Great idea must have great presentation. Math is a very condensed language for expressing great idea and -in my humble opinion- math suffers from bad presentation. A lot of people find that (math) equation is mind-bending and not easy to grasp. Why? Because, in math, you try to cram multi-dimension into single-dimension. In my term, math is HTML page source code. It is ugly. You need to render it in order to view. Thus, I try to present and render math by adding one more dimension which is color. FYI, color is a dimension that can cause chaos if not use properly as mention in Flatland.
Here is the first attempt:
The Doppler effect (or Doppler shift), named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. The received frequency is higher (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower during the recession. The frequency is decreased if source or receiver is moving away from the other.
The total Doppler effect may therefore result from motion of the source, motion of the observer, or motion of the medium. Each of these effects are analyzed separately. For waves which do not require a medium, such as light or gravity in general relativity, only the relative difference in velocity between the observer and the source needs to be considered.
The formula works for sound waves if and only if the speeds of the source and receiver relative to the medium are slower than the speed of sound in the medium.