SELF

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S.B. Karavashkin and O.N. Karavashkina

3.2. Transverse Doppler effect

Above we saw, if we calculated the change of frequency of received signal by the observer moving non-centrally as to the signal source, the classical formalism will also reveal the transverse effect, but in distinct from relativistic formalism, this effect has a negative sign and essentially depends on the aimed distance.

It is easy to yield the relation in frequencies of source and receiver from (9) at the condition

In this case (9) takes the form

As we see, in case of transverse Doppler effect, the square bracket in (28) is strongly positive, due to which the value of source frequency will always be more than frequency received by moving observer, and this regularity will essentially change with respect to the aimed distance. In Fig. 8 we see the plots of this regularity at different frequencies of source signal.

Fig. 8. The change of relation of frequency ' of signal registered by the receiver to the frequency of source signal against the aimed distance H at different frequencies of source and v/c = 0,5  in the transverse Doppler effect, built on the basis of classical formalism

This regularity shows that from the view of classical formalism, the transverse Doppler effect rapidly decreases with the distance, but not from zero value of effect ( = '), as we could expect, but from the value (1 - v/c) . This is unexpected, as the condition at which the aimed distance vanishes is the condition to find the longitudinal Doppler effect. This means, even in this particular case the transverse Doppler effect is present.

The cause of this unexpectedness becomes clear if we look at the general formula (9) and recall that when finding the longitudinal effect with the aimed distance lessened to zero, we accounted the change of quadratic bracket of denominator in transition through the point of signal radiation. Just due to this, there appeared a sign-alternating term in the expression for longitudinal Doppler effect (15). But to find the transverse effect, we at once in general case indicated the value of angle , which lifted the alternation in sign in the denominator of (9). But with it the very pattern of effect's regularity did not change, and this caused non-zero value of effect. With it we should emphasise, this differentia cannot be revealed in a particular solution yielded before, as for it we would have to know the type of received frequency change over the aimed distance and the change of angle between the source and receiver. While when considering the central motion, this information in its full amount was absent.