Which Doppler formula if light were to travel at non-relativistic speed?

Question

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)

Answer 1

You should be aware of the derivation of these equations and what the different terms mean. The first fraction in your last equation is the classical Doppler effect. The square root expression stems from the relativistic time dilation. This time dilation is between the reference frames of the source and the receiver. If their relative velocity is small compared to the speed of light, the root term is approximately equal to one and hence one obtains the classical expression. That is very important for the following two questions.

1. If the sound would travel at about $0.8 c$, the Doppler effect is negligible unless the velocity reaches about the speed of sound and hence the speed of light. For those high velocities the time dilation is not negligible and thus you need to apply the latter equation.
2. This is a bit more tricky, because there is a problem with light having a medium but one may break it down: If there was a medium for light, Einstein's postulate would be wrong and hence there would not be a time dilation and you would have to apply the classical Doppler effect formula.

Hope this helps