Presets
Playback
Plate
Drive
Sand
Camera
Layers
Quality
Model check
The science
What a Chladni figure is
Sand on a vibrating plate is thrown off wherever the plate moves strongly and stays where it hardly moves. A plate driven near one of its resonances vibrates in a standing-wave pattern, and the sand collects on the nodal lines where the displacement u(x,y) is zero.
Plate equation and frequencies
A thin elastic plate obeys D ∇4u + ρh ∂2u/∂t2 = 0 with bending stiffness D = E h3 / (12(1 - ν2)). A standing wave u = φ(x,y) cos ωt needs ∇4φ = k4φ with ω = k2 √(D/ρh). For a square plate of side a and shapes built from cos(mπx/a) cos(nπy/a) one gets fmn = (π/2a2) √(D/ρh) (m2 + n2).
Mode shapes and mixing
The two degenerate products combine as φmn± = cos(mπx)cos(nπy) ± cos(nπx)cos(mπy). When you drive at frequency f, each mode responds with the resonance amplitude |H| = 1/√((1-r2)2 + (r/Q)2), r = f/fmn, and the simulator sums the shapes that respond by more than 4 % of the strongest one. A low Q makes neighbouring modes mix.
Sand model
Each grain hops in a random direction when the local drive A|u| exceeds a static friction threshold; the hop length grows with the overdrive. Grains therefore random-walk until they sit where A|u| is below the threshold: on the nodal lines. This reproduces the qualitative figure, not the detailed physics of grain bouncing.
Honest simplifications
The shapes are the classic cosine idealisation (exact for a simply supported plate in the sense of the frequency law, approximate for a free square plate whose true modes need a Ritz solution), so real plates differ in frequency by roughly 10 to 30 %. The splitting slider only mimics the way real plates lift the degeneracy of the ± modes. The plate is driven at its centre in all cases; air damping, sand mass loading and bouncing are ignored. Educational model, not a plate-design tool.
Model checks in this page
(1) finite differences of mode (2,3)- satisfy ∇4φ = k4φ; (2) the mean |u| over uniformly scattered grains equals (2/π)2 for a product mode; (3) the number of sign changes along a line equals m.
What the 3D scene shows
The plate surface is displaced by u(x,y) cos ωt (exaggerated about 100 times and slowed down so that you can see it); red and blue tint mark the two signs of the displacement, and every grain rides on the surface and hops with the same friction rule as in the 2D page. The display on the signal generator and the knob angles follow the sliders. The shaker, generator, amplifier, microphone and lab were modelled and baked in Blender.