Chladni Figures
Tune mode numbers m and n, mix degenerate modes with β, and watch sand-grain patterns bloom on a vibrating square plate.
🎵 Acoustic Resonance
When a plate vibrates at a natural frequency, it forms a standing wave. Sand migrates away from antinodes (maximum displacement) to nodal lines (zero displacement), revealing the underlying eigenfunction.
𝛹 Mode Numbers
m and n count interior nodal lines parallel to each axis. Mode (m,n) has m−1 lines along x and n−1 lines along y, giving (m−1)×(n−1) nodal intersection points.
♾ Degeneracy & Mixing
When m≠n, modes (m,n) and (n,m) vibrate at the same frequency. The mixing parameter β controls their superposition, sweeping through an infinite family of nodal patterns at a single frequency.
🧪 History
Ernst Chladni first demonstrated these patterns in 1787 using a violin bow on a metal plate. His work inspired modern acoustics and later the development of vibration analysis and non-destructive testing.
The Eigenvalue Problem of the Plate
The transverse displacement w(x,y,t) of a thin elastic plate satisfies the biharmonic wave equation:
where ρ is the density, h the thickness, and D = Eh³/[12(1−ν²)] the flexural rigidity. For a simply-supported square plate (w = 0 and M = 0 on all edges), the eigenfunctions are products of sine functions:
Each eigenfunction has a corresponding natural frequency ωmn:
so ω ∝ √(m²+n²). The simulation's HUD shows this proportionality factor for the current mode.
Degeneracy and Mode Mixing
Whenever m ≠ n, the two modes (m,n) and (n,m) have exactly the same natural frequency (because m²+n² = n²+m²). Any linear combination
is equally valid at that frequency. The mixing parameter β ∈ [−1, +1] smoothly transitions between nodal patterns:
| β | Nodal pattern (m≠n) |
|---|---|
| 0 | Pure (m,n): (m−1) lines ∥ y-axis, (n−1) lines ∥ x-axis |
| +1 | Symmetric superposition diagonal or curved nodal lines |
| −1 | Anti-symmetric superposition orthogonal diagonal pattern |
| intermediate | Continuous family of mixed patterns |
For m = n (e.g. (3,3)), the two “modes” are identical, degeneracy does not arise, and β has no effect.
Nodal Lines as Zeros of a Harmonic Function
The amplitude function u(x,y) is a finite sum of sine modes. Its zero-set the nodal lines is an algebraic curve of degree m+n in disguise. By Courant’s nodal domain theorem, eigenfunction uk of the Laplacian has at most k nodal domains. The simply-supported plate eigenfunctions saturate this bound exactly: mode (m,n) has exactly m·n nodal domains (rectangular cells).
Physical Analogues
| System | Analogue |
|---|---|
| Vibrating plate — Chladni (1787) | Sand collects at nodal lines |
| Cymatics (Jennie 1967) | Water surface waves; sand/salt/powder patterns |
| Drum head | Bessel function nodal circles and diameters |
| Atomic orbitals | Nodal surfaces of hydrogen wave functions |
| Quantum billiards (chaotic) | |
| Seismic plates | Resonant modes of Earth’s free oscillations |
Display Modes
Wave shows the instantaneous displacement cos(ωt)·u(x,y). Positive displacement is coloured with the “positive” palette colour, negative with the “negative” colour, and the nodal lines are rendered white. Amplitude shows |u(x,y)|, the static envelope, which is what you would see in a long-exposure photograph of the vibrating plate with luminescent powder. Nodal shows only the zero-set of u, on a dark background, matching the appearance of a real Chladni sand pattern.
Frequently Asked Questions
What are Chladni figures?
Chladni figures are the geometric nodal patterns that form on a vibrating plate. Sand sprinkled on the plate migrates away from maximum-vibration areas and collects along the nodal lines, where displacement is zero, revealing the eigenfunction zero-set. First studied by Ernst Chladni in 1787, they are now fundamental to acoustics, structural vibration analysis, and musical instrument design.
Why do some modes produce diagonal patterns?
Diagonal or curved nodal lines arise from mixing two degenerate modes. When m≠n, modes (m,n) and (n,m) share the same natural frequency. The superposition sin(mπx)sin(nπy)+βsin(nπx)sin(mπy) for β≠0 has a zero-set that breaks the axis-aligned symmetry and can form diagonal lines, chevrons, or smooth curves depending on β.
How do I find patterns with many nodal domains?
Set m and n to larger values (e.g. 6, 7). The number of rectangular nodal cells is m×n, so higher modes produce finer, more intricate patterns. Try the (5,6) or (6,7) presets, then vary β to explore the full family of degenerate patterns at each frequency.
How were Chladni figures used historically?
Chladni demonstrated his figures across Europe, famously before Napoleon in 1809, inspiring the Emperor to offer a prize for a mathematical theory of elastic plates. The prize was eventually awarded to Sophie Germain in 1816 after several attempts, marking a milestone in mathematical physics. Today Chladni patterns are used in musical instrument making (violin plates, guitar tops) and non-destructive testing to locate delaminations and cracks.