Acoustic Resonance Modes
Standing wave patterns in 2D — Chladni figures and drum harmonics
Mode (1,1) — fundamental
A 2D rectangular membrane (drum) has eigenmodes described by ψₘₙ(x,y,t) = sin(mπx/L) · sin(nπy/L) · cos(ωₘₙ t), where the resonant frequencies are ωₘₙ = πc√(m²/Lx² + n²/Ly²). Chladni figures (1787) are the nodal lines where the membrane is stationary — sand collects there. Ernst Chladni discovered these patterns by bowing metal plates with a violin bow. These modes are the foundation of acoustic engineering, instrument design, and quantum billiards.