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Doppler shift has different forms if the observer or the source are in motion. I consider Doppler shift in the case of sound.

I tried to find an answer on the non symmetry of Doppler effect, especially on its apparent contrast with Galilean relativity.

What I found is that, in the case of Doppler shift for sound, there is a privileged frame of reference, which is the one where the sound wave travels, air for istance, and this is the reason why the simmetry is broken and it is not possible to talk about Galilean relativity.

I think I did not get this completely, in particular how is the existance of the privileged frame translated in a different physical phenomenon in the cases in which the observer or the source move?

I got the reason why the two phenomenon occur but it is not clear what this has to do with the presence of a privileged frame (the frame of air).

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The only thing which matters is relative positions or velocities of two bodies. If you have only two bodies in a vacuum, then it doesn't matter which one is moving - what matters is their relative velocity. If you now have three bodies, say A, B, C then still what matters is their relative velocities. "A is moving 10 km/h while B and C are staying" is not the same as "B is moving 10 km/h while A and C are staying". Here relative velocities between A and B are the same, but these between A and C are not! So these represent different physical situations (rather than the same observed in different frame of reference). In your case (Doppler shift) A=emitter, B=observed, C=air.

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I think I did not get this completely, in particular how is the existance of the privileged frame translated in a different physical phenomenon in the cases in which the observer or the source move?

Imagine a case where one of the two is at rest with respect to the medium and the other is moving away at $c$. If the emitter is moving, the receiver still gets the (Doppler shifted) signal. If the receiver is moving away, no signal is ever received.

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