Vertically vibrate a cylindrical container filled with liquid, and the surface answers at half the driving frequency.

The figure below is that response, computed live in your browser - a standing wave pattern on the water surface in a cylindrical container. It is not a video. Move the drive frequency and watch the surface hand over from one pattern to the next.

What you are seeing

In 1831 Michael Faraday noticed that a tray of liquid, shaken up and down, grows standing wave patterns on its surface - and that they oscillate at half the shaking frequency. Shaking straight up and down cannot push the surface sideways; what it does is strengthen and ease the pull that keeps the surface flat, once for every shake. A rocking pattern reaches its extreme twice in each cycle - once as a crest, once as a trough - and the pull grips it hardest at both, so it takes two shakes to carry the pattern through one full cycle. That two-for-one is why the surface answers at half the drive: an effect called parametric resonance. The water in a cylindrical container can only vibrate in a family of natural patterns, each labelled by two counts (m, n): m spokes (still lines running across the surface) and n rings (still circles around its centre). Drive the container at frequency f and the surface picks whichever pattern has a natural frequency nearest f/2; sweep the slider and the patterns hand over one after another - a mode ladder - each saturating in colour as f/2 lands on its resonance.

This page shows the shape of the waves only; how hard you must shake to summon them - the onset thresholds - lives in the sandbox. How the patterns and their frequencies are computed is explained in the notes on the method below.

This live figure needs JavaScript; nothing else on the page does. The physics: Faraday waves respond subharmonically at half the drive frequency, as Bessel modes of the cylindrical cell.

Readout

drive f: -
response f/2: -
dominant (m, n): -
mode f₀: -
detuning f₀ − f/2: -
playback: -

drive frequency f

1-250 Hz. Arrow keys adjust in 0.1 Hz steps.

view top-down perspective

The live cell. Faraday waves in a cylindrical cell, viewed from above. Water: R = 50 mm, depth h = 10 mm, σ = 72.8 mN/m, ρ = 1000 kg/m³, g = 9.81 m/s². Surface elevation is the resonant Bessel mode η ∝ J_m(k_mnr) cos mθ cos ω_r t with a free contact line, J′_m(kR) = 0, and natural frequencies from Rayleigh-Ritz eigenmode calculations after Shao et al. (2021), the research pipeline's ~1% method, over the finite-depth dispersion operator ω² = (gk + σk³/ρ) tanh kh. Linear standing waves; playback slowed (factor in the readout) so the subharmonic oscillation is visible. Computed in-browser at load - not a recording. Notes on the method below.

mode catalogue - every pattern at its resonance drive; each diagonal rail is one ring count n

shape selector - click any (m, n) combination; each owns exactly one frequency

Notes on the method

Where the catalogue comes from. Every pattern the container can hold has one natural frequency, set the way the pitch of a mass on a spring is: by a restoring pull working against inertia. The pull is gravity and surface tension - bending the surface away from flat lifts water against gravity and stretches the surface, which behaves like a lightly stretched skin, and both act to pull it back toward flat. The inertia is the water that has to move with the shape, and the deeper the pattern reaches the more of it moves. A stronger pull rings faster, more water in motion rings slower, and the natural frequency is where the two settle into balance. Working this out properly for the actual cylinder and its contact line is the Rayleigh-Ritz eigenmode method, following Shao et al. (2021) - the most accurate frequency model in the research behind this site, agreeing with laboratory measurements to about 1% - and your browser runs it afresh when the page loads.

Why this pattern. A vertically shaken surface answers at half the shaking frequency. The page therefore takes half the drive frequency, compares it with every natural frequency in the catalogue, and shows the single closest match - one pure pattern, never a blend. As half the drive closes in on a pattern's own frequency the colours saturate; as it moves away, the next pattern takes over.

What is left out. Friction. This page treats the fluid as frictionless, so it can show each pattern's shape and natural frequency but not how hard you must shake to summon it. Those thresholds - with friction included, and the honestly flat surface below them - are the sandbox. The numbers this page computes are tested against the project's research code.