Sand That Draws a Sound
A thin metal plate is ringing. Sand on it is thrown off wherever the plate moves and comes to rest on the lines where it does not. Each pitch draws a different figure.
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A Violin Bow, 1787
In 1787 the German physicist Ernst Chladni published Entdeckungen über die Theorie des Klanges (Discoveries on the Theory of Sound) in Leipzig. He held a glass or metal plate with his fingers at one point and drew a violin bow across its edge at a right angle. The plate gave a pure tone.
He had sprinkled sand on the plate. As it rang, the sand was thrown off the parts that moved and settled on the parts that did not, in sharp lines. The book has eleven copperplates of these figures.
Model: a black-anodised aluminium plate, 360 mm square and 2 mm thick, held at its centre on a shaker
Sand Stops Where the Plate Is Still
A vibrating plate does not move the same everywhere. Along some lines it hardly moves at all: the nodal lines. On either side of one, the plate goes up while the other side goes down.
Where the plate’s acceleration is more than gravity, a grain is thrown up and lands a little way off. It is thrown again and again until it lands on a nodal line. There the plate barely moves, the grain is no longer thrown, and it stays.
The plate’s movement is drawn larger than life and slowed down
Tuning to Resonance
Now a shaker under the plate drives it, at a frequency that can be turned up and down. At most frequencies the plate just moves gently up and down with it, and the sand hardly stirs.
When the frequency meets one of the plate’s own natural frequencies, the plate’s motion grows tens of times over and the sand quickly gathers into a figure. Driven from the centre, this plate has 12 such frequencies below 3,000 Hz: 43, 175, 336, 543, 872 Hz, and on.
Finer and Finer Figures
Each way the plate can vibrate is a mode, with its own frequency and its own nodal lines. The higher the frequency, the more nodal lines and the finer the figure.
A plate is not a string. Double the nodes on a string and the frequency roughly doubles; double the nodal lines on a plate and it roughly quadruples. The frequency climbs with about the square of the number of lines, because a plate bent into tighter waves is stiffer.
A Finger Picks the Figure
Driven from the centre, only figures as symmetric as the square itself appear: the same left and right, top and bottom, and across the diagonals. With a bow, Chladni could call up the others.
The plate must move where the bow touches it and stay still where a finger holds it. Of all the modes that meet both conditions, the lowest one sounds. Hold a different point and a different figure appears.
Two Figures in One
Some of a square plate’s modes come in pairs: the same shape turned through 90 degrees, at exactly the same frequency. Such a pair can be mixed in any proportion, and the mix is still a vibration at that frequency.
Different mixes have different nodal lines. The same pair can give a figure lined up with the middle of the plate, or one lined up with a diagonal, with a run of skewed shapes in between. Many of the classic figures in Chladni’s book are mixes like these. With a bow, where the finger holds the plate sets the mix.
A Prize Question
In 1809 Chladni showed his figures to Napoleon in Paris. That year the Paris Academy of Sciences offered a prize of 3,000 francs for a mathematical theory of thin elastic plates that would explain them.
In 1811 Sophie Germain was the only entrant, and no prize was given; one of the judges, Lagrange, derived the correct fourth-order equation from her hypothesis. In 1813 she received an honourable mention. Her third entry won the prize in January 1816.
Her derivation was not rigorous, and her conditions at the plate’s edge were wrong. In 1850 Gustav Kirchhoff gave the right conditions for a free edge, worked out the vibrations of a round plate and compared them with the pitches Chladni had measured. The modes on this page are computed from Kirchhoff’s plate equations.
The Top of a Violin
Violin makers use the same trick. They lay a carved top, not yet glued on, on soft pads over a loudspeaker, sprinkle it with fine particles, and slowly raise the speaker’s pitch to see where figures form.
They watch modes 2 and 5 most. In Carleen Hutchins’s method, mode 5 is aimed at around 350 to 370 Hz, an octave above mode 2. If a tone is off, they scrape a little wood from one area and try again.
Model: a flat spruce plate of even thickness, with a simplified violin outline, no arching and no f-holes
The Bench
The knob sweeps from 20 Hz to 3,000 Hz, with the frequencies that make figures marked on its scale. The plate can be swapped for the violin top, whose thickness can be set. Or draw the bow across the square plate’s edge and hold it with a finger anywhere. Sound is off until switched on.
Try thisTurn the frequency slowly from 1,000 Hz to 1,500 Hz: how many figures does the sand make, and at what frequencies?
What the model leaves out
The modes are approximated from Kirchhoff’s thin-plate equations by the Rayleigh–Ritz method (products of Legendre polynomials up to degree 25 each way). For a square plate free on all four edges, the first frequencies agree with Leissa’s tables to better than 0.1 %. The clamp at the centre is an 18 mm washer held in an average sense; the rocking modes are sensitive to it, and their frequencies may be off by up to a tenth. The sand follows a simplified model: each hop stands for many of the plate’s cycles, slowed down, and goes further the more the acceleration exceeds gravity. Real sand falls off the edge; here it stays on the plate. The violin top is flat spruce of even thickness, so its tones are lower than a carved, arched top’s and its figures simpler.
A common mistake
Chladni figures are often drawn from products of sines. Those belong to a plate simply supported along its edges, not a free one, and their figures are different. Also, sand gathers on the nodal lines, but very fine powder (such as lycopodium) gathers where the plate moves most. In 1831 Faraday showed that air currents set up by the plate carry it there: with the air pumped out, the powder went to the nodal lines like sand.
Sources
- Chladni: Entdeckungen über die Theorie des Klanges (Leipzig, 1787; scan)
- Neue Deutsche Biographie: Chladni
- MacTutor: Sophie Germain
- The Paris Academy’s elastic surfaces prize, 1809–1816 (arXiv 2303.13615)
- Kirchhoff (1850): Über das Gleichgewicht und die Bewegung einer elastischen Scheibe
- Leissa: Vibration of Plates (NASA SP-160, 1969), table 4.65
- Faraday (1831): On a peculiar class of acoustical figures
- Hutchins (1981): The Acoustics of Violin Plates, Scientific American
- Hutchins & Voskuil (1993): Mode Tuning for the Violin Maker
- UNSW Music Acoustics: Chladni patterns for violin plates
- USDA Forest Products Laboratory: Wood Handbook (elastic constants of wood)
Data
The figures are Ernst Florens Friedrich Chladni’s, from Entdeckungen über die Theorie des Klanges (Leipzig, 1787). The prize story follows MacTutor and the account of the 1809–1816 competition after Bucciarelli and Dworsky (1980); the free-edge theory is Gustav Kirchhoff’s (J. reine angew. Math. 40, 1850). Free square plate frequencies for the check are Iguchi’s, as tabulated in A. W. Leissa, Vibration of Plates (NASA SP-160, 1969), table 4.65. Free-plate tuning targets are from C. M. Hutchins and D. Voskuil, Mode Tuning for the Violin Maker (CAS Journal, 1993). Spruce stiffness ratios are from the USDA Forest Products Laboratory’s Wood Handbook. The plate model, its modes, the sand and the words are the site’s own.
Follow the question
Plate 15How Sound Crosses the AirSound is a push passed along through the air. The air doesn’t travel to you; the push does.
Plate 37When Two Ripples MeetDrop two stones in water and their rings of ripples pass straight through each other. In some places they cancel out, and the water there barely moves.
