### The most basic classical Doppler effectformulae:

We describe below the most basic formulae of the Doppler effect in a (transmission) medium in the classical limit: i.e., when relativistic effects are negligible. Hereafter, for brevity, we just say the classical Doppler effect instead of "the Doppler effect in a medium ... "

1. The most basic formula the classical Doppler effect is
`  f' = f(1 - v/v_ph)  or  f = f'/(1 - v/v_ph) , `
where v is the velocity relative to the medium of an observer moving in the direction of wave propagation, v is negative if the observer is actually moving opposite to the direction of wave propagation, f ' is the frequency observed by the moving observer, f is the frequency observed by an observer at rest in the medium and v_ph is the (medium) phase velocity (i.e., the velocity of wave propagation relative to the medium).

Note f ' < f for a redshift and f ' > f for a blueshift.

From the most basic formula, all other formulae for observer motion in the direction of wave propagation can be derived.

2. A very important one of these other formulae is the classical Doppler effect formula for the Doppler shift in the wavelength for a source with a fixed emission frequency: i.e., one independent of the source velocity. When the source is at rest in the medium, this wavelength is
`  λ_source = v_ph/f_source .  `
When the source is moving relative to the medium at velocity v_source, all obsersers on the wave propagation direction no no matter what their velocity observe
`  λ = λ_source(1-v_source/v_ph)  , `
where v_source is taken as positive/negative when the observer is ahead/back of the source.

The basie reason for the invariance of wavelength λ is that in the classical limit length is an invariant quantity.

Credit/Permission: © David Jeffery, 2022 / Own work.