Doppler effect: an understandable explanation for a mathematician

Question

I have curiosity by

Question. Can someone explain me, and to the audience too, the mathematical essence behind the so called Doppler effect? Thanks in advance.

Then you have the ability to translate for example from [1], or another reference in physics, the essentials in mathematical language. I excuse this questions because is a nice topics including functions, derivatives, frecuencies, and I believe too functional equations.

Please, feel free to update this post, including tags as you precise.

References:

[1] Wikipedia, Doppler effect https://en.wikipedia.org/wiki/Doppler_effect

Answer 1

This seems a proper reference on the issue: http://onlinelibrary.wiley.com/doi/10.1002/prop.2190400104/abstract