I am an undergraduate student and I can barely find a resource that shows the math involved with the doppler effect such as calculus, change in frequency, etc. I already asked so many questions about this topic but your help is very useful.
In addition to the suggestions in the comments, I think a good starting point is to study the wave equation (see Wikipedia for example). Look at the simplest case of a one dimensional plane wave travelling in one direction, say positive $x$, and then ask yourself what happens when you change from a stationary reference frame to one travelling with constant speed $v_0$ in the $+x$ and $-x$ directions, i.e. $x'=x \mp v_0 t$.