Chladni Plate Simulator
Bow a virtual metal plate and watch sand collect into the geometric figures Ernst Chladni published in 1787. Sweep the (n, m) vibration modes, change drive intensity, and see why nodal lines appear exactly where the plate stops moving.
What you are looking at
In 1787 the German physicist Ernst Chladni sprinkled sand on a thin metal plate and drew a violin bow across its edge. The sand jumped on the parts of the plate that vibrated hardest and came to rest on the lines that did not move — the nodal lines of a standing wave. The figures that appeared depended on which natural frequency the plate was ringing at. Those drawings are the founding image of what we now call cymatics.
This simulator models a square plate with free edges. Each particle is given a small random kick whose size is proportional to the local amplitude of the plate's motion. Particles starting near a moving region get bounced around until they wander into a quiet stripe — the nodal line — where they stop. Run the simulation long enough and the population of particles is sculpted into the eigenfunction shape.
The Chladni equation
For an ideal square plate of side L with free boundaries, the vibration amplitude at point (x, y) for mode numbers (n, m) is well-approximated by a symmetric combination of two cosines:
The minus sign is what makes the figure interesting. It enforces the symmetry of a freely vibrating square and produces curved nodal lines instead of a plain checkerboard. Wherever Z = 0 the plate is standing still; wherever |Z| is large the plate is heaving up and down. Sand — and the digital particles in this simulator — gets shaken off the loud parts and onto the quiet curves.
Mode numbers (n, m) and what they look like
| (n, m) | Visual signature | Symmetry |
|---|---|---|
| (2, 4) | Quartered cross — the entry-level Chladni figure. | 4-fold mirror |
| (3, 5) | Eight-petalled flower with curved arms. | D4 |
| (4, 6) | Star with internal diamond grid. | D4 |
| (5, 7) | Dense web; lines start to braid. | D4 |
| (6, 8) | Tight lattice of small cells. | D4 |
| n = m | Z ≡ 0 — the equation cancels, no figure. | degenerate |
Higher integers correspond to higher resonant frequencies. On a real plate the modes are not perfectly spaced; small defects, asymmetric clamping, and material anisotropy split degenerate modes apart and produce the slightly wobbly figures you see in lecture demonstrations.
Why physicists still care
- Eigenfunctions made visible. Chladni figures are arguably the most photogenic demonstration of an eigenvalue problem in all of physics. The plate's shape selects which (n, m) modes exist; bowing at a given frequency picks one out.
- Instrument design. Violin and guitar makers tune top and back plates by tapping them and watching Chladni patterns appear — they shave wood until the modes line up properly.
- Modal analysis. Engineers use the same idea (accelerometers replacing sand) to find the natural modes of bridges, turbine blades, and spacecraft panels before resonance destroys them.
- Quantum analogues. The 2D Helmholtz equation that produces these patterns is mathematically identical to the time-independent Schrödinger equation in a 2D box; Chladni figures are a visual proxy for the probability lobes of a confined particle.
Try this
Walk the modes. Hold m fixed and step n from 1 to 12. The figure's complexity climbs in jumps, not smoothly — that's the discrete nature of standing waves.
Bump the vibration. A small drive lets particles slowly find the deepest nodal valleys; a large drive shakes them right over shallow ones and only the strongest nodes hold sand. This is exactly why a louder bow produces a cleaner figure on a real plate.
Toggle the wave field. The coloured background is |Z(x, y)|. Compare it with the sand: every black ridge in the sand sits on a zero of the field.