Where the Sand Stands Still
Scatter sand on a vibrating plate and it draws a figure — fleeing the parts that shake, gathering on the lines that don't. Almost every explanation models the plate as a drumhead. That is the wrong physics, and the real one is a problem so hard it forced a new method into being.
Ernst Chladni, in 1787, drew a bow across the edge of a sand-strewn brass plate and made sound visible. The sand abandons the antinodes — where the plate flings it up — and piles along the nodal lines, the curves that stand perfectly still. Change the pitch and the figure changes. The patterns are real, reproducible, and strange, and the story everyone tells about them is subtly false.
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Click a mode to retune the plate; the sand walks to the new nodal lines. Flip to Drumhead to see the model the textbooks use — and watch the same low modes come out as a tidy straight grid instead of curved stars.
The drum is the wrong model
A drumhead is a membrane: a thin sheet held flat by tension, like the skin of a drum pulled tight over its rim. Its motion obeys the second-order wave equation, and on a square clamped at the edge its modes are exactly sin(mπx)·sin(nπy), with frequencies proportional to √(m²+n²) and nodal lines that form a straight grid. It is clean, closed-form, and in every textbook.
A Chladni plate is none of those things. It is a stiff sheet of metal, restored not by tension but by its own bending stiffness, and — lying free on a support, bowed at the rim — its edges are free, not clamped. That is a different equation: the fourth-order biharmonic D∇⁴w = ρh ω²w, with the edge conditions that the bending moment and the shear both vanish. For a square it has no closed-form solution at all. You cannot write the modes down; you can only approximate them.
That wall is where a piece of mathematics took shape. Walther Ritz published his variational method in 1908 and applied it to these square-plate figures in 1909: guess the shape as a sum of simple functions, then let the energy pick the best coefficients. It became the Rayleigh–Ritz method, a direct ancestor of the finite element method now used to model everything from bridges to bones. The instrument above runs Ritz's method live, in your browser, on a basis of free–free beam shapes, and solves the eigenvalue problem on the spot.
The tell: it depends on the material
Is "the drum makes straight lines, the plate makes curves" the real difference? No — and this is the kind of half-truth this place exists to refuse. A membrane can make curves too: when two modes like (3,2) and (2,3) share a frequency, any blend of them is also a mode, and the blend's nodal lines bend. (Switch to Drumhead, pick a mode marked deg., and drag the degenerate mix slider: straight lines curl into curves in front of you.)
The honest difference is the physics, and it leaves one clean fingerprint. A membrane with uniform tension and density has frequency ratios set by its shape and edge conditions; its physical pitches also depend on tension, density and size. A free plate's normalized frequencies depend on Poisson's ratio: how much the metal, bent one way, curls the other. Change the setting in the menu and the normalized spectrum shifts: this solver's fundamental moves from 14.12 at Ritz's ν = 0.225 to 13.48 at ν = 0.300, the nominal steel setting. A uniform membrane's frequency ratios have no such Poisson-ratio dependence.
The check — show, don't assert
The plate's pitches are reported as the dimensionless frequency parameter λ = ω·a²·√(ρh/D) (side a, flexural rigidity D) — the convention Leissa tabulates. The instrument computes them live by Rayleigh–Ritz; an offline verifier recomputes the same numbers from the same code and checks the first five against the published literature. For a completely free square plate at ν = 0.3:
| mode | this page | Leissa 1973 | Δ |
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Leissa's (1973) beam-function values (13.49, 19.79, 24.43, 35.02, 35.02), reproduced in Huang, Leissa and Liao (2008), Table 1, are Rayleigh–Ritz upper bounds. This page's larger basis gives values ~0.1–0.5% lower. The verifier also compares the ν = 0.225 fundamental with Leissa's NASA SP-160, Table 4.67. The plate has exactly three modes at λ = 0 (it floats: one slide, two tilts), the fundamental's nodal lines cross at the centre, the next mode's lie on the diagonals, the third is a ring: figures Chladni and Mary Waller catalogued. The nodal checks sample the first mode's vertical centre-line, the second mode's main diagonal and the third mode's centre amplitude; they do not establish the full cross, X or closed ring. research/chladni-figures/verify.mjs runs 25 checks. A separate development check compared the eigensolver with LAPACK.
How the sand finds the nodes
The grains are not drawn onto the lines — they walk there. Each frame every grain is kicked by an amount set by the local vibration amplitude: violent on the antinodes, much slower near the nodes. A grain that stumbles onto a quiet line slows nearly to a stop, but keeps wandering. Watch long enough and the figure emerges from a uniform scatter, exactly as it does on a real plate — the simulation encodes only "shake hard where the plate moves," and the pattern is the plate's own, not ours.
Corrections
Corrected 2026-10-14: The material menu labelled ν = 0.225 as brass. It is now labelled as Ritz's historical parameter; Lakes's table lists brass at ν = 0.33 (found by the assay line, 2026-10-14). Source.
Corrected 2026-10-14: The membrane mix was said to make hyperbolas. For the equal (3,2)+(2,3) mix, the curved branch obeys cos(πx)cos(πy) = 1/4, rather than a hyperbola in the displayed coordinates (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The page said a membrane's frequencies depend only on shape and never on material. For a membrane with uniform tension and density, frequency ratios depend on shape and edge conditions; physical pitches also depend on tension, density and size (found by the assay line, 2026-10-14). Source.
Corrected 2026-10-14: The fundamental was given as 13.46 for the ν = 0.225 preset. This solver displays 14.12 at ν = 0.225 and 13.48 at ν = 0.300; Leissa's Table 4.67 gives corresponding upper bounds of 14.111 and 13.464 (found by the assay line, 2026-10-14). Source.
Corrected 2026-10-14: The comparison attributed six frequency parameters to Young (1950) and Warburton (1954). The five retained values are Leissa's (1973), as reproduced by Huang, Leissa and Liao (2008), Table 1. That table does not establish the sixth value, 61.53, which has been removed from the comparison (found by the assay line, 2026-10-14). Source.
Corrected 2026-10-14: The animation was described as leaving grains still at nodes. Its random kick retains a 0.004 floor at zero amplitude, so grains keep moving slowly there (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The page dated the invention of Ritz's method to 1909. Ritz published his method in 1908 and applied it to the free square plate in 1909 (found by the assay line, 2026-10-14). Source.
Corrected 2026-10-14: The related-layer note called the Hessian determinant discrete Gaussian curvature and intrinsic. For z = w(x,y), Gaussian curvature is K = (w_xx·w_yy − w_xy²)/(1 + w_x² + w_y²)²; the Hessian determinant approximates K at small slopes. Gauss's theorem applies to K (found by the assay line, 2026-10-14). Source.
Corrected 2026-10-14: The related-layer note said a clamped or simply supported edge makes frequencies forget the material. For a rectangular plate with every edge clamped or simply supported, the dimensionless frequency parameters lose explicit Poisson-ratio dependence; physical frequencies still depend on material through flexural rigidity and density (found by the assay line, 2026-10-14). Source.