# B Help with understanding the Doppler effect

#### frostysh

Summary
Why it is so 'absolute' and why the case of source of waves is different from the observers at concept level?
My answer on this question for now is that producing a waves in the medium is an event which is basically must be invariant in the any of frame of references. For an example: a brick is freely falling, then the brick suddenly splinted into two pieces — no matter from which frame will we observe this situation, the brick will not glue off together because of that. That is means an event such as destroying of bricks is absolute to the any frame of references. I have a poor understanding of the subject so in the bottom I will describe my primitive model and viewing of the situation, maybe it will help to clarify something at this level.

The article about discovering of the effect has been published by Christian Andreas Doppler in 1842, which born in Austrian Empire. The meaning of the effect is in waves properties changing due to moving of the observer or the source of waves considering to the medium.

First we may imagine very simplified picture of the model of wave process which can be described by the equation of $$\large A \left(t \right) = \left|A_{max} \right| \sin{t},$$ where $\large A(t)$ — is somekind of property value of that 'waver' during time, for an example it can be the pressure in medium, $\large \left|A_{max} \right|$ — the maximum value of the property, which is constant. And we have set up frame and measure of distance in the way of sinus period, so the function is simple.

First we may answer on question: how to imagine the wave process in the simple case? Well, it can be barely done with the next modeling: imagine an observer, a source and the medium. The source is a straight plank with the length of $\large 2 \left|A_{max} \right|$, placed perpendicular to the distance line between observer and the source, that line hit the middle of the plank. Then let's imagine the holes on that plank, from which we can shoot with a small cores, which moving with constant velocity on trajectory of a straight line parallel to the distance line. The every core have some particular value inscribed on it, $\large A \nearrow$ or $\large A \searrow$, the arrow up when the closest previous by time neighborhood of the core have a lover value $\large A$ than a core, and closest next neighborhood have larger value than a current core, the arrow down when the opposite situation. In the case of the shooting holes placed on the edge of the plank, there will be no any arrow. We should consider that $\large A \left(t \right) \nearrow \neq \large A \left(t \right) \searrow$ even the values of our property is exactly the same.
Now we can launch the first core from the port on the middle, after some time the second core from the lower edge port and after the same period of time the middle port will shoot again, and so on. Cores is moving with the same and constant speed so the distance between them will be the same in the any time. When the first core will travel the some characterizing distance $\large 2cT$, where $\large c$ — is the speed of the core, $\large T$ — some characterizing time of launchung of the absolutely identical neighborhood in time core to the particular one, $\large A \nearrow = \large A \nearrow$ if inscribed $\large A$ is equal, the nine cores will be shot at all.

If we can make a very large amount of ports, and decrease the time interval between shots then we can obtain a smooth line that build from cores, this curve which consist of the values of $\large A$ will be dedicated to the timeline and a projection of this curve on a distance line between source and observer of waves in the medium will give to us somekind a picture of the wave propagation process.

For now will be interesting to imagine what will happs when the observer of the waves is resting due to frame of reference hard connected to the medium and the source of the waves (in our model it's a plank) is moving dedicated to this frame. The direction of this movement is straight into observer with the trajectory of the distance line between them.
Let imagine a shooting of the first core from the middle port, in the moment of time $\large t = 0$, in the zero point of our coordinates, after some time we shoot second core, but the problem is that by this time the plank has traveled distance to the observer which is equal $\large vt$, where $\large v$ — is the speed of the plank related to the frame of medium and it is constant. The first core will be on distance $\large ct$ from the zero point, where $\large c$ — is the speed of the core, and in the our case $\large c > 0$, $\large v > 0$, and $\large c > v$. When some characterizing time $\large 2T$ has passed, again will be launched a nine cores but the position of the plank will be far from the zero point (in the middle of the picture).

As we can see, in this case the place of the cores will be different compared to the situation when the source is in rest to the medium (and observer), and the characterizing distance between two closest in time, absolutely identical cores, fro and example cores number $\large 1 — 5 — 9$ will be different too. But this is continuous process actually, there is no any cores with inscribed values, so how we can imagine it . . . ? It's can be imagine like the sources is chasing the waves and in the same time producing the waves continuously.

We have done some thing with an imagination, so for now we should look what can give to use some formals and sign dedicated to it, we mean algebra of course. The characterizing time of absolute repeating is called period of wave process $\large T$, dedicated distance between two peaks of the value of the amplitude $\large A$ in waves is called the wavelength $\large \lambda$, the wave propagation speed $\large c$ is the property of medium itself, it's not depends on the frame of reference we choose. The frequency $\large \nu$ is the how many period of waves was in the unit of time, for an example we have forty periods on twenty seconds, how many periods will be on the one second? The answer is two periods per second so the frequency is two hertzs in SI system of units, $\large \nu = \dfrac{2} {1 \thinspace \text{sec}} = 2 \cdot 1 \thinspace \left(\text{sec} \right)^{-1}$. Esily to see that the frequency is reciprocal to the value of period measured in the time units.
From the picture we can notice that $\large \lambda = cT$ will decrease in case of moving source of the waves, on which value it will decrease? $$\large \bigtriangleup \lambda = \lambda_{0} - \lambda = cT - \left(c - v \right)T = vT$$ The $\large \lambda_{}0$ — is a wavelength in stationary source case. But if the wavelength is changed, and the speed of wave propagation is the same then due to their law of relationship must be changed the period $$\large \lambda = \lambda_{0} - \bigtriangleup \lambda = \left(c - v \right)T_{0}$$ $$\large \left(c - v \right)T_{0} = cT,$$ $\large T$ — period of waves when source is resting due to medium and observer, from this point we can show the new period in terms of the stationary case $$\large T = \dfrac{\left(c - v \right)T_{0}}{c} = \left(1 - \dfrac{v}{c} \right)T_{0},$$ and corresponding to, the new frequency will be $$\large \nu = \dfrac{1}{ \left(1 - \dfrac{v}{c} \right)T_{0}} = \dfrac{\nu_{0}}{\left(1 - \dfrac{v}{c} \right)},$$ as easily to see the frequency enlarging comparing to the resting source situation. If the source moving in the opposite direction of the observer which is resting in the frame of medium, the distance between them is continuously increasing, so the characterizing wavelength is increasing too, and will be $$\large \lambda = \left(c + v \right)T_{0},$$ and the corresponding frequency $$\large \nu = \dfrac{\nu_{0}}{\left(1 + \dfrac{v}{c} \right)},$$it will decrease corresponding to the rest case as easily to see.

Surprising and even shoking (to me at least) in so called Doppler's effect is that the case of resting source of waves due to frame of references of medium, and moving observer in this frame is totally different! Again our observer will be a plank, transparent plank that providing measurements without any interactions with a medium, the observer is moving trough medium right to the source of wave propagation, moving with constant speed with trajectory of line of distance between them.

As we can see from the picture the characterizing distance between cores after and before the observer plank is the same. The moving of the observer is not changing wavelength. The the period of time between two absolutely same values of waver property which observer will detect will change corresponding to the rest case. Why it will change? — Because observer is moving towards wave, and the each value of property is moving like a shooting cores with constant distance between them because the speed of propagation is the very property of the medium, so the characterizing time between two absolutely the same values will change, after the detection of the first peak, the observer will travel some distance further 'towards to closing' second peak by some time, and second peak (or the core in our case) will travel some distance too by the same time. It is easily to see that distance traveled by observer and by the core after detection first peak to the time of detection the second should be related in the next way $$\large vT_{r} + cT_{r} = \lambda,$$ where $\large T_{r}$ — is the time passed from registration of the first peak to the registration of the second. We know that $\large \lambda = cT$, so we can find the new 'period' reflated to the period in rest case, from this equation like that $$\large T_{r} = \dfrac{T}{1 + \dfrac{v}{c}},$$ the corresponding frequency which is based on period $$\large \nu_{r} = \left(1 + \dfrac{v}{c} \right) \cdot \nu,$$ and this is wondering asymmetry of the Doppler's effect. Because in general $$\large \nu_{r} \neq \nu$$ where $\nu$ — is the frequency from the case of resting observer and the moving source.

So the point is, if we have medium and Doppler's effect, we can distinguish which movement corresponding to the medium is taking place, observer or source. Also it's we should note that movement of observer is actually same as the movement of the medium in this effect, but movement of source of waves is not. Or I am talking something wrong?

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#### Dale

Mentor
My favorite reference on the Doppler effect is

First we may imagine very simplified picture of the model of wave process which can be described by the equation of $$\large A \left(t \right) = \left|A_{max} \right| \sin{t},$$ where $\large A(t)$ — is somekind of property value of that 'waver' during time
What you have described here is an oscillation, not a wave. To get a wave you need a function of time and space:
$$A(t,x)=A_{max}\sin(\omega t-k x)$$
Your whole plank and core thing confused me. I can’t help you with that. If you feel confused and you are using a confusing personal concept, the best approach is to discard the personal concept and learn standard concepts instead.

Without reference to the planks, in brief, what is your question?

#### frostysh

My favorite reference on the Doppler effect is

I have looked at it briefly and understand that it is for relativistic case, not the classical one, but I will read, thanx.

What you have described here is an oscillation, not a wave.
Me apologizing... Somehow not even saw such mistake. I just trying to self education and repeat my university program from long ago, so I may make such mistakes.
Your whole plank and core thing confused me. I can’t help you with that. If you feel confused and you are using a confusing personal concept, the best approach is to discard the personal concept and learn standard concepts instead.
I thought it will be more easily to imagine than a wave from a moving source.
Without reference to the planks, in brief, what is your question?
Is wavelength in the Doppler's effect changing or not? Is it changing in all variants of moving source, moving observer referred to the medium? Is Doppler effect absolute and not relative and using this effect we can distinguish what is actually moving referred to medium, observer or source?

For me understand for now, there is two cases, first moving source and there is actually appear physical wavelength change. Second case is a moving observer, which is not changing any wavelength. So the Doppler's effect is not relativistic in terms of classical relativism.

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#### Dale

Mentor
I have looked at it briefly and understand that it is for relativistic case, not the classical one
It is for both and shows how both can be treated identically. That is why it is my favorite.

Is wavelength in the Doppler's effect changing or not? Is it changing in all variants of moving source, moving observer referred to the medium?
Yes, the wavelength changes, in general, in all cases.

Is Doppler effect absolute and not relative and using this effect we can distinguish what is actually moving referred to medium, observer or source?
All laws of physics are relative in the sense that they are the same in all reference frames. Therefore, you are always free to work the problem in the reference frame where it is simplest. For waves in a medium, this is usually the rest frame of the medium. Then, the question becomes: “in the frame of the medium, can we use the Doppler effect to unambiguously determine the speed of the emitter and the speed of the receiver?” Is this a fair restatement of the question?

#### A.T.

“in the frame of the medium, can we use the Doppler effect to unambiguously determine the speed of the emitter and the speed of the receiver?” Is this a fair restatement of the question?
Since it's the velocities that are frame dependent, not the Doppler shift, maybe better phrased as

Can we use the Doppler shift to unambiguously determine the velocities of the emitter and the receiver relative to the medium?

If that's the question, then the answer is no, because there are different combinations of emitter and receiver velocities relative to the medium, that give the same Doppler shift. But these combinations are not simply swapped velocities (no symmetry).

#### frostysh

It is for both and shows how both can be treated identically. That is why it is my favorite.
There is nothing about classic, as I can see, just a bold formula which is not giving any answers of the physical meaning of that, well at least me cannot see it there.
Yes, the wavelength changes, in general, in all cases.
It is impossible to change the wavelength without interraction with medium. Observer has no any interraction, it's a very different from the source of waves. We can imagine that like we are moving in medium near a wave in which the different values of the pressure is colored in the different colors, as we look from aside, there is distance between a colors will be the same as we are in rest correspondig to medium, but if the source of waves start to move corresponding to medium, the will be actual difference in wavelengths.
All laws of physics are relative in the sense that they are the same in all reference frames. Therefore, you are always free to work the problem in the reference frame where it is simplest. For waves in a medium, this is usually the rest frame of the medium. Then, the question becomes: “in the frame of the medium, can we use the Doppler effect to unambiguously determine the speed of the emitter and the speed of the receiver?” Is this a fair restatement of the question?
Actually I don't asked about the values of speed (which is obviously can be obtained from the formula), but something like that, you understand right.
The point is Doppler effect is not relative, it is absolute corresponding to medium, me think so, we can change frame of reference and the Doppler's effect will be not changed, the wavelength is absolute distance, it is an invariant. Actually me start to understand. The movement of source of waves is absolute in medium, like an 'ether' or something, using this movement we can distinguish absolute rest corresponding to the medium.

#### Dale

Mentor
It is confusingly phrased
Sorry about that. As near as I can tell the questions are identical, so if you think mine is confusing then I am happy using yours.

#### Dale

Mentor
It is impossible to change the wavelength without interraction with medium.
This is incorrect. The wavelength can be changed simply by changing reference frame of the analysis.

The point is Doppler effect is not relative, it is absolute corresponding to medium
Unfortunately, the word “absolute” has all sorts of negative connotations here. Can you express your question without it? I am pretty sure you do not intend it to have the negative connotation that it evokes.

#### A.T.

The point is Doppler effect is not relative, it is absolute corresponding to medium,
You have simply replaced "relative to..." with "absolute corresponding to...". What is the difference?

#### frostysh

If that's the question, then the answer is no, because there are different combinations of emitter and receiver velocities relative to the medium, that give the same Doppler shift. But these combinations are not simply swapped velocities (no symmetry).
Can you prove it? Me think it's terribly wrong, the point is we are measuring the wavelength, and definitely know the speed of the source of waves corresponding to medium. The speed of observer can do no change to the wavelength, only to detected period of wave.

Emmiter and reciever is not summetric, we cannot "switch" them in Doppler's effect and obtain the same effect. Or maybe me wrong... But it is interesting, indeed.

#### Dale

Mentor
Can we use the Doppler shift to unambiguously determine the velocities of the emitter and the receiver relative to the medium?

If that's the question, then the answer is no,
Can you prove it?
That the answer to the question formulated by @A.T. is “no” is almost trivial to prove. There are two unknowns and only one equation. There is generally no unique solution to one equation in two unknowns.

#### frostysh

This is incorrect. The wavelength can be changed simply by changing reference frame of the analysis.

Unfortunately, the word “absolute” has all sorts of negative connotations here. Can you express your question without it? I am pretty sure you do not intend it to have the negative connotation that it evokes.
It's simple: Doppler's effect is an invariant (if the medium is not changing), like a mass of a body, no matter the speed of frame of reference of the observer in which we will measure it.

Also I found that the Doppler's effect is an invariant to Galilean transformation, but the wavelength changing only in case of the source of waves moving.

#### A.T.

the point is we are measuring the wavelength,
When I say Doppler shift, then I mean the shift in frequency, and no info about the wavelength. If you additionally know the wavelength, then you can find both velocities for co-linear movement.

#### frostysh

That the answer to the question formulated by @A.T. is “no” is almost trivial to prove. There are two unknowns and only one equation. There is generally no unique solution to one equation in two unknowns.
I am not understanding of what you saying, but we can measure the wavelength in case of rest observer, and if we know it (and know it because we know the very feature of the medium $\large c$), and the achieving of the speed of the source of the wave is becoming trivial, there not only mathematics, there is physics. Sec I will post the prove which is I not understand but I googled, that Doppler's effect is an invariant.

#### A.T.

Sorry about that. As near as I can tell the questions are identical, so if you think mine is confusing then I am happy using yours.
I'm sorry for sounding too critical. I rephrased it as a suggestion.

#### A.T.

I am not understanding of what you saying, but we can measure the wavelength ...
See post #13. Doppler shift usually just means the shift in frequency. I did not assume we know the wavelength in that question.

#### frostysh

When I say Doppler shift, then I mean the shift in frequency, and no info about the wavelength. If you additionally know the wavelength, then you can find both velocities for co-linear movement.
If we know the characterizing feature of the medium, which is the speed of wave propagation there — $\large c$, we can found that source moving or not corresponding to medium, it is easily first we found the frequency (simple measuring), then we can measure wavelength of mechanical waves (it's simply too), and then just comparing wavelengths in case of rest to our case, if there are difference — source is moving, if there not, maybe moving observer or maybe medium. Something like that, but me can be wrong...

#### frostysh

This is incorrect. The wavelength can be changed simply by changing reference frame of the analysis.
Me think you are totally, terribly wrong (me actually thought long ago too that change appear), and I have googled the prove of it. Well, me no good in mathematics and there, but even me can read English: "the wavelength is the same in both frames" — Galilean Transformation of Wave Velocity (page 2). The point is the wavelengths is only changing when source of waves is moving corresponding to the medium — this is mean the Doppler's effect is breaking relativity of frame of references of classical mechanics because wavelength is an invariant! This is can be feel on intuitive level, the length is actually invariant like a mass or like, I don't know, the volume.
See post #13. Doppler shift usually just means the shift in frequency. I did not assume we know the wavelength in that question.
Me stupid then, not understand, of course then in equation will be many unknowns. But in case of physics I have a no idea if we are in the medium, why we cannot measure it's features like characterizing speed of wave propagation in rest? Just take some medium in bottle and use sound waves on it... But if there medium with which we cannot interact, so there is no point to talk about any medium at all.

Mentor

#### A.T.

this is mean the Doppler's effect is breaking relativity of frame of references of classical mechanics because wavelength is an invariant!
How does the invariance of wavelengths break the relativity of frame of references of classical mechanics?

#### vanhees71

Gold Member
I'm a bit puzzled by this thread, because it simply answers the question about the Doppler effect for light in a vacuum. This is pretty simple when just writing a plane em. wave in a relativistic covariant way and then Lorentz boost to another frame of reference (or you write it covariantly to begin with by introducing the four-velocities of the source and the observer; I've to think about the details, how to really do this).

But since in the OP the question is about a medium, maybe the question was about the Doppler effect for sound? This can of course also be formulated relativistically, and it's quite illuminating since in contradistinction to em. waves in the vacuum here we indeed have a medium (the air for usual acoustic phenomena). Thus in this case we have even three four-velocities to consider: the four-velocities of the sound source, the observer, and of the medium.

#### frostysh

That is for the Galilean transformation. However, physics is governed by the Lorentz transform, not the Galilean transform.
What Lorentz's transformation? The same Lorentz's transformation which in the Special Relativity Theory kinematics? Well, me no understand what is mean "governed" in that case, I think old uncle Newton's model working perfectly in most of cases, in case of a classical Doppler's effect especially.
How does the invariance of wavelengths break the relativity of frame of references of classical mechanics?
I am apologizing, no good word 'break'... I mean we have two body moving, we don't know which body is 'actually' moving (with constant velocities relative to each other, so called inertial frames), this questions in relativity have no any sense, we can suggest that the first body moving and describe this movements and etc, you know... But in case of Doppler's effect appear, the movement of medium corresponding of source of waves and the movements of the source of waves corrpesponding to medium, is not the same, we cannot jump so easily, despite it's both inertial frames of references. At least on my present level of understanding of this curious phenomenon.
I'm a bit puzzled by this thread, because it simply answers the question about the Doppler effect for light in a vacuum. This is pretty simple when just writing a plane em. wave in a relativistic covariant way and then Lorentz boost to another frame of reference (or you write it covariantly to begin with by introducing the four-velocities of the source and the observer; I've to think about the details, how to really do this).

But since in the OP the question is about a medium, maybe the question was about the Doppler effect for sound? This can of course also be formulated relativistically, and it's quite illuminating since in contradistinction to em. waves in the vacuum here we indeed have a medium (the air for usual acoustic phenomena). Thus in this case we have even three four-velocities to consider: the four-velocities of the sound source, the observer, and of the medium.
Well I am totally not understood what you talking about...

1) There is classical kinematics, no any relativistic effects on model.

2) Medium is not a vacuum of course, this is true, despite in this case we characterizing the medium only by a speed of wave propagation $\large c$ and saying it is constant despite of frame of reference, this is still not a relativistic case, and this is still not the vacuum physical meaning, vacuum cannot moving for an example, we cannot connect to it any frame of reference, but to medium we can, to object we can, but I am not heard 'vacuum frame of reference'.

3) We have only three velocities, only three frame of references, the all is an inertial frames. We can make only two, hardly connect the observer to the medium for an example. The third frame of reference in formula in my topic included by constant $\large c$, as you may know saying 'speed' in classical relativity have no any sense, when we saying 'speed' we means corresponding to something, we means at least two frame of reference, this is because speed of propagation of waves in medium giving addition frame to the observer and the source frames.

4) If you about philosophical question: "How we can introduce the single inertial frame without defining another, and this another can be defined only by next one frame, and so on..." — this is the question from the mathematical logic and so called 'reversed mathematics' which is super incredibly abstract, and which almost totally unknown to me, so this question I cannot answer.

Just to clarify first post about Doppler's effect. I just curious if my states a true or not:

• Wavelength is the invariant corresponding to the observer inertial frame of references, so we cannot distinguish which movement appear, the medium moving or the observer of waves moving.
• The wavelength is changing by changing the velocity of the source of the waves (look at first post pictures) corresponding to the medium, and due to it's invariantcy in the frames of observer we can say that this is somekind of invariant property by which we can distinguish what is actually moving, medium or the source corresponding to the medium.
• This is means Doppler's effect in classical kinematics is not relative it is 'absolute' in medium.

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#### A.T.

This is means Doppler's effect in classical kinematics is not relative it is 'absolute' in medium.
"Absolute in medium" is just a confusing way to say "relative to the medium", or simply "relative". It's not absolute if you have to reference it to something.

#### frostysh

"Absolute in medium" is just a confusing way to say "relative to the medium", or simply "relative". It's not absolute if you have to reference it to something.
The abstraction there is next: if medium exist then Doppler's effect can define motion in this medium regardles of the inertial frames, and this is me calling 'an absolute'.

#### Ibix

The abstraction there is next: if medium exist then Doppler's effect can define motion in this medium regardles of the inertial frames, and this is me callind 'an absolute'.
The wind allows you to detect motion with respect to a medium. It doesn't make the rest frame of the medium absolute - it's just the velocity with respect to the medium.

"Help with understanding the Doppler effect"

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