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frostysh
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- TL;DR 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?
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?
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.
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.
(Cannot find facepalm smile.) The invariant is something that is not changing due to change of intertial frame. The speed of wind will change, this is not an invariant.