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.

1. EOF

2. Derivation of the Doppler effect in a medium in the classical limit:

   1. We assume a homogeneous, isotropic medium at rest in an inertial frame with a constant medium phase velocity: i.e., velocity of wave propagation.

      Note the phase velocity when spoken or given without qualification is relative to the rest frame of the medium and is a property of the medium.

   2. Quantities measured in the rest frame of the medium are unsubscripted: i.e., frequency f, period p, wavelength λ, and phase velocity v_ph.

      Subscripted quantities are measured by observers moving with respect to the medium. They are observer-frame quantities: e.g., for observer i, frequency f_i, period p_i, and phase velocity v_ph_i.

      To recapitulate from section The basic classical Doppler effect formulae above, wavelength λ has NO observer subscript since it is invariant. ALL observers NO matter how they are moving measure the same wavelength. It is subscripted for a source (i.e., λ_source) as we have done above in section The basic classical Doppler effect formulae.

   3. The derivation of the Doppler effect for the motion of an observer (moving at velocity v_i relative to the rest frame of the medium) in the direction of wave propagation is simple:

      1.   First, f_iλ = v_ph_i   and   fλ = v_ph    are general formulae.

      2.   Next v_ph_i = v_ph - v_i  .

      3.   Thus, f_iλ = v_ph - v_i = v_ph(1 - v_i/v_ph) = fλ(1 - v_i/v_ph)  .

      4.   Finally, f_i = f(1 - v_i/v_ph)  .  

      The formula

        f_i = f(1 - v_i/v_ph)  or  f = f_i/(1 - v_i/v_ph) 

      is the most basic (classical) Doppler effect formula for motion in the direction of wave propagation. We illustrate its use in the items below.

   4. The cases of f_i = f(1 - v_i/v_ph) for qualitatively distinct v_i/v_ph are given above in The basic classical Doppler effect formulae.

   5. The Doppler shift between two moving observers 1 and 2 is given by

        f_2 = f_1[(1 - v_2/v_ph)/(1 - v_1/v_ph)]  .

   6. 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 = -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.

   7. What if one has source with a fixed emission frequency: i.e., one independent of the source velocity. Call this frequency f_source.

      The basic formula still applies, taking the source as the observer moving with source velocity v_source:

        f_source = f(1 - v_source/v_ph)  or  f = f_source/(1 - v_source/v_ph) 

      where f_i = f_source and f is still the frequency measured by an observer at rest in the medium.

   8. The intrinsic wavelength λ_source (which the wavelength of emission when the source is at rest in the medium) is given by

                  λ_source = v_ph/f_source  . 

   9. What is the wavelength observed for the source moving with velocity v_source by an observer at rest in medium? Behold:

        λ = v_ph/f = (v_ph/f_source)(1-v_source/v_ph) = λ_source(1-v_source/v_ph)  .

   10. Now as argued above in The basic classical Doppler effect formulae, wavelength is invariant in the classical limit, and so all observers (including one moving with the source) observe

         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.

   11. Electromagnetic radiation (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 (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns, and so is always extremely relativistic.

       In the case of relativistic Doppler effect, the invariant is, of course, vacuum light speed c, NOT wavelength.

       We will NOT give the derivation here, but the formulae are actually simpler than for the classical Doppler effect for motion in in the direction of wave propagation:

        
         f_2 = f_1*sqrt[(1-v/c)/(1+v/c)]
       
         λ_2 = λ_1*sqrt[(1+v/c)/(1-v/c)] 

       for two observers 1 and 2 and where v is their line-of-sight relative velocity taking v as positive/negative for increasing/decreasing separation.

       Note the formulae depend just on relative velocity in this case NOT the individual velocities.

   12. Making use of a Taylor expansion, we find the 1st order formulae in simplified form to be

         Δf/f = -v/c
       
         Δλ/λ_1 = v/c  ,  

       where Δf is the change in frequency and f is either f_1 or f_2 or an average of these since which is used does NOT matter to 1st order and wavelength formula is described in the same way, mutatis mutandis.

       Note he first formula is exactly like the corresponding 1st order classical formula. But the second although similar in appearance to the corresponding classical formula Δλ/λ_source = -v_source/v_ph has a very different meaning. For the EMR case, the two observers measure different wavelengths and, for the classical case, all observers measure the same wavelength which is just different from intrinsic wavelength for the source (i.e., the wavelength when the source is is at rest in the medium). This is a tricky point.