Maxwell's equations become

Consider equations (6.13) and (6.15). If we take, of (6.13) we obtain

Now we use the curl of (6.15) to rewrite the right hand side in terms of alone. Hence,

Finally, using (6.14) we obtain

where

is the speed of light squared. Thus, the magnetic field and the electric field support electromagnetic waves that propagate at the speed of light. Note, that this represents very rapid time variations so that the usual MHD approximation does not apply.

In Cartesian coordinates, the wave equation has constant coefficients
and solutions can be represented in the form

where

and the dispersion relation is

or

Electromagnetic waves, such as gamma rays, x-rays, UV, visible
light, infra-red waves and radio waves, have wavelengths ranging
from
to
, all propagate at the
speed of light. (6.12) and (6.14) imply that

and so the electromagnetic waves are

so that

and the electric and magnetic fields associated with the waves are

where is a unit vector in the direction of the magnetic field,

and so

__Example 6. .2__Suppose that
,
then (6.12) and (6.14) are satisfied identically
and (6.13) and (6.15) imply

and so

One possible solution is

and