Electromagnetic wave equation

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field $E$ or the magnetic field $B$ , takes the form:

{\begin{aligned}\left(v_{\mathrm {ph} }^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {E} &=\mathbf {0} \\\left(v_{\mathrm {ph} }^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {B} &=\mathbf {0} \end{aligned}}

where

v_{\mathrm {ph} }={\frac {1}{\sqrt {\mu \varepsilon }}}

is the speed of light (i.e. phase velocity) in a medium with permeability $μ$ , and permittivity $ε$ , and $\nabla 2$ is the Laplace operator. In a vacuum, $v ph = c 0 = 299792458 m/s$ , a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In most older literature, $B$ is called the magnetic flux density or magnetic induction. The following equations

{\begin{aligned}\nabla \cdot \mathbf {E} &=0\\\nabla \cdot \mathbf {B} &=0\end{aligned}}

predicate that any electromagnetic wave must be a transverse wave, where the electric field $E$ and the magnetic field $B$ are both perpendicular to the direction of wave propagation.

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