Standing Waves & Normal Modes

Explore the fundamental vibration patterns of strings and membranes. Standing waves form when waves reflect between fixed boundaries, creating nodes and antinodes.

Fundamental Mode (n=1): The simplest standing wave with one antinode at center and nodes at boundaries. Frequency: f₁ = v/(2L) where v is wave speed and L is string length.

Standing Waves Theory

Wave Equation: ∂²u/∂t² = c² ∇²u with boundary conditions u = 0 at edges.

Standing waves are formed by interference of traveling waves reflecting between boundaries:

1D String: u(x,t) = A·sin(nπx/L)·cos(ωt) where n = 1, 2, 3, ...
2D Rectangular: u(x,y,t) = A·sin(mπx/L)·sin(nπy/W)·cos(ωt)
2D Circular: u(r,θ,t) = A·J_m(k_mn·r)·cos(mθ)·cos(ωt) (Bessel functions)

Applications: Musical instruments (guitars, drums, bells), resonance phenomena, quantum mechanics (particle in a box).