Wave Superposition

How it works

The superposition principle states that when two or more waves overlap in the same medium, the resulting displacement at any point is simply the algebraic sum of the individual displacements. This is one of the most fundamental ideas in wave physics and holds true for any linear system — from water waves and sound to light and quantum mechanical wave functions.

Joseph Fourier showed in 1822 that any periodic function can be decomposed into a sum of sinusoidal waves of different frequencies, amplitudes, and phases. This insight — Fourier analysis — is one of the most powerful tools in mathematics and physics. The "Square Wave" preset above demonstrates this: by adding odd harmonics (1, 3, 5, 7 ...) with amplitudes proportional to 1/n, the sum converges toward a square wave. The "Sawtooth" preset uses all harmonics.

When two waves of nearly equal frequency are added together, their sum exhibits a phenomenon called beats: the amplitude slowly oscillates at a frequency equal to the difference between the two component frequencies. Musicians use beats to tune instruments — when two notes are perfectly in tune, the beats disappear. The "Beat Frequency" preset demonstrates this with two waves whose frequencies differ by a small amount.

Constructive interference occurs when waves are in phase (crests aligned with crests), producing a larger combined amplitude. Destructive interference occurs when waves are out of phase (crests aligned with troughs), reducing or cancelling the amplitude. Noise-cancelling headphones exploit destructive interference by generating an anti-phase copy of ambient sound.

Standing waves form when two waves of equal frequency travel in opposite directions. The superposition produces a pattern of fixed nodes (zero displacement) and antinodes (maximum displacement). Standing waves explain resonance in musical instruments, vibrating strings, organ pipes, and even the electron orbitals of atoms in quantum mechanics.