- 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 medium phase velocity is relative to the rest frame of the medium and is a property of the medium.

- Quantities measured in the
rest frame
are unprimed: i.e.,
frequency f,
period p,
wavelength λ,
and
medium
phase velocity v_ph.
Primed quantities are measured by observers moving with respect to the medium. The are observer-frame quantities: i.e., frequency f ', period p', wavelength λ', and phase velocity v_ph'.

- In the classical limit,
length (the distance
between 2
points
at one instant in time)
is invariant
for all observers no matter how they are moving.
Thus, the wavelength for a given wave phenomenon is invariant.

Thus, λ' = λ in all cases, and we no longer need the symbol λ' or the expression observer-frame wavelength.

- The derivation of
the Doppler effect for
the motion of an observer (moving at
velocity v
relative to the rest frame of
the medium)
in the direction of
wave propagation
is now simple:
f'λ = v_ph' and fλ = v_ph are general formulae and v_ph' = v_ph - v thus f'λ = v_ph - v = v_ph(1 - v/v_ph) = fλ(1 - v/v_ph) f' = f(1 - v/v_ph) .

The formulaf' = f(1 - v/v_ph) or f = f'/(1 - v/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. - Note
- For 0 < v < v_ph,
f' = f(1 - v/v_ph) gives 0 < f' < f ,

and the moving observer sees a redshift. - For v = v_ph, f '=0, and the moving observer sees the waves at rest.
- For v > v_ph, f ' < 0, and the moving observer is plowing through the waves from back to front.
- For v < 0,
f' = f(1 - v/v_ph) gives f' > f ,

and the moving observer sees a blueshift.

- For 0 < v < v_ph,
- The Doppler shift
between two moving observers 1 and 2 is described by
f_1'/(1 - v_1/v_ph) = f_2'/(1 - v_2/v_ph) .

If v_1/v_ph << 1 and v_2/v_ph << 1, then we have (making use of a Taylor expansion) the 1st order formulaf_2' - f_1' = -f_1'(v_2 - v_1)/v_ph

or in the simplifed relative Doppler shift formula formΔf/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 does**NOT**matter to 1st order, and Δv 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. - 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 ' = f_source and f is still the frequency measured in the rest frame. - The explication of
wavelength
for the case of fixed emission frequency
source is a bit tricky.
The wavelength does

**NOT**depend on observer velocity since wavelength is invariant with respect to observer velocity as emphasized above. But it does depend on the source velocity. This trickiness is often completely obscured in the explication of the Doppler effect resulting in complete incomprehension even if one thinks one comprehends.The intrinsic source wavelength λ_source is given by

λ_source = v_ph/f_source

which is the wavelength all observers would measure if the source were at rest. This much is clear.What is the wavelength observed for the source moving with velocity v_source by an observer moving at velocity v? Behold:

λ' = v_ph'/f' = (v_ph-v)/[f(v-v/v_ph)] = (v_ph-v)(1-v_source/v_ph)/[f_source(1-v/v_ph)] λ' = λ_source(1-v_source/v_ph) = λ ,

which is independent of the observer velocity as foretold.However, λ' does depend on the source velocity v_source as also foretold.

Fortunately, the formula for λ is simple even if the derivation is tricky:

λ = λ_source(1-v_source/v_ph) .

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

Δλ/λ_source = -v_source/v_ph .

- Note
- For 0 < v_source < v_ph,
λ = λ_source(1-v_source/v_ph) gives 0 < λ < λ_source ,

and the observer (moving with any velocity or none) sees blueshift (compressed) waves. - For v_source = v_ph, λ = 0, and all the crests of the waves are piled on top of each other. For sound, this is the sonic boom case.
- For v_source > v_ph, λ < 0. What does this mean? The interpretation is just that waves are moving in the source direction, but trailing behind it. You could still measure them as positive if you like: i.e., λ_measured = |λ|. All observers (moving with any velocity or none) measure the same wavelength.
- For v_source < 0,
λ = λ_source(1-v_source/v_ph) gives λ > λ_source ,

and the observer (moving with any velocity or none) sees redshifted (expanded) waves. In this case, the source is moving opposite to the direction of wave propagation.

Note if the source is emitting isotropically as it usually would, then there would be blueshifted (compressed) waves in the direction of source motion and redshifted (expanded) waves in the direction opposite of the source motion.

- For 0 < v_source < v_ph,
- The
electromagnetic radiation (EMR)
(in vacuum: i.e., with
**NO**medium) is the (extreme) relativistic Doppler effect is 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.In the case, of relativistic Doppler effect the invariant is, of course, vacuum light speed c,

**NOT**wavelength.We will

**NOT**give a 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. We do**NOT**use primes here since they have no use when there is**NO**medium.Note the formulae depend just on relative velocity in this case

**NOT**the individual velocities. - Making use of a Taylor expansion,
we find the
1st order
formulae in simplified form to be
Δf/f = -v/c Δfλ/λ_1 = v/c .

The first is exactly like the corresponding 1st order classical formula. But the second although similar in appearance to the corresponding 1st order classical formula has a very different meaning. Here the two observers measure different wavelengths and there all observerse measure the same wavelength which is just different from wavelength for the source when it is at rest in the medium. This is another tricky point.

Caption: An animation illustrating traveling waves.

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

EOF

Image link: Wikimedia Commons: File:Travelling wave animated plot.gif.

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