Vibration modes of a rectangular membrane
Standing waves on a 2D membrane satisfy z(x,y,t) = sin(mπx/L)·sin(nπy/L)·cos(ωt). Integer pairs (m,n) are the normal modes; superposition creates complex interference patterns. Chladni mode shows the nodal lines where displacement is always zero.
Resonant frequencies are f(m,n) = (v/2L)√(m²+n²). Degenerate modes occur when different (m,n) pairs share the same frequency, producing beating in superposition.