In physics the term plane wave denotes a wave whose wavefronts (surfaces of equal phase) are infinite, parallel planes. As an idealized model it simplifies many problems by removing dependence on transverse coordinates and permitting solutions that travel without change of shape. A plane wave can represent fields in optics, sound pressure in acoustics, or probability amplitudes in quantum mechanics.
Mathematical description
The simplest monochromatic plane wave is expressed as a sinusoidal function of the form f(r,t) = A cos(k·r - ωt + φ), where k is the wavevector, ω the angular frequency, and A the amplitude. The constant-phase surfaces k·r = constant are planes perpendicular to k. Plane waves may be scalar or vector-valued; electromagnetic plane waves also satisfy Maxwell's equations with electric and magnetic field vectors orthogonal to k.
Properties and common forms
- Monochromatic vs. broadband: monochromatic plane waves have a single frequency; more general fields can be built by superposing plane waves.
- Homogeneous plane waves propagate without spatial decay; evanescent plane-wave components decay exponentially and are not true propagating plane waves.
- Plane waves are characterized by wavevector direction, polarization (for vector waves), amplitude, and phase.
A plane wave is often introduced in the context of a wave packet decomposition: any sufficiently regular field can be represented as an integral (Fourier transform) of plane waves. This makes plane waves a convenient basis for solving linear partial differential equations and scattering problems.
Applications include modeling light in the far field, simplifying antenna analysis, serving as trial solutions in quantum scattering theory, and forming the basis of plane-wave expansions in solid-state physics. Practical systems never produce true infinite plane waves; real beams have finite extent and energy.
Limitations and distinctions: plane waves are idealizations with infinite total energy and extent, unlike spherical waves that originate from a point source. In boundary-value problems one often uses superpositions of plane waves to match physical boundary conditions. For further general background see parallel planes and related wave concepts.