Sound

Chladni figures

Sand on a sounding plate runs off the shaking parts and gathers on the nodal lines, where the plate stands still. A note picks the figure; a chord adds several.

  • GPU shader
  • seamless tile
  • looping animation
  • works with sound
Open in the editor →

When a bow is drawn across the edge of a sand-strewn metal plate clamped at its centre, the plate sings and the sand leaps, gathering within a second into crosses, stars or rings. These are Chladni figures: the sand leaves the shaking parts and collects on the nodal lines, where the plate stands still. Every note has its own figure, and the higher the note, the finer it is.

How it works

A sounding plate vibrates in its eigenmodes, standing waves that fit its shape. The generator uses the classic approximation, with cosines on the square and a Bessel function on the disc.

square, x and y from 0 to 1: f = cos(nπx)·cos(mπy) ∓ cos(mπx)·cos(nπy) frequency ∝ m² + n² disc of radius 1: f = Jₙ(k·r)·cos(nθ), Jₙ(k) = 0 frequency ∝ k² sand gathers where f = 0

Modes (m, n) and (n, m) share a frequency, so their sum and difference are equally valid; “Symmetric mode” switches between them. Unlike a string, a plate's frequency grows as the square of the wavenumber: a figure twice as fine needs a note two octaves higher. “Notes” picks the mode with the nearest frequency: C2 matches m² + n² = 5, and C5, eight times higher, gets m² + n² = 40, the mode (2, 6). A chord overlays several modes, and in the animation the sand walks through them in turn.

A bit of history

Ernst Chladni described his figures in 1787 and demonstrated them at courts across Europe. In 1809 an impressed Napoleon had the Paris Academy of Sciences offer a prize for a mathematical theory of vibrating plates. Sophie Germain won it in 1816 at her third attempt, the first woman to win an Academy prize. Her theory still had flaws: Gustav Kirchhoff found the right conditions for a free edge in 1850, and Walther Ritz computed the figures of a square plate in 1909. Chladni was also one of the first to argue that meteorites fall from space.

What to tweak

  • Try “Notes” such as “C5”, “A4 E5”, “Am” or “440”; the ▶ button beside the field plays them.
  • “Mode m” and “Mode n” set the figure by hand; on a “round” “Plate” m counts diameters and n rings.
  • “Draw” “nodal lines” gives clean line art; “displacement” shows which parts move up and which down.
  • “Plates per row” at 3 in tile mode makes a seamless tile of figures.
  • Modulate “Sand” with sound (source “sound”, band “beat”) and load a track or turn on the microphone: the sand pulses in time. A song from the Studio arrives via “Visualise in Patterns”, and a video with sound (“Download” → “Video” → “With sound”) keeps it all together.

Parameters

Plate
square · round
Notes
Notes, a chord or frequencies: “C5”, “A4 E5”, “Am”, “440”. Higher notes draw finer figures
Mode m
Set the mode by hand (0 = from the notes). On the disc: the number of diameters
Mode n
On the disc: which zero of the Bessel function (how many rings)
Symmetric mode
The sum instead of the difference of two degenerate modes — another figure for the same note
Draw
sand · nodal lines · displacement
Sand
How thick the sand lines are
Grain size
Line width
Sand colour
one ink · multicoloured · by amplitude
Sand ink
Plate colour
Plates per row
A grid of identical plates; repeats seamlessly in tile mode
Motion
How many times per loop the sand walks through the chord's notes (one note flows between its degenerate modes)