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
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Note the
medium
phase velocity
is relative to the
rest frame of
the medium
and is a property of the
medium.
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'.
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.
Note only to
1st order
does the Doppler shift
depend only on the
relative velocity
and NOT on v_1 and v_2 individually.
The basic formula still applies,
taking the source as the observer moving with source
velocity v_source:
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
What is the wavelength
observed for the
source moving with velocity v_source
by an observer moving at
velocity v?
Behold:
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:
Making use of a Taylor expansion,
we find the
1st order
formulae in simplified form to be
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.
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:
Note the formulae depend just on
relative velocity
in this case NOT the individual
velocities.
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.
Derivation of
the Doppler effect
in a medium
in the
classical limit:
Credit/Permission: ©
User:Abhinav P B,
2021 /
Creative Commons
CC BY-SA 4.0.
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 formula
f' = 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.
f' = f(1 - v/v_ph) gives 0 < f' < f ,
and the moving observer sees a
redshift.
f' = f(1 - v/v_ph) gives f' > f ,
and the moving observer sees a
blueshift.
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
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 does NOT
matter to
1st order,
and Δv is the
relative velocity
between the observers.
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.
λ_source = v_ph/f_source
which is the
wavelength
all observers would measure if the
source were at rest.
This much is clear.
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.
λ = λ_source(1-v_source/v_ph) .
Δλ/λ_source = -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.
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.
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.
Δf/f = -v/c
Δfλ/λ_1 = v/c .
Image link: Wikimedia Commons:
File:Travelling wave animated plot.gif.
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