into modes whose natural frequency is half of the drive. That sub-harmonic response is the signature of Faraday waves.
What sets the pattern (stripes vs squares vs hexagons)?: The selection comes from weakly nonlinear interactions of competing modes — damping, fluid depth, and meniscus boundary conditions favour different symmetries. Our simulator only chooses a few Fourier modes, so the patterns we draw are illustrative cartoons rather than predictions.
How is this related to the swing-pumping demo?: Pumping a swing by squatting at twice the swing frequency is the same parametric resonance: the natural oscillator is excited by a periodic modulation of one of its parameters (effective length here, restoring force there).