For sound waves, the Doppler's formula for frequency shift differs slightly between the two situation :
(i) source at rest , observer moving (ii) source moving , observer at rest.
The exact Doppler formulae for the case of light waves in vacuum, are however, strictly identical for the two situations in case of light travelling in a medium ?

Explanation

Sound waves require a medium for propagation. Thus even though the situations (i) and (ii) may correspond to the same relative motion (between the source and the observer), they are not identical physically since the motion of the observer relative to the medium is different in the two situations. Therefore, we cannot expect Doppler formulas for sound to be identical for (i) and (ii). For light waves in vacuum, there is clearly nothing to distinguish between (i) and (ii). Here only the relative motion between the source and the observer counts and the relativistic Doppler formula is the same for (i) and (ii). For light propagation in a medium, once again like for sound waves, the two situations are not identical and we should expect the Doppler formulas for this case to be different for the two situations (i) and (ii).