SELF

6

Supplement to the study of classical transverse Doppler effect in respond to received criticism

Let us continue our analysis and consider the event next for stationary frame, when the source has emitted the next pulse of light. At this instant the source was located at the point having the coordinate

The receiver at that instant was located at the point with coordinate

Given the time intervals cannot be negative, the delay time of signal between the instant of emission and instant of reception will be

In (30) we also have two variants to solve the problem, dependently on the sign of numerator in the right-hand part. In the analysis fully similar to the above, we will yield two solutions. At

we have

at

we have

General solution of the second stage of problem will be

where

Having determined the delay times of both signals reception, we can determine also the interval between these signals reception. It will be determined as

Substituting (27) and (35) into (37) and at the same time transforming the relation of periods between the radiated and detected pulses into the relation between the frequencies of radiation and reception, we yield

where

But this still is not the final solution, although we seemingly yielded the relation between the frequencies of radiation and reception which interested us. Pay our attention to the fact that we determined the frequency of reception after only two sequential pulses, which added a third summand to the denominator of final expression. Now let us think: if the mutual location of source and receiver has an influence on the change of period between two sequentially radiated and, accordingly, received pulses, for two next pulses mutual location of source and receiver will change in general case. This means, for the next two pulses we will have to substitute already a new value of initial location of the source and receiver. From this, to yield final solution, we have to substitute the difference of locations by more general value that will account the mutual displacement of source and receiver. With it (38) will take the following form:

where x_A0 and x_B0 determine already the location of source and receiver at the initial instant of time.

At the same time, expressions for ₀ and ₁ (39) determining the signs in (40) will also change. Now they will be

As we see, the general solution (40) noting (41) is a multi-factored regularity that changes with respect not only to the initial mutual location of the source and receiver but to the time, which is basically different from your regularity (1). Though under condition of your particular problem

and negative value of the source speed that means source motion toward negative x, we yield

and immediately come to (1 of your work) as to a very particular case. But the fact that namely and exceptionally in particular case our formula coincides with yours only adds questions to your study.

In particular, to all above described problems this fact puts under question, how did your model appear consistent with the experiment on whose basis you tried to prove the bankruptcy of classical formalism? From the full solution (40) we see that dependently on the initial location of source and receiver, as well as on speed relation, the value of frequency shift can considerably vary also in time. But if in the experiment which you are meaning you studied the transverse frequency shift while you check it with the longitudinal frequency shift, and even not simply with longitudinal shift but with its very particular case, then your conclusions are irrelevant at all to the construction and modelling equations on whose basis you were trying to conclude. There remain only most general questions like, how could you compare doing not understanding so simple and obvious things? How dared you to claim our work incorrect without grounds, without any analysis? Perhaps, just because you have no idea of unbiased thorough study. Our above analysis of your work showed it clearly. And we would like to emphasise, we are concluding without any regard to the fact that we also have a study on this subject. We did not touch our work when analysing yours. You have made gross mistakes in your study, which show you being ignorant in the subject and unscrupulous in studying. Of course, your results appeared so much discrepant and inconsistent. We think, it would be for great help for you if you took it into account, in order to avoid such failures in the future. We hope much that you will understand this what we showed you and will make use of it.

Regards,

Sergey and Olga Karavashkin