Faraday Waves a live instrument

The full instrument.

Everything is adjustable here - drive frequency and acceleration, cell size, fluid, contact line. The onset thresholds are real: below the critical acceleration the surface stays flat, exactly as the parametric instability demands. Computed live; not a video.

Sandbox. Faraday waves in a cylindrical cell, viewed from above. The cell, fluid, depth and contact line are as set in the controls. Surface elevation is the resonant Bessel eigenmode η ∝ Φmn(r) cos mθ cos ωr t; natural frequencies and mode shapes 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. Onset Ac is the calculated critical shaking strength for the displayed pattern (damped-Mathieu model, Lamb (1932) damping γ = 2νk²); below it the flat surface is stable, so the figure honestly shows none. Above onset the displayed amplitude grows as √(A − Ac) with a soft display cap - a display scaling, stated plainly. Far above onset a real surface steepens, breaks and throws droplets; that nonlinear regime is outside this linear model, and the state row says so. Playback is slowed (factor in the readout). Notes on the method below.

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

Logarithmic scale, 0.05 to 50 m/s².

cell display
contact line

Where the water meets the wall: free - the waterline slides up and down; pinned - it stays attached to the rim; robin - it meets the wall at the set contact angle θ (90° behaves as free).

dissipation

Friction inside the fluid: Lamb (1932) is one classical formula for every pattern; Wilson et al. (2022) computes each pattern's own friction from its shape, as in the research pipeline. Both leave out the walls and the waterline.

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

What the acceleration does

Shake the container gently and nothing happens: the flat surface is stable, and friction inside the fluid kills small ripples faster than the shaking can feed them. Every pattern has a critical acceleration Ac - cross it and that pattern grows out of the flat surface, oscillating at half the drive frequency, until it settles at a finite size. Plotted against frequency, these thresholds trace valley-shaped curves - the instability tongues of parametric excitation: each valley bottoms out where half the drive frequency meets a pattern's natural frequency, which is why the Ac readout dips and rises as you sweep. A thicker (more viscous) fluid raises the whole landscape - it needs a harder shake. Just above the threshold the pattern is faint; drive harder and its peaks and valleys deepen, here as √(A − Ac). Push far beyond and a real surface stops being polite: the waves steepen, break and throw droplets. That regime is beyond the linear theory running this page - the figure keeps growing the same pattern, and the state row tells you when you have left the model's territory.

Notes on the method

Natural frequencies and shapes. Each pattern's natural frequency is set like the pitch of a mass on a spring, 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, a lightly stretched skin, and both act to pull it back toward flat. The inertia is the water set moving beneath the shape - this is where the depth of the fluid enters, since a shallower layer has less to move. A stiffer pull rings faster, more water in motion rings slower, and the natural frequency is where the two balance; the same calculation yields the pattern's exact shape for this cell and contact line. It 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 it reruns in your browser whenever you change the cell, the fluid or the contact line.

Onset thresholds. Shaking the container rhythmically strengthens and eases the pull that holds the surface flat, and a pattern grows when that rhythm runs at twice its own - which is why the response comes at half the drive frequency. The equation for such an oscillator working against friction - known as the damped Mathieu equation - takes two numbers for each pattern: its natural frequency, and how quickly friction inside the fluid drains its motion. It returns the critical acceleration Ac, the gentlest shake at which the flat surface first gives way to that pattern. The friction figure is set by the dissipation toggle: Lamb (1932) is the classical smooth-wall estimate γ = 2νk², one formula for every pattern; Wilson et al. (2022) computes each pattern's own friction from its shape, through the research pipeline's viscous model, and nudges its rhythm down accordingly. Either way the count covers friction in the body of the fluid only - not the extra rubbing at the walls and the waterline.

Which pattern is shown. Near half the drive there are usually several patterns with nearly the right natural frequency. The sandbox picks the one that is easiest to excite - the lowest threshold Ac among the closest few - which is why the display can hand over to a neighbouring pattern as you change the fluid or the cell, not only the frequency.

How far to trust the numbers. The frequencies are the strong part: about 1% against laboratory measurements, and tested against the project's research code. The thresholds are honest but rougher. A real container also loses energy at its walls and waterline, so whichever friction model is chosen it generally needs a somewhat harder shake than the calculated Ac; and the research behind this site finds that reducing onset to a single pattern with a single damping number, however tuned, cannot reproduce measured thresholds exactly - a limit of the reduction itself, not of the frequencies fed into it. When two patterns have nearly equal thresholds, a real container may pick the other one. Read Ac as a guide to which patterns are easy or hard to summon, not as a laboratory-grade prediction.