SELF

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

Now see in your work one more corroboration that we were right telling you our criticism as to your modelling. See, after you compared your formula with that Einsteinian, you intended to account also the aberration. It would be not for worse, of course, should you wish to compare with the experiment in which the source is moving and the observer is at rest. For it you have introduced the relation between the dotted and undotted angles (14 of your paper):

We will not discuss now, is this formula true, although Pauli gives this relation in considerably different appearance. We are taking this formula as the fact and substituting to it your condition (3 of our post), for which this formula is destined. And we yield

Given the statement of problem says, the dotted and undotted axes are parallel, from (8 of our post) it straight follows that

Two questions arise from (9). First, undistorted angles of frames are inherent in central motion of source and observer. This also corroborates that at your condition (3 of our post) you are studying the longitudinal effect. Thus, we approached your text from three different sides and yielded the same result corroborating your erroneous approach to the problem and its solution. And second, what would be the sense to account additionally the aberration angle in your case, if the cosines of angles were equal?

By the way, when we did not discuss your formula of angle relation (7 of our post), we did so not in vain. As we revealed above, your formula (8 of our post) leads us to (9 of our post). But is it correct in case of central motion of the source and observer? In Fig. 5 we show the graph for the case when the observer's frame goes away from the stationary frame of receiver toward the positive values of axis x (for visualisation, the axes x and x' are some shifted).

Fig. 5

As we can make sure immediately from the graph, in this case your formula had to give

and

With it, (15 of your work)

had to come straight to Einstein's equation (shutting the eyes to your incorrect manipulation with the parameter c), not to your 'either- or' in (16 of your work). The very fact, how did you record your (16), says that you saw a discrepancy in this particular place but did not take a trouble to clear and to lift it. This reflects all your way of approaching to your work, as well as the 'scrupulosity' of your study. All our above criticism factually evidences it.

And one more example. As far as we understand, you in your work have put just the same target as we did - to check the results that Relativity and classical formalism predict concerning the transverse Doppler effect. But we in our work proceeded from the fact that relativistic formula of transverse Doppler effect is known while that classical is not, and there are many evidences. While you proceeded from the opposite. You have typed without derivation some strange formula (1 of your work), which you attributed to the transverse effect in frames of classical formalism:

<< Consider a frame of reference in which the medium of signal propagation is assumed to be at rest, and suppose a transmitter ("out") and receiver ("in") are located on the x axis, with the transmitter moving to the left at a speed of v and the receiver moving to the right at a speed of u. Let c₀ denote the speed at which the signal propagates with respect to the medium. Then, according to the classical (non-relativistic) treatment, the Doppler frequency shift is

But this is not the expression for Doppler effect in classical formalism. The formula (13 of our post) describes the longitudinal effect in presence of the speed of source and observer as to some stationary frame. And we can easily show it.