Partial differential equation used in physics

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:

- ${\begin{aligned}\left(v_{ph}^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {E} &=\mathbf {0} \\\left(v_{ph}^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {B$
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_{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, v_{ph}=*c*= 299,792,458 meters per second, a fundamental physical constant._{0}^{[1]}The electromagnetic wave equation derives from Maxwell's equations. In most older literature,**B**is called the*magnetic flux density*or*magnetic induction*.## Contents

- 1 The origin of the electromagnetic wave equation
- 2 Covariant form of the homogeneous wave equation
- 3 Homogeneous wave equation in curved spacetime
- 4 Inhomogeneous electromagnetic wave equation
- 5 Solutions to the homogeneous electromagnetic wave equation
- 6 See also
- 7 Notes
- 8 Further reading
- 8.1 Electromagnetism
- 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, v_{ph}=*c*= 299,792,458 meters per second, a fundamental physical constant._{0}^{[1]}The electromagnetic wave equation derives from Maxwell's equations. In most older literature,**B**is called the*magnetic flux density*or*magnetic induction*.## Contents

- 1 The origin of the electromagnetic wave equation
- 2 Covariant form of the homogeneous wave equation
- 3 Homogeneous wave equation in curved spacetime
- 4 Inhomogeneous electromagnetic wave equation
In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In

*Part VI*of his 1864 paper titled*Electromagnetic Theory of Light*,^{[2]}Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

^{[3]}Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern '

**Heaviside' form of Maxwell's equations**. In a vacuum- and charge-free space, these equations are:- ${\begin{aligned}\nabla \cdot \mathbf {E} &=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \cdot \mathbf {B} &=0\\\nabla \times \mathbf {B} &=\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partia$
The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

^{[3]}

- ${\begin{aligned}\nabla \cdot \mathbf {E} &=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \cdot \mathbf {B} &=0\\\nabla \times \mathbf {B} &=\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partia$

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.

To obtain t

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern '

**Heaviside' form of Maxwell's equations**. In a vacuum- and charge-free space, these equations are:These are the general Maxwell's equations specialized to the case with charge and current both set to zero. Taking the curl of the curl equations gives:

- $\begin{array}{rl}\mathrm{\nabla}\times \left(\mathrm{\nabla}\times \mathbf{E}\right)& =\mathrm{\nabla}\times \left(-\frac{\mathrm{\partial}\mathbf{B}}{\mathrm{\partial}t}\right)=-\frac{\mathrm{\partial}}{\mathrm{\partial}t}\left(\mathrm{\nabla}\times \mathbf{B}\right)=-{\mu}_{0}{\epsilon}_{0}\frac{{\mathrm{\partial}}^{2}\mathbf{E}}{\mathrm{\partial}{t}^{2}}\\ <\end{array}$
We can use the vector identity

- $\nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V}$

where

**V**is any vector function of space. And- $\nabla ^{2}\mathbf {V} =\nabla \cdot \left(\nabla \mathbf {V} \right)$

where ∇

**V**is a dyadic which when operated on by the divergence operator ∇ ⋅ yields a vector. Since- ${\begin{aligned}\nabla \cdot \mathbf {E} &=0\\\nabla \cdot \mathbf {B} &=0\end{aligned}}$

then the first term on the right in the identity vanishes and we obtain the wave equations:

- ${\begin{aligned}{\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} &=0\\{\frac {1}{c_{0}^{2}}}{\fr$
where

**V**is any vector function of space. And- $\nabla ^{2}\mathbf {V} =\nabla \cdot \left(\nabla \mathbf {V} \right)$

where ∇

**V**is a dyadic which when operated on by the divergence operator ∇ ⋅ yields a vector. Since- $}$
where ∇

**V**is a dyadic which when operated on by the divergence operator ∇ ⋅ yields a vector. Since- $\begin{array}{rl}\mathrm{\nabla}\cdot \mathbf{E}& =0\\ \mathrm{\nabla}\cdot \mathbf{B}& =0\end{array}$
- $\begin{array}{r}\frac{1}{{c}_{0}^{2}}\frac{{\mathrm{\partial}}^{2}\mathbf{E}}{\mathrm{\partial}{t}^{2}}\end{array}$
where

- $c_{0}={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}=2.99792458\times 10^{8}\;{\textrm {m/s}}$

is the speed of light in free space.

## Covariant form of the homogeneous wave equation

These relativistic equations can be written in contravariant form as

- $\Box A^{\mu }=0$

where the electromagnetic four-potential is

- ${\di$
is the speed of light in free space.

## Covariant form of the homogeneous wave equation

These relativistic equations can be written in contravariant form as

- $\u25fb{A}^{\mu}=0$
- $\Box A^{\mu }=0$

where the electromagnetic four-potential is

where the electromagnetic four-potential is - ${A}^{\mu}=\left(\frac{\varphi}{c},\mathbf{A}\right)}<$
with the Lorenz gauge condition:

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

and where

- $$$\u25fb={\mathrm{\nabla}}^{2}-\frac{1}{{c}^{2}}\frac{{\mathrm{\partial}}^{2}}{}$
is the d'Alembert operator.

## Homogeneous wave equation in curved spacetime

Main article: Maxwell's equations in curved spacetimeThe electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

- $-{{A}^{\alpha ;\beta}}_{;\beta}+{{R}^{\alpha}}_{}$
The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

- $-{{A}^{\alpha ;\beta}}_{}$
where $\scriptstyle {R^{\alpha }}_{\beta }$ is the Ricci curvature tensor and the semicolon indicates covariant differentiation.

The generalization of the Lorenz gauge condition in curved spacetime is assumed:

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

## Inhomogeneous electromagnetic wave equation

Main article: InhomogeneousThe generalization of the Lorenz gauge condition in curved spacetime is assumed:

Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.

## Solutions to the homogeneous electromagnetic wave equation

Main article: Wave equationThe general solution to the electromagnetic wave equation is a linear superposition of waves of the form

- $k=|\mathbf {k} |={\omega \over c}={2\pi \over \lambda }$

- $-{{A}^{\alpha ;\beta}}_{}$

- $-{{A}^{\alpha ;\beta}}_{;\beta}+{{R}^{\alpha}}_{}$

- ${A}^{\mu}=\left(\frac{\varphi}{c},\mathbf{A}\right)}<$

- $\u25fb{A}^{\mu}=0$

- $\begin{array}{r}\frac{1}{{c}_{0}^{2}}\frac{{\mathrm{\partial}}^{2}\mathbf{E}}{\mathrm{\partial}{t}^{2}}\end{array}$

- $\begin{array}{rl}\mathrm{\nabla}\cdot \mathbf{E}& =0\\ \mathrm{\nabla}\cdot \mathbf{B}& =0\end{array}$

- speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇

- 8.1 Electromagnetism