[Halc]

For wave propagation in medium, Doppler effect can still occur even without change of distance between source and observer, if they move through the medium.

You have an example of that with light?  I can do it with sound, but sound doesn't obey Galilean relativity:
Imagine a pair of supersonic jets always staying 1 km apart, but one circling the other.  The sound travels only one-way from the lead jet to the rear one.  As it moves from directly behind to the shock wave of the leading jet, the tone will change.
I suppose that sort of thing could be done by shining light through a high refractive index material moving fast, but unclear how one would shine light into such material.

[hamdani yusuf]

You have an example of that with light? 

https://en.wikipedia.org/wiki/Fizeau_experiment

[Halc]

You have an example of that with light?
...
I suppose that sort of thing could be done by shining light through a high refractive index material moving fast

https://en.wikipedia.org/wiki/Fizeau_experiment

OK, he's using water here.

There is still no Doppler effect in that picture.  Both output beams shine with the same frequency as it would if the source was observed directly without the intervening apparatus.  All it does is a phase shift on both sides.

[hamdani yusuf]

OK, he's using water here.

There is still no Doppler effect in that picture.  Both output beams shine with the same frequency as it would if the source was observed directly without the intervening apparatus.  All it does is a phase shift on both sides.

The Doppler effect (or the Doppler shift) is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.

https://en.wikipedia.org/wiki/Doppler_effect
From the definition above, Doppler effect can happen if wavelength changes even if the frequency stays the same, which means that the propagation speed also changes accordingly.

From the diagram, at the right end of water column, water velocity at light direction is 0, hence the Doppler effect is canceled at that points, and the light frequency and wavelength of the top light becomes the same as the bottom light as they come out of water before observed. But the Doppler effect has occured along the top and bottom horizontal columns by changing propagation speed, frequency, and wavelength, which generate changes in interference pattern at the detector. Hence the changes of each individual parameters can't be directly measured, but either frequency or wavelength must have been changed, thus Doppler effect must have happened.

[Halc]

From the diagram, at the right end of water column, water velocity at light direction is 0, hence the Doppler effect is canceled at that points, and the light frequency and wavelength of the top light becomes the same as the bottom light as they come out of water before observed. But the Doppler effect has occured along the top and bottom horizontal columns by changing propagation speed, frequency, and wavelength, which generate changes in interference pattern at the detector. Hence the changes of each individual parameters can't be directly measured, but either frequency or wavelength must have been changed, thus Doppler effect must have happened.

I agree that the wavelength is shorter in the water, but it would even if it wasn't moving.  This is due to a changed speed of light in a non-vacuum, not a change in frequency.
So if the observer looks at one beam or the other (or observes from within the water) he'll find them all at the exact same frequency.  All the device does is a phase shift, not a frequency change.  The only way to change the frequency is to move the light source (depicted by a little sun in the picture) or move the observer.

[Halc]

The Doppler effect (or the Doppler shift) is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.

https://en.wikipedia.org/wiki/Doppler_effect
From the definition above, Doppler effect can happen if wavelength changes even if the frequency stays the same, which means that the propagation speed also changes accordingly.

I agree that the definition above says that, but it is wikipedia, and I think they mean frequency change and associated wavelength change and not the case of wavelength change associated only with refraction and not frequency change.  I'm saying wiki is wrong here.

From oxford dictionary (top of list if you google "what is doppler effect"):
"[physics:] an increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move toward (or away from) each other. The effect causes the sudden change in pitch noticeable in a passing siren, as well as the redshift seen by astronomers."

Britanica:  "Doppler effect, the apparent difference between the frequency at which sound or light waves leave a source and that at which they reach an observer, caused by relative motion of the observer and the wave source."

webster:
"a change in the frequency with which waves (as of sound or light) from a given source reach an observer when the source and the observer are in motion with respect to each other so that the frequency increases or decreases according to the speed at which the distance is decreasing or increasing"

http://physics.bu.edu/~duffy/py105/Doppler.html
"The Doppler effect describes the shift in the frequency of a wave sound when the wave source and/or the receiver is moving."

Pretty much every place except wiki says it's a frequency shift and does not consider a wavelength change without frequency change to be an example of Doppler effect.

[hamdani yusuf]

The Doppler effect (or the Doppler shift) is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.

https://en.wikipedia.org/wiki/Doppler_effect
From the definition above, Doppler effect can happen if wavelength changes even if the frequency stays the same, which means that the propagation speed also changes accordingly.

I agree that the definition above says that, but it is wikipedia, and I think they mean frequency change and associated wavelength change and not the case of wavelength change associated only with refraction and not frequency change.  I'm saying wiki is wrong here.

From oxford dictionary (top of list if you google "what is doppler effect"):
"[physics:] an increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move toward (or away from) each other. The effect causes the sudden change in pitch noticeable in a passing siren, as well as the redshift seen by astronomers."

Britanica:  "Doppler effect, the apparent difference between the frequency at which sound or light waves leave a source and that at which they reach an observer, caused by relative motion of the observer and the wave source."

webster:
"a change in the frequency with which waves (as of sound or light) from a given source reach an observer when the source and the observer are in motion with respect to each other so that the frequency increases or decreases according to the speed at which the distance is decreasing or increasing"

http://physics.bu.edu/~duffy/py105/Doppler.html
"The Doppler effect describes the shift in the frequency of a wave sound when the wave source and/or the receiver is moving."

Pretty much every place except wiki says it's a frequency shift and does not consider a wavelength change without frequency change to be an example of Doppler effect.

You are correct. In order to get frequency change, we have to change the number of waves in transit between source and observer, such as when distance between source and observer changes. This can also be done by accelerating medium.

[hamdani yusuf]

You are correct. In order to get frequency change, we have to change the number of waves in transit between source and observer, such as when distance between source and observer changes. This can also be done by accelerating medium.

It makes me wonder if both source and observer accelerate uniformly. Does the light received by observer have the same frequency as the light emitted by the source? How much is the difference?

[Halc]

It makes me wonder if both source and observer accelerate uniformly. Does the light received by observer have the same frequency as the light emitted by the source? How much is the difference?

The proper distance between them changes if one is in front of the other, but not side-by-side.  So no redshift in the latter case.
In the former case, the lead ship will outdistance the trailing one in the frame of either ship, so there will be a small Doppler effect as the proper distance between them grows.
If, on the other hand, the two observers are in the same ship but opposite ends, the proper distance between the two would be fixed and the acceleration of each would not be the same and the one in front would see a red-shifted light from the rear and a blue shift looking the other way.  This is a pure relativistic effect and not Doppler.

[hamdani yusuf]

It makes me wonder if both source and observer accelerate uniformly. Does the light received by observer have the same frequency as the light emitted by the source? How much is the difference?

The proper distance between them changes if one is in front of the other, but not side-by-side.  So no redshift in the latter case.
In the former case, the lead ship will outdistance the trailing one in the frame of either ship, so there will be a small Doppler effect as the proper distance between them grows.
If, on the other hand, the two observers are in the same ship but opposite ends, the proper distance between the two would be fixed and the acceleration of each would not be the same and the one in front would see a red-shifted light from the rear and a blue shift looking the other way.  This is a pure relativistic effect and not Doppler.

If they accelerate uniformly, their distance should not change.

In special relativity, lengths only contracts while time only dilates when an inertial system is observed by other inertial systems moving at constant velocity relative to the observed one. So I assume you are talking about general relativity there.

[Halc]

It makes me wonder if both source and observer accelerate uniformly. Does the light received by observer have the same frequency as the light emitted by the source? How much is the difference?

The proper distance between them changes if one is in front of the other, but not side-by-side.  So no redshift in the latter case.
In the former case, the lead ship will outdistance the trailing one in the frame of either ship, so there will be a small Doppler effect as the proper distance between them grows.
If, on the other hand, the two observers are in the same ship but opposite ends, the proper distance between the two would be fixed and the acceleration of each would not be the same and the one in front would see a red-shifted light from the rear and a blue shift looking the other way.  This is a pure relativistic effect and not Doppler.

If they accelerate uniformly, their distance should not change.

True, but their proper distance will change, so one observing the other will potentially observe a shifted light.  It is arguably a Doppler shift because the proper distance between source and observer is constantly changing.  Accoring to an inertial observer, the distance between the two identically accelerating things does not change, but the frequency at which the light is emitted does change.

In special relativity, lengths only contracts while time only dilates when an inertial system is observed by other inertial systems moving at constant velocity relative to the observed one. So I assume you are talking about general relativity there.

Pretty much just SR, since no gravity is involved in my statements above. SR is not just about inertial systems. One can always integrate inertial solutions to derive solutions for accelerating systems.