Image 1 Caption: An animation illustrating traveling waves.

The animation helps to illustrate the Doppler effect in a medium at rest in an inertial frame in the classical limit: i.e., when relativistic effects are negligible.

The classical Doppler effect is a shift in frequency depending on the MOTION OF AN OBSERVER relative to a medium at rest in an inertial frame in the classical limit: i.e., when relativistic effects are negligible.

For the same physcial conditions, there is also a shift in wavelength depending on the MOTION OF A SOURCE relative to a medium with the source having a fixed emission frequency: i.e., one independent of the source velocity.

Features:

The Doppler shift between two moving observers 1 and 2 is given by
```
              f_1 
  f_2 =  --------------  (1 - v_2/v_ph)  ,
         (1 - v_1/v_ph)   
```
where for generic observer i (f_i) is the frequency, (v_i) is the velocity in the direction of wave propagation (which is negative for wave propagation opposite the direction of wave propagation), and v_ph is the phase velocity (i.e., the phase velocity relative to the inertial frame in which the medium is at rest).
Note, the two observers are observing the SAME set of waves. They are NOT looking at waves in different places or times or under different conditions. They are really, really looking at the SAME set of waves.
The above formula follows from the general result
```
        f_i        v_ph 
  -------------- = ----  ,
  (1 - v_i/v_ph)    λ  
```
where λ is the wavelength of the set of waves observer i is observing. In the classical limit, λ is invariant: all observers observing the set of waves observe the SAME λ. Since the right-hand side is invariant in the general result, so is the left-hand side, and thus the Doppler shift between two moving observers 1 and 2 follows as advertised.
If v_1/v_ph << 1 and v_2/v_ph << 1, then we have (making use of a Taylor expansion) the 1st order formula
```
  (f_2 - f_1)_1st = f_1(1 - v_2/v_ph)(1 + v_1/v_ph) - f_1 = -f_1(v_2 - v_1)/v_ph 
```
or in the simplifed relative Doppler shift formula form
```
  Δf/f = -Δv/v_ph  , 
```
where Δf is the change in frequency, f is either f_1 or f_2 or an average of these since which is used does NOT matter to 1st order and Δv = (v_2 - v_1) is the relative velocity between the observers.
Note, only to 1st order does the Doppler shift depend only on the relative velocity and NOT on v_1 and v_2 individually.
Note:
1. f_2 > f_1 is a blueshift for observer 2 relative to observer 1.
2. f_2 < f_1 is a redshift for observer 2 relative to observer 1.
3. However, a blueshift can be considered a negative redshift and usually is in astronomy.
A key feature of the classical Doppler effect is that wavelength is invariant for ALL observers: i.e., they all measure the same invariant wavelength NO matter how they are moving.
But recall, they are really, really looking at the SAME set of waves.
The situation is very different for Electromagnetic radiation (EMR). EMR in vacuum (i.e., with NO medium) experiences the (extreme limit) relativistic Doppler effect since EMR is always moving at the vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns.
For EMR, wavelength depends on the observer velocity. It is the vacuum light speed that is invariant for ALL observers: i.e., they all measure the same invariant vacuum light speed NO matter how they are moving.
Note, EMR travelling in a transparent media is moving at less than the vacuum light speed, and so its behavior is NOT in the extreme relativistic limit and said behavior requires a more complex analysis. But if the light speed is close to the vacuum light speed, the behavior can usually be approximated as in the extreme relativistic limit.
Now there is a wavelength shift for the classical Doppler effect, but it does NOT depend on the velocity of the observer. As aforesaid, it depends on the motion of a source relative to a medium with the source having a fixed emission frequency: i.e., one independent of the source velocity.
Image 2 Caption: An animation showing the classical (i.e., non-relativistic) Doppler effect for sound for a siren from a car at rest and in motion relative to the air.
Qualitatively, the animation explains much of the behavior. Along the car direction of motion, frequency is increased/decreased (blueshifted/redshifted) in the forward/backward direction from the car relative to the car at rest.
Similarly, wavelength is decreased/increased (blueshifted/redshifted or scrunched/stretched) in the forward/backward direction from the car relative to the car at rest.
However, one still needs formulae for knowing quantitative behavior and to bring out the distinction between frequency and wavelength behavior for the classical Doppler effect.
We present the formula for the frequency Doppler shift above.
We present the formula for the wavelength Doppler shift below.
The formula for the wavelength Doppler shift is
```
  λ = λ_source(1-v_source/v_ph)  .
```
Rewriting the last equation, we have
```
  Δλ/λ_source = -v_source/v_ph  , 
```
where Δλ = λ - λ_source.
Note, all observers in the medium measure the same invariant wavelength and no one measures λ_source, unless v_source = 0 in which case λ = λ_source and all observers in the medium measure λ_source.
However, inside the car is a different medium that is at rest relative to the car. Thus, an observer inside the car always measures just frequency f_source and wavelength λ_source.
The classical Doppler effect formulae given above are derived and further explicated in Waves file: doppler_effect_classical_derivation.html.
But the quick derviation of the wavelength Doppler shift formula starting from the Doppler effect frequency formula given above mutatis mutandis is as follows:
```
  f_ob =  f_so(1 - v_ob/v_ph)/(1 - v_so/v_ph) = (v_ph/λ_so)[(1 - v_ob/v_ph)/(1 - v_so/v_ph)]

  (v_ph-v_ob)/λ_ob = (v_ph/λ_so)[(1 - v_ob/v_ph)/(1 - v_so/v_ph)]

  (1-v_ob/v_ph)/λ_ob = (1/λ_so)[(1 - v_ob/v_ph)/(1 - v_so/v_ph)]

  1/λ_ob = (1/λ_so)[1/(1 - v_so/v_ph)]

  λ_ob = λ_so(1 - v_so/v_ph)] 
```
where subscript "ob" is for observer and "so" is for source and λ_so is the source wavelength when the source is at rest in the medium (which is NOT a wavelength observed by anyone when the source is motion).

Images:

Credit/Permission: © User:Abhinav P B, 2021 / Creative Commons CC BY-SA 4.0.
Image link: Wikimedia Commons: File:Travelling wave animated plot.gif.
Credit/Permission: © User:Charly Whisky / Creative Commons CC BY-SA 3.0.
Image link: Wikipedia: File:Dopplerfrequenz.gif.

Local file: local link: doppler_effect_classical_formulae.html.
File: Waves file: doppler_effect_classical_formulae.html .