The Chladni Plate

A Square Plate and Some Sand

Where the Numbers Come From

A thin plate in bending obeys the Kirchhoff equation, and for a rectangle with simply supported edges it has a closed-form solution. This is the one classical edge condition for which it does.

The plate is 300 mm square and 1.5 mm thick, in aluminium: E = 70 GPa, ν = 0.3, ρ = 2700 kg/m³. That gives D = 21.64 N·m, a fundamental at 80.7 Hz, and 23 resonances between 60 and 2700 Hz. Move the frequency slider a few hertz off one and watch Γ collapse: these are sharp.

The formula was checked against a published worked example rather than against itself. Guguloth, Singh and Ranjan give six analytical frequencies for a 600 × 400 × 6.25 mm aluminium plate. Running the same expression used in this panel:

The Edge Chladni Actually Used

Chladni’s plates were free at the edges, not supported. That problem has no closed-form solution. Rayleigh called it “one of great difficulty, and has for the most part resisted attack,” and Ritz produced the first accurate computation of square-plate Chladni figures in 1909, which is the work that made the Rayleigh-Ritz method famous.

This panel does not attempt it. It solves the case that is exactly solvable, so that every line the sand lies along is a true nodal line of a real plate equation rather than a numerical approximation to one. The figures are genuine. They are not Chladni’s. The most visible difference is at the boundary: a simply supported edge is itself a node, so the sand frames the plate. On a free-edged plate the rim is moving, and it does not.

Which Way the Sand Goes

Grains do not simply fall into troughs. Above about one g of plate acceleration they are thrown clear on every cycle, and the collisions that land them cost them energy fastest where the plate is not moving, so they accumulate on the nodes. Van Gerner and colleagues showed that below one g the grains never leave the surface, the force averaged over a cycle points the other way, and the figure inverts: the sand collects on the antinodes instead. They also showed this is a rolling effect and not air drag, since large grains do it too.

Γ in the readout is that ratio, the plate’s peak acceleration in units of g. Turn the drive strength down until Γ falls below 1 and the figure turns inside out. It is the same plate, the same frequency and the same nodal lines; only the sand’s relationship to them has reversed.

Why This Is an Attractlet

There are two objects here. The panel is a simulation, which the kernel’s Attractlet Model class (ATTRACTLET, § 7.4) covers outright. The physical plate is an authored reading, which is why the card tag carries no citation.

The reading turns on what happens when you switch off the shaker. Press Stop the drive. Nothing collapses. The sand sits exactly where it was, and the figure is still there, perfectly legible, with no process of any kind maintaining it. Ferrofluid spikes vanish the instant the field goes; this does not. That is worse for the pattern, not better. What is left is a heap of sand in the shape of something that used to be held. The figure outlived the process by not being a process: it was a record of one, lying on a plate.

The Shape of the Basin

Press Scatter the sand a few times at a fixed setting and the same figure comes back every time, from any starting arrangement. Measured here across four independent random scatters, the settled sand distributions agree to a correlation of 0.96. The basin is essentially everything.

Then press Move the shaker without touching the frequency. At 2017 Hz three different modes share one resonance, so any mixture of them is equally a solution and what the plate actually does is decided by which of them the shaker can reach from where it sits. Against the same figure, the same four scatters that agreed to 0.96 with each other fall to 0.64 at one other drive point and 0.24 at two more. The frequency does not pick the figure. The supply does. Two of those drive points sit on the diagonal and give figures that match each other exactly, which is the symmetry doing what it should.

Put the shaker at the exact centre and the effect is at its starkest. A mode with an even index has a node running through the centre, so a shaker there cannot raise it at all: 16 of the 23 resonances in this range stop answering altogether. The 2017 Hz resonance still answers, but only with one of its three modes, the (5,5), so the same frequency draws a completely different figure depending only on where the shaker sits.

What This Model Is Not

The edges are wrong for Chladni, deliberately and for the reason above.

The clock is arbitrary. The direction the sand travels is the published criterion. The speed is not: grains drift along the gradient of the plate’s motion at a rate this panel picks. Only the destination is a claim.

The grains do not interact. Real sand piles up, shadows itself and avalanches. Here each grain moves as though it were alone, so the ridges are as thin as the nodal lines rather than as thick as a heap of sand.

The response is truncated. The sum runs over the first hundred modes, m and n up to 10. That is well past the top of the slider, so what it costs is a slightly wrong tail far off resonance, where the plate is barely moving anyway.

Damping is a choice. Q = 500, which sets how sharp the resonances are. A real plate’s damping depends on its mounting, its alloy and how much sand is sitting on it, and loading a plate with sand changes it as you watch.

The grains do not touch each other, so nothing here would stop them all piling onto the same point. In the ordinary figure it does not matter, because the sand stops as soon as the plate under it is doing less than a g and that is a band of real width. In the inverted figure it matters a great deal: an antinode is a single point, and without crowding every grain would end up on it. The heaps you see are a jostle term standing in for grain-on-grain contact. Where each heap sits is a prediction. How big it is is that number.

The drive stops at 15 g. The band the sand lies in is the part of the plate doing less than one g, so its width goes as 1/Γ and driving harder sharpens the figure. Past 15 g it sharpens past what this model can honestly resolve: the band gets thinner than the background left by truncating the modal sum, and the figure breaks into dashes that are arithmetic rather than plate. Measured across all 23 resonances and all four drive points, the worst case at 15 g leaves the still region in 3 connected pieces holding 43% of it. At 20 g it is 172 pieces holding 4%.

Kirchhoff, G. (1850). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik, 40, 51–88.
Rayleigh, J. W. S. (1877). The Theory of Sound, Vol. 1.
Ritz, W. (1909). Theorie der Transversalschwingungen einer quadratischen Platte mit freien Rändern. Annalen der Physik, 333(4), 737–786.
Gander, M. J. & Kwok, F. (2012). Chladni figures and the Tacoma bridge: motivating PDE eigenvalue problems via vibrating plates. SIAM Review, 54(3). Preprint. Source of the Rayleigh quotation and of the free-edge boundary conditions.
van Gerner, H. J., van der Hoef, M. A., van der Meer, D. & van der Weele, K. (2010). Inversion of Chladni patterns by tuning the vibrational acceleration. Physical Review E, 82(1), 012301. Source of the Γ = 1 threshold and the rolling mechanism.
Guguloth, G. N., Singh, B. N. & Ranjan, V. (2019). Free vibration analysis of simply supported rectangular plates. Vibroengineering PROCEDIA, 29, 270–273. Source of the worked example in the table above.