Faraday Waves
Shake a shallow tray of fluid up and down and its surface erupts into standing wave patterns — stripes, squares, hexagons, and quasicrystalline order. Michael Faraday described this parametric resonance in 1831: the surface waves oscillate at half the driving frequency, and their geometry is selected by the competition between surface tension, viscosity, and inertia.
∂²h/∂t² + 2γ∂h/∂t − [g + aω²cos(ωt)]∇²h + (σ/ρ)∇⁴h = 0
Click or drag on the surface to disturb it. Sliders update in real time.
Parametric resonance in a vibrating fluid
When a container of fluid is vibrated vertically at frequency f, the gravitational acceleration experienced by the fluid oscillates: g_eff(t) = g + aω²cos(ωt). The flat surface becomes unstable above a critical amplitude, and standing waves appear spontaneously at exactly half the driving frequency (f/2). This subharmonic response is a hallmark of parametric resonance — the same mechanism that makes a playground swing pump higher when you stand and crouch at the right moment.
The geometry of the pattern — stripes, squares, or hexagons — is determined by which wavevectors satisfy the dispersion relation most efficiently. Surface tension sets the wavelength: λ ∝ (σ/ρf²)^(1/3). Low viscosity favors hexagons; high viscosity pushes toward stripes. Two simultaneous frequencies can produce quasicrystalline 10-fold or 12-fold symmetric patterns — spatial quasicrystals, fluid analogs of Penrose tilings.
- Stripes — one dominant wavevector; appears near onset at high viscosity
- Squares — two orthogonal wavevectors of equal amplitude
- Hexagons — three wavevectors at 60°; the lowest-energy configuration at low viscosity
- Quasicrystal — two incommensurate frequencies drive incompatible symmetries, creating 10-fold order with no repeating unit cell