Chladni patterns
Sprinkle sand on a vibrating plate and it migrates to the nodal lines — the curves where the plate stands still. The geometry of resonance, made visible.
Sand on brass, 1787
In 1787, the German physicist Ernst Florens Friedrich Chladni drew a violin bow across the edge of a brass plate sprinkled with fine sand. The sand migrated away from vibrating regions and collected along the nodal lines — curves where the plate’s displacement is zero. The resulting geometric patterns, different for each resonant frequency, are now called Chladni figures.
Napoleon was so impressed by a demonstration in 1809 that he offered a prize to anyone who could explain the mathematics. Sophie Germain took up the challenge and, after several attempts, produced the first correct theory of vibrating elastic plates — a foundational contribution to mathematical physics.
Standing waves on plates
A vibrating plate obeys the biharmonic equation: the time-harmonic solutions satisfy ∇&sup4;w = ω²w (in dimensionless form). For a simplified rectangular plate with free edges, the mode shapes approximate products of cosines: cos(nπx/L)·cos(mπy/W), where n and m are integer mode numbers and L, W are the plate dimensions.
The resonant frequencies scale as f ∝ n² + m² for a square plate. The nodal lines are the zero-level sets of the displacement function — the curves where z(x,y) = 0. Sand collects there because the plate is stationary along those lines while oscillating everywhere else.
For a circular plate, the solutions involve Bessel functions of the first kind: Jₙ(k·r)·cos(n·θ), where the radial wavenumber k is determined by the boundary conditions. The nodal patterns become circles and radial lines.
Why the patterns are so complex
On a square plate, modes (n, m) and (m, n) have the same frequency whenever n ≠ m. This is called degeneracy. Because both modes vibrate at the same frequency, the plate can vibrate in any linear combination of the two: z = A·cos(nπx)cos(mπy) + B·cos(mπx)cos(nπy).
The mixing ratio A/B determines which pattern appears. When A = B, you get the classic Chladni figure with its characteristic curved nodal lines. When A = −B, a different figure appears. Between these extremes, the nodal lines morph continuously. The “superposition mix” slider above controls this ratio.
Higher-order degeneracies occur when n² + m² = p² + q² for distinct pairs (n,m) and (p,q). These accidental degeneracies produce even richer patterns, though they require more careful frequency tuning to observe experimentally.
Kac’s question (1966)
In 1966, Mark Kac posed a famous question: “Can one hear the shape of a drum?” If you know all the resonant frequencies of a vibrating membrane, can you deduce its shape? The frequencies are eigenvalues of the Laplacian, and the question asks whether the spectrum determines the geometry.
For simple shapes — rectangles, circles, equilateral triangles — the answer is yes. But in 1992, Carolyn Gordon, David Webb, and Scott Wolpert constructed a pair of isospectral drums: two different polygonal shapes that produce exactly the same set of frequencies. You literally cannot hear the difference.
Chladni patterns make this tangible. Each pattern corresponds to an eigenvalue, and the full set of patterns encodes the plate’s geometry — almost. The Gordon-Webb-Wolpert counterexample shows the encoding is not always unique.
From violin plates to acoustic metamaterials
Luthiers have used Chladni patterns for centuries to tune the top and back plates of violins. By observing where sand collects at specific frequencies, a maker can thin the wood in the right places to achieve the desired tonal balance. Stradivarius may not have known the mathematics, but the method is ancient.
In engineering, modal analysis — the study of vibration patterns in structures — uses the same principles at industrial scale. Aircraft wings, bridge decks, and circuit boards are all tested for their resonant modes to avoid catastrophic vibration.
Acoustic metamaterials exploit Chladni-like physics to create materials with extraordinary properties: sound-focusing lenses, vibration isolators, and noise barriers that work by engineering the local resonant structure. The geometry of nodal lines, once a curiosity of 18th-century salons, now informs the design of next-generation acoustic devices.
Nodal lines all the way down
The mathematics of Chladni patterns — eigenfunctions of differential operators and their nodal sets — appears identically in quantum mechanics. The wavefunctions of electrons in atoms have nodal surfaces where the probability of finding the electron is zero. The familiar s, p, d, f orbital shapes are exactly the quantum-mechanical Chladni patterns of the hydrogen atom.
The quantum billiard problem, where a particle bounces inside a closed region, produces eigenfunctions whose nodal lines form patterns strikingly similar to Chladni figures on plates of the same shape. In chaotic billiards, these nodal lines become tangled and fractal-like — a topic of active research connecting quantum chaos, random matrix theory, and the Riemann hypothesis.
Berry (1977) conjectured that for classically chaotic systems, the high-energy eigenfunctions behave statistically like random superpositions of plane waves. The nodal lines in this regime resemble percolation clusters — another connection between vibrating plates and deep mathematics.