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:
where
is the speed of light in a medium with permeability,
and permittivity, and ∇2 is the Laplace
operator. In a vacuum, c = c0 = 299,792,458
meters per second, which is the speed of light
in free space. The electromagnetic wave equation
derives from Maxwell's equations. It should
also be noted that in most older literature,
B is called the magnetic flux density or magnetic
induction.
The origin of the electromagnetic wave equation
In his 1864 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, 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.
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:
where ρ = 0 because there's no charge density
in free space.
Taking the curl of the curl equations gives:
We can use the vector identity
where V is any vector function of space. And
where ∇V is a dyadic which when operated
on by the divergence operator ∇ ⋅ yields
a vector. Since
then the first term on the right in the identity
vanishes and we obtain the wave equations:
where
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
where the electromagnetic four-potential is
with the Lorenz gauge condition:
and where
is the d'Alembert operator.
Homogeneous wave equation 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.
where is the Ricci curvature tensor and the
semicolon indicates covariant differentiation.
The generalization of the Lorenz gauge condition
in curved spacetime is assumed:
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
The general solution to the electromagnetic
wave equation is a linear superposition of
waves of the form
for virtually any well-behaved function g
of dimensionless argument φ, where ω is
the angular frequency, and k = is the wave
vector.
Although the function g can be and often is
a monochromatic sine wave, it does not have
to be sinusoidal, or even periodic. In practice,
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
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:
where k is the wavenumber and λ is the wavelength.
The variable 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:
where
i is the imaginary unit,
ω = 2π f  is the angular frequency in
radians per second,
 f  is the frequency in hertz, and
is Euler's formula.
Plane wave solutions
Consider a plane defined by a unit normal
vector
Then planar traveling wave solutions of the
wave equations are
where r = is the position vector.
These solutions represent planar waves traveling
in the direction of the normal vector n. If
we define the z direction as the direction
of n. and the x direction as the direction
of E, then by Faraday's Law the magnetic field
lies in the y direction and is related to
the electric field by the relation
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 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 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
where
t is time,
ω is the angular frequency,
k = is the wave vector, and
is the phase angle.
The wave vector is related to the angular
frequency by
where k is the wavenumber and λ is the wavelength.
The electromagnetic spectrum is a plot of
the field magnitudes as a function of wavelength.
Multipole expansion
Assuming monochromatic fields varying in time
as , if one uses Maxwell's Equations to eliminate
B, the electromagnetic wave equation reduces
to the Helmholtz Equation for E:
with k = ω/c as given above. Alternatively,
one can eliminate E in favor of B to obtain:
A generic electromagnetic field with frequency
ω can be written as a sum of solutions to
these two equations. The 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 E or B will give
solutions that are not generically divergence-free,
and therefore require additional restrictions
on the coefficients.
The multipole expansion circumvents this difficulty
by expanding not E or B, but r · E or r · B
into spherical harmonics. These expansions
still solve the original Helmholtz equations
for E and B because for a divergence-free
field F, ∇2 = r ·. The resulting expressions
for a generic electromagnetic field are:
,
where and are the electric multipole fields
of order, and and are the corresponding magnetic
multipole fields, and aE(l, m) and aM(l, m)
are the coefficients of the expansion. The
multipole fields are given by
,
where hl(1,2)(x) are the spherical Hankel
functions, El(1,2) and Bl(1,2) are determined
by boundary conditions, and
are vector spherical harmonics normalized
so that
The multipole expansion of the electromagnetic
field finds application in a number of problems
involving spherical symmetry, for example
antennae radiation patterns, or nuclear gamma
decay. In these applications, one is often
interested in the power radiated in the far-field.
In this regions, the E and B fields asymptote
to
The angular distribution of the time-averaged
radiated power is then given by
Other solutions
Other spherically and cylindrically symmetric
analytic solutions to the electromagnetic
wave equations are also possible.
In spherical coordinates the solutions to
the wave equation can be written as follows:
and
These can be rewritten in terms of the spherical
Bessel function.
In cylindrical coordinates, the solutions
to the wave equation are the ordinary Bessel
function of integer order.
See also
Theory and experiment
Applications
Biographies
Notes
^ Current practice is to use c0 to denote
the speed of light in vacuum according to
ISO 31. In the original Recommendation of
1983, the symbol c was used for this purpose.
See NIST Special Publication 330, Appendix
2, p. 45
^ Maxwell 1864, page 497.
^ See Maxwell 1864, page 499.
References
Further reading
Electromagnetism
Journal articles
Maxwell, James Clerk, "A Dynamical Theory
of the Electromagnetic Field", Philosophical
Transactions of the Royal Society of London
155, 459-512.
Undergraduate-level textbooks
Griffiths, David J.. Introduction to Electrodynamics.
Prentice Hall. ISBN 0-13-805326-X. 
Tipler, Paul. Physics for Scientists and Engineers:
Electricity, Magnetism, Light, and Elementary
Modern Physics. W. H. Freeman. ISBN 0-7167-0810-8. 
Edward M. Purcell, Electricity and Magnetism.
ISBN 0-07-004908-4.
Hermann A. Haus and James R. Melcher, Electromagnetic
Fields and Energy ISBN 0-13-249020-X.
Banesh Hoffmann, Relativity and Its Roots.
ISBN 0-7167-1478-7.
David H. Staelin, Ann W. Morgenthaler, and
Jin Au Kong, Electromagnetic Waves ISBN 0-13-225871-4.
Charles F. Stevens, The Six Core Theories
of Modern Physics, ISBN 0-262-69188-4.
Markus Zahn, Electromagnetic Field Theory:
a problem solving approach, ISBN 0-471-02198-9
Graduate-level textbooks
Jackson, John D.. Classical Electrodynamics.
Wiley. ISBN 0-471-30932-X. 
Landau, L. D., The Classical Theory of Fields,.
ISBN 0-08-018176-7.
Maxwell, James C.. A Treatise on Electricity
and Magnetism. Dover. ISBN 0-486-60637-6. 
Charles W. Misner, Kip S. Thorne, John Archibald
Wheeler, Gravitation, W.H. Freeman, New York;
ISBN 0-7167-0344-0.
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
ISBN 0-393-92516-1.
External links
