The main difference between light and sound is that to get a detectable Doppler effect you have to be moving at a speed which isn't tiny relative to the speed of the waves. On the other hand, a very sensitive device could measure this change, which is just what happens when a speeding motorist is caught in a radar speed trap. Distant galaxies are moving away from us extremely fast. So fast in fact that it is relatively easy to measure the shift in their spectral lines. This is evidence that the Universe is expanding, which is one of the most important pieces of evidence in support of the Big Bang picture.
More details Consider a distant galaxy emitting f wave crests per second.
Suppose the galaxy is moving away at speed V . This formula for redshift is only true for velocities V that are small compared to the speed of light. Between 1910-1920, Slipher found that many extragalactic nebulae showed systematic redshifts. However without a convincing estimate of distance to these nebulae, no further progress could be made.