After having consulted numerous physics books, forums and videos, in all cases I found that the electromagnetic wave equation is derived from taking as a premise that the medium is a vacuum. This causes the charge density and current density to be zero in Maxwell's equations, which leads to the electromagnetic wave equation. Unfortunately, the only explanations I have found for making such a premise are:
- "Electromagnetic waves can travel in a vacuum."
- "The electric and magnetic fields of a wave are perpendicular to each other and, in turn, the wave vector is perpendicular to these."
- "Not taking this premise into account causes the electromagnetic wave equation to be inhomogeneous."
Personally, I find these explanations insufficient, since if I were Maxwell trying to derive the electromagnetic wave equation in the 19th century, how would I go about knowing these facts beforehand? That said, I would love to know what Maxwell realized that allowed him to make that assumption.