Spread sand on a thin metal plate, draw a violin bow along its edge, and the sand jumps into sharp, symmetrical lines. Change the pitch and the picture changes. The sand is mapping where the plate refuses to move.
Below is a simulated square plate sprinkled with sand. Drag left and right on it, or use the slider, to change the pitch. Most pitches do almost nothing. At certain special pitches the plate rings and the sand rushes into a pattern.
When a plate rings, it does not move as one piece. It splits into patches. While one patch bulges up, its neighbour dips down, and a moment later they swap. Between two opposite patches there is a line that stays still the whole time. Those still lines are called nodal lines.
Now think about a grain of sand on the plate. On a moving patch it is thrown into the air again and again and lands somewhere random. On a nodal line it is hardly kicked at all. Grains wander off the busy regions by chance, and once one lands on a still line it tends to stay. After a few seconds nearly all the sand has been shaken off the moving parts and piled on the lines.
That is why the pattern is not a drawing of the vibration. It is a drawing of the stillness.
A plate has a set of favourite ways to vibrate, called modes, and each mode has its own pitch. Slide up from the bottom and you will meet simple patterns first: one line across, a cross, a ring. Higher modes split the plate into more and smaller patches, so the sand shows more lines and finer detail. The pitches are not evenly spaced either. Between two resonances there are gaps where the plate barely responds and the sand just sits.
In the simulation each mode is built from two waves with different numbers of ripples along the plate's sides, added with opposite signs so that the corners can move freely, like a plate held loosely. Where the sum is zero, there is a nodal line.
Ernst Chladni, a German physicist (1756–1827), published this technique in 1787 in a book called Entdeckungen über die Theorie des Klanges, or “Discoveries in the Theory of Sound”. He drew a violin bow along the edge of a metal plate until it rang, with sand lightly scattered on top. Then he showed the figures to audiences around Europe. In 1808 he demonstrated them to the Paris Academy, where Napoleon was in the audience, and Napoleon set a prize for the best mathematical explanation.
The maths was hard. Plates are far more awkward than strings,. Sophie Germain’s entry had the right basic approach, though it was rejected at first because of flaws. The patterns that Chladni made visible launched the study of how plates and shells vibrate.
The same physics decides how a violin or guitar top sounds. Instrument makers still sprinkle powder on plates and bow them to see the nodal lines, then thin the wood in particular places to tune the pattern. Engineers use the same idea to find where a vibrating part is quiet and where it shakes itself apart.
Try one last thing. Press “Sweep up” and watch the picture dissolve and re-form as the pitch passes through each mode. Every pattern you see is a mode of the plate, and every one of them is mostly empty space where the sand has been shaken away.