Chladni patterns and drum modes

Sprinkle sand on a plate and make the plate ring. The sand does not stay where you put it. It runs into lines, and the lines are a picture of the sound.

Square plate, mode (2,3). Computed live from the formula, not recorded. Yellow and blue are the two directions the sheet moves, the pale lines do not move.

Chladni patterns

In 1787 the German physicist Ernst Chladni published patterns made by drawing a violin bow along the edge of a metal plate covered in sand. At certain pitches the plate vibrates in sections. Some parts of the surface move up and down, and some stay still. The sand is thrown off the moving parts and collects along the places that do not move. Those places are called nodal lines.

Each pitch at which the plate rings strongly has its own pattern. That pattern and its pitch together are called a mode. Higher modes have more nodal lines, and the figures become more intricate.

The same experiment works today with a plate, fine sand or salt, and a speaker or tone generator driving the plate at a steady pitch. You find the patterns by sweeping the pitch slowly and listening for the plate to speak up.

A drum skin has modes too

A drum membrane is a stretched skin held at its rim. When it vibrates, it divides itself into regions that move and lines that do not. For an ideal circular membrane, clamped at the edge and uniform, the modes are described by two whole numbers.

n is the number of nodal diameters, straight lines through the centre.
m is the number of nodal circles, counting the rim, which never moves.

The shape of mode (n, m) is a Bessel function of order n across the radius, multiplied by a cosine of n times the angle. The Bessel function is the part that makes the circles. It is standard textbook mathematics for a circular drum.

(0,1)
1.000
(1,1)
1.593
(2,1)
2.136
(0,2)
2.295
(3,1)
2.653
(1,2)
2.917

The figures above show the first six modes of an ideal membrane, seen from above. The bright lines are the nodal lines, the places that stay still. The pale circle is the rim. The number under each figure is its pitch as a ratio to the lowest mode.

Why the pitches are not whole number ratios

The pitch of each mode is proportional to a zero of a Bessel function, a number where the function crosses zero. Those numbers are not whole multiples of each other, which is why a drum does not give a clear musical note the way a string does.

Mode (n, m)	Pitch ratio
(0, 1)	1.000
(1, 1)	1.593
(2, 1)	2.136
(0, 2)	2.295
(3, 1)	2.653
(1, 2)	2.917

Where the ideal ends

A real drum is not the ideal membrane. The air on both sides of the skin changes the pitches. The skin has some stiffness. The tension is never perfectly even, and the shell and the hoop move too. So the measured modes of a real drum are close to these, and not equal to them.

A metal plate is a different problem again. A plate resists bending, which a perfectly flexible membrane does not, and its edges can be free instead of clamped. A free square plate has no simple exact solution. People use approximations, and the closed form cosine pattern used for square Chladni figures is one of them.

What I do with it

The Cymatic Wavetable Pack reads closed paths, circles and out and back lines, through these standard mode shapes and turns each path into one cycle of a wave. The tables are calculated from the formulas. Nothing was recorded from a real drum or plate.

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