Overview
One of the surprises of electromagnetic theory was that when the charge density given in Maxwells equations is set to zero, the equations reduce to the wave equation. That is, the equations predict that the electro-magnetic field can propagate as a wave. When the speed of the wave is calculatd, it was found to equal the experimentally determined speed of light.
This was one of the profound successes of electro-magnetic theory, the prediction that light is an electro-magnetic wave.
Maxwells Equations in Vacuum
Maxwell's equations take the following form in a vacuum.
{% \nabla \cdot E = 0 %}
{% \nabla \cdot B = 0 %}
{% \nabla \times E = - \frac{\partial{B}}{\partial{t}} %}
{% \nabla \times B = \mu_0 \epsilon_0 \frac{\partial{E}}{\partial{t}} %}
Derivation
Starting with Maxwells equations in a vacuum, we have
{% \nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} = \nabla \times -\frac{\partial B}{\partial t}
= - \frac{\partial}{\partial t} (\nabla \times \vec{B}) = -\mu_0\epsilon_0 \frac{\partial ^2 \vec{E}}{\partial t ^2}
%}
but
{% \nabla \cdot \vec{E} = 0 %}
this then implies that
{% \nabla^2 E = \mu_0 \epsilon_0 \frac{\partial^2{E}}{\partial{t}^2} %}
A similar calculation also yields
{% \nabla^2 B = \mu_0 \epsilon_0 \frac{\partial^2{B}}{\partial{t}^2} %}
Both these equations are versions of the
wave equation.