# Plane wave

File:Onde plane 3d.jpg
The wavefronts of a plane wave traveling in space.
File:Plane wave.gif
The real part of a plane wave travelling up.

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 phase) are infinite parallel planes of constant amplitude normal to the phase velocity vector.

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. Equivalently, the "rays" in the limit where ray optics is valid (i.e. for propagation in a homogeneous medium over lengthscales much longer than the wavelength) correspond locally to approximate plane waves.

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

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

where i is the imaginary unit, k is the wave vector, ω is the angular frequency, and a is the (complex) amplitude. (The above form of the plane wave uses the physics time convention; in the engineering time convention, ${\displaystyle -j}$ is used instead of ${\displaystyle +i}$ in the exponent.) The physical solution is usually found by taking the real part of this expression.

This is the solution for a scalar wave equation in a homogeneous medium. For vector wave equations, such as the ones describing electromagnetic radiation or waves in an elastic solid, the solution for a homogeneous medium is similar: ${\displaystyle e^{i({\vec {k}}\cdot {\vec {x}}-\omega t)}}$ multiplied by a constant vector a. (For example, in electromagnetism a is typically the vector for the electric field, magnetic field, or vector potential.) A transverse wave is one in which the amplitude vector is orthogonal to k (e.g. for electromagnetic waves in an isotropic medium), whereas a longitudinal wave is one in which the amplitude vector is parallel to k (e.g. for acoustic waves in a gas or fluid).

In this equation, the function ω(k) is the dispersion relation of the medium, with the ratio ω/|k| giving the magnitude of 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).

The form of the planewave solution is actually a general consequence of translational symmetry. More generally, for periodic structures (i.e. with discrete translational symmetry), the solutions take the form of Bloch waves, most famously in crystalline atomic materials but also in photonic crystals and other periodic wave equations. As another generalization, for structures that are only uniform along one direction x (such as a waveguide along the x direction), the solutions (waveguide modes) are of the form ${\displaystyle e^{i(kx-\omega t)}}$ multiplied by some amplitude function ${\displaystyle a(y,z)}$. (This is a special case of a separable partial differential equation.)

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

## References

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