Presets
Playback
Drive
Shaking frequency and acceleration.
Liquid
Fluid properties and the pattern shown.
Layers
Model checks
The science
Parametric resonance of surface waves
A liquid layer shaken vertically with acceleration ad cos(Ωt) sees an oscillating effective gravity. Each surface mode of wavenumber k obeys a damped Mathieu equation
η″ + 2γη′ + ωk² [1 + ε cos(Ωt)] η = 0, ωk² = (g k + σk³/ρ) tanh(k h), ε = ad g k tanh(kh) / ωk²The first instability tongue is at Ω = 2ωk: the surface responds at half the drive frequency (subharmonic). Growth appears when ε exceeds εc = 4γ/ωk with viscous damping γ = 2νk². The page computes the Floquet exponent by integrating two basis solutions over one drive period with RK4 and bisects the exact threshold numerically. The selected wavelength comes from ω(k) = Ω/2.
The nonlinear saturation is a cubic Landau law, da/dt = μa − c a³. Squares and hexagons are superpositions of 2 or 3 plane waves at 90 and 120 degrees; which pattern appears in a real experiment depends on viscosity and frequency (the page lets you choose).
Honest simplifications
- Linear damping of a free-surface wave only (no boundary layers at the walls or contact-line effects), so thresholds are lower than in real dishes.
- The picture is a rendering of the selected pattern, not a solved nonlinear free-surface flow.
- The pattern phase oscillation is slowed down about 50 times so that the subharmonic flip is visible.
What to try
- Raise the drive acceleration slowly in Stripes at 60 Hz: the surface stays flat until the pink threshold line is crossed, then the Floquet curve turns positive and the pattern grows.
- Switch to the oil preset: damping is 20 times larger and so is the threshold.
- Increase the frequency: the wavelength shrinks as omega(k) = pi f is solved at larger k and capillarity takes over.
- Compare stripes, squares and hexagons on the same liquid.