The Fizeau experiment is discussed as a concrete example for investigating wave propagation in nonsimple moving media. Exact special relativistic formalism is used throughout, and first-order approximations are developed from the exact forms. No Doppler frequency shifts occur to an observer in the laboratory frame of reference, because in Fizeau's experiment the moving fluid is contained within stationary boundaries. Consequently, only phase shifts are measurable. The results show that in order to measure the velocity effects from the interference fringes, one has to adequately modify the construction of the original Fizeau experiment. A relativistically exact model for first-order in velocity was developed for the chiral medium giving a relatively simple formalism and enabling an easy solution to propagation and scattering of electromagnetic waves in the presence of moving chiral media. ¿ American Geophysical Union 1995