Harald88, you are mixing up everything again like an autodidact, and you don't want to listen. You know once again  everything in advance and much better than anybody else. What you are stating here, however, is nonsense again. The classical forerunner of SRT has been  optics of moving bodies, a discipline dealing with the observation of  point source from a moving platform, which is governed by Bradley's law of velocity aberration. A point source emits spherical waves, not plane waves.

In none of the references stated in the nonsensible Wikipedia article on the transverse Doppler effect, to which you have contributed, a proof is delivered, but it is simply asserted there that the transverse Doppler effect has been unknown in classical physics. In dozens of textbooks this nonsense is repeated without proof because authors tend to write off from one another.

'Proofs' of the nonsense are delivered in Chapter 11 of J. D. Jackson's book on clsssical electrodynamics as well as in a German book on SRT by Greiner/Rafelski who probably have copied the 'deduction' offered in Jackson's book (who himself probably got that 'deduction' from somewhere else). These 'deductions' boil down to direct betrayal of naive students, because the students are made to believe that a point source, such as a fixed star, does emit plane waves rather than spherical waves. Nobody tells them that the plane wave approximation is only an approximation and that it is, therefore, completely useless to 'prove' something ia a context of paramount importance. 

The nonsense of the 'non-existing classical Doppler effect' spreads in the literature since Einstein published in 1907 a short note, where he had stated that the transverse Doppler effect, which already Stark wanted to measure (why, if it is unknown in classical optics of moving bodies?), is "important", but not that it is "unknown" in classical physics. It apparently has to do with the hystery which seems to capture most people who get in touch
with the special relativity nonsense.

In reality SRT departs from classical optics of moving bodies solely for the case of longitudinal observation.

This I was going to demonstrate in the article Modern Galilean relativity.

Because of your unprofessional interference with that article I stopped today contributing to it.

Since you have the big mouth and know everything better, I suggest that you finish the article
now.