A standing wave forms when two traveling waves of equal frequency and amplitude move in opposite directions. On a fixed string of length L, boundary conditions require nodes at both ends, restricting wavelengths to λ_n = 2L/n.
The harmonics (n = 1, 2, 3, ...) are the natural resonant frequencies. The fundamental (n=1) has one antinode; n=2 has two, etc. The superposition of all harmonics is a Fourier series — any periodic function on [0,L] can be represented this way.
Applications: musical string instruments (violin, guitar — bridge and nut create fixed-end BCs), quantum particle in a box (same eigenfunctions ψ_n = √(2/L)sin(nπx/L), with E_n = n²π²ℏ²/2mL²), acoustic resonators, microwave cavities. The spectrum of active harmonics forms the timbre of a sound.