A Geometry of Sound

In 1787 Chladni published Entdeckungen über die Theorie des Klanges — Discoveries in the Theory of Sound. The method was plain: dust a glass or metal plate with fine sand, pin it at one point, draw a violin bow across the edge. The plate sings. The sand moves.

Within seconds the grains leave the parts that move and gather along the curves that do not — the nodal lines. Chladni toured Europe with the demonstration; in 1809 he performed it in Paris for Napoleon, who set a prize for its mathematical explanation.

A drumhead is pulled taut; tension restores it. A plate needs no pulling: it resists being bent, and that bending stiffness, D = Eh³ / 12(1 − ν²), grows with the cube of the thickness h — twice as thick, eight times as stiff. This is the thin-plate, or Kirchhoff–Love, description.

Hence a different law of pitch. For a thin plate f = C·k², with C = (1/2π)√(D/ρh): frequency grows with the square of the wavenumber k, which measures how tightly the pattern is folded. Halve the spacing of the lines and the pitch rises fourfold; on a drumhead it would only double.

A grain leaves the surface when the plate accelerates upward faster than gravity pulls down: Γ = Aω²/g > 1, with A the local amplitude and ω the angular frequency. Above that threshold every landing throws the grain into a short hop in a random direction, nudged downhill by the tilt of the surface beneath it.

Each grain therefore performs a random walk whose step is set by the motion under it. Over the antinodes, where the plate swings widest, grains are hurried along; at the nodes, where it is still, they stop being kicked and stay. The figure resolves in a few seconds.

Rayleigh’s approximation for a free square plate adds or subtracts two products of cosines: cos(mπx/a)cos(nπy/a) ± cos(nπx/a)cos(mπy/a). The integers m and n count the half-waves along each edge; the second product is the first with the two roles swapped.

The sign is not bookkeeping. It sets the symmetry about the diagonals: the minus combination is still along the diagonal, so sand gathers there; the plus combination is not. Same m, same n, two different figures (a real plate also splits their pitch slightly; this model does not).

Modern setups retire the bow. A shaker, or a screw fixed to the centre, drives one point at a frequency you choose. The plate answers with a weighted sum of all its modes, weighted most toward those whose natural frequency lies nearest the drive.

Sweep the drive and one figure morphs continuously into the next; nothing snaps. A resonance is where a single mode takes over the sum and the lines go sharp. On the 240 × 240 × 1 mm aluminium plate modelled here, the first resonances were measured at 630, 1023, 1240, 1368 and 1795 Hz.

A round plate’s nodal lines are n straight diameters and s concentric circles. Chladni noticed that adding one circle raises the pitch about as much as adding two diameters — in symbols, f ∝ (n + 2s)².

Rayleigh explained why: n + 2s stands in for the wavenumber, and pitch goes as its square. Rossing tested the law against measured plates in 1982 and mapped where it holds and where it drifts — it is loosest for the lowest modes. An approximation, and one that has aged well.

A drive at the exact centre can only wake the modes that move there; modes with a node at the centre stay silent. Shift the point and the push no longer matches the plate’s symmetry: the silent modes join in, and the figure tilts, splits or loses a mirror line.

That is the entire instrument: a plate, a frequency, a place to push. Everything on screen follows from those three. The Lab hands you all of them, and the sand decides the rest.