The Doppler effect: Now you see it, now you don't

Two main classes of problems are identified in the theory of electromagnetic scattering in velocity-dependent systems. The first involves transformation of space and time coordinates and field components from the laboratory system of reference to the comoving system of the scatterer, solution of the scattering problem, and inverse transformations. In general, this method displays the Doppler frequency shifts. The second class involves the substitution of Minkowski's constitutive relations into Maxwell's equations for harmonic time variation, heuristically stipulating the absence of Doppler frequency shifts. The interrelation between the two methods is investigated here. It is argued that the second method is a limiting case for very low, as well as very high frequencies, and provided the mean square fluctuation of the dielectric constant is small, and the geometrical boundaries defining the scatterers are fixed. Canonical problems of plane, cylindrical, and spherical stratification are discussed and analytical results for the scattered fields are derived. If the parameters of the problem do not meet the above conditions, the first method should be used, giving rise, in general, to a whole spectrum of frequencies due to the Doppler effect.