Figure: Doppler effect basics explicated

Image 1 Caption: An illustration of traveling waves in a medium at rest in an inertial frame in the classical limit: i.e., when relativistic effects are negligible.

There is an observer observing the Doppler effect for various velocities relative to the medium.

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 physical 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.

Further explication:

1. Note in Image 1:
   1. f_obs > f_0 for the observer plowing in the waves.
   2. f_obs = f_0 for the observer at rest in the medium.
   3. f_obs < f_0 for the observer moving in the direction of wave propagation, but with v < v_0.
   4. f_obs = 0 for the observer moving in the direction of wave propagation, but with v = v_0.
   5. f_obs < 0 for the observer moving in the direction of wave propagation, but with v > v_0. We do NOT usually think of negative frequencies, but in this context we can.
   6. f_obs < f_0 is called a redshift in astro-jargon.
   7. f_obs > f_0 is called a blueshift (or a negative redshift) in astro-jargon.

3. 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 generic observer i has frequency f_i and velocity v_i in the direction of wave propagation (which is negative for motion opposite the direction of wave propagation), and v_ph is the travel velocity of the wave forms (i.e., phase velocity relative to the inertial frame in which the medium is at rest).

   The v_1 and v_2 can be positive or negative

   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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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 traveling 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.

9. 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. The situaion is illustrated in Image 2 below.

   

10. Image 2 Caption: A generic classical Doppler effect image for a point source that generates spherical waves when at rest in the medium.

11. Qualitatively, Image 2 explains much of the behavior. Along the point source direction of motion, frequency is increased/decreased (blueshifted/redshifted) in the forward/backward direction from the point source relative to when the point source is at rest.

12. Similarly, wavelength is decreased/increased (blueshifted/redshifted or scrunched/stretched) in the forward/backward direction from the point source relative to the point source at rest.

13. Off the direction of motion, the situation is actually rather tricky. Each wavefront can be viewed a spherical wavefront expanding a trailing set of point sources.

14. Along the direction of motion, there is a simple formula for the wavelength Dopper shift:

      λ= λ_source(1∓v_source/v_ph)  ,

    where λ_source is the wavelength of the source in all directions for source when it is at rest and upper/lower case is for ahead/behind of the source.

    Note, all ahead/behind observers measure the same wavelength since that is invarient no matter how they are moving, but the frequency shift they measure is given frequency Doppler formula above. For observers at rest in the medium,

      f_obs= f_source/(1∓v_source/v_ph)  .

15. Rewriting the last wavelength equation (i.e., the second to last equation), we have

      Δλ/λ_source = ∓v_source/v_ph  , 

    where Δλ = λ - λ_source.

16. For the student who wishes to go beyond the scope of Introductory Astronomy Lectures (IAL), the classical Doppler effect formulae given above are derived and further explicated in Waves file: doppler_effect_classical_derivation.html.