It is important to realize that the frequency of the sounds that the source emits does not actually change. To understand what happens, consider the following analogy. Someone throws one ball every second in a man's direction. Assume that balls travel with constant velocity. If the thrower is stationary, the man will receive one ball every second. However, if the thrower is moving towards the man, he will receive balls more frequently because the balls will be less spaced out. The converse is true if the thrower is moving away from the man. So it is actually the wavelength which is affected; as a consequence, the perceived frequency is also affected. It may also be said that the velocity of the wave remains constant whereas wavelength changes; hence frequency also changes.

If the moving source is emitting waves through a medium with an actual frequency f₀, then an observer stationary relative to the medium detects waves with a frequency f given by

where v is the speed of the waves in the medium and v_{s, r} is the speed of the source with respect to the medium (positive if moving away from the observer, negative if moving towards the observer), radial to the observer.

With a relatively slow moving source, v_{s, r} is small in comparison to v and the equation approximates to

A similar analysis for a moving observer and a stationary source yields the observed frequency (the observer's velocity being represented as v_o):

where the same convention applies : v_o is positive if the observer is moving away from the source, and negative if the observer is moving towards the source.

These can be generalized into a single equation with both the source and receiver moving. However the limitations mentioned above still apply. When the more complicated exact equation is derived without using any approximations (just assuming that everything: source, receiver, and wave or signal are moving linearly) several interesting and perhaps surprising results are found. For example, as Lord Rayleigh noted in his classic book on sound, by properly moving it is possible to hear a symphony being played backwards. This is the so-called "time reversal effect" of the Doppler effect. Other interesting cases are that the Doppler effect is time dependent in general (thus we need to know not only the source and receivers' velocities, but also their positions at a given time) and also in some circumstances it is possible to receive two signals or waves from a source (or no signal at all). In addition there are more possibilities than just the receiver approaching the signal and the receiver receding from the signal.

All these additional complications are for the classical?i.e., nonrelativistic Doppler effect. However, all these results also hold for the relativistic Doppler effect as well.

The first attempt to extend Doppler's analysis to light waves was soon made by Fizeau. In fact, light waves do not require a medium to propagate and the correct understanding of the Doppler effect for light requires the use of the Special Theory of Relativity. See relativistic Doppler effect.