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

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)

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.
• If you have the speed of light in some strange medium $c_m$ and editor and absorber move with non-relativistic speed, then we have the non-relativistic case which is the classical or sound doppler formula. Commented May 27, 2020 at 10:59