The basic classical Doppler effect formulae:

We present and explicate below the most basic formulae of the Doppler effect in a medium in the classical limit: i.e., when relativistic effects are negligible. The medium is at rest in an inertial frame.

Hereafter, for brevity, we usually just say the classical Doppler effect instead of "the Doppler effect in a medium ... "

The derivation of the most basic formulae and other basic formulae is given in Waves file: doppler_effect_classical_derivation.html: Derivation of the Doppler effect in a medium in the classical limit ---which may be this file.

Presentation and Explication:

The most basic formula the classical Doppler effect is
```
  f_i = f(1 - v_i/v_ph)  or  f = f_i/(1 - v_i/v_ph) , 
```
where v_i is the velocity relative to the medium of observer i moving in the direction of wave propagation, v_i is negative if observer i is actually moving opposite to the direction of wave propagation, f_i is the frequency observed by observer i, 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).
From the most basic formula, all other basic classical Doppler effect formulae for observer motion in the direction of wave propagation can be derived.
For the Doppler shift from observer 1 (which may be a source) to observer 2, the formula is just
```
  f_2/(1 - v_2/v_ph) =  f = f_1/(1 - v_1/v_ph)  or  

  f_2 f_1[(1 - v_2/v_ph)/(1 - v_1/v_ph)]  ,
```
where frequency f is again just the frequency observed by an observer at rest in the medium.
Note for observer i moving with velocity v_i relative to the medium:
1. ```
v_i/v_ph = 0 gives f_i = f
```
  and their is NO Doppler shift since the observer is at rest in the medium.
2. ```
1 > v_i/v_ph > 0 gives f_i < f
```
  for redshift. The observer is running away from the waves.
3. ```
v_i/v_ph = 1 gives f_i = 0
```
  and observer is keeping pace with the waves and they are at rest with respect to the observer. For sound waves, the observer is transonic.
4. ```
v_i/v_ph > 1 gives f_i < 0
```
  and the observer is running in to the back end of the waves. For sound waves, the observer is supersonic.
5. ```
v_i/v_ph < 0 gives f_i > f
```
  for a blueshift. The observer is plowing into the waves. Note a blueshift is can be considered a negative redshift and often is in astronomy.
A very important one of these other basic classical formulae is the classical Doppler effect formula for the Doppler shift in the wavelength for a source with a fixed emission frequency f_source: 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 observers in the wave propagation direction NO matter what their velocity observe
```
  λ = λ_source*(1-v_source_i/v_ph)  
```
including an observer moving with velocity v_source_i. Note λ has NO observer subscript i because it is the same for ALL observers. It is subscripted for source as we have already done.
In fact, the source is also an observer. Thus, v_source_i is positive/negative when the source is moving in/opposed to the wave propagation direction.
Just to emphasize again, no one measures λ_source if v_source ≠ 0 and everyone does if v_source = 0 since λ = λ_source in this case.
The reason for the invariance of wavelength λ with respect to ALL observers is that in the classical limit, length is an invariant quantity.
Note what we mean by length is a difference in spatial position that can be measured at one instant in time. You may often measure length in some other way, but it is NOT a length if it CANNOT be measured at one instant in time.
Note for observer at rest and a source with fixed f_source moving at velocity v_source_i relative to the medium: In this case,
```
  f = f_source/(1 - v_source_i/v_ph) and λ = λ_source*(1-v_source_i/v_ph)  .  
```
Note v_source_i is positive/negative for source motion in/opposite the direction of wave propagation. It is easiest to understand the case for sound waves and when viewing the animations for sound waves in Waves file: doppler_effect_sonic.html. Behold:
1. ```
v_source_i/v_ph = 0 gives f = f_source and λ = λ_source
```
  and their is NO
  Doppler shift since the source is at rest in the medium.
2. ```
1 > v_source_i/v_ph > 0 gives f > f_source and λ < λ_source 
```
  for a blueshift relative to the source.
3. ```
v__source_i/v_ph = 1 gives f = ∞ and λ = 0
```
  and source is keeping pace with the waves and they are piling on top of each other. For sound waves, the source is transonic and creating a sonic boom right where it is. The observer right on the line of source motion (as we have been assuming), only receives one super wave pulse moving in the direction of source motion when the source moves by them: e.g., they receive a sonic boom for sound waves. It helps to view the animations for sound waves in Waves file: doppler_effect_sonic.html.
4. ```
v_source_i/v_ph > 1 gives f < 0 and λ < 0
```
  and the source is running ahead of the waves. The negative frequency f and wavelength simply mean that the observer observes the waves moving in the direction of the source motion in reverse order from emission. It helps to view the animations for sound waves in Waves file: doppler_effect_sonic.html. For sound waves, the source is supersonic and is creating sonic boom along a Mach cone.
5. ```
v_source_i/v_ph < 0 gives f < f_source and λ > λ_source
```
  and the observer observes a redshift relative to f_source since the source is moving away from the observer.

Credit/Permission: © David Jeffery, 2022 / Own work.
Local file: local link: doppler_effect_classical_formulae.html.
File: Waves file: doppler_effect_classical_formulae.html .