How to use it
Chladni figures are the patterns that sand draws on a vibrating plate. The plate above starts with sand scattered at random and is shaken at one frequency. Where the plate moves, the grains are thrown about; where it stays still, they come to rest, so within a few seconds the sand marks the quiet lines of the vibration. Move the frequency slider, or tap one of the marks above it, and the sand rearranges into a new figure. Press Sound to hear the driving tone.
Between the marks the plate hardly responds, so the figure is weak or mixed. The real experiment behaves the same way: clear figures appear only at resonances.
Controls
- Frequency: drag the slider, type a value or use the ±1 and ±10 buttons. Prev, Next and a tap on the scale of marks jump to a resonance. Auto sweep raises the frequency slowly and waits on every resonance.
- Pick a mode: square or round plate, the two mode numbers, the symmetric (+) or antisymmetric (−) pattern of the square plate, and a gallery of twelve modes.
- Sand and drive: drive amplitude, number of grains and simulation speed. Scatter sand throws the grains back to random places.
- Plate: drag with a mouse or a finger to pour sand there; the grains come from the rest of the plate.
- View and share: nodal lines, amplitude map, Save PNG and Copy link to the current mode.
- Keyboard: Space plays or pauses, S scatters, M switches the sound. With the plate focused, the left and right arrows change the frequency by 1 Hz (10 Hz with Shift); up and down jump between resonances.
How it works
A plate shaken at one frequency carries a standing wave. Some regions swing up and down; the points that move most are the antinodes. The nodal lines between them do not move at all, and that is where the sand stays. For a square plate with free edges this page uses the classic textbook approximation, on a square of side 1:
The numbers m and n count the half-waves along the two sides, and the sign gives the symmetric or the antisymmetric pattern. In bending waves the frequency grows with the square of the number of waves, so mode (m, n) resonates at 20 × (m² + n²) Hz here. That is also why high tones give fine figures: a higher frequency means a shorter wavelength, so more nodal lines fit on the plate.
At a driving frequency f, each mode answers with the weight 1 / (1 + ((f − fmode) / 4 Hz)²), a bell-shaped resonance curve, and the plate takes the weighted sum of the shapes. Every frame each grain makes a random hop. Its length is proportional to the local amplitude |u| and to the drive amplitude, plus a very small jitter that keeps the sand lines from shrinking to the width of one grain. Grains that reach the edge bounce back. The round plate uses the shapes Jn(kr) cos(nθ), where Jn is a Bessel function and k makes the slope zero at the rim.
This is a simplified model. The true shapes of a free plate are only close to these formulas, and real resonant frequencies depend on the size, thickness and material of the plate. The frequencies shown come from the model, not from a particular plate. A real plate also answers only in modes that move at the point where it is driven; here you can excite all of them.
The figures are named after the German physicist Ernst Chladni, who published them in 1787 in his book Entdeckungen über die Theorie des Klanges. He strewed sand on a plate and drew a violin bow along its edge. The mathematics proved hard: Sophie Germain won the prize of the Paris Academy of Sciences in 1816 for her work on the theory of elastic plates. Makers of violins and guitars still use the method to test the wooden plates of an instrument.
Things to try
- Pick mode (1, 3) and switch between symmetric and antisymmetric: the same two waves give two different figures. With (1, 2) the two are mirror images.
- Set 1000 Hz with the slider. Modes (1, 7) and (5, 5) share this frequency in the model, so they combine. Then pick (5, 5) directly.
- Step 3 Hz away from a resonance: the plate response drops to 64 % and the figure builds more slowly.
- Start Auto sweep at a low frequency and listen as the figures get finer.
- On the round plate, set the diameters to 0 for pure rings, or the circles to 0 for a star of straight lines.
Frequently asked questions
Why does the sand collect on lines?
The grains are kicked around wherever the plate moves. On a nodal line the plate is still, so grains that land there stay.
Are these the frequencies of a real plate?
No. The scale of 20 Hz per unit was chosen so that the tones are comfortable to hear. A real plate has its own resonances, set by its size, thickness and material.
Why does nothing happen between the marks?
Away from a resonance the plate barely vibrates. Use Prev or Next to land exactly on one.
Why can I not hear the lowest tones?
Small phone and laptop speakers reproduce tones around 100 Hz and below very weakly. Use headphones or larger speakers.