Alexandra- That's an extremely perceptive question. (I assume you meant "relative speed of receiver and transmitter").
Let's talk about sound first. Sound is a disturbance in some medium. You can figure out the laws for the sound propagation just by looking at the force between the parts of the medium. It certainly makes sense in any such model that the sound, once it has left its source, propagate at a fixed speed relative to the medium. If two objects are at rest with respect to each other, but moving faster than the speed of sound through the medium, you can easily see that sound can propagate from one to the other but not the other way. One way the receiver is running up to meet the sound; the other way it's running away too fast to be reached. So motion through the medium matters, not just relative motion of the two objects. The Doppler effect is just a milder case of this, where the wave can reach the receiver either way, but the rate of arrival of the crests depends on the motion through the medium.
What about light? Here, there is, so far as we currently know, no medium. There are simple basic laws governing the relations between electrical and magnetic fields (Maxwell's equations). From those laws you can derive a speed of propagation of the waves, just like Maxwell did. If the general principle of relativity (that the laws look the same regardless of your state of motion) is correct, then the speed must look the same regardless of your reference frame. The formula for the Doppler shift in turn must be expressed purely in terms of the relative motion, since there's no medium to compare with.
What happens if someday we develop a theory of, say, stringy space-time in which the electromagnetic waves can be viewed as a disturbance in some underlying medium? The arguments for why such theories must also have the same laws in all the moving reference frames are over my head, but if Lee or another colleague comes up with one we'll add it in.
(published on 02/15/11)