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Why does the classical Doppler formula make a distinction between movement of the source and movement of the receiver?

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I’ve tried rewriting the Doppler formula to include only the relative velocity between the source and the receiver of sound waves. However, when I compare the results with the results of the formula that includes “velocity of the receiver” and “velocity of the source”, they are not the same. Is that because the medium (the air) constitutes an absoulte reference frame when it comes to sound waves or is there a different explanation?


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