Chladni figures: sound that draws forms

Scatter sand on a vibrating plate and it gathers into symmetrical patterns. In 1787 Ernst Chladni made sound visible, and the patterns follow whole numbers.

In 1787 the physicist and musician Ernst Florens Friedrich Chladni published a book in Leipzig titled Entdeckungen über die Theorie des Klanges (“Discoveries on the theory of sound”). He had clamped a thin metal plate at its centre, scattered sand on it and stroked its edge with a violin bow. The plate began to ring, and the sand jumped away from the vibrating parts and gathered on the lines that stayed at rest. Depending on the tone, different, often astonishingly regular figures appeared.

Copper plate with thirty small squares, each with a different pattern of lines and curves — Sound figures of a square plate, plate II from Chladni’s works. Each field shows the lines along which the sand gathers for a particular note. The small numbers above the fields say how many nodal lines run in each direction.

Nodal lines

A vibrating plate does not move the same everywhere. It divides into fields that swing up and down in turn, and between them lie nodal lines on which it rests. For a square plate the vibration shapes can be described approximately by two whole numbers n and m, which count how often the wave repeats in each direction. The higher the numbers, the higher the tone and the finer the pattern.

n m

Chladni figures of a square plate in the classical approximation. The sliders choose the two whole numbers n and m. The sand gathers on the nodal lines where the plate is at rest.

The figures were a sensation. Chladni demonstrated them at the courts of Europe, including before Napoleon, who in 1809 offered a prize for their mathematical explanation. It was won in 1816 by the mathematician Sophie Germain, with a theory of elastic plates that remains the foundation to this day.

A mathematician against all odds

The solution of the prize question is a story in itself. As a woman Sophie Germain was not allowed to study and had taught herself mathematics from books and from lecture notes she obtained under another name. Her first attempt in 1811 was the only entry and was rejected, her second received an honourable mention, and only the third in 1816 won the prize. Her derivation and the conditions at the edge were not yet quite right. The correct boundary conditions were found in 1850 by Gustav Kirchhoff, and a usable calculation for the square plate was achieved only by Walther Ritz in 1909. But Germain’s approach, describing the bending of the plate by its curvature, holds to this day.

Stones from the sky

Chladni made a second discovery that has nothing to do with sound and yet fits the attitude of this website. In 1794 he published a small treatise claiming that the masses of iron and stone reported by farmers and travellers had fallen from the sky and came from space. The learned world laughed. Stones falling from the sky were considered superstition. But Chladni had collected and compared the reports carefully, and when in 1803 thousands of stones came down near L’Aigle in Normandy before many witnesses, the physicist Jean-Baptiste Biot investigated the case and proved him right. Chladni has since been regarded as the founder of meteoritics.

The story shows that scepticism itself must be tested (see Dialectic). The scholars were right to doubt many reports of wonders. With meteorites, though, the farmers were right, and only careful examination of the evidence showed it.

From Chladni to cymatics

In the 20th century the Swiss physician Hans Jenny continued the experiments with liquids, pastes and powders and called the field cymatics, the study of waves. His pictures, and the later water sound images of Alexander Lauterwasser (see Links), show how vibration brings forth form, often in patterns reminiscent of flowers, mandalas and stars.

Physically there is nothing mysterious about it, and that is precisely what makes it remarkable. Where a sound meets a bounded body, standing waves arise, and their shapes can be numbered with whole numbers, like the divisions of the string on the monochord (see The monochord). Unlike with the string, though, the tones of a plate do not stand in simple ratios, it sounds more like a bell than like a chord. Chladni’s plate shows in two dimensions what the Pythagoreans heard on the string: number becomes sound, and sound becomes form.

Sources and further reading

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