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Emission theories propose that the velocity of light depends on the velocity of the light source. But the ordinary Doppler effect assumes the velocity of light remains as ##c## with respect to the ether medium, even when the light source is moving at speed ##v## with respect to the ether. They are saying different things, so how could they be consistent?
Explicitly, the original-source emission theory suggests that the velocity of light measured in the ether frame is ##c+v## when the source moves ahead at speed ##v## with respect to the ether, while the Doppler effect says the velocity of light is still ##c##.
Further, for a mirror moving at speed ##v## (with respect to the ether) towards a light source (stationary with respect to the ether), the Doppler effect gives the frequency of the reflected light as ##f=\frac{c+v}{c-v}f_o##, where ##f_o## is the frequency of the incident light. I don't understand how the original-source emission theory could come up with the same formula.
Explanations of the various emission theories are in the text below.
Source: Introduction to Special Relativity by Robert Resnick