For sound waves, the Doppler formula for frequency shift differs slightly between the two situations: (i) source at rest; observer moving, and (ii) source moving; observer at rest. The exact Doppler formulas for the case of light waves in vacuum are, however, strictly identical for these situations. Explain why this should be so. Would you expect the formulas to be strictly identical for the two situations in case of light travelling in a medium?

(i) No, the properties of propagation of light and sound are very different from each other. Sound waves can propagate only in the presence of a medium. The velocities do not just simply add up when they approach the speed of light.

(ii) The two situations whether it is source at rest or observer at rest are not identical because the sound waves are mechanical and thus depends on the properties of the medium through which it is travelling.

In case of the observer moving the apparent change in the frequency can be given as

In case of source moving the apparent change frequency can be given as,

Where,

f is the frequency of waves from source

f’ is the apparent frequency observed.

ν’ is the speed of the observer and ν is the speed of the wave from the source

The negative sign is for the observer or source moving away from the one at rest. The positive direction is for when the distance between them is decreasing.

Hence, The Doppler’s formulas for the two situations cannot be the same.

(iii) In case of the light waves, In vacuum, the above situations (a) and (b) are identical because the speed of light is a universal constant and does on depend on the motion of the observer or the source and thus the value always remains the same. But when light travels in a medium, according to the wave theory of light the speed depends on the nature of the medium.