A thin elastic plate vibrates according to the Kirchhoff–Love equation. For a square plate of side a with simply supported edges the mode shapes are products of sines and the frequency follows exactly:
D ∇⁴w + ρh ∂²w/∂t² = 0, D = E h³ / (12(1 − ν²)) wmn = sin(mπx/a) sin(nπy/a), fmn = (π/2) √(D/ρh) (m² + n²) / a²Modes (m,n) and (n,m) share one frequency, so any mixture φ = wmn + s wnm is also a resonance: the slider s moves between the symmetric and antisymmetric family. Sand bounces off the vibrating plate with a kick roughly proportional to the local amplitude, so grains random-walk with a position-dependent step and pile up where |φ| is zero: the nodal lines.
Checks: the finite-difference biharmonic of φ against π⁴(m²+n²)²φ, the mode energy integral, the thickness scaling of f, and a statistical test that settled grains sit near nodes.
Simplifications: real Chladni plates are clamped at the centre with free edges, whose modes are not exactly sines (this page uses simply supported edges); the sand model is a diffusion caricature without air drag or grain collisions.