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:
( v p h 2 ∇ 2 − ∂ 2 ∂ t 2 ) E = 0 ( v p h 2 ∇ 2 − ∂ 2 ∂ t 2 ) B = 0 {\displaystyle {\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 p h = 1 μ ε {\displaystyle 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 ∇2 is the Laplace operator. In a vacuum, vph = c0 = 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 ∇ ⋅ E = 0 ∇ ⋅ B = 0 {\displaystyle {\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.