We can use Maxwell's equations to derive the wave equation that electromagnetic waves obey.
We start from the equation
Substitute into this the equation
rearranged to give
and the equation
We obtain
We take the curl of both sides of this equation to obtain
Assuming
and
to be constant allows us to write
Now use the vector identity
to obtain
Free space has no charge so
to give
A similar procedure can be followed to obtain
These are both wave equations, with the wave speed give by
In a vacuum
and
They are given the symbols
and
respectively. In other materials
This means that the speed of electromagnetic radiation is less in these materials.