The Phase Velocity and Doppler Effect Associated with Moving Electromagnetic Sources and Moving Observers

This article provides general expressions for the phase velocity and the Doppler shift of the electromagnetic fields radiated from a uniformly moving Hertzian dipole measured by a uniformly moving observer. The results show that the phase velocity of the electromagnetic wave is always equal to when measured exactly in the direction pointing to the birthplace of the field. The expression for the Doppler effect is of the same form of the Newtonian type classical formula, which implies that it might be not proper to consider that the classical formula for the Doppler shift is the low speed approximation of the conventional relativistic formula.