The ripples that answer at half the shaking

Only some notes fit found that a string or a column of air held at both ends can vibrate only at the frequencies whose waves fit a whole number of half-wavelengths between the ends. The chain that swings in Bessel’s notes found the same counting for a hanging chain whose tension varies along its length, with notes that are no longer harmonics. In both, the medium was set ringing directly: plucked, bowed, pushed sideways at the frequency it answers to. A standing wave can also be set ringing indirectly, by changing something the medium depends on, and then the rules change — including the frequency at which it answers.

In 1831 Michael Faraday, in an appendix to a paper on the patterns sand makes on vibrating plates, described what happens to a shallow layer of liquid on a plate shaken up and down. Above a certain strength of shaking, the liquid’s surface broke into a regular pattern of crests, and the crests rose and fell at half the frequency of the plate’s vibration. Lord Rayleigh confirmed the halving half a century later, and in 1954 Brooke Benjamin and Fritz Ursell showed that each wave on the surface obeys the equation Émile Mathieu had written down in 1868 for the vibrations of an elliptical drum.

Gravity that rises and falls #

A dish shaken vertically with acceleration $a\cos\Omega t$ is, for the liquid in it, a dish in which gravity itself oscillates: in the dish’s frame the liquid feels an effective gravity $g + a\cos\Omega t$. The liquid’s surface waves are restored by gravity and by surface tension, and the frequency of a wave of wavenumber $k$ on a layer of depth $h$ is set by

$$\omega_k^2 = \left(gk + \frac{\sigma k^3}{\rho}\right)\tanh kh,$$

the capillary–gravity dispersion relation that the speed that depends on the length drew for waves on the sea. Shaking the dish changes the $g$ in that formula periodically, so each wave becomes an oscillator whose stiffness is modulated at the drive frequency. That is exactly the situation the swing that is pumped, not pushed found on a playground: a child who stands and crouches changes the swing’s effective length, and with it its stiffness, twice per swing, and the swing grows. Faraday’s waves are a playground swing for every ripple at once.

Each mode of the surface, with viscous damping $\gamma_k = 2\nu k^2$ for a liquid of kinematic viscosity $\nu$, obeys

$$\ddot\eta + 2\gamma_k\dot\eta + \left[\omega_k^2 + a\,k\tanh(kh)\cos\Omega t\right]\eta = 0,$$

a damped Mathieu equation. Its solutions either decay or grow, depending on the drive’s strength and on how the wave’s own frequency compares with the drive’s.

Where the surface gives way #

The figure was computed one wave at a time. For each wavenumber and each strength of shaking, the Mathieu equation was integrated over one period of the drive from two independent starting states, and the growth over that period read off from the result — the Floquet multiplier. Where it exceeds one, the wave grows from any small disturbance; where it is less, the wave dies. The unstable regions form tongues reaching down towards the wavenumber axis.

The largest tongue sits where the wave’s own frequency is half the drive’s. Its tip, at the lowest shaking that excites anything, is the threshold. A second, narrower tongue sits where the wave’s frequency equals the drive’s, and further ones at higher multiples, each needing harder shaking. The ordering is the parametric resonance found in the swing: a stiffness modulated at frequency $\Omega$ pumps most effectively an oscillation at $\Omega/2$, because then the stiffness is high at both extremes of each swing and low at both crossings of the middle, and every stroke of the drive does work on the wave in the same sense.

Without damping, the tongues would reach right down to zero shaking, and the gentlest vibration would excite waves. Viscosity lifts each tongue’s tip to a finite threshold, and lifts the tips of short waves most, because viscous damping grows as the square of the wavenumber. The threshold at the tip is $a_c \approx 4\gamma_k\omega_k/(k\tanh kh)$, and the computed tongue’s tip agrees with that formula to within a few per cent.

The wavelength the water chooses #

The pattern’s wavelength is chosen by the liquid, not by the dish. Of all the waves the surface could carry, the drive pumps only those whose own frequency is close to half its own, and the dispersion relation says which wavelength that is. The figure reads it off directly: draw half the drive frequency as a horizontal line, and where it crosses the dispersion curve is the wave that grows. Shake slowly and long gravity waves appear; shake fast and the waves are short ripples held by surface tension.

On logarithmic axes the wavelength falls along two straight lines and turns from one to the other. For slow shaking the waves are gravity waves, with $\omega^2 = gk$, and the wavelength falls as the square of the drive frequency. For fast shaking they are capillary waves, with $\omega^2 = \sigma k^3/\rho$, and it falls as the two-thirds power. The bend comes at the wavelength where the two terms in the dispersion relation are equal, $2\pi\sqrt{\sigma/\rho g}$, which is seventeen millimetres for water — the same capillary length that the skin that is not a skin found setting the scale at which surface tension matters. A dish of water shaken at a few tens of hertz sits right in the crossover, and its ripples are a few millimetres to a centimetre long, which is why Faraday’s patterns are easy to see by eye.

A wavelength chosen by the liquid, not the basin #

The contrast with other standing waves on water is instructive. The tide a lake makes by itself found a lake sloshing from end to end in a seiche whose period is set by the basin: the wave has to fit the lake, as a note has to fit a string, so the basin chooses the wavelength and the wavelength chooses the frequency. A Faraday pattern runs the other way. The shaking supplies the frequency, the dispersion relation turns it into a wavelength, and the pattern fits itself into the dish as best it can, with only the edges disturbed. Shrink the dish to a few wavelengths across and the two choices compete: only the dish’s own modes can appear, and the drive picks whichever of them lies nearest to half its frequency, so that changing the frequency smoothly makes the pattern jump from one mode to the next.

The choice is also unlike the one the thread that cannot stay a thread found for a jet of water breaking into drops. There, every wavelength longer than the jet’s circumference is unstable, and the drops form at the wavelength that grows fastest, which a balance of surface tension and inertia sets. In Faraday’s dish, almost no wavelength is unstable — only a narrow band near the resonance, inside a tongue — and the narrowness of that band is why the patterns are so regular.

Two ways to pump a pendulum #

The Mathieu equation behind Faraday’s waves is the same one behind a stranger effect. Held up by a force that averages to nothing found that a pendulum whose pivot is shaken up and down fast enough can balance upside down: Kapitza’s pendulum. That is the Mathieu equation too, with gravity modulated by the shaking, but in a different region of the stability chart. Faraday’s waves live inside the tongues, where the drive is resonant with the oscillator and pumps it. Kapitza’s pendulum lives far outside them, where the drive is much faster than the pendulum’s own motion, and the averaged effect of the rapid shaking is an extra restoring force that holds the pendulum up. The same equation, the same modulated gravity, produces instability at resonance and stability far from it.

A dish of water shaken at a few hundred hertz is in the first regime for its short ripples and in the second for its long sloshing modes, whose natural frequencies are far below the drive. The long modes are quietly stiffened while the short ones erupt, and both effects are present at once in the same liquid.

How hard the dish must be shaken #

The threshold rises steeply with frequency. Faster shaking excites shorter waves, viscous damping rises as the square of the wavenumber, and the shaking has to overcome it. For water, a hundredth of gravity is enough at twenty hertz; at four hundred hertz the dish must be shaken nearly three times harder than gravity pulls. An oil twenty times more viscous needs twenty times more shaking at any given wavelength. That is why experiments on Faraday waves choose their liquid and their frequency together, and why, in an ultrasonic humidifier, where a vibrating plate drives water at over a megahertz, the shaking is violent enough to fling droplets off the crests: the droplets are a fraction of the Faraday wavelength, a few micrometres across, which is how such devices make a fine mist.

Once for every two strokes #

Followed in time, a single mode above threshold grows exponentially while oscillating at exactly half the drive frequency: thirty wave cycles in sixty strokes of the dish. Its growth rate is the pumping minus the damping. With the modulation written as a fraction $\epsilon = ak\tanh(kh)/\omega_k^2$ of the wave’s own stiffness, the pump adds energy at a rate $\epsilon\omega_k/4$ while viscosity removes it at $\gamma_k$, so the wave grows as $e^{(\epsilon\omega_k/4 - \gamma_k)t}$; at the threshold the two balance exactly, and at four times the threshold the growth is three times the damping rate. A water ripple at fifty hertz is damped in a little over a second, so even well above threshold the pattern takes a second or so to appear — a delay easily seen when the shaking is switched on. The halving is not an approximation. The Mathieu equation’s growing solution at the first tongue is a periodic function with twice the drive’s period multiplied by an exponential, and no amount of nonlinearity later changes the period, which is locked to the drive. Above threshold the growth is eventually limited by nonlinear effects — the crests steepen, the waves interact, energy spills into other modes — and the surface settles into a steady pattern whose crests rise and fall at half the drive frequency for as long as the shaking continues.

Squares, stripes and quasicrystals #

The linear theory chooses the wavelength. It says nothing about the pattern, because every direction on a flat surface is equivalent, and any combination of waves of that wavelength is equally unstable. The pattern is chosen by the nonlinear interactions among them, and it is where Faraday waves become a laboratory for pattern formation in general. In a viscous liquid with a single drive frequency the commonest pattern is squares, two standing waves at right angles; in thinner liquids stripes and hexagons appear; in a dish driven by two frequencies at once, patterns with twelve-fold symmetry and no repeating unit at all — quasicrystalline patterns — were found in 1993 by Wendy Edwards and Stephan Fauve. A layer of sand or of fine beads, shaken in the same way, forms the same family of patterns, and isolated localised peaks called oscillons that bounce at half the drive frequency while the surface around them stays flat.

The same mechanism appears wherever something restoring is modulated in time. A Bose–Einstein condensate held in an elongated trap whose tightness was modulated developed Faraday ripples along its length in 2007, at half the modulation frequency, in a cloud of atoms a few hundred micrometres long. Fuel in a rocket’s tank, shaken by the vibration of the engines, can slosh parametrically if the vibration contains a component at twice a sloshing frequency, and tanks are baffled against it.

Droplets that walk on their own waves #

The most curious use of Faraday waves came from Yves Couder and Emmanuel Fort in Paris in 2005. A bath of silicone oil is shaken just below its Faraday threshold, so that no waves grow on their own but any disturbance rings for a long time before dying. A droplet of the same oil placed on the surface does not coalesce, because a thin film of air beneath it is renewed with each bounce, and it bounces at half the drive frequency, each impact launching a ring of Faraday waves. Under the right conditions the droplet lands each time on the slope of the waves its earlier bounces made, and is kicked sideways: it walks, guided across the bath by its own wave field.

The walkers attracted great attention because their motion resembles, in some ways, the pilot-wave picture of quantum mechanics that Louis de Broglie proposed in the 1920s. Walkers passing through a slit were reported to scatter into patterns like diffraction; walkers in rotating baths settle into a discrete set of orbits; a walker confined in a corral visits places with a probability that follows the corral’s wave modes. Later experiments have qualified some of those results — careful repetitions of the single- and double-slit experiments did not reproduce the original interference-like statistics — and the analogy has clear limits, since a walker’s wave lives in two dimensions and carries the walker’s whole history. What is not in doubt is the mechanism: a particle steered by a standing wave of exactly the Faraday wavelength, which it keeps alive by bouncing at half the bath’s shaking frequency.

A single mode, a deep layer and a sharp threshold #

The figures treat each surface mode as independent and linear, with damping from viscosity in the bulk of a deep layer, $2\nu k^2$, and they place the dish’s walls far enough away not to matter. Real dishes have walls that pin the contact line and fix a slightly different spectrum of allowed modes, and in a small dish the excited wavelength is quantised by the dish, as a string’s notes are; thin layers add damping at the bottom that the deep-layer formula omits, and surfactants on water add a much larger damping at the surface. The threshold drawn is the tip of the linear tongue; in practice slight imperfections let waves appear a little below it or require a little more. The tongue diagram is drawn on a grid of wavenumbers and accelerations, so its outline is accurate to the grid’s spacing. The growth figure follows one mode in the linear equation, and real waves saturate long before they would grow without bound. The domain is liquids a centimetre or more deep, shaken at frequencies from a few hertz to a few hundred, in dishes much wider than the Faraday wavelength.

Still open: what picks the pattern #

The wavelength and the threshold are predicted by the linear theory drawn here, to within the accuracy of the damping. The pattern is not, except in limits. Which of squares, stripes, hexagons or quasipatterns appears depends on the viscosity, the depth, the frequency and the strength of shaking through nonlinear interactions among many waves, and computing them requires either a weakly nonlinear theory valid only just above threshold or full simulations of the free surface that are still expensive. Experiments find patterns the theories do not predict, and the transitions between patterns as the shaking grows, through disorder and into spatiotemporal chaos, are mapped experimentally more completely than they are understood. How a liquid chooses among patterns that are all equally unstable to begin with is the question Faraday’s dish still poses.

The linear story is short. A liquid shaken vertically feels gravity modulated at the drive frequency $\Omega$, so each surface wave is a damped Mathieu oscillator; the first instability tongue sits where the wave’s own frequency is $\Omega/2$ — a wavelength the dispersion relation chooses, 9.9 mm for water at 50 Hz and 55.5 mm at 10 Hz — with its tip lifted by viscous damping to $a_c \approx 4\gamma\omega/(k\tanh kh)$, 0.081 g at 50 Hz, and above it the wave grows while rising and falling once for every two strokes of the drive. The water answers the shaking with half its frequency and a length of its own choosing.