The non-Relativistic Doppler shift equation is $f = \left( \frac{c + v_\text{r}}{c + v_\text{s}} \right) f_0 $ where c is the speed of the medium (346.4 m/s for sound at 25 C temperature). I tried calculating the Doppler shift for the case when the source was moving towards the observer, and the case where the observer is moving with the same velocity towards the source, but I get different answers by about 0.75 m/s. Why does the frame I choose make a difference?
http://www.wolframalpha.com/input/?i=200*%28346.4+%2B+20%29%2F%28346.4+-+0%29
I am running the calculation for a wave with 200Hz frequency and a source(or an observer) moving with 20 m/s. $v_s$ is velocity of the source and $v_r$ is the velocity of the observer.
Answer
Sound waves propagate at 340 m/s relative to the medium (air). In both cases, the relative motion of the source and receiver are the same, but the relative motions of the source and receiver with respect to the medium are different. This is what breaks the symmetry of your two setups, and why you're getting different answers.
It may help to look at a derivation of the doppler shift to better understand how that difference plays a role. Here is one direvation I came across while searching. There may be better ones.
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