Doppler47

V.5 No 1	47
Classical transverse Doppler-effect

2. Central motion of the source and observer in classical formalism

In order to compare correctly the solutions of classical and relativistic models, let us state the problem as Pauli [2] did. Only we will not move the source away to the infinity but will think the distance between the source and receiver finite. This will not a least change the par, since the relativistic reduction of moving reference frames is independent of the distance between them; this makes excessive the Pauli's limitation. It was rather an attempt to make a relation with a real situation in which usually just very distant objects are chosen as stationary. We may suppose so because Pauli's general solution is analysed at different momentary angles between the source and receiver, which would be logically impossible with the considerable distance between the source and receiver.

Thus, let in the origin of some stationary reference frame XOY be located a spherical source of periodic light pulses S. The between-pulse interval is T in XOY. Noting we are working in classical formalism, let us suppose the light pulses velocity to be constant in the stationary frame and equal to the light velocity in free space c.

Furthermore, let some moving frame X'O'Y' be related to the observer N receiving the signals of source S. Let this frame be moving with axis x, in the positive x direction, and having the velocity v, as is shown in Fig. 2.

Fig. 2. Doppler effect calculation graph for the non-central motion of receiver as to the observer

Also we will think the trajectory of observer's motion crossing the axis Y of the stationary frame XOY at the distance H from the frame origin.

To determine the degree of transformation of the time interval between the pulses received by the moving observer, consider the sequence, how the observer meets the fronts of waves propagating from the source.

From this view, suppose that at some instant t₁ the observer's trajectory crosses at the point N₁ with some wave front of the pulse of source. They to be able to meet, the source has radiated its pulse before, at the instant

It radiated the next pulse in the time interval T; the wave front of this pulse will cover the distance r₂ to meet the observer at the point N₂. Thus, the time balance with whose help we can calculate, when the observer meets the front of second pulse, is

At the same time, the observer together with his dotted frame is shifted from the position N₁ to the position N₂. The time indicated by observer will naturally be equal to the interval T' between pulses which he receives and which generally are not equal to T. So from the view of observer we have the following balance of time t₂:

Joining (4) and (5), we will yield the modelling equation of the studied problem:

Proceeding from Fig. 2, we can write down

Substituting (7) into (6), we yield

On the basis of (8), we can easily determine the relation between the frequency of pulses radiated by stationary source, and frequency ' received by moving observer:

Yielded solution (9) is just the sought regularity between the frequencies of stationary source and moving receiver in most general case. And the inverse relation of frequencies will be

Formula (10) is true in the whole range of aimed distances H, with the exception of the domain of very small aimed distances, where we have to write () before the root.

Now on the basis of complete solution we can correctly compare the results yielded by classical and relativistic approaches.