If i have 2 coordinate systems (CS) which are travelling one towards another. CS

Question

If i have 2 coordinate systems (CS) which are travelling one towards another. CS xy with an observer in its origin and CS x?y? with a source in its origin. Correct me if I am wrong, but i think that I have to use this variation of an equation for a relativistic Doppler effect:

Here ? is a frequency that observer receives and ?? is a frequency which source transmits. Correct me if I am wrong.

Question: Do I have to swap ? and ?? in the equation above if I put observer in CS x?y? and source in CS xy? A brief explanation will do just fine.

Explanation / Answer

The equation is correct, though whether it's (c+u)/(c?u) or (c?u)/(c+u) depends on your sign convention. As it stands your equation is correct if u is positive when the source is approaching you and negative when the source is moving away from you. The Wikipedia article you cite actually uses the opposite convention.

Anyhow, provided u^' is always the moving frame and ? is the stationary frame the equation applies to both sets of observers. To make this clearer let's take an example. Suppose both frames have a source emitting the same frequency of 1KHz and their relative speed is 0.6c then both observers will see the other source to be 2KHz when approaching and 0.5KHz when receding (I chose 0.6c to give a factor of 2).

The situation has to be symmetric otherwise it would be possible for the observers to tell which frame was moving and which was standing still.

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