Chladni PlateBeta

About Chladni Plate

This tool computes the amplitude on a unit square plate as A(x, y) = cos(nπx)·cos(mπy) ± cos(mπx)·cos(nπy). On every frame each grain of sand hops a random distance in a random direction, with the hop size proportional to |A| at its current spot. Grains in strongly shaking regions keep getting thrown out; grains that land on a nodal line, where A = 0, stay put. Given enough time the sand traces the nodal lines.

The |A| figure in the stats line is the average of |A| over every grain position. Right after a fresh, even sprinkle it is about 0.58 for any mode, and it drops toward zero as the figure forms. Treat it as a progress meter for how complete the pattern is.

Chladni's plates, Napoleon's prize and Sophie Germain

In his 1787 book Entdeckungen über die Theorie des Klanges (Discoveries in the Theory of Sound), Ernst Chladni bowed the edges of metal plates with a violin bow and recorded dozens of patterns traced by sand sprinkled on top. It is usually counted as the first systematic way of seeing sound. After Chladni demonstrated the plates in Paris in 1808, Napoleon asked for a prize to be offered for a mathematical explanation, and in 1809 the Academy of Sciences set the challenge: give the mathematical theory of the vibration of an elastic surface and compare it with experiment.

Lagrange remarked that solving it would require a new branch of analysis, and almost nobody entered. Sophie Germain did. Her first entry, in 1811, was judged not to have established the true equations of motion. The second, in 1813, received only an honorable mention because of errors in the mathematics. Her third paper won the prize on 8 January 1816, making her the first woman to win a prize from the Paris Academy. The boundary conditions at the plate edge were still wrong, though. Gustav Kirchhoff sorted them out in 1850 by showing that a free edge can carry only two conditions, and the first calculation of the actual figures on a square plate with free edges, approximate as it was, came from Walther Ritz in 1909. His paper was also one of the first demonstrations of what is now called the Ritz method.

How the formula here differs from a real plate

A product like cos(nπx)cos(mπy) has zero slope at the edges. A real Chladni plate, clamped at the center and free around the rim, instead needs the bending moment and shear force to vanish at the edge, and the solutions to that problem cannot be written as plain cosines. So the figures here share the overall character of real Chladni figures, including their symmetries and the way nodal lines multiply with mode number, but the exact positions and curvature of the lines differ.

The reason for adding or subtracting two terms is the symmetry of a square. Modes (n, m) and (m, n) have the same frequency (they are degenerate), so a real plate vibrates in some mixture of the two. With the minus sign, swapping x and y flips the sign of A, so A must be zero along the diagonal x = y: the diagonal is always a nodal line. With n = m and a minus sign the whole plate would be zero, so the tool switches to plus automatically.

The frequency shown is 60 Hz × (n² + m²) / 2, a relative scale that puts mode (1, 1) at 60 Hz. For a thin plate, bending stiffness makes frequency grow with the square of the wavenumber (ω ∝ k²), so doubling both n and m multiplies the frequency by four. A membrane such as a drumhead would only double. Real plate frequencies in hertz depend on thickness, material and size, so the numbers on screen do not transfer to any particular plate.

Sand goes to the nodes, fine powder goes to the antinodes

Gathering on nodal lines is what heavy, coarse sand does. In 1831 Michael Faraday reported that very light powders such as lycopodium do the opposite and collect in little heaps at the antinodes, the most violently moving parts of the plate. He traced the cause to the air: small whirlwinds form above the vibrating regions and trap the light powder, while heavier sand is thrown straight through them. This tool has no air, so every grain behaves like heavy sand.

The grain motion itself is a summary rule rather than physics. Real sand only leaves the plate when the upward acceleration exceeds g, and it slides down the local slope. Here, the hop distance is proportional to |A|, and the maximum hop is divided by the larger of n and m because nodal lines get closer together at higher modes. The end state, sand on the nodal lines, is the same, but how fast the pattern forms or how crisp it looks tells you nothing about a real plate's damping or driving strength.

Worked examples

How to use it

1. Tap a frequency preset or choose n, m and the sign directly.
2. Adjust the vibration strength and the number of grains.
3. Turn on the amplitude map to check that the sand lines sit on the boundary between red and blue regions.
4. Re-sprinkle the sand to watch the figure form again from scratch.

Frequently asked questions