Two types of relativistic transformation for the four-vector (ω/c, k) of waves (including light waves in media, or acoustic waves) are obtained based on the differential Lorentz transformation. One of them is just the usual Lorentz transformation, whereas the second one is not. The second one is required to secure the frequency of waves from being negative, in the case that the speed of waves in a medium is less than the speed of that medium moving in the direction opposite to the propagation direction of waves. Therefore, the four-vector (ω/c, k) of waves is in general not Lorentz-covariant. The invariance of the phase of waves among inertial frames is questionable.

The classical and the relativistic Doppler effect for both acoustic and light waves are formulated exactly by a systematic method. This method differs only in its usage of classical and relativistic transformation laws in the formulation of the classical and relativistic Doppler effect, respectively. The method is straightforward, and much more logically rigorous than the typical demonstrations using graphical illustration. In the formulation of the classical Doppler effect, as expected, no aberration and no transverse classical Doppler effect are found. In the formulation of the relativistic Doppler effect, one important discovery is that the transverse relativistic Doppler effect depends only on the speed of the source relative to the observer, irrespective of the nonskew velocities of the source and the observer relative to the medium. PACS No.: 03.30.+p

A new scheme for relativistic transformation of the electromagnetic fields is formulated through relativistic transformation in the wavevector space, instead of the space–time space. Maxwell's equations of electrodynamics are shown to be form-invariant among inertial frames in accordance with this new scheme of relativistic transformation. This new perspective on relativistic transformation not only fulfills the principle of relativity, but is also compatible with quantum theory.

The present expression of radiation of an accelerated point charge is only approximately valid. The exact expression of radiation of an accelerated point charge is derived based on special relativity, and using the Larmor formulation for the radiation of an charged particle being accelerated, but instantaneously at rest. The totaled radiation power obtained by the exact expression is the same as Liénard's generalization of the Larmor formula.

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