I was going through this article about Doppler formulas and what it says is, that we only really have 1 Doppler formula, not 2. I only want if someone can confirm if I am understanding what it implies correctly.
Here goes:
The classical non-relativistic formula for sound where $f_e$ is emitter frequency, $f_a$ is absorber frequency, they are moving away from each other, medium of propagation is not moving and speeds of absorber and emitter are subsonic is:
$$f_a=f_e \frac{c_s-v_a}{c_s+v_e}$$
Then for light in a vacuum, we have a different more symetric formula, where only the relative speed matters:
$$f_a=f_e \sqrt\frac{1-v_{diff}}{1+v_{diff}}=f_e \sqrt\frac{1-|v_e-v_a|}{1+|v_e-v_a|}$$
Now, the article says that both these formulas can be derived from a more general formula where $c$ is the universal constant and is used as such in the equation and $c_s$ is the speed of signal, which can be speed of sound in the medium or speed of light in a (non-moving) medium or any kind of a signal for that matter in the medium of which speed depends solely on the medium.
$$f_a=f_e {\frac{1-\frac{v_a}{c_s}}{1+\frac{v_e}{c_s}}\sqrt{\frac{1-\frac{v_e}c}{1+\frac{v_e}c}}}$$
This means, that for any slow speeds, the first equation is good enough, but for speeds close to the speed of light in vacuum, the second equation is a good enough approximation.
Questions:
1: If there was a medium at which sound could travel 0.8 the speed of light, would that mean that I would be better off using the relativistic Doppler equation?
2: If there was a medium which would slow down the phase speed of light to non-relativistic speed and the source would be able to move in it, would that mean that the "sound Doppler equation" would be the one to use? (Ergo - light does not have the same speed for all observers in the medium, just like sound doesn't)