What you are looking at
In 1831 Michael Faraday shook shallow dishes of water, mercury and egg white up and down and watched their surfaces. Above a certain strength of shaking, the flat surface broke up into a lattice of standing waves, crests and troughs swapping places in place. He noticed something odd: the waves rose and fell only once for every two shakes. The shaking doesn't push the waves directly. It changes the effective gravity twice per wave cycle, the way a child on a swing pumps by standing up and squatting at the right moments. Physicists call it parametric resonance.
The shaking only wins if it adds energy faster than viscosity takes it away, so there is a sharp threshold, the Faraday threshold. Below it the surface stays flat; above it the waves grow until they saturate. Their wavelength isn't set by the dish. It follows from the frequency: the waves pick the length whose natural frequency is half the shaking frequency, from the dispersion relation of capillary-gravity waves.
Waves that start to run
In 2023 Jian Guan, Connor Magoon, Matthew Durey, Roberto Camassa and Pedro Sáenz at UNC Chapel Hill and Glasgow reported something new (Traveling Faraday waves, Phys. Rev. Fluids 8, 110501). Put the liquid in a narrow ring-shaped channel whose width is about one wavelength and about one capillary length (2.7 mm for water), and shake harder than the Faraday threshold. The standing pattern starts to move, round and round the ring at about 10 mm/s, a thousand times faster than the slow drift seen before in wider, shallower channels. Each run picks clockwise or counter-clockwise at random, and then keeps going that way.
Ratchets and pumps
The motion comes with a steady streaming flow of the liquid under the waves. The team found that surface tension and the menisci at the walls make it much stronger. A sawtooth wall, like a ratchet, decides the direction in advance. Two ratcheted rings side by side, turning opposite ways and both touching a straight channel, make a pump with no moving parts. Drops of dye show the liquid in the channel being carried one way. The same waves can carry a small floating object around the ring.
Things to try
Start the Ring channel preset, wait for the pattern to start running, then press Restart a few times: the direction changes from run to run. Turn the drive down to 1.4 × the threshold and the waves stand still again. In the Twin-ring pump, click in the straight channel to drop dye and watch it carried along. The Open dish shows Faraday's original standing patterns, which don't travel. Slow motion shows the waves rising and falling once for every two shakes; Strobe samples one moment per wave cycle, as a strobe lamp in the lab would, so only the slow motion of the pattern is left.
Keys: Space pause · R restart · D drop dye · C clear dye · K kick the surface · M shaker sound.
What is computed and what is modelled
- Computed: the wave field. The surface height η and surface potential φ obey linear capillary-gravity wave equations on a grid: ηt = −h∇²φ and φt = −g(t)η + (σ/ρ)∇²η. The effective depth h = tanh(kH)/k makes the dispersion exact at the Faraday wavelength. Gravity is modulated by the shaker, g(t) = g + A cos Ωt. Viscous damping is 2νk², with walls that keep the liquid in.
- The Faraday threshold shown in the panel is the instability point of that model's damped Mathieu equation, found by Floquet analysis. The page's tests check that the grid simulation decays just below it and grows above it, answers at exactly half the drive frequency, and conserves the liquid's volume.
- The damping beyond bulk viscosity (walls, menisci, contact lines) is one tuned number. It puts the threshold for water at 140 Hz near 3 g.
- Modelled, not derived: how big the waves get. Two simple saturating terms (a stiffening and an amplitude-dependent damping) stop the growth at about 0.1–0.3 mm, a few percent of a wavelength.
- Modelled, not derived: the streaming flow. Computing it properly needs 3D boundary layers and moving contact lines, which this page doesn't have. Instead, a slow 2D flow under the waves is amplified wherever the waves are strong, held back by drag and viscosity, and kept divergence free. The flow carries the wave pattern and the dye. That is enough to give what the experiment shows: no motion until a second threshold, then a random, persistent direction at about 10 mm/s. The ratchet's push is an imposed bias along the toothed rings. I first tried to read a push from the simulated waves at the teeth, but at this grid resolution it didn't reliably follow the tooth direction. Which way real teeth send the waves depends on details this model lacks.
- The pump is laid out for a 2D model. The liquid can't pile up at one end, so the straight channel is one side of a closed racetrack, and the return leg runs below the lower ring. Only the rings' waves drive the flow, and the quiet racetrack keeps a fifth of the drag that the wavy rings feel.
- The real liquid in a 3D-printed channel also has a curved meniscus, wets the walls and loses energy at a moving contact line. None of that is here.