Plate modes, and the sand that finds them
Choose rectangle, disc, triangle, hexagon or ring. Compare the six displayed mode patterns with the driven sand response; sweep the drive frequency or reshake the particles.
This demo uses separable string modes for the rectangle and a Bessel basis for the disc. For the triangle, hexagon and ring it chooses a masked-grid numerical solve; this does not rule out analytic solutions for those geometries. The disc’s frequencies do not form a harmonic series. Near a resonance one mode dominates; between resonances the driven response combines modes. Pattern detail changes the six gallery images, which redraw only when needed; it does not increase the eigensolve or particle-simulation budget.
Model scope. The rectangle uses separable discrete-string modes; the disc uses a Bessel basis. The other shapes use a small masked-grid Laplacian eigensolve. These are scalar membrane teaching models, not native vector elasticity or bending-plate FEM. The particle display is a nodal-pattern illustration.