See electromagnetic wave equation for a discussion of this important discovery.

Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation.

The electromagnetic wave equation derives from Maxwell's equations.

Sinusoidal plane-wave solutions are particular solutions to the electromagnetic wave equation.

It can be shown, that, under these conditions, the electric and magnetic fields satisfy the electromagnetic wave equation:

For example, consider the electromagnetic wave equation:

The electromagnetic wave equation describes the propagation of electromagnetic waves through a medium or in a vacuum.

Other spherically and cylindrically symmetric analytic solutions to the electromagnetic wave equations are also possible.

In fact, symmetric equations can be written when all charges are zero, and this is how the Electromagnetic wave equation is derived (see immediately above).

Therefore, these equations allow self-sustaining "electromagnetic waves" to travel through empty space (see electromagnetic wave equation).

Time dependence of a positive-frequency wave (see, e.g., the electromagnetic wave equation):

From classical equations of motion and field equations; mechanical and electromagnetic wave equations can be derived.

Instead, it must be analyzed as an electromagnetic structure, by solution of Maxwell's equations as reduced to the electromagnetic wave equation.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations.

AM Bork Maxwell and the Electromagnetic Wave Equation (1967)

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (cgs units)

Slowly varying envelope approximation (SVEA) - a method in optics to solve the electromagnetic wave equation.

Maxwell however drops the term from equation "D" when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame.

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.

In the frequency domain, with an assumed (engineering) time convention of , the homogeneous electromagnetic wave equation is known as the Helmholtz equation and takes the form:

Maxwell's equations can be written in the form of a inhomogeneous electromagnetic wave equation (or often "nonhomogeneous electromagnetic wave equation") with sources.

The general solution of the electromagnetic wave equation in homogeneous, linear, time-independent media can be written as a linear superposition of plane-waves of different frequencies and polarizations.

Maxwell had eliminated the Coulomb electrostatic force from his derivation of the electromagnetic wave equation since he was working in what would nowadays be termed the Coulomb gauge.

A general solution to the homogeneous electromagnetic wave equation in rectangular coordinates may be formed as a weighted superposition of all possible elementary plane wave solutions as:

The concept of displacement current, which he had introduced in his 1861 paper "On Physical Lines of Force", was utilized for the first time, to derive the electromagnetic wave equation.