# Doppler shift formula and the assumptions behind it

Einstein in his "On the electrodynamics of moving bodies" (1905) starts the derivation of the Doppler shift formula by stating: In the system K, very far from the origin of coordinates let there be a sourcce of electromagnetic waves..." . Taking into account his advise "never stop thinking" we could ask if the Doppler shift formula he derives holds in the case of scenarios which do not fulfil the "very large source-receiver" assumption.
Consider a scenario proposed by A.P.French, Special Relativity" (Nelson 1968) pp 141-143 that involves a satellite (plane) that moves at a constant altitude emitting light signals at constant time intervals which are received by a stationary observer located on Earth. In order to recover Einstein's formula, French makes the assumption: ' The satellite travels a very small distance during one cycle of its transmitter signal which is equivalent with Einstein's assmption or with the assumption that the period at which the successive liht signals are emitted is very small (the frequency is very high). Take into account that we can emit successive light signals at lowmechanical frequencies.
My question is: Before using a ready derived formula following a given scenario, should we investigate if the assumtions made deriving it fit in the case of the scenario we follow.
My oppinion is that we should, deriving shift formulas in accordance with the scenario we follow showing how flexible special relativity theory is. What is your oppinion?
Sine ira et studio

Ich
French's assumption is not equivalent to Einstein's - there's no assumption about absolute distance, only the justification of differential calculus.
Einstein himself meant to examine plane waves, that's where the "very far" comes from.
Both approaches lead to exact results, independent of the chosen frequency. You only should bear in mind that if the doppler effect changes during one oscillation, the identification f=1/T becomes misleading, as T is discrete.

Doppler

French's assumption is not equivalent to Einstein's - there's no assumption about absolute distance, only the justification of differential calculus.
Einstein himself meant to examine plane waves, that's where the "very far" comes from.
Both approaches lead to exact results, independent of the chosen frequency. You only should bear in mind that if the doppler effect changes during one oscillation, the identification f=1/T becomes misleading, as T is discrete.

Please explain what do you mean by the Doppler effect changes during one oscillation. Thanks

Ich
I mean that the frequency of the incoming wave changes during one oscillation.
There's no conceptual problem with it. It's the same as angular velocity changing during one cycle in a rotating machine.
It may be hard to measure if you take only one count per cycle/oscillation. But that could be overcome.

doppler

I mean that the frequency of the incoming wave changes during one oscillation.
There's no conceptual problem with it. It's the same as angular velocity changing during one cycle in a rotating machine.
It may be hard to measure if you take only one count per cycle/oscillation. But that could be overcome.

IMHO the problem is not with the emission. Consider the scenario proposed by French. The receiver is stationary and the satelite emits successive light signals at constant proper period. During the emission of two successive wave crests, the angle under which the stationary observer receives two successive wave crests, as well as the radial component of the velocity of the source changes, important facts which should be taken into account. If we apply in that case the Doppler shift formula in large use, we consider that the receiver could perform a continuos recording of the period (frequency) what is physically (and physically impossible.
Thanks for the participation at the discussion.

Ich
IMHO the problem is not with the emission. Consider the scenario proposed by French. The receiver is stationary and the satelite emits successive light signals at constant proper period. During the emission of two successive wave crests, the angle under which the stationary observer receives two successive wave crests, as well as the radial component of the velocity of the source changes, important facts which should be taken into account. If we apply in that case the Doppler shift formula in large use, we consider that the receiver could perform a continuos recording of the period (frequency) what is physically (and physically impossible.
Thanks for the participation at the discussion.

Why should it be impossible? And why should it be important?
dphi = omega(t)*dt (tex still doesn't work)
gives you all you can know about the incoming wave. If you want to know the cycle duration T, you integrate over 2pi. If you then plug in (naively) f=1/T, you get those discrete values for f, with all the changing velocities and angles that one should take into account.
I'm not an HF-freak, but I see no reason why one should not be able to measure the actual field of a, say, kHz wave. If you have waves with circular polarisation, that's a perfect encoder signal, you just have to read off the phase. That gives you the changing frequency during one cycle.

Edit: maybe you're concerned about time delays due to the finite signal speed? Of course a general doppler formula connects different times of emitter and receiver.

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doppler

Why should it be impossible? And why should it be important?
dphi = omega(t)*dt (tex still doesn't work)
gives you all you can know about the incoming wave. If you want to know the cycle duration T, you integrate over 2pi. If you then plug in (naively) f=1/T, you get those discrete values for f, with all the changing velocities and angles that one should take into account.
I'm not an HF-freak, but I see no reason why one should not be able to measure the actual field of a, say, kHz wave. If you have waves with circular polarisation, that's a perfect encoder signal, you just have to read off the phase. That gives you the changing frequency during one cycle.

Edit: maybe you're concerned about time delays due to the finite signal speed? Of course a general doppler formula connects different times of emitter and receiver.
As I see we do not arrive to consense. But that is not surprising at all.
Please answer my last question: Consider a stationary source which emits short light signals at constant time intervals and a receiver in uniform motion (not compulsory longitudinal). Can the observer perform a
continuos recording of the period at which he receives the light signals. Take please into account that in the time interval between the reception of two successive light signals he is not able to evaluate the period at which he receives them? Avoid please any other complication which are not involved in the definition of the Doppler Effect according to which: What we have to compare in a Doppler Effect experiment are the period at which a source emits successive light signals measured in the rest frame of the source with the period at which an observer receives them measured in his rest frame. Thanks for the participation on my thread.

Ich
What we have to compare in a Doppler Effect experiment are the period at which a source emits successive light signals measured in the rest frame of the source with the period at which an observer receives them measured in his rest frame.
I do not understand where we differ. The Doppler formula is exact for the limit, where the period goes to zero. If you want to evaluate finite periods, you integrate the formula.
If velocity or angle change during the experiment, you have to use a more complicated formula which (smoothly, continuously) ties emission times to reception times. Kind of a generalized doppler formula.
The limit from discrete to continuous events is in principle feasible, given a suitable singal form.

doppler

I do not understand where we differ. The Doppler formula is exact for the limit, where the period goes to zero. If you want to evaluate finite periods, you integrate the formula.
If velocity or angle change during the experiment, you have to use a more complicated formula which (smoothly, continuously) ties emission times to reception times. Kind of a generalized doppler formula.
The limit from discrete to continuous events is in principle feasible, given a suitable singal form.

Thanks. Now we share the same oppinion!:rofl: 