Chladni Patterns: How Sand Draws the Shape of Sound

Sprinkle sand on a metal plate, draw a violin bow down its edge, and the sand does something that looks like magic. It jumps off some parts of the plate and piles up in thin, sharp lines, forming crosses, stars and nested rings. Change the note and the picture changes. These are Chladni patterns — maps of places that stay still. The same nodal sand piles show up in our Kundt's Tube poster, where grains gather at the nodes of a standing wave in a tube. This post builds on our explainer What Is Quantum Measurement?, because the same mathematics of allowed shapes, called modes, returns in quantum physics.

What is a Chladni pattern?

A Chladni pattern is a map of a vibration mode. When a plate rings at one of its natural frequencies, most of it moves up and down, but some lines stay still. Those still lines are called nodal lines. Sand bounces off the moving parts and comes to rest on the lines that do not move, so the sand draws the nodes for you.

The first recorded version of the experiment is older than Chladni. According to APS News, on 8 July 1680 Robert Hooke sprinkled sand on a metal plate, bowed its edge and noted that the grains gathered along the nodes (APS News, 2017^[1]). The German physicist and musician Ernst Chladni turned this into a systematic study of circular, square and rectangular plates. His 1787 book Entdeckungen über die Theorie des Klanges (Discoveries on the Theory of Sound) contained 11 plates showing 166 figures, and his 1802 Die Akustik earned him the nickname “father of acoustics” (APS News, 2017^[1]). The Linda Hall Library describes his method: brass plates dusted with sand and set ringing with a rosined bow, which revealed the nodal lines (Linda Hall Library^[2]).

Why does the sand collect on the still lines?

Where the plate moves, it throws the grains into the air again and again, and each landing nudges them sideways. Only on a nodal line does the surface stay still, so that is where the grains stop. APS News adds a twist that Chladni himself noticed: very fine dust, such as rosin shed by the bow, does the opposite and drifts to the most strongly moving regions, the antinodes. The cause is acoustic streaming, a flow of air set up by the vibrating surface, first observed by Michael Faraday in 1831 (APS News, 2017^[1]).

What is a vibration mode?

The simplest place to see modes is a string fixed at both ends. It can only vibrate in patterns that fit a whole number of half-wavelengths between the ends. Each pattern is a normal mode, with nodes that never move and antinodes that move the most (OpenStax University Physics, §16.6^[3]). For a string of length L, the allowed frequencies are

fₙ = n v / (2L)

where fₙ is the frequency of mode n in hertz, n = 1, 2, 3 … is the mode number, v is the wave speed on the string in metres per second and L is the length in metres. That neat whole-number ladder is why a string sounds musical. It is the physics of our Oscillayne tee, the character who personifies resonance.

How does a drumhead vibrate?

A drumhead is a two-dimensional version of the string: a thin membrane under tension, fixed at its rim. Its small vertical displacement u(x, y, t), in metres, obeys the two-dimensional wave equation

∂²u/∂t² = v² ∇²u

Here t is time in seconds, ∇²u = ∂²u/∂x² + ∂²u/∂y² measures how curved the membrane is, and v = √(T/σ) is the wave speed, set by the tension per unit length T (in newtons per metre) and the mass per unit area σ (in kilograms per square metre). The rule is simple: the more sharply the skin is curved at a point, the harder it is pulled back.

For a circular drum of radius a, the solutions are Bessel functions in the radius times a cosine in the angle. The allowed frequencies are

f_mn = j_mn v / (2πa)

where m = 0, 1, 2 … counts nodal diameters, n = 1, 2, 3 … counts nodal circles (including the fixed rim), and j_mn is the n-th zero of the Bessel function J_m. The lowest is j₀₁ ≈ 2.405. Penn State acoustician Daniel Russell’s animations label the modes the same way and list their frequency ratios: the (1,1) mode rings at 1.593 times the fundamental and the (2,1) mode at 2.135 times (Russell, Penn State^[4]).

Those ratios are not whole numbers, and that is the key difference from a string. Our own Bessel-zero calculation gives the next few as 2.295 for (0,2) and 2.653 for (3,1). Because the overtones do not form a harmonic ladder, a plain drum sounds more like a thud than a note. The (3,2) mode looks like u = J₃(kr) cos 3θ, with three nodal diameters and one nodal circle inside the rim — the kind of vibration art gathered in our Waves, Chaos & Matter collection.

Are Chladni plates the same as drums?

Not quite. A metal plate is stiff, while a drumhead is floppy and held in shape only by tension. A stiff plate follows a more complicated equation, so its frequencies follow a different pattern, and its edges are often free rather than clamped. But the core idea is identical: at each natural frequency the surface splits into regions moving in opposite directions, separated by nodal lines that stay still. Square drumheads, which our diagrams use, also show how two different modes with the same frequency can add together into new crossed patterns, which is why Chladni figures on square plates can look so intricate.

Can you hear the shape of a drum?

If you know a drum’s shape, you can work out every frequency it can play. In 1966 the mathematician Mark Kac asked the reverse question in a famous paper titled “Can One Hear the Shape of a Drum?” (Kac, 1966^[5]). In other words: does the full list of frequencies fix the shape?

The answer, at least for flat drums in the plane, is no. In 1992 Carolyn Gordon, David Webb and Scott Wolpert built two differently shaped polygons with exactly the same set of frequencies. Their paper, “One cannot hear the shape of a drum”, describes these as isospectral but non-congruent plane domains, answering Kac’s question negatively (Gordon, Webb & Wolpert, 1992^[6]; arXiv version^[7]). That is the punchline in plain English: some drums sound identical, even when their shapes do not match.

Physicists then tested it. In 1994 S. Sridhar and A. Kudrolli built thin microwave cavities in the two shapes, which obey the same wave equation as a drumhead, and reported in “Experiments on not ‘hearing the shape’ of drums” that at least 54 of the low-lying frequencies they measured in the two shapes agreed to a few parts in 10⁴ (Sridhar & Kudrolli, 1994^[8]).

Why does this matter beyond sand and drums?

Chladni’s method became a practical tool. APS News notes that violin and other instrument makers used it to see where the vibration modes fall on the back plates of instruments, and that it is still used today (APS News, 2017^[1]). The deeper legacy is mathematical. A system with boundaries has a discrete set of allowed shapes, each with its own frequency. The same structure gives the energy levels of an electron in an atom, which is why modes keep returning in our posts on quantum physics, such as Why Can’t Two Electrons Share a State?

If you want more vibration art, the Waves, Chaos & Matter collection gathers our wave and resonance designs.

Quick FAQ

Who discovered Chladni patterns?

Robert Hooke recorded sand gathering on nodes in 1680, and Ernst Chladni made the first systematic study, published in 1787.

Why does the pattern change with the note?

Each natural frequency has its own mode shape with its own nodal lines. A higher frequency usually means more nodal lines.

Is a Chladni plate the same as a drum?

No. A plate is stiff and a drumhead is held by tension, so they follow different equations, but both form nodal lines.

Can two drums sound exactly the same?

Yes. In 1992 Gordon, Webb and Wolpert found two different flat shapes with identical sets of frequencies.

References

1. APS News (July 2017), “This Month in Physics History: Ernst Chladni and his figures”. https://www.aps.org/apsnews/2017/07/first-experiments-chladni-figures
2. Linda Hall Library, Scientist of the Day: “Ernst Chladni”. https://www.lindahall.org/about/news/scientist-of-the-day/ernst-chladni/
3. W. Moebs, S. J. Ling & J. Sanny, University Physics Volume 1 (OpenStax, Rice University), §16.6 “Standing Waves and Resonance”. https://openstax.org/books/university-physics-volume-1/pages/16-6-standing-waves-and-resonance
4. D. A. Russell, “Vibrational Modes of a Circular Membrane,” Acoustics and Vibration Animations, Penn State University. https://www.acs.psu.edu/drussell/demos/membranecircle/circle.html
5. M. Kac (1966), “Can One Hear the Shape of a Drum?,” The American Mathematical Monthly 73, 1–23 (DOI 10.1080/00029890.1966.11970915; metadata verified via Crossref). https://doi.org/10.1080/00029890.1966.11970915
6. C. Gordon, D. L. Webb & S. Wolpert (1992), “One cannot hear the shape of a drum,” Bull. Amer. Math. Soc. 27, 134–138. https://www.ams.org/journals/bull/1992-27-01/S0273-0979-1992-00289-6/
7. C. Gordon, D. Webb & S. Wolpert (1992), “One cannot hear the shape of a drum,” arXiv:math/9207215. https://arxiv.org/abs/math/9207215
8. S. Sridhar & A. Kudrolli (1994), “Experiments on not ‘hearing the shape’ of drums,” Phys. Rev. Lett. 72, 2175–2178 (DOI 10.1103/PhysRevLett.72.2175; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.72.2175

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.