Chladni figures

Frequency

Frequencies are relative: a real plate's depend on its size, thickness and material.

What's going on

In 1787 Ernst Chladni sprinkled sand on a thin metal plate and bowed its edge with a violin bow. At certain pitches the sand jumped into sharp, symmetric patterns, and for the first time you could see how a vibrating surface moves.

At those pitches, the plate's resonances, it splits into patches that move up and down in opposite directions. Between them run lines that stay perfectly still. The sand is thrown off the moving patches and settles on the still lines, drawing them for you. Higher pitches give more, smaller patches and more lines.

Why it matters: the same vibration patterns shape the sound of drums, guitar bodies and loudspeakers, and acousticians still measure them to design and tune instruments. Violin makers compare the patterns of the top and back plates, and adjust the wood until they match what they want.

The maths behind itFor the curious · formulas and details

For a square plate, a classic approximation of the vibration pattern (the mode shape) with n and m half-waves across the two directions is:

u(x, y) = cos(nπx)·cos(mπy) − cos(mπx)·cos(nπy)

The still lines are where u = 0, the nodal lines. For a thin plate the frequency grows roughly with n² + m², so the frequencies shown are relative; a real plate's depend on its size, thickness and material.

The simulation moves each grain by a random step proportional to |u| where it sits: grains wander off the moving regions and get stuck where the plate does not move, just like real sand.