Doppler Effect and Relativity The equation for the Doppler effect is given by
$$f_L = \frac{v+v_L}{v+v_S}f_S$$
where the velocities of both the source and the listener matter. My question is, how does this fit into Galilean relativity? For instance, if a source was moving and the listener is also moving, if I take the reference frame to be that of the listener, change the velocity of the source to a relative velocity I will get a different result. Why is that? Does the equation just not work like that?
 A: Most waves travel in a medium, which has a preferred reference frame. Both velocities are measured with respect to this medium. This breaks the equivalence of the reference frames: in the reference frame of the listener, the medium itself is moving.
A: The equation you have provided is incorrect . You should have a difference of velocities in the denominator of the fraction. This is important because it means that in Special Relativity the frequency of a moving wave source, measured from a stationary observer
is always larger than the frequency of the emitted wave.
To answer your question, the Doppler Effect can have applications in Astronomy, when measuring the shift in frequency of EM waves produced by moving stars or galaxies, in order to get more information about them, and determine whether they are moving/rotating towards (blue shift) or away (red shift) from Earth. This method has also been used as an early proof of the expansion of the universe. 
I am not sure what you mean in the second part of your post by saying that if you change frames of reference you will get a different result. These conclusions should apply only to Special Relativity not to Galilean Relativity.
