A plane wave satisfies the one-dimensional wave equation in Cartesian coordinates.
To obtain plane waves, the position vector must remain perpendicular to a give plane. Pick a point on the plane
The wave then satisfies the generalization of the one-dimension equation,
where is the angular frequency, k is the wave vector, r is the radius vector, and is the phase of the wave.
References
Bekefi, G. and Barrett, A. H. Electromagnetic Vibrations, Waves, and Radiation. Cambridge, MA: MIT Press, pp. 150-154, 1987.
