In free space, do Maxwell's equations contain the same amount of information regarding electric and magnetic fields as is contained in the wave equations derived from them? If so, how?
Answer
No, they're not. The wave equations for the force fields contain a strict subset of the information contained in the full set of Maxwell's equations. In particular, it's important to note that you need the Gauss-type equations, $$ \nabla\cdot \mathbf E = 0 = \nabla\cdot\mathbf B, $$ to ensure the transversality of the waves. If all you had to go was the wave equations in the form $$ \left[\partial_t^2 - c^2 \nabla^2 \right]\mathbf E = 0 $$ then you'd have no way of knowing that longitudinal EM waves are forbidden. (Though, to be clear, the transversality conditions are not sufficient, either.)
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