Physics - Faraday Waves

In one paragraph

Shake a dish of liquid straight up and down. Below a critical shake strength the surface stays flat; cross it, and a standing pattern of crests and hollows blooms across the surface, rocking at half the frequency of the drive (Faraday, 1831). The pattern is not imposed by the shaking - it is one of the container's own natural shapes, picked out because its natural frequency lies nearest to half the drive. Everything on this site - the catalogue of shapes, their frequencies, the onset thresholds - is the working-out of that one sentence for a cylindrical cell, computed live.

Why half the frequency

Vertical shaking cannot push a flat surface sideways; a perfectly flat surface would stay flat at any drive. What the shaking does is retune gravity in rhythm: in the container's frame the surface feels an effective gravity g + A cos(2πft), so the force that pulls the surface flat strengthens and weakens f times per second. Driving a system by modulating one of its parameters, rather than pushing on it directly, is called parametric forcing.

Each natural pattern behaves as an oscillator whose stiffness is modulated this way; its motion obeys the damped Mathieu equation (Benjamin & Ursell, 1954). Such an oscillator absorbs energy fastest when its stiffness is worked at twice its own frequency, and the reason is the pattern's own rhythm: it reaches full tilt twice in every cycle, once as a crest and once as a trough, and drawing the surface's pull tight at each of those turning points adds a little energy every time. Two strengthenings of the pull for every cycle of the pattern means two shakes of the dish for every cycle - so the surface answers at f/2.

The experiment. A platform shakes the vessel straight up and down at the drive frequency f, here 16.0 Hz; the surface answers with the (0, 3) ring pattern - the worked example of the figure below - at half that, 8.0 Hz: two bobs of the vessel for every rock of the surface. The pattern, its frequency, and the two-to-one rhythm are the real catalogue entries for this cell (water, 50 mm radius, 10 mm deep, free contact line), and the dish is drawn at its true proportions. Two display scalings, stated plainly: the motion is slowed about 16× so the rhythm can be counted, and the wave height and platform stroke are exaggerated to be seen - a real stroke at this pattern's onset is about 4 thousandths of a millimetre.

Friction pushes back, and the contest draws a map. For each pattern, onset traces a valley-shaped curve in the plane of drive frequency against drive strength - an instability tongue. The valley floor sits where half the drive meets that pattern's own frequency f₀, and its depth is set by friction, roughly A_c ∝ γ: the more the fluid dissipates, the harder you must shake. Below every valley the flat surface survives. The sandbox's onset row is exactly this curve, evaluated live as you sweep.

The instability tongue. Onset for a single pattern, drawn across drive frequency. The curve is the drive acceleration at which the flat surface first gives way: below it the surface stays flat, above it the pattern grows. Its floor sits at twice the pattern's own frequency, 2f₀ (24 Hz here), where onset is cheapest, and the depth of the valley is set by how much the fluid dissipates.

The catalogue of shapes

A cylinder cannot hold just any ripple. Its admissible surface shapes are J_m(kr) cos(mθ) - Bessel functions dressed around the circle - where m counts spokes (still lines running across the surface) and n counts rings (still circles around the centre). The pair (m, n) names the pattern; it is the label on every figure here.

Spokes and rings. Three separate patterns, side by side, each read by its own still lines - the nodes that never move. (4, 1) is four still diameters and the rim; (0, 3) is three still circles; (3, 2) is three diameters crossed by two circles. The first number counts spokes, the second counts rings. These are three examples, not a sequence and not multiplied together - the pattern each one names is the whole pattern.

Where do those still lines come from? Each part of the formula owns one family. The angular part cos(mθ) is a pinwheel of wedges, and its quiet lines are the m spokes. The radial part J_m(kr) is a bullseye of nested rings, and n is its radial harmonic: only certain radial waves fit the dish, the ones that meet the wall correctly, and n says which - the first that fits, the second, the third, and so on - each higher n adding another ring. Multiply the two and the pattern carries both, which is why the figure below builds it in the order the label (m, n) counts: spokes first, then rings.

One point catches everyone, so it is worth stating outright: the subscript on the Bessel function is the spoke count, not the ring count. A three-spoke pattern always uses J₃, however many rings it has. So the same m does two jobs - it is the order of the Bessel function and the number of spokes - while n only chooses which of that function's zeros the rim falls on. (The formula is conventionally written radial part first, the reverse of the (m, n) label; this site follows the label.) It is also why the bullseye is pale at its centre: that is J_m starting from zero, the point where the spokes meet, not an extra ring.

One pattern, two factors. Taken in the order the pair (m, n) counts them. The cosine factor (left) is a pinwheel of six wedges, and its three quiet diameters are the spokes: m = 3. The Bessel factor (middle) is a bullseye, and its two quiet circles are the rings: n = 2 - the pale centre is not a third one, it is the point where the spokes meet. Multiplied (right), the pattern carries both, and that is (3, 2). Mind the subscript: the pattern uses J₃, not J₂ - the order of the Bessel function is the spoke count m, never the ring count. The two rings come instead from reading that one function out to its second zero at the rim. Drawn in the same clamped-rim convention as the schematic above; the colours are the site's own field scale.

Which wavelengths fit is decided at the rim, by how the waterline meets the wall. Three rules cover it, the three classic boundary conditions: free (Neumann), the waterline slides up and down and the surface meets the wall level; pinned (Dirichlet), the waterline stays attached to the rim; and the in-between robin (Robin), where the surface meets the wall at a set contact angle θ. Real dishes live between the ideals - measured frequencies fall between the pinned and free predictions (Wilson et al., 2022) - which is why the sandbox offers all three. The low, wall-hugging patterns care the most: across the wetting range the fundamental (1, 1) can shift by half its frequency, while higher patterns barely notice.

One consequence is worth saying out loud, because the live dish shows it and the figures above do not. Those figures are drawn as a pinned drum - the wall held still, so the rim is itself a still ring, and the drawing carries the full n of them. The overview and sandbox default to a free wall, where the waterline rides up and down, so the rim is no longer still and no longer a ring. A pattern with spokes therefore shows one ring fewer than its label, n − 1, on the live dish; only the ring-only patterns, the m = 0 family, still show the full n. The two differ by where the free waterline settles: it comes to rest at the nearest crest or trough of the ripple. For a ring-only bullseye that point falls just outside the outermost ring, so the ring stays inside the wall; for a spoked pattern it falls just inside where the last ring would be, and carries that ring off past the wall. Switch the sandbox to a pinned wall and each spoked pattern's rim ring returns.

Frequencies from an energy balance

Every admissible shape has one natural frequency, set by the same tug-of-war that fixes the pitch of a mass on a spring: a restoring pull against inertia. The pull is gravity and surface tension - bending the surface away from flat lifts liquid against gravity and stretches the surface, a lightly stretched skin, and both act to pull it back. The inertia is the liquid set moving beneath the shape, and this is where depth enters: a shallower layer has less to move. A stiffer pull rings faster, more liquid in motion rings slower, and f₀ for each (m, n) is where the two balance. Carried out properly for the actual cylinder and its rim condition, this is the Rayleigh-Ritz eigenmode method (Shao et al., 2021), and in the research behind this site it agrees with laboratory measurements to about 1% across every dataset tested.

But which shape? Every shape you could draw on the surface carries its own stiffness and its own inertia, and so its own √(K/M) - a candidate frequency. The method never assumes the mode shapes in advance; it finds them, by one rule: the natural modes are the shapes that make √(K/M) as low as it will go. Any other shape overestimates - the frequency it hands back is a blend of the true modes' frequencies, and a blend can never fall below the lowest one - so the real fundamental is the single shape with nothing else mixed in, sitting at the floor. Swept over a family of trial shapes, that search is exactly the eigenvalue calculation the site runs: the shapes where √(K/M) stops changing, and the solutions of K a = ω² M a, are the same shapes.

pattern to find view 2D profile 3D

trial shape deformation +1.00

Finding a mode. Pick a pattern to find, then drag the trial shape; the pitch it produces, √(K/M), reads off the curve. Every shape overestimates, so the curve is a valley - and its floor is that pattern, its shape and its natural frequency (3.58 Hz here) found together. Deform away in either direction and you stir in the next ring, so the blended pitch climbs. Finer patterns also tilt the K bar from gravity toward surface tension. The solver does not walk down the valley, it jumps straight to the bottom - but the bottom is what it finds.

There is a tempting shortcut: pretend the dish is an endless tray, take the textbook dispersion relation ω₀² = (gk + σk³/ρ) tanh(kh), and simply quantise the wavenumber at the Bessel roots. In real cylinders the shortcut misses by 2-45%, worst exactly for the low patterns that hug the walls, and it misidentifies the observed pattern far more often than the energy balance does. In a dish, the walls are not a detail.

Friction and its price

Thresholds are where friction enters. The classical estimate is Lamb's (1932) γ = 2νk² - one formula for every pattern, from the viscosity alone. Wilson et al. (2022) sharpen it: each shape dissipates by its own internal shearing, so each earns its own damping rate, computed from the same eigenmode machinery. These are the two dissipation models behind the sandbox's toggle.

Both count only friction in the body of the fluid, and that is a known lower bound. A real dish also rubs where modelling is hardest: in the thin layers on the walls and floor, in any film sitting on the surface, and at the moving waterline itself. Careful experiments measure 1.2 to 3.1 times the clean prediction (Henderson & Miles, 1994), and a 74-condition survey found 3.6 to 6.7 times the bulk estimate (Howell et al., 2000). More friction than modelled means a real dish needs a somewhat harder shake than computed - the sandbox says so wherever thresholds appear.

Two functions, two jobs

The mathematics behind all of the above runs on two named function families, and they divide the work cleanly: one owns space, the other owns time.

Bessel draws the shape. Every admissible pattern is built from J_m(kr) cos(mθ): the Bessel function lays the rings along the radius, the cosine folds the spokes around the circle, and the rim rule - free, pinned or robin - selects the wavenumbers k that fit the dish. Everything about where the surface stands still is settled here, before any shaking is switched on; the shaking never changes a shape, it only decides which one appears. And because each allowed k fixes a natural frequency, the space side also decides where every pattern's valley will sit. (Under a pinned or robin waterline the true shape is a blend of several Bessel profiles - that is what the energy balance above computes - but the building blocks are the same.)

Mathieu runs the clock. Shake the dish and each shape keeps its geometry while its amplitude a(t) obeys the damped Mathieu equation (Benjamin & Ursell, 1954): a″ + 2γa′ + [ω₀² + Ak tanh(kh) cos(2πft)] a = 0. The bracket is the mechanism of the second section written out: the shape's own stiffness plus the drive's rhythmic retuning of gravity. Mathieu's equation is the mathematics of exactly that situation - an oscillator whose stiffness is modulated in time - and its solutions split the drive plane into flat and growing regions: the instability tongues. The floor of the first tongue is the onset A_c, and its position at 2f₀ is the halving itself.

Two functions, drawn. The same panel carries each function's own graph, and the dish collects what it contributes. J_m unrolls along the radius and every zero becomes a still ring; the cosine unrolls around the circle and every zero becomes a still diameter; their product is pattern (3, 2), drawn here in the same clamped-rim convention as the schematics above (in the real free-walled cell the rim moves). Then Mathieu takes the clock: the trace is the damped Mathieu equation integrated live, its stiffness worked at twice its own rate - drive at 0.4 of onset and the bloomed pattern dies away; at 1.7 and it grows from a whisper until the pattern stands. In the standard cell this pattern is a real catalogue entry: natural frequency - Hz, principal drive - Hz, onset near - m/s² with its own damping. Display scalings, stated plainly: the trace oscillates at a slowed 0.8 Hz so the two-to-one rhythm can be counted, and its damping and drive are chosen so one run fits a few seconds.

The two sides meet in three handover numbers. From the space side the clock receives the shape's natural frequency ω₀, its grip on the shake k tanh(kh) - long waves over shallow water take hold of the shaking only weakly - and its friction γ. That is the whole pipeline of this site, run live: Bessel and the rim rule say what can appear, the energy balance prices each shape's frequency, Lamb or Wilson prices its friction, and Mathieu says how hard you must shake and which pattern rises first. The chain is also why frequencies matter so much: shift a shape's f₀ by a few percent and its whole valley moves sideways - the workbench notes measure what that does to thresholds.

Where prediction ends

Feed each pattern's frequency and damping into the Mathieu equation and the thresholds follow. What survives contact with experiment splits cleanly - this split is the subject of the research paper behind the site. The conservative predictions hold: the catalogue of admissible shapes, their ordering, and the positions of the tongues are good to about 1%. The threshold magnitudes do not: across the datasets tested they miss by factors of 2 to 8, and where two patterns' valleys nearly tie, the predicted winner can be the wrong one of the pair.

The deeper finding is that this failure is structural, not a matter of a better friction number. Back-calculate the damping each measured tongue would need, and it varies by tens of percent within a single tongue - 27% on average, approaching 180% in the worst case - so no single per-pattern damping value can reproduce the measured shapes. Theory says the same from first principles: no constant damping coefficient matches the rigorous viscous threshold (Chen & Viñals, 1999), and a pinned waterline couples the modes, so a single-mode equation is an approximation from the start (Kidambi, 2013). The honest summary: frequencies are a precision tool; thresholds are a guide.

The complete linear theory - the viscous Floquet framework of Kumar & Tuckerman (1994), keeping every mode and every harmonic - closes much of this gap where it has been carried out, but for a standard dish with a pinned waterline it remains an open computation. This site therefore runs the validated part: Rayleigh-Ritz frequencies (Shao et al., 2021), a stated dissipation model, and the damped Mathieu threshold, with the limits above attached wherever numbers appear. The full study will be available on the research page; the practical version of this trust guidance lives in the sandbox notes. And the arithmetic itself is yours to audit: the checks page recomputes the site's numbers in your own browser, beside their references.