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{{Use American English|date = March 2019}}
{{Short description|Partial differential equation used in physics}}
The '''electromagnetic wave equation''' is a second-order [[partial differential equation]] that describes the propagation of [[electromagnetic wave]]s through a [[Medium (optics)|medium]] or in a [[vacuum]].  It is a [[Wave equation#Scalar wave equation in three space dimensions|three-dimensional form of the wave equation]]. The [[Homogeneous differential equation|homogeneous]] form of the equation, written in terms of either the [[electric field]] {{math|'''E'''}} or the [[magnetic field]] {{math|'''B'''}}, takes the form:

:<math>\begin{align}
\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} &= \mathbf{0}
\end{align}</math>

where

:<math> v_{ph} = \frac{1}{\sqrt {\mu\varepsilon}} </math>

is the [[speed of light]] (i.e. [[phase velocity]]) in a medium with [[Permeability (electromagnetism)|permeability]] {{mvar|μ}}, and [[permittivity]] {{mvar|ε}}, and {{math|∇<sup>2</sup>}} is the [[Vector Laplacian|Laplace operator]].  In a vacuum, {{math|v<sub>ph</sub> {{=}} ''c<sub>0</sub>'' {{=}} 299,792,458}} meters per second, a fundamental [[physical constant]].<ref>Current practice is to use {{math|''c''<sub>0</sub>}} to denote the speed of light in vacuum according to [[ISO 31]]. In the original Recommendation of 1983, the symbol {{mvar|c}} was used for this purpose. See [http://physics.nist.gov/Pubs/SP330/sp330.pdf NIST ''Special Publication 330'', Appendix 2, p. 45 ]</ref>  The electromagnetic wave equation derives from [[Maxwell's equations]]. In most older literature, {{math|'''B'''}} is called the ''magnetic flux density'' or ''magnetic induction''.

==The origin of the electromagnetic wave equation==
[[File:Postcard-from-Maxwell-to-Tait.jpg|thumb|right|175px|A postcard from Maxwell to [[Peter Guthrie Tait|Peter Tait]].]]

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'',<ref>[//upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf Maxwell 1864], page 497.</ref> 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:

<blockquote>''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.''<ref>See [//upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf Maxwell 1864], page 499.</ref></blockquote>

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:

:<math>\begin{align}
 \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}} {\partial t}\\
\end{align}</math>

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:

:<math>\begin{align}
\nabla \times \left(\nabla \times \mathbf{E} \right) &= \nabla \times \left( -\frac{\partial \mathbf{B}}{\partial t} \right) = -\frac{\partial}{\partial t} \left( \nabla \times \mathbf{B} \right) = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \\
\nabla \times \left(\nabla \times \mathbf{B} \right) &= \nabla \times \left(\mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right) =\mu_0 \varepsilon_0 \frac{\partial}{\partial t}  \left( \nabla \times \mathbf{E} \right) = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2}
\end{align}</math>

We can use the [[Vector calculus identities#Curl of curl|vector identity]]

:<math>\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}</math>

where {{math|'''V'''}} is any vector function of space. And

:<math>\nabla^2 \mathbf{V} = \nabla \cdot \left( \nabla \mathbf{V} \right)</math>

where {{math|∇'''V'''}} is a [[Dyadics|dyadic]] which when operated on by the divergence operator {{math|∇ ⋅}} yields a vector. Since

:<math>\begin{align}
\nabla \cdot \mathbf{E}  &= 0\\
\nabla \cdot \mathbf{B}  &= 0
\end{align}</math>

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

:<math>\begin{align}
\frac{1}{c_0^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} - \nabla^2 \mathbf{E} &= 0\\
\frac{1}{c_0^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} - \nabla^2 \mathbf{B} &= 0
\end{align}</math>

where

:<math>c_0 = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = 2.99792458 \times 10^8\;\textrm{m/s}</math>

is the speed of light in free space.

==Covariant form of the homogeneous wave equation==
[[File:Time dilation02.gif|right|frame|Time dilation in transversal motion. The requirement that the speed of light is constant in every [[inertial frame|inertial reference frame]] leads to the [[Special relativity|theory of Special Relativity]].]]
These [[Formulation of Maxwell's equations in special relativity|relativistic equations]] can be written in [[Covariance and contravariance of vectors|contravariant]] form as

:<math>\Box A^{\mu} = 0</math>

where the [[electromagnetic four-potential]] is

:<math>A^{\mu}= \left ( \frac{\phi}{c}, \mathbf{A} \right )</math>

with the [[Lorenz gauge condition]]:

:<math>\partial_{\mu} A^{\mu} = 0,</math>

and where

:<math>\Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}</math>

is the [[d'Alembert operator]].

==Homogeneous wave equation in curved spacetime==
{{main|Maxwell's equations in curved spacetime}}

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.

:<math> -{A^{\alpha ; \beta}}_{; \beta} + {R^{\alpha}}_{\beta} A^{\beta} = 0 </math>

where <math> \scriptstyle {R^\alpha}_\beta </math> is the [[Ricci curvature tensor]] and the semicolon indicates covariant differentiation.

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

:<math> {A^\mu}_{; \mu} = 0. </math>

==Inhomogeneous electromagnetic wave equation==
{{main| Inhomogeneous electromagnetic wave equation }}

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 ==
<!-- [[Thumb right|This 3D diagram shows a plane linearly polarized wave propagating from left to right with the same wave equations where {{math|'''E''' {{=}} ''E''<sub>0</sub> sin(-''ωt'' + '''k''' ⋅ '''r''')}} and {{math|'''B''' {{=}} ''B''<sub>0</sub> sin(-''ωt'' + '''k''' ⋅ '''r''')}}]] -->

{{main|Wave equation }}
The general solution to the electromagnetic wave equation is a [[Superposition principle|linear superposition]] of waves of the form

:<math> \mathbf{E}( \mathbf{r}, t ) = g(\phi( \mathbf{r}, t )) = g( \omega t - \mathbf{k} \cdot \mathbf{r} ) </math>
:<math> \mathbf{B}( \mathbf{r}, t ) = g(\phi( \mathbf{r}, t )) = g( \omega t - \mathbf{k} \cdot \mathbf{r} ) </math>

for virtually ''any'' well-behaved function {{mvar|g}} of dimensionless argument {{mvar|φ}}, where {{mvar|ω}} is the [[angular frequency]] (in radians per second), and {{math|'''k''' {{=}} (''k<sub>x</sub>'', ''k<sub>y</sub>'', ''k<sub>z</sub>'')}} is the [[wave vector]] (in radians per meter).

Although the function {{mvar|g}} can be and often is a monochromatic [[sine wave]], it does not have to be sinusoidal, or even periodic. In practice, {{mvar|g}} cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of [[Fourier transform|Fourier decomposition]], a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.

In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the [[dispersion relation]]:

:<math> k = | \mathbf{k} | = { \omega \over c } =  { 2 \pi \over \lambda } </math>

where {{mvar|k}} is the [[wavenumber]] and {{mvar|λ}} is the [[wavelength]]. The variable {{mvar|c}} can only be used in this equation when the electromagnetic wave is in a vacuum.

===Monochromatic, sinusoidal steady-state===
The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:

:<math>\mathbf{E} ( \mathbf{r}, t ) = \Re \left \{ \mathbf{E}(\mathbf{r}) e^{i \omega t} \right \}</math>

where
:{{mvar|i}} is the [[imaginary unit]],
:{{math|''ω'' {{=}} 2''π''&thinsp;''f''&thinsp;}} is the [[angular frequency]] in [[radians per second]],
:{{math|&thinsp;''f''&thinsp;}} is the''' [[frequency]] in [[hertz]], and
:<math> \scriptstyle e^{i \omega t} = \cos(\omega t) + i \sin(\omega t)</math> is [[Euler's formula]].

===Plane wave solutions===
{{main|Sinusoidal plane-wave solutions of the electromagnetic wave equation}}
Consider a plane defined by a unit normal vector

:<math> \mathbf{n} = { \mathbf{k} \over k }. </math>

Then planar traveling wave solutions of the wave equations are

:<math> \mathbf{E}(\mathbf{r}) = \mathbf{E}_0 e^{ -i \mathbf{k} \cdot \mathbf{r} } </math>
:<math> \mathbf{B}(\mathbf{r}) = \mathbf{B}_0 e^{ -i \mathbf{k} \cdot \mathbf{r} } </math>

where {{math|'''r''' {{=}} (''x'', ''y'', ''z'')}} is the position vector (in meters).

These solutions represent planar waves traveling in the direction of the normal vector {{math|'''n'''}}.  If we define the z direction as the direction of {{math|'''n'''}}. and the x direction as the direction of {{math|'''E'''}}, then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation

:<math>c^2{\partial B \over \partial z} = {\partial E \over \partial t}.</math>

Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.

This solution is the linearly [[polarization (waves)|polarized]] solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.

===Spectral decomposition===
Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of [[sine|sinusoids]]. This is the basis for the [[Fourier transform]] method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form

:<math> \mathbf{E} ( \mathbf{r}, t ) = \mathbf{E}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0 ) </math>
:<math> \mathbf{B} ( \mathbf{r}, t ) = \mathbf{B}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0 ) </math>

where
:{{mvar|t}} is time (in seconds),
:{{mvar|ω}} is the [[angular frequency]] (in radians per second),
:{{math|'''k''' {{=}} (''k<sub>x</sub>'', ''k<sub>y</sub>'', ''k<sub>z</sub>'')}} is the [[wave vector]] (in radians per meter), and
:<math> \scriptstyle \phi_0 </math> is the [[phase angle]] (in radians).
The wave vector is related to the angular frequency by

:<math> k = | \mathbf{k} | = { \omega \over c } =  { 2 \pi \over \lambda } </math>

where {{mvar|k}} is the [[wavenumber]] and {{mvar|λ}} is the [[wavelength]].

The [[electromagnetic spectrum]] is a plot of the field magnitudes (or energies) as a function of wavelength.

===Multipole expansion===
Assuming monochromatic fields varying in time as <math>e^{-i \omega t}</math>, if one uses Maxwell's Equations to eliminate {{math|'''B'''}}, the electromagnetic wave equation reduces to the [[Helmholtz Equation]] for {{math|'''E'''}}:

:<math> (\nabla^2 + k^2)\mathbf{E} = 0,\, \mathbf{B} = -\frac{i}{k} \nabla \times \mathbf{E},</math>

with ''k = &omega;/c'' as given above. Alternatively, one can eliminate {{math|'''E'''}} in favor of {{math|'''B'''}} to obtain:

:<math> (\nabla^2 + k^2)\mathbf{B} = 0,\, \mathbf{E} = -\frac{i}{k} \nabla \times \mathbf{B}.</math>

A generic electromagnetic field with frequency {{mvar|ω}} can be written as a sum of solutions to these two equations. The [[Helmholtz equation#Three-dimensional solutions|three-dimensional solutions of the Helmholtz Equation]] can be expressed as expansions in [[spherical harmonics]] with coefficients proportional to the [[spherical Bessel functions]]. However, applying this expansion to each vector component of {{math|'''E'''}} or {{math|'''B'''}} will give solutions that are not generically divergence-free ({{math|'''∇''' · '''E''' {{=}} '''∇''' · '''B''' {{=}} 0}}), and therefore require additional restrictions on the coefficients.

The multipole expansion circumvents this difficulty by expanding not {{math|'''E'''}} or {{math|'''B'''}}, but {{math|'''r · E'''}} or {{math|'''r · B'''}} into spherical harmonics. These expansions still solve the original Helmholtz equations for {{math|'''E'''}} and {{math|'''B'''}} because for a divergence-free field {{math|'''F'''}}, {{math|∇<sup>2</sup> ('''r · F''') {{=}} '''r ·''' (∇<sup>2</sup> '''F''')}}. The resulting expressions for a generic electromagnetic field are:

:<math>\mathbf{E} = e^{-i \omega t} \sum_{l,m} \sqrt{l(l+1)} \left[ a_E(l,m) \mathbf{E}_{l,m}^{(E)} + a_M(l,m) \mathbf{E}_{l,m}^{(M)} \right]</math>
:<math>\mathbf{B} = e^{-i \omega t} \sum_{l,m} \sqrt{l(l+1)} \left[ a_E(l,m) \mathbf{B}_{l,m}^{(E)} + a_M(l,m) \mathbf{B}_{l,m}^{(M)} \right]</math>,

where <math>\mathbf{E}_{l,m}^{(E)}</math> and <math>\mathbf{B}_{l,m}^{(E)}</math> are the ''electric multipole fields of order (l, m)'', and <math>\mathbf{E}_{l,m}^{(M)}</math> and <math>\mathbf{B}_{l,m}^{(M)}</math> are the corresponding ''magnetic multipole fields'', and {{math|''a<sub>E</sub>''(''l'', ''m'')}} and {{math|''a<sub>M</sub>''(''l'', ''m'')}} are the coefficients of the expansion. The multipole fields are given by

:<math>\mathbf{B}_{l,m}^{(E)} = \sqrt{l(l+1)} \left[B_l^{(1)} h_l^{(1)}(kr) + B_l^{(2)} h_l^{(2)}(kr)\right] \mathbf{\Phi}_{l,m}</math>
:<math>\mathbf{E}_{l,m}^{(E)} = \frac{i}{k} \nabla \times \mathbf{B}_{l,m}^{(E)}</math>
:<math>\mathbf{E}_{l,m}^{(M)} = \sqrt{l(l+1)} \left[E_l^{(1)} h_l^{(1)}(kr) + E_l^{(2)} h_l^{(2)}(kr)\right] \mathbf{\Phi}_{l,m}</math>
:<math>\mathbf{B}_{l,m}^{(M)} = -\frac{i}{k} \nabla \times \mathbf{E}_{l,m}^{(M)}</math>,

where ''h''<sub>l</sub><sup>(1,2)</sup>(''x'') are the [[Spherical Bessel function#Spherical Hankel functions|spherical Hankel functions]], ''E''<sub>l</sub><sup>(1,2)</sup> and ''B''<sub>l</sub><sup>(1,2)</sup> are determined by boundary conditions, and

:<math>\mathbf{\Phi}_{l,m} = \frac{1}{\sqrt{l(l+1)}}(\mathbf{r} \times \nabla) Y_{l,m}</math>

are [[vector spherical harmonics]] normalized so that

:<math>\int \mathbf{\Phi}^*_{l,m} \cdot \mathbf{\Phi}_{l', m'} d\Omega = \delta_{l,l'} \delta_{m, m'}.</math>

The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae [[radiation pattern]]s, or nuclear [[gamma decay]]. In these applications, one is often interested in the power radiated in the [[Near and far field#Radiation zone, including radiating far-field|far-field]]. In this regions, the {{math|'''E'''}} and {{math|'''B'''}} fields asymptote to

:<math>\mathbf{B} \approx \frac{e^{i (kr-\omega t)}}{kr} \sum_{l,m} (-i)^{l+1} \left[a_E(l,m) \mathbf{\Phi}_{l,m} + a_M(l,m) \mathbf{\hat{r}} \times \mathbf{\Phi}_{l,m} \right]</math>
:<math>\mathbf{E} \approx \mathbf{B} \times \mathbf{\hat{r}}.</math>

The angular distribution of the time-averaged radiated power is then given by

:<math>\frac{dP}{d\Omega} \approx \frac{1}{2k^2} \left| \sum_{l,m} (-i)^{l+1} \left[ a_E(l,m) \mathbf{\Phi}_{l,m} \times \mathbf{\hat{r}} + a_M(l,m) \mathbf{\Phi}_{l,m} \right] \right|^2.</math>

==See also==

===Theory and experiment===
{{Col-begin}}
{{Col-break}}
* [[Maxwell's equations]]
* [[Wave equation]]
* [[Partial Differential Equations]]
* [[Electromagnetic modeling]]
* [[Electromagnetic radiation]]
* [[Charge conservation]]
* [[Light]]
* [[Electromagnetic spectrum]]
* [[Optics]]
{{Col-break}}
* [[Special relativity]]
* [[General relativity]]
* [[Inhomogeneous electromagnetic wave equation]]
* [[Photon polarization]]
* [[Larmor formula|Larmor power formula]]
* [[Theoretical and experimental justification for the Schrödinger equation]]
{{col-end}}

===Applications===
{{Col-begin}}
{{Col-break}}
* [[Rainbow]]
* [[Cosmic microwave background radiation]]
* [[Laser]]
* [[Laser fusion]]
* [[Photography]]
* [[X-ray]]
* [[X-ray crystallography]]
* [[Radar]]
{{Col-break}}
* [[Radio waves]]
* [[Optical computing]]
* [[Microwave]]
* [[Holography]]
* [[Microscope]]
* [[Telescope]]
* [[Gravitational lens]]
* [[Black-body radiation]]
{{col-end}}

===Biographies===
{{Col-begin}}
{{Col-break}}
* [[André-Marie Ampère]]
* [[Albert Einstein]]
* [[Michael Faraday]]
* [[Heinrich Hertz]]
* [[Oliver Heaviside]]
* [[James Clerk Maxwell]]
{{col-end}}

==Notes==
<references/>

== Further reading ==

===Electromagnetism===

====Journal articles====
* Maxwell, James Clerk, "''[//upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf A Dynamical Theory of the Electromagnetic Field]''", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)

====Undergraduate-level textbooks====
*{{cite book|author=Griffiths, David J.|title=Introduction to Electrodynamics (3rd ed.)|publisher=Prentice Hall|year=1998|isbn=0-13-805326-X|url=https://archive.org/details/introductiontoel00grif_0}}
*{{cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.) | publisher=W. H. Freeman | year=2004 | isbn=0-7167-0810-8}}
* Edward M. Purcell, ''Electricity and Magnetism'' (McGraw-Hill, New York, 1985). {{ISBN|0-07-004908-4}}.
* Hermann A. Haus and James R. Melcher, ''Electromagnetic Fields and Energy'' (Prentice-Hall, 1989) {{ISBN|0-13-249020-X}}.
* Banesh Hoffmann, ''Relativity and Its Roots'' (Freeman, New York, 1983). {{ISBN|0-7167-1478-7}}.
* David H. Staelin, Ann W. Morgenthaler, and Jin Au Kong, ''Electromagnetic Waves'' (Prentice-Hall, 1994) {{ISBN|0-13-225871-4}}.
* Charles F. Stevens, ''The Six Core Theories of Modern Physics'', (MIT Press, 1995) {{ISBN|0-262-69188-4}}.
* Markus Zahn, ''Electromagnetic Field Theory: a problem solving approach'', (John Wiley & Sons, 1979) {{ISBN|0-471-02198-9}}

====Graduate-level textbooks====
*{{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|isbn=0-471-30932-X}}
* [[Lev Davidovich Landau|Landau, L. D.]],  ''The Classical Theory of Fields'' ([[Course of Theoretical Physics]]: Volume 2),  (Butterworth-Heinemann: Oxford, 1987). {{ISBN|0-08-018176-7}}.
*{{cite book | author=Maxwell, James C. | title=A Treatise on Electricity and Magnetism | publisher=Dover | year=1954 | isbn=0-486-60637-6}}
* Charles W. Misner, [[Kip Thorne|Kip S. Thorne]], [[John Archibald Wheeler]], ''Gravitation'', (1970) W.H. Freeman, New York; {{ISBN|0-7167-0344-0}}. ''(Provides a treatment of Maxwell's equations in terms of differential forms.)''

===Vector calculus===
*P. C. Matthews ''Vector Calculus'', Springer 1998, {{ISBN|3-540-76180-2}}
*H. M. Schey, ''Div Grad Curl and all that:  An informal text on vector calculus'', 4th edition (W. W. Norton & Company, 2005) {{ISBN|0-393-92516-1}}.

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