Maxwell and the Wave Equation:

Let's start with an engineers perspective on Maxwell's equations. This is a very nice discussion and very much on point.

The next step would be to understand the connection between Coulomb's Law and Gauss' Law as well as Faraday's Law and the form of Faraday's Law articulated by Maxwell. These sites are from a geophysics website and use slightly different symbols for things to be careful. None the less this is one of the better presentations of these connections that I have seen.

There are a variety of ways to get from Maxwell's Laws to the Wave Equation for E fields and B fields. This is a very direct approach that is pretty consistent with math tools that you will have seen and used in MTH 254 and MTH 255.

If you want to go all in on this investigation you can do no better than going to the Feynman Lectures. Here is a link to Chapter 18 which discusses Maxwell's Laws and from section 18.5 onwards explores the connection to the wave equation.