Plane wave
An EM wave whose phase is constant on planes perpendicular to the propagation direction k. Written E(r,t) = E₀ cos(k·r − ωt + φ), with ω = c|k| in vacuum. The simplest solution of the wave equation.
Definition
A plane wave is a solution of the electromagnetic wave equation in which the phase is constant on planes perpendicular to the propagation direction. The canonical form is E(r,t) = E₀ cos(k·r − ωt + φ), where k is the wavevector (|k| = 2π/λ, pointing along the propagation direction), ω is the angular frequency (ω = c|k| in vacuum), and E₀ is a constant vector perpendicular to k specifying amplitude and polarisation.
Plane waves are idealisations — a true plane wave has infinite transverse extent and infinite duration — but they are the natural basis for building all other solutions. Any real wave (a laser pulse, a radio broadcast, sunlight through a window) can be decomposed into a Fourier superposition of plane waves via ∫ dk Ẽ(k) e^{i(k·r − ωt)}. This decomposition underlies the entire theory of wave propagation in linear media: each Fourier component propagates independently with its own dispersion relation ω(k), and the output field is the reassembled superposition. Plane-wave analysis is why we can reason about refraction, diffraction, and interference using the trigonometry of a single k-vector: each component handles itself.