Normal modes of beams, plates, and membranes — Chladni figures
Every elastic structure has characteristic normal modes — spatial patterns that oscillate at a fixed frequency. A forced structure responds most strongly at resonance.
Chladni figures: sand collects on nodal lines (zero displacement). Used since 1787 to visualize plate modes. Modern analogy: acoustic levitation.
For a circular membrane, modes are indexed by (m,n) with Bessel functions J_m(β_mn r/R) cos(mθ).