Inconsistency in the Formulae for the Doppler Effect of Moving Electromagnetic Sources

The inconsistency between the classical formula and the relativistic formula for the Doppler effect in electromagnetic waves is revisited. The classical formula may be derived based on the hypothesis of phase invariance of a plane wave under the Galilean transformations, while the relativistic formula is derived based on that hypothesis under the Lorentz transformations. In this paper, we propose to derive a relativistic formula for the Doppler effect basing on the Lorentz transformations. The resultant formula is exactly the same as the classical one. Our analysis shows that the inconsistency is due to the fact that the angular frequencies defined in different frames have different bases because of time dilation. It may be more proper to convert the value of the angular frequency of the source from the rest frame to the laboratory frame so that we can evaluate the Doppler effect between the angular frequency of the source and that of the fields received by the observer in the same time base. We have derived the fields of a moving Hertzian dipole by rigorously applying the Lorentz transformations. The results exactly agree with those we have obtained with the Liénard-Wiechert potentials. We have derived a general expression for the Doppler effect from the far field and found that the classical formula and the modified relativistic formula are simply the approximate forms of the general formula by representing the far field with a plane wave.