When a dish of fluid is shaken up and down, the flat surface can spontaneously break into beautiful standing-wave patterns. These are Faraday waves, first reported by Michael Faraday in 1831. This simulation models the surface as a set of resonant modes driven by vertical vibration, so you can watch patterns ignite above a critical amplitude.
a''(t) + 2γ·a'(t) + ω₀²·[1 − F·cos(ω·t)]·a(t) = −β·a³
Subharmonic response: ω_response = ω_drive / 2. Mode dispersion: ω₀² = (g·k + (σ/ρ)·k³)·tanh(k·h).
Vibrated grains of sand show the very same Faraday patterns — and acoustic-levitation rigs use Faraday-style instabilities to atomise droplets for fuel injectors and inkjet printers.
Faraday waves are standing surface waves that appear on a layer of fluid when its container is vibrated vertically. Above a critical driving amplitude the flat surface becomes unstable and ripples self-organise into regular patterns.
The instability is subharmonic. Energy is pumped into the surface twice per drive cycle, so the response settles into a standing wave that completes one oscillation for every two drive oscillations — at f/2. This is the hallmark of parametric resonance.
Each surface mode behaves like a pendulum whose effective gravity is modulated by the vertical shaking. Its amplitude obeys a Mathieu-type equation, whose unstable solutions grow exponentially inside resonance bands called Faraday tongues.
A Faraday tongue is a V-shaped region in the drive-frequency / drive-amplitude plane where a particular surface mode is unstable. The tip of the tongue marks the smallest amplitude that can excite that mode. The diagram overlaid in the simulation shows these tongues.
The selected pattern depends on drive amplitude, frequency and fluid properties. Low forcing favours stripes; moderate forcing favours squares; strong nonlinear coupling between modes at 120-degree angles favours hexagons. The simulation switches pattern symmetry with the controls.
Below the critical amplitude, viscosity damps every perturbation and the surface stays flat. At the threshold the growth rate of the most unstable mode reaches zero; above it that mode grows until nonlinearity saturates it into a steady pattern.
Viscosity damps the modes and raises the critical amplitude. More viscous fluids need stronger shaking to form patterns, and they select different wavelengths. In the simulation, higher damping makes patterns harder to ignite.
Michael Faraday described them in 1831 while studying vibrating surfaces. He noted that crispations on a shaken fluid oscillate at half the frequency of the support — a result later explained through parametric resonance theory.
No. It is a fast educational model: a grid of standing-wave modes whose amplitudes are integrated with the parametric (Mathieu) growth law, then summed and rendered as a height field. It captures threshold, subharmonic response and pattern symmetry without solving full Navier-Stokes equations.
They appear in inkjet printing, atomisation and droplet ejection, granular media (vibrated sand), acoustic levitation, and as a classic laboratory model for pattern formation and nonlinear dynamics far from equilibrium.