One function generator, one knob, three materials. A steel plate covered in sand, a tray of water on a loudspeaker, and a crystal wine glass. Turn the frequency and each one answers in its own way: the sand gathers where the steel stands still, the water breaks into a lattice at half the note, and the glass sings only within a fifth of a hertz of its own.
What does a sound look like?
A tone is a push repeated hundreds of times a second. Bolt a 24 cm steel plate to a vibration driver and feed it 332 Hz: the plate bends up and down 332 times a second, about 67 micrometres at its corners for a push of 0.3 newton. That is invisible to the eye: on the bench the steel looks perfectly flat, and that is how we draw it while sand lies on it. Its effect is not invisible: those corners accelerate at thirty times gravity.
Bending waves run across the steel at about 56 m/s, reflect off the edges and interfere. At only certain frequencies do the reflections line up into a standing wave. Those are the resonances, 3 Hz wide at this pitch. Between them, the same push barely moves the plate: the same shake takes ten to a hundred times more force.
So our generator closes the loop, as laboratory shakers do, in an idealised way: it watches the whole plate (as a scanning vibrometer could) and holds the shake of its loudest spot steady (14 cm/s here) at every frequency. A real controller regulates one sensor point. The force it needs, shown under the acceleration readout and on the generator's amplitude knob, plunges at every resonance. The sand keeps dancing as you turn the dial, and the still lines drift from one shape to the next. Near each resonance the dial turns finer, so a casual turn parks on a figure.
f = C · k² bending waves: frequency grows with the square of the wavenumber C = √(D / ρh) / 2π ≈ 0.242 m²/s for 1 mm steel, D = Eh³ / 12(1 − ν²)Where does the sand go?
Sand goes where nothing moves. A standing wave has lines that never rise or fall: the nodal lines. A grain of fine sand (0.2 mm) on a shaking patch is thrown up whenever the plate drops away faster than gravity. At the loudest spots it leaves at about 30 cm/s, rises 4 mm on average, lands a centimetre or two away 50 thousandths of a second later, and is thrown again. Seen from above, 97 % of the grains there are in the air at any moment: a grey, shimmering haze. Near a nodal line the plate never reaches 1 g, the hopping stops and the grains stay.
One more effect sharpens the lines. A bending plate's top surface also slides sideways by a micrometre, in step with the bending, fastest on the nodal lines. Grains resting there are nudged by friction toward the middle of the line, so each line ends up single and crisp. Everything here runs in real time: half the sand reaches the lines in about a second, 80 % in three. Turn the dial and the old lines explode into haze within a tenth of a second, then the new shape draws itself. Between resonances the plate still has still lines, those of the mix of its two nearest modes, so the figure morphs as you turn; on a resonance it snaps into a clean, symmetric shape.
Γ = |W| ω² / g local peak acceleration in g: grains lift only where Γ > 1Ernst Chladni published these figures in 1787 using a violin bow. After Chladni demonstrated them in Paris in 1808, Napoleon set a prize for the mathematics. Sophie Germain was the only entrant, and on her third attempt, in 1816, she won it, the first woman to win a prize from the Paris Academy of Sciences.
Why does water answer at half the note?
Shake a tray of water up and down 60 times a second, and gravity inside the tray seems to swell and fade 60 times a second. The water's own ripples get pumped like a swing whose rider stands and squats twice per swing. The ripples that win oscillate at exactly half the drive, 30 times a second. Michael Faraday first saw this half-frequency response in 1831. Lord Rayleigh explained it in 1883 as parametric resonance.
(πf)² = (gk + σk³/ρ) · tanh(kh) the wave at half the drive, f/2, picks its wavelengthBelow a threshold (about half a g at 60 Hz) friction wins and the surface stays a mirror, apart from small ripples running in from the wall at the drive's own frequency. Above it, ripples grow out of the noise everywhere at once. After a few seconds a texture of crests appears, then patches of squares at different angles compete, and the boundaries between them sweep out through the wall. At 60 Hz one square lattice fills the tray after 15 to 20 seconds; closer to threshold, and for the hexagons below about 25 Hz, ordering takes longer. The spacing, 8.6 mm, comes from the tug-of-war between gravity and surface tension, not from any “shape” of the sound.
The crests are only 0.2 mm tall, their slopes under 6°. You see them because the water mirrors a light panel hung behind and above the tray: each tilted patch throws the panel's light toward you, a lattice of glints. The slow-motion view (the default here, labelled on screen) shows the tray beating twice for every rise and fall of the water; while you turn the dial it holds still at a crest, as a strobe would, so the lattice never flattens out of view. At real speed the 30 rises and falls per second blur into a still, shimmering grid.
Can sound really break a glass?
Yes, with loudness and precision. Flick a good wine glass and it rings for seconds. That long ring means a quality factor near 3000, so at 630 Hz the glass responds strongly only within about 0.2 Hz of its note. Hold a loud tone there and every cycle adds a little energy. Within a few seconds the rim is swinging by millimetres, turning oval twice per cycle, with its edge accelerating at more than 1000 g.
Δf = f₀ / Q ≈ 630 / 3000 = 0.21 Hz how close you must tuneGlass softens as it bends, so its note slides down by about a hertz and you must follow it. Somewhere between about 0.17 % and 0.4 % bending strain a hidden flaw lets go. Pour water in and the note drops (half full: about 586 Hz). When the glass sings, the water at its rim ripples at half the note.
On this bench the dial spans 3 Hz below the glass's note to 2 Hz above it. On the note the rim's swing is drawn at one fixed magnification for each loudness (×10 at 110 dB, ×2 at 120–125 dB, true scale at 130 dB); away from it the swing is drawn on a logarithmic scale, because the real one falls under a tenth of its peak only a hertz away and would look frozen. Both are labelled, against a thin ring that marks the rim at rest, and the readouts give the real swing: hertz away the rim and the water at its wall barely breathe, within a fraction of a hertz they come alive, and on the note the swing builds up over a second and a half. Near breaking, the oval you see is the real one.
At real speed the rim does not visibly wobble: it swings 630 times a second, and the eye sees a doubled, ghosted rim where the motion is largest. When it breaks, the cracks run round the bowl in about a tenth of a millisecond, far less than one swing. Thin glass shells break into a power-law spray of sizes: many small pieces from the rim, where the strain was highest, and a few large ones lower down, while the stem survives.
The numbers
| Quantity | Note | Value |
|---|---|---|
| Steel plate | flexural rigidity D = Eh³/12(1−ν²) | 240 × 240 × 1mm |
| Flexural rigidity | steel, E = 200 GPa, ν = 0.29 | 18.2N·m |
| Mode (2,2) | guided-edge model, centre drive · bending wavelength 170 mm | 332Hz |
| Plate resonance width | at 332 Hz (Q ≈ 110), damping fitted by Tuan et al. | 3Hz |
| Sand lifts off | plate acceleration above 1 g · 67 µm at 332 Hz gives 30 g | > 1g |
| Sand | fine quartz, 0.2 mm grains, a teaspoon (7 g) sieved over the plate | 630 000grains |
| Grain hops at the antinodes | shake 14 cm/s (0.3 N on (2,2)): mean apex, 1 % reach 19 mm · 24 cm/s sideways | 4mm |
| Force for that shake | on mode (2,2) · between (0,2) and (2,2), at 230 Hz: 8.3 N | 0.3N |
| Lines appear | half the sand near the lines (model, real time) · 80 % at 3 s | ≈ 1s |
| Sand ridges | (2,2): volume of the sand on the lines at a 30° slope (height × width) | 1.4 × 4.8mm |
| Faraday waves, 60 Hz drive | wave frequency 30 Hz · measured ≈ 8 mm (Francois et al.) | 8.6mm |
| Faraday threshold, 60 Hz | model; 0.6 g measured with surfactant, ~0.11 g predicted for perfectly clean water | ≈ 0.45g |
| Crest height, 60 Hz at 0.6 g | steepest slope 5.8° · ripples from 1 µm noise reach it in ≈ 4 s | 0.2mm |
| One lattice | patches order into one square lattice (amplitude-equation model) | 15–20s |
| Wine glass note | D♯5, Skeldon et al. wineglasses | 630Hz |
| Glass quality factor | width 0.21 Hz, ring time 1.5 s (measured Q 3500–4200) | 3000 |
| Level to break | typical thin-walled glass, at the glass (Skeldon) | ≈ 125dB |
| Half-full glass | French's formula, α = 1.25 | 586Hz |
| Lead crystal | refractive index (LLF1 as a stand-in), Abbe number 45.8 | 1.548 |
| Shards | sizes follow p(m) ∝ m−1.35 (thin brittle shells) · smallest 4 mm, largest 24 mm | 120 |
What's simplified
- A steady shake. The generator holds the plate's loudest spot at a set speed with up to 25 N of force. That controller is idealised: it watches the whole plate, as a scanning vibrometer could, while a real closed-loop shaker regulates one sensor point (the natural one, the drive point at the centre, sits near an antiresonance between modes and would ask for very different forces). A plain amplifier at a fixed setting would fling the sand off the plate on a resonance and leave it still between them. Likewise the tray is driven at a set multiple of its Faraday threshold at each frequency (within the speaker's 5 mm stroke and 10 g), the way experiments report their drive; below about 35 Hz that multiple is raised (up to 1.9×, 4.5× for the dish modes) so the waves grow within seconds, and the drive control says so. While the dial swings from one material's band into another's, the output is off.
- Guided edges. The plate's free edges are modelled as “guided” edges. That makes simple cosine shapes exact. Real resonances sit within roughly 10 % of ours, and real plates break the perfect symmetry.
- Grains that hop. Each grain hops like a ball thrown by the plate: its take-off speed (about twice the plate's) and bounce angle (50°) are calibrated on grain-tracking experiments, and grains don't collide. The plate's surface also slides back and forth a micrometre as it bends; with friction (assumed, μ = 0.5) this nudges grains toward the centre of each line. Piles slump at a 30° angle of repose (assumed). The teaspoon of sand is a choice for legibility (a light sprinkle of 3 g draws the same lines, under a millimetre high); on a phone each drawn particle stands for several grains. Very fine powder really gathers at the antinodes, carried by air currents; this sand is too coarse for that. Above about 100 g (the high modes) the friction nudge is extrapolated beyond the range it was computed for. When more than a quarter of the teaspoon has bounced off, a little more is sieved on at random spots, as a demonstrator would.
- Water. The onset uses the linear theory with an effective damping four times that of clean water, standing in for surface contamination. The pattern grows from noise and orders itself through amplitude equations whose angle couplings are tuned to the patterns seen in experiments. When you turn the dial, the lattice takes the new spacing at once (a real one re-spaces through defects over a few seconds; this is labelled on screen while you turn), and when the frequency crosses into another kind of pattern (a line taken from the literature, not a computed transition) the old lattice steps back and the new one grows out of the waves already there rather than from flat water. Right at threshold below about 18 Hz, the envelope diffusion is floored by the simulation's time step (a numerical floor; there the tray's wall sets the shape anyway). The rim ripples are drawn, not computed, and so are the droplets a real tray would spit.
- Light. The light panel the water mirrors is part of the bench (it hangs behind and above the tray, in the mirror direction of the bench view); the rest of what the water mirrors (the bench, the generator, the sky) is captured once from the tray. The bowl of the glass is drawn as a thin crystal wall (it shifts what is behind it by under a millimetre) lit by two softbox strips, as in a product photograph; refraction is computed for the solid stem and the water. The ripples at the glass's rim are finer than a pixel from the bench, so they are drawn as the sheen they put on the water there, breathing with the rim's swing a quarter cycle after it (the wall stirs the water hardest when it moves fastest); that sheen follows the rim's drawn swing, so it fades smoothly away from the note instead of switching off.
- One oscillator per glass. Real glasses have two near-identical modes a few hertz apart, and liquid adds damping. The coupling to sound is calibrated to 125 dB, and each glass's breaking point is drawn at random because flaws decide. The shards' sizes follow the power law measured for thin brittle shells (120 pieces from 4 to 24 mm; 60 on phones); how much of the stored energy becomes motion (half) is estimated, the crack lines are straight cuts between the pieces, and the water spill is drawn. The first moment of the break is slowed ×1/20.
- Slowed and magnified. The plate's bending (vibration map, the M key), the water in slow motion (the default view) and the glass are slowed to about one cycle per second (the glass to one and a half), and the plate, the tray and the glass are magnified so you can see them (the glass at one fixed factor for each loudness on its note, on a logarithmic scale away from it); every factor is labelled on screen. The sand, the growth and ordering of the water waves and the glass's build-up run in real time, at true scale. One exception, labelled while it shows: when you switch to the water, the tray has already been shaking for 6 seconds at its own frequency (those seconds are computed step by step, as in real time, but within about one second of the camera move). While you turn the dial, the slowed water holds still at a crest, as a strobe would show it.
- Quiet speakers. Your speakers play the tone quietly. The real 125–130 dB would harm your hearing.
Myths
Sources
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- Abramian, Protière, Lazarus, Devauchelle, “Chladni patterns explained by the space-dependent diffusion of bouncing grains” — Phys. Rev. Research 7, L032001 (2025)
- Devauchelle, Popović, Szymczak, Abramian, Lazarus, “Thermodynamics of bouncing grains” — arXiv:2606.05930 (2026)
- PASCO, “Chladni Plates Kit WA-9607” — instruction manual
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