Derivation of the Classical Doppler Effect

The classical Doppler effect is the Doppler effect in the classical limit. It can apply only to mechanical waves since only they can be in the classical limit. They can be relativistic too, but that is another story.
electromagnetic radiation (EMR) waves are always relativistic since they always move at the vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns at least at the microscopic scale and so are always relativistic. In transmission medium they usually move at some large fraction of that speed, and so are almost always relativistic in that sense too.
Quantum mechanical waves also exhibit the Doppler effect, but they takes special treatment beyond the scope of this insert.
First note that mechanical waves are always waves in a (transmission) medium. It is the medium that oscillates.
The properties of the medium fix the (intrinsic) phase velocity which is velocity at which the waves propagate RELATIVE to medium. The phase velocity is INDEPENDENT of the wavelength and frequency.
For our derivations, we assume any observer velocity is in the propagation direction of the waves. If the velocity is negative, the observer is moving opposite the propagation direction .
In fact, any transverse velocity (i.e., velocity perpendicular to the propagation direction) causes NO classical Doppler effect. There is a relativistic transverse Doppler effect, but it it very small and usually NOT so important.

To start our derivation consider a light wave cycle (i.e., a spatial pattern that repeats over and over again) propagating to the right:

Time 1: The wave cycle is just starting
to pass point A moving at speed v_ph_i which is the phase velocity relative
to an observer moving in the progation direction a velocity v_i.
__________
| |
|__________|

---------- λ ---------

The length of the wave or cycle is the wavelength
which universally symbolized by the small Greek letter
lambda λ.

Time 2: The wave is just past point A
a time P_i later.
__________
| |
|__________|

The speed of the wave passing A is

v_ph_i=λ/P_i which gives P_i=λ/v_ph_i .

If N waves pass a point A in N periods,
the frequency of the waves is

N 1
f_i = ______ = _______ which is the just reciprocal of the period.

NP_i P_i

Substituting for P_i gives the general relationship that everyone remembers

f_iλ=v_ph_i .

If v_i = 0, then v_ph_i reduces to v_ph the (intrinsic) phase velocity and f_i reduces
to f, the frequency in the frame
in which the medium is at rest (i.e., the medium frame). Thus, we have

fλ=v_ph .

Note in the above derivations, we did NOT adorn λ with a subscript i. This is because in the classical limit length is reference frame invariant. This invariance is actually the key point of the derivation.

Now note that v_ph_i = v_ph - v_i by our ordinary understanding of relative motion. Therefore

  f_iλ =v_ph_i = v_ph(1-v_i/v_ph) = fλ(1-v_i/v_ph)  .

The basic frequency shift formula follows from the last expression:
```
  f=f_i/(1-v_i/v_ph)  .  
```
The basic frequency shift formula It allows you to relate the frequency f_i measured in any reference frame to the frequency f measured in the medium frame.
What if there are two observers in relative motion: one with v_1 and the other with v_2. From the basic frequency shift formula, we obtain
```
 
  f_1/(1-v_1/v_ph) = f = f_2/(1-v_2/v_ph) which leads to

  f_2 = f_1[(1-v_2/v_ph)/(1-v_1/v_ph)] 
```
Note that in general, the total shift in frequency does NOT depend on the relative velocity Δv = v_2 - v_1. But if you Taylor expand to 1st order in small v_1/v_ph and v_2/v_ph, you obtain
```
  f_2 = f_1[1-(v_2-v_1)/v_ph)] = f_1(1-Δv/v_ph)   or

  (f_2-f_1)/f_1 = -Δv/v_ph  which is abbreviated as the memorable formula

  Δf/f = -Δv/v_ph 
```
Note that a positive/negative Δv leads to a redshift/blueshift in frequency in astro jargon.
What if in the basic frequency shift formula, v_i ≥ v_ph? Rewriting the basic frequency shift formula
```
  f_i=f(1-v_i/v_ph)  .  
```
Now f is considered always positive since the direction that waves go by an observer at rest in the medium frame is always considered the positive direction.
If v_i = v_ph, then f_i = 0 which just means the waves are at rest with respect to the observer moving at velocity v_i. Being at rest with respect to waves is easy in some cases: e.g., waves in a pool.
If v_i > v_ph, then f_i < 0 which just means the observer moving at velocity v_i is observing the the waves moving opposite to their direction of motion, but they are moving positively relative to the propagation direction relative to the medium frame. This situation can easily be oberved for waves in a pool.
Now what if an observer is also a source and the source has a fixed frequency f_s in the observer's reference frame. Well, the source waves just move by the observer (who is moving with velocity v_i = v_s) with frequency f_i = f_s or, if the observer is at rest in the medium frame, with frequency f = f_s. So one can just use the basic frequency shift formula straighforwardly.
How does wavelength change with the classical Doppler effect?
Well, as discussed above wavelength does NOT change with the motion of the observer.
However, if you define a source to have an intrinsic wavelength λ_s=v_ph/f_s, then the wavelength observed by all observers will be different from λ_s if the source is moving with respect to the medium frame.
Using the basic frequency shift formula, the observed wavelength
```
           λ = v_ph/f = (v_ph/f_s)(1-v_s/v_ph) = λ_s*(1-v_s/v_ph) 
```
depending on motion.
Features:
1. Note that in motion, wavelength (measured outside of the car) reference frame) is decreased in the direction of motion and wavelength (measured outside of the car) and increased/decreased opposite the direction of motion. The waves are scrunched, the waves are stretched.
2. Note that the change in wavelength depends on the motion of the source in the transmission medium.
  But---and this is a key point---wavelength as measured by an observer is the SAME for all observers no matter how they are moving since we are in the classical limit where length is invariant with respect to reference frame.
3. frequency is does depend on both on the motion of the source in the transmission medium and on the observer's motion.
4. There is a simple derivation of the classical Doppler effect frequency shift following from the invarianc of wavelength for observers:
```
 
  λ/p' = v_ph - v 
  f'λ = v_ph(1 - v/v_ph)
  f' = f(1 - v/v_ph)  ,
       where 
       λ is the
                  invariant
                  wavelength,
        p is the period (i.e., the time that it takes 
                  a wavelength to 
                  go past a point)
        v_ph is the (intrinsic)
                 phase velocity
                 of sound
                 in air
                 (i.e., the velocity
                 of  sound
                 in air
                 when you are at rest
                 in air) 
         f is the intrinsic 
```
  On the other hand, frequency is does depend on reference frame. driver detects the transmission medium rest-frame frequency since they are at rest in the transmission medium inside the car.
  Actually, inside the car, the wavelength
  However, an outside observer An observer in Medium frame observer (i.e., one at rest with respect to air and ground) has frequency is high before the car and low after it.
  This the sound Doppler effect.
  The electromagnetic radiation is similar, but the formulae are different in the two cases.
  Credit/Permission: © User:Charly Whisky / Creative Commons CC BY-SA 3.0.
  Image link: Wikipedia: File:Dopplerfrequenz.gif.
  Local file: local link: doppler_effect_derivation_classical.html.
  File: Waves file: doppler_effect_derivation_classical.html .