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In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations make the partial differential equations inhomogeneous, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

Maxwell's equations

For reference, Maxwell's equations are summarized below in SI units and Gaussian units. They govern the electric field E and magnetic field B due to a source charge density ρ and current density J:

Name	SI units	Gaussian units
Gauss's law	${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}$	${\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }$
Gauss's law for magnetism	${\displaystyle \nabla \cdot \mathbf {B} =0}$	${\displaystyle \nabla \cdot \mathbf {B} =0}$
Maxwell–Faraday equation (Faraday's law of induction)	${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$	${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$
Ampère's circuital law (with Maxwell's addition)	${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)}$	${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)}$

where ε₀ is the vacuum permittivity and μ₀ is the vacuum permeability. Throughout, the relation

${\displaystyle \varepsilon _{0}\mu _{0}={\dfrac {1}{c^{2}}}}$

is also used.

SI units

Covariant form of the inhomogeneous wave equation

Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of relativity

The relativistic Maxwell's equations can be written in covariant form as

${\displaystyle \Box A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\beta }\partial ^{\beta }A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {A^{\mu ,\beta }}_{\beta }=-\mu _{0}J^{\mu }\quad {\text{SI}}}$

${\displaystyle \Box A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\beta }\partial ^{\beta }A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {A^{\mu ,\beta }}_{\beta }=-{\frac {4\pi }{c}}J^{\mu }\quad {\text{cgs}}}$

where

${\displaystyle \Box =\partial _{\beta }\partial ^{\beta }=\nabla ^{2}-{1 \over c^{2}}{\frac {\partial ^{2}}{\partial t^{2}}}}$

is the d'Alembert operator,

${\displaystyle J^{\mu }=\left(c\rho ,\mathbf {J} \right)}$

is the four-current,

${\displaystyle {\partial \over {\partial x^{a}}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{a}\ {\stackrel {\mathrm {def} }{=}}\ {}_{,a}\ {\stackrel {\mathrm {def} }{=}}\ (\partial /\partial ct,\nabla )}$

is the 4-gradient, and

${\displaystyle A^{\mu }=(\varphi /c,\mathbf {A} )\quad {\text{SI}}}$

${\displaystyle A^{\mu }=(\varphi ,\mathbf {A} )\quad {\text{cgs}}}$

is the electromagnetic four-potential with the Lorenz gauge condition

${\displaystyle \partial _{\mu }A^{\mu }=0\,.}$

Curved spacetime

The electromagnetic wave equation is modified in two ways in curved spacetime, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears (SI units).

${\displaystyle -{A^{\alpha$ ;\beta }}_{\beta }+{R^{\alpha }}_{\beta }A^{\beta }=\mu _{0}J^{\alpha }}

where

${\displaystyle {R^{\alpha }}_{\beta }}$

is the Ricci curvature tensor. Here the semicolon indicates covariant differentiation. To obtain the equation in cgs units, replace the permeability with 4π/c.

The Lorenz gauge condition in curved spacetime is assumed:

${\displaystyle {A^{\mu }}_{;\mu }=0\,.}$

Solutions to the inhomogeneous electromagnetic wave equation

Retarded spherical wave. The source of the wave occurs at time t'. The wavefront moves away from the source as time increases for t > t'. For advanced solutions, the wavefront moves backwards in time from the source t < t'.

In the case that there are no boundaries surrounding the sources, the solutions (cgs units) of the nonhomogeneous wave equations are

${\displaystyle \varphi (\mathbf {r} ,t)=\int {{\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}\rho (\mathbf {r} ',t')d^{3}r'dt'}$

and

${\displaystyle \mathbf {A} (\mathbf {r} ,t)=\int {{\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}{\mathbf {J} (\mathbf {r} ',t') \over c}d^{3}r'dt'}$

where

${\displaystyle {\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)}}$

is a Dirac delta function.

These solutions are known as the retarded Lorenz gauge potentials. They represent a superposition of spherical light waves traveling outward from the sources of the waves, from the present into the future.

There are also advanced solutions (cgs units)

${\displaystyle \varphi (\mathbf {r} ,t)=\int {{\delta \left(t'-{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}\rho (\mathbf {r} ',t')d^{3}r'dt'}$

and

${\displaystyle \mathbf {A} (\mathbf {r} ,t)=\int {{\delta \left(t'-{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}{\mathbf {J} (\mathbf {r} ',t') \over c}d^{3}r'dt'\,.}$

These represent a superposition of spherical waves travelling from the future into the present.

See also

• Wave equation
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Larmor formula
• Covariant formulation of classical electromagnetism
• Maxwell's equations in curved spacetime
• Abraham–Lorentz force
• Green's function

References