KlangfigurenSound figures · E. F. F. Chladni, 1787
Bow a metal plate and it rings. Scatter sand on it first and the sand runs into lines, a different drawing for every note. This page solves the plate equation in your browser and pours up to 131,072 grains of sand on the answer.
Every pattern you see is a computed vibration mode of the plate.
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Part one · the figures
Sand finds the stillness
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01 · 1680–1787
A plate that sings
On 8 July 1680 Robert Hooke ran a violin bow along the edge of a glass plate dusted with flour, and the flour gathered into lines. A century later Ernst Chladni, a law graduate from Wittenberg who had turned to acoustics, made the trick into a science. His Entdeckungen über die Theorie des Klanges (1787) drew 166 of these figures from brass and glass plates held at one point and bowed at the edge.
Here the plate is 24 cm of steel, 0.8 mm thick, bolted on a post at its centre and shaken by it at …. Watch where the sand goes.
peak Γ …
02 · the shape of a note
Where the plate stands still
Here the camera is lifted and time slowed about a thousand times. The plate bends in one fixed shape: the copper regions rise while the teal ones fall, then the other way round, … times a second. Between them run lines that never move, the nodal lines. Sand thrown off the moving parts lands there and stays.
The shape solves the equation of a thin elastic plate, which took sixty years to get right:
D ∇4w + ρh ∂2w/∂t2 = 0, D = Eh3/12(1−ν2)
w deflection · h thickness · E stiffness · ρ density · ν Poisson’s ratio
Napoleon watched Chladni’s figures in 1808 and had the Paris Academy offer a prize for their theory. Sophie Germain, self-taught, was its only serious entrant; Lagrange, one of the judges, used her first memoir of 1811 to write down the fourth-order equation, and she won on her third attempt in 1816. The right conditions for a free edge (no bending moment and no net shear at the rim) came from Gustav Kirchhoff in 1850.
03 · 1909
Ritz’s zoo
Even with the right equation, nobody could solve the free square plate exactly, and nobody has since. In 1909 Walther Ritz attacked this very problem with a new idea: write the shape as a sum of simple functions and pick the mixture that makes the plate’s energy stationary. The finite element method descends from it.
This page does the same with 576 Legendre polynomials when it loads. Here are the first 24 modes; tap one to drive it. The dim ones have a nodal line through the post, so shaking the post cannot excite them, and the drive moves to where it can.
solved in your browser in …
04 · resonance
Only the right notes
Between resonances the plate barely moves and the sand lies still. Sweep the frequency slowly and the plate wakes only near its natural frequencies, each a sharp peak on the curve below it. Each new peak redraws the sand.
The sharpness is set by damping. With a quality factor Q of 150, a peak is about 1/Q of its frequency wide. Drag along the curve to play the plate yourself. The gold ticks under it mark modes the post can drive; the grey ones it cannot reach.
width of this peak …
05 · threshold
One g
A grain resting on the plate leaves it only when the plate drops faster than gravity can pull the grain after it, which happens when Γ = Aω2/g exceeds 1 (A is the local amplitude, ω the angular frequency). At 1 kHz that takes a quarter of a micrometre.
The colours show Γ. Grains in the warm regions are thrown up again and again and land a few millimetres away in random directions. Once they land inside the white line, where Γ < 1, they stay. Turn the drive up and the quiet band narrows: louder notes draw thinner lines.
peak Γ …largest amplitude …
06 · 1831
Faraday’s dust
Chladni noticed that fine dust from his bow went the other way, to the parts that moved most. In 1831 Michael Faraday showed why. The vibrating plate drives slow, steady currents in the air, running from the still lines toward the moving regions, and light powder rides them into little heaps. Under an air pump the heaps did not form: with the air gone, his lycopodium went to the quiet parts like sand.
Yellow is lycopodium, the spores of club moss; the pale grains are sand.
07 · circles
Round plates and Chladni’s law
A round plate shaken at its centre can only take shapes that look the same all the way round, so the sand draws rings. Move the drive toward the rim and diameters appear, cutting the rings into petals.
Chladni counted the lines, m diameters and n circles, and found that the pitch rises roughly as (m + 2n)2. The round plate has an exact solution in Bessel functions; the dots are its frequencies, against Chladni’s count.
08 · instruments
The luthier’s plate
Spruce is about twelve times stiffer along the grain than across it, so bending waves run faster lengthwise and the figures stretch along the grain, here top to bottom. This is a free, unbraced guitar top, 2.8 mm thick, with its soundhole cut.
Instrument makers use the figures to see inside their wood. From the 1960s Carleen Hutchins and the Catgut Acoustical Society tuned free violin plates by scattering glitter on them over a loudspeaker and shaving wood until the patterns, and their pitches, fell where they wanted.
mode …
09 · the bench
Your plate
Everything is live: change the plate and the modes are solved again. Touch the plate to move the drive, pour sand or wipe it away.
The plate. A thin plate obeys Kirchhoff’s equation, with bending stiffness D growing as the cube of the thickness. All edges are free: no bending moment and no net shear. Every frequency of a plate of side L therefore scales as h/L2 times √(E/ρ), and doubling the thickness doubles every note.
Square and rectangle. The deflection is written as a double sum of normalised Legendre polynomials, 24 in each direction for the square (for other plates the 576 are shared out by how many half-waves fit each way). Minimising the energy turns the equation into a 576×576 matrix eigenproblem. Because the Legendre polynomials are orthonormal the mass matrix is the identity, and the four symmetry classes (even or odd across each centre line) split it into four problems of 144, solved in your browser with Householder reduction and the QL algorithm. Spruce enters through four separate stiffnesses (Dx, Dy, D12, D66) instead of one.
Circle. The round plate is solved exactly: w = [Jn(kr) + C In(kr)] cos nθ. The two free-edge conditions give a 2×2 determinant whose roots are the frequencies; the Bessel functions are summed from their power series.
Guitar top. The same polynomial basis (18×26) over the outline with the soundhole removed, integrated on a masked grid and solved offline in Python; only the coefficients ship with the page. These frequencies are converged to a few per cent; the square’s to better than one part in a million.
Driving. A point force F cos Ωt at the post excites each mode in proportion to its deflection there, through a resonance with damping 1/2Q. The plate’s complex amplitude is the sum over all modes below 4 kHz. The post itself is taken to be massless; it holds the plate up but keeps the plate’s rigid-body motion out of the picture.
Sand. Up to 131,072 grains live in float textures on the GPU. A grain where Γ > 1 is thrown up with a take-off speed growing with Γ − 1 (hops of a few millimetres, under real gravity), in a random direction with a slight lean down the amplitude slope. Below Γ = 1 it no longer leaves the plate but can creep slowly downhill in amplitude. Grains that land off the plate fall. Lycopodium follows the air currents, with a drift proportional to ∇Γ2; in vacuum it hops like sand. Grain collisions, the weight of the sand, air loading and large-amplitude effects are left out.
What you hear. A sine at the drive frequency, its loudness following how hard the plate is moving, so it swells at each resonance.
Plate and free edge
D∇4w + ρh ẅ = F δ(x − x0) cos Ωt
Mn = 0, Qn + ∂Mnt/∂s = 0
Ritz
w = ΣΣ cij P̂i(x/a) P̂j(y/b), K c = ω2ρh c
Forced response
w = Σk Fφk(x0)φk(x) / m(ωk2 − Ω2 + iωkΩ/Q)
Threshold
Γ = Ω2|w|/g > 1
Checks
Computed when the page loads.
Quantity
This page
Reference
Part three · people
Three centuries of figures
Sources
E. F. F. Chladni, Entdeckungen über die Theorie des Klanges, Leipzig (1787).
S. Germain, Recherches sur la théorie des surfaces élastiques, Paris (1821).
M. Faraday, “On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces”, Phil. Trans. R. Soc. 121, 299 (1831).
G. Kirchhoff, “Über das Gleichgewicht und die Bewegung einer elastischen Scheibe”, J. reine angew. Math. 40, 51 (1850).
W. Ritz, “Theorie der Transversalschwingungen einer quadratischen Platte mit freien Rändern”, Ann. Phys. 333, 737 (1909).
A. W. Leissa, Vibration of Plates, NASA SP-160 (1969).
M. D. Waller, Chladni Figures: A Study in Symmetry, Bell (1961).
C. M. Hutchins, “The acoustics of violin plates”, Scientific American 245(4) (1981).
T. D. Rossing and N. H. Fletcher, Principles of Vibration and Sound, 2nd ed., Springer (2004).
M. J. Gander and G. Wanner, “From Euler, Ritz, and Galerkin to modern computing”, SIAM Review 54, 627 (2012).
Q. Zhou, V. Sariola, K. Latifi and V. Liimatainen, “Controlling the motion of multiple objects on a Chladni plate”, Nat. Commun. 7, 12764 (2016).