The Doppler shift occurs whenever a source of sound (or light or any other kind of wave) and a receiver of that sound move relative to one another. It is easiest to think of the sound source as motionless and the listener as moving. Assume that the siren on a factory roof is wailing continuously, all on one note. The sound is broadcast outwards as a series of waves. The waves can't be seen, because they are waves of air pressure. If they could be seen they would resemble the concentric circles spreading outwards when we throw pebbles into the middle of a still pond. Imagine that a series of pebbles is being dropped in quick succession into the middle of a pond, so that waves are continuously radiating out from the middle. If we moor a tiny toy boat at some fixed point in the pond, the boat will bob up and down rhythmically as the waves pass under it. The frequency with which the boat bobs is analogous to the pitch of a sound. Now suppose that the boat, instead of being moored, is steaming across the pond, in the general direction of the centre from which the wave circles are originating. It will still bob up and down as it hits the successive wavefronts. But now the frequency with which it hits waves will be higher, since it is travelling towards the source of the waves. It will bob up and down at a higher rate. On the other hand, when it has passed the source of the waves and is travelling away the other side, the frequency with which it bobs up and down will obviously go down.
For the same reason, if we ride fast on a (preferably quiet) motorbike past a wailing factory siren, when we are approaching the factory the pitch will be raised: our ears are, in effect, gobbling up the waves at a faster rate than they would if we just sat still. By the same kind of argument, when our motorbike has passed the factory and is moving away from it, the pitch will be lowered. If we stop moving we shall hear the pitch of the siren as it actually is, intermediate between the two Doppler-shifted pitches. It follows that if we know the exact pitch of the siren, it is theoretically possible to work out how fast we are moving towards or away from it simply by listening to the apparent pitch and comparing it with the known 'true' pitch.
The same principle works when the sound source is moving and the listener is still. That is why it works for ambulances. It is rather implausibly said that Christian Doppler himself demonstrated his effect by hiring a brass band to play on an open railway truck as it rushed past his amazed audience. It is relative motion that matters, and as far as the Doppler Effect is concerned it doesn't matter whether we consider the sound source to be moving past the ear, or the ear moving past the sound source. If two trains pass in opposite directions, each travelling at 125 m.p.h., a passenger in one train will hear the whistle of the other train swoop down through a particularly dramatic Doppler Shift, since the relative velocity is 250 m.p.h.
