Relativity and mathematical tools: Waves in moving media

We discuss the electromagnetic wave equation and dispersion relation for moving media as an example of the challenge of integrating vectors and dyadics and their covariant counterparts into the teaching of special relativity. We discuss two methods of deriving the dispersion relation and its associated wave equation. One is a direct approach for which the starting point is Maxwell's equations combined with the Minkowski constitutive relations in the laboratory frame Γ in which the medium is moving uniformly. The second approach, starting in the medium's rest frame Γ', establishes the invariance properties of the dispersion relation, and then proceeds to derive its counterpart in Γ. Application to the calculation of the group velocity in each frame is also discussed.