# Formula for Doppler effect in moving medium?

• I
• xxxyyy
In summary, the source and receiver can move at different speeds with respect to the medium, and the relativistic doppler formula must be used.
xxxyyy
TL;DR Summary
Contradictory results in Doppler effetc formulas
Hi,
I know the usual formula for both moving source and receiver in a static medium (from wiki):

Is ir correct?

What about when the medium is moving too?
I can't seem to find an answer, and worst, I'm finding contradicting ones.
For example, when the source and the receiver are moving at the same speed in the same direction, someone says there should be no Doppler shift in frequency, and so does the prevoius formula.

But then, picture this: source and receiver both stationary in moving medium (wind blowing from the source to the receiver, to fix the ideas), then the receiver hears a higher frequency because wavefronts are pushed towards it, and the formula to be used is that of a fixed source and moving receiver, moving towards the source at the speed of the wind in a stationary medium.

The two experiments are physically the same... but the formulas give differtent results.
So, which formula is right?
What's going on here?

Thank you!

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Search for "Fizeau drag effect"

We can use the relativistic addition of velocity formula, taylor expanded for v<<c and obtain
## u = u' + v\left( 1- \dfrac{1}{n^2}\right) ##

Remember that if light travels in a medium, its speed is not equal to c and you can not use the standard doppler formula (which assumes vacuum)

The relativistic doppler formula, for light in vacuum is
## f = f' \sqrt{\dfrac{c-v}{c+v}} ##
The formula you qouted is not relativistic

drmalawi said:
Search for "Fizeau drag effect"

We can use the relativistic addition of velocity formula, taylor expanded for v<<c and obtain
## u = u' + v\left( 1- \dfrac{1}{n^2}\right) ##

Remember that if light travels in a medium, its speed is not equal to c and you can not use the standard doppler formula (which assumes vacuum)
I think OP is asking about sound not light since wind is mentioned.
To @xxxyyy : The formula you quote assumes that the medium is at rest and the source and receiver are moving with velocities ##v_s## and ##v_r## relative to it. Can you figure out the relative velocities when all three are moving relative to the ground?

Delta2 and malawi_glenn
xxxyyy said:
Summary: Contradictory results in Doppler effetc formulas

Hi,
I know the usual formula for both moving source and receiver in a static medium (from wiki):
View attachment 303424
Is ir correct?
What about when the medium is moving too?
I can't seem to find an answer, and worst, I'm finding contradicting ones.
For example, when the source and the receiver are moving at the same speed in the same direction, someone says there should be no Doppler shift in frequency, and so does the prevoius formula.
But then, picture this: source and receiver both stationary in moving medium (wind blowing from the source to the receiver, to fix the ideas), then the receiver hears a higher frequency because wavefronts are pushed towards it, and the formula to be used is that of a fixed source and moving receiver, moving towards the source at the speed of the wind in a stationary medium.
The two experiments are physically the same... but the formulas give differtent results.
So, which formula is right?
What's going on here?
Thank you!
The above formula is for the rest frame of the medium. If the medium is "moving", then that is equivalent to the source and receiver moving at the same velocity relative to the medium.

That is the Gallilean principle of relativity, which allows you always to consider a scenario from any reference frame.

Note that this principle applies to non-relativistic physics.

Delta2
kuruman said:
I think OP is asking about sound not light since wind is mentioned.
I could swear this thread was in the "relativity" section. Too hot outside (and inside), need a drink

kuruman said:
I think OP is asking about sound not light since wind is mentioned.
To @xxxyyy : The formula you quote assumes that the medium is at rest and the source and receiver are moving with velocities ##v_s## and ##v_r## relative to it. Can you figure out the relative velocities when all three are moving relative to the ground?
You mean I should move to a new reference frame in which the medium is at rest and look at the velocities of source and receiver in this new reference frame?
But then, if originally the source and receiver are at rest with the wind blowing, and then I move to a reference frame with no wind, the source and receiver move at the same speed in the same direction, and so no Doppler shift... but that's wrong, because there is, and it's like the receiver is moving toward the source at the velocity of the wind, when there is no wind... at least that's what my undergrad text is saying.

vanhees71
xxxyyy said:
You mean I should move to a new reference frame in which the medium is at rest and look at the velocities of source and receiver in this new reference frame?
But then, if originally the source and receiver are at rest with the wind blowing, and then I move to a reference frame with no wind, the source and receiver move at the same speed in the same direction, and so no Doppler shift... but that's wrong, because there is, and it's like the receiver is moving toward the source at the velocity of the wind, when there is no wind... at least that's what my undergrad text is saying.
This is wrong. One of the foundations of modern physics since Galileo and Newton is that we can study physics in any (inertial) reference frame.

In your scenario the wind is moving past the source as well as the receiver.

kuruman, Orodruin and vanhees71
True, but you have to stick to one frame for each study ;-)).

vanhees71 said:
True, but you have to stick to one frame for each study ;-)).
Depends on what you study …

vanhees71
xxxyyy said:
You mean I should move to a new reference frame in which the medium is at rest and look at the velocities of source and receiver in this new reference frame?
And then what? Perhaps you should consider if there is a frequency shift when there is no wind but both source and receiver are moving relative to the ground and are at rest relative to each other. Of course, this is equivalent to source and receiver at rest with respect to the ground and a wind blowing.

All this is answered in my article mentioned in #6...

Sound waves are in a medium (air), and can't be treated by relativity, which depends only on relative velocity with no difference between source and receiver. For sound, a good experiment is to listen to an outdoor concert on a windy day. The orchestra is not out of key.

Meir Achuz said:
Sound waves are in a medium (air), and can't be treated by relativity, which depends only on relative velocity with no difference between source and receiver.
It is perfectly possible to treat waves in a medium using relativity (see #6). Of course the wave speed relative to the medium and the medium rest frame will enter into the treatment and the result will not be the ”standard” relativistic Doppler effect, but it is certainly possible.

Meir Achuz said:
Sound waves are in a medium (air), and can't be treated by relativity, which depends only on relative velocity with no difference between source and receiver. For sound, a good experiment is to listen to an outdoor concert on a windy day. The orchestra is not out of key.
Of course, you can describe soundwaves relativistically. Why shouldn't you be able to do so? The most simple derivation is as an approximate solution of the relativistic perfect-fluid hydro equations as in the non-relativistic case.

For a relativistic treatment of the general wave equation (not only for em. waves in a vacuum) see

https://itp.uni-frankfurt.de/~hees/pf-faq/rela-waves.pdf

The important point to realize is that em. waves in a vacuum is a special case, because the vacuum is Poincare invariant, while if you consider wave propagation in a medium (eg., em. waves in a dielectric or sound waves) you have the (local) rest frame(s) of the medium as a preferred frame. That makes the difference: While for em. waves in the vacuum the only relevant velocity for the Doppler effect is the relative velocity between the sender and the receiver, while for waves related to wave propagation in a medium you have in addition the velocities of the medium relative to observer and the receiver.

The above manuscript describes only the approximation for a monochromatic wave in the region of normal dispersion, where the in-medium phase velocity of the wave is less than ##c##. For the also interesting case of anomalous dispersion you need to take into account the corresponding in-medium Green's function (##\omega##-dependent refractive index). For that see, e.g., Sommerfeld, Lectures on Theoretical Physics, vol. 4 (optics).

PeroK
"But then, picture this: source and receiver both stationary in moving medium (wind blowing from the source to the receiver, to fix the ideas)," I guess I was referring to the original question.

## 1. What is the formula for Doppler effect in moving medium?

The formula for Doppler effect in moving medium is given by:
f' = f (v ± u) / (v ± vs)
where f' is the observed frequency, f is the emitted frequency, v is the speed of sound in the medium, u is the velocity of the source relative to the medium, and vs is the velocity of the observer relative to the medium.

## 2. How does the Doppler effect in moving medium differ from the formula for stationary medium?

The main difference between the two formulas is the addition of the velocities of the source and observer relative to the medium in the formula for moving medium. This is because the motion of the source and observer also affects the perceived frequency of the sound waves in addition to the speed of sound in the medium.

## 3. Can the Doppler effect in moving medium be applied to any type of wave?

Yes, the Doppler effect in moving medium can be applied to any type of wave, including sound waves, light waves, and electromagnetic waves. The formula remains the same, but the values for the speed of the wave in the medium and the velocities of the source and observer may vary depending on the type of wave.

## 4. How is the Doppler effect in moving medium used in real life?

The Doppler effect in moving medium has many practical applications in various fields such as astronomy, meteorology, and medical imaging. For example, it is used to measure the speed and direction of objects in space, to track weather patterns, and to create images of internal body structures using ultrasound technology.

## 5. What are the limitations of the Doppler effect in moving medium?

The Doppler effect in moving medium assumes that the source and observer are moving in a straight line relative to each other and that the medium is uniform and unchanging. In reality, these conditions may not always be met, which can lead to inaccuracies in the calculated frequency shift. Additionally, the effect is only applicable to relative motion between the source and observer, and cannot account for other factors that may affect the perceived frequency, such as the shape or size of the source or the medium through which the wave is traveling.

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