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The science
Standing-wave modes of a plate
A square plate on the domain -1 ≤ x, y ≤ 1 vibrates in standing-wave shapes. For the symmetric-pair combination used here u(x,y) = cos(mπx) cos(nπy) - cos(nπx) cos(mπy), with integers m ≠ n. The set u = 0 is the nodal pattern. Because swapping x and y flips the sign, the diagonal x = y is always nodal.
Why sand finds the nodes
Where the plate moves, grains are bounced about; where it is still, they stay. Averaged over many bounces a grain drifts toward lower vibration amplitude. The model uses the overdamped drift dr/dt = -K u ∇u (descent of u²/2) plus random jostling that is strongest where |u| is large, so grains pile up on u = 0. For an ideal simply supported square plate the resonant frequencies scale as f ∝ m² + n², which the third graph shows relative to the (1,2) mode.
Honest simplifications: a real free plate has different boundary conditions (its exact figures are not these cosines), real grains also feel air streaming and plate edges, and the drift rule is a phenomenological stand-in for full bouncing-grain dynamics.
What to try
- Classic (2,3): watch the first graph fall as sand reaches the nodes.
- Raise the shake slider: the lines blur because jostling balances the drift.
- Tap the plate to drop new sand on an antinode and see it slide away.
- Move to (6,8): finer lattices need a faster sliding rate.