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https://www.mathpages.com/rr/s2-04/2-04.htm

Note that here to author works explicitly " frame of reference in which the

**medium of signal propagation**is assumed to be at rest", when he treats the acoustic Doppler effect (sound waves) relativistically. It should be easy to derive the more general formula in an arbitrary frame of reference, where also the medium moves.

It is shown clearly that in the case for em. waves in a vacuum (optical Doppler effect), where the phase speed of the waves goes to the speed of light, the Doppler formula correctly depends only on the relative velocity of the source and the observer (which the author calls absorber) as it must be since there is no aether (within relativistic theory).