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# Plane wave

In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts[?] (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction[?].

By extension, the term is also used to describe waves that are approximately plane waves in a localized region of space. For example, a localized source such as an antenna produces a field that is approximately a plane wave in its far-field region.

Mathematically, a plane wave is a solution to the wave equation of the following form:

$u(\vec{x},t) = a e^{i(\vec{k}\cdot\vec{x} - \omega t)}$

where i is the imaginary unit, k is the wavevector, ω is the angular frequency, and a is the (complex) amplitude. (In some conventions, this expression is conjugated.) The physical solution is usually found by taking the real part of this expression. For the vector wave equation of electromagnetism, a is the vector for the electric or magnetic field (and is orthogonal to k, for an isotropic medium).

In this equation, the function ω(k) is the dispersion relation[?] of the medium, with the ratio ω/|k| giving the phase velocity and dω/dk giving the group velocity. For electromagnetism in an isotropic medium with index of refraction n, the phase velocity is c/n (which equals the group velocity only if the index is not frequency-dependent). For the same reason, the ratio of the phase velocity to c is called the effective index and is proportional to the characteristic impedance of the medium.

(The term is used in the same way for telecommunication, e.g. in Federal Standard 1037C and MIL-STD-188.)

• J. D. Jackson, Classical Electrodynamics (Wiley: New York, 1998).

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