Derivation of general Doppler eﬀect equations

In this paper we will derive the general equations for Doppler eﬀect. It will be proved that regardless of the nature of the emitted waves and the medium through which the waves propagate the formula always has the same form and is identical to the general Doppler eﬀect formula for sound. We will also show that Doppler eﬀect can be used to establish a relationship between the local time of the source and the local time of the receiver. In addition, some new features of the Doppler eﬀect have been presented that have not been discussed in the literature so far.


Introduction
In the general formula for Doppler effect for sound waves, a reference frame is the body of air through which these waves travel. This means that we measure the speed of a source of sound waves and the speed of the observer of those waves relative to the body of air. The emitted frequency f and the detected frequency f , are related by following formula [1].
where f -the frequency observed by the observer f -the frequency of sound which the observer emits v -the speed of sound in the medium (air) v o -the velocity of the observer v s -the velocity of the sender The positive direction is directed from the observer toward the sender.
For the electromagnetic waves there is no medium that we can use as a reference frame to measure velocities. We Journal of Advances in Physics Vol 18 (2020) ISSN: 2347-3487 https://rajpub.com/index.php/jap use a frame of reference in which is the observer at the rest. The relativistic Doppler effect for electromagnetic waves depends only on the the relative velocity v between source and observer [1], [2].
From (2) it follows that relative velocity v is equal to:

Derivation of general Doppler effect equations
Suppose that the source (Z) and the receiver (S) of the signal move uniformly at velocities u and v respectively with respect to the coordinate system (K) Figure (1). The coordinate system (K) is not defined but we will assume that there is such a coordinate system. Let a signal be emitted in all directions at some point, which we will denote by τ 1 .
The speed of the signal is denoted by c. The time when this signal is registered by the receiver (S) is denoted by t 1 .
In a similar way we can assume that a second signal was emitted from (Z) at some point τ 2 , and that this signal was received by the receiver (S) at time t 2 . (K) S 1 S 2 = v∆t (10) The velocity u can be decomposed into two components, radial and tangential, which we usually denote by u r and u t respectively. In the same way the velocity v can be decomposed into components v r and v t .
Now we have that: From (27) and (31) it follows that Suppose that we have a quadratic equation : Equation (33) has two solutions ∆t 1 and ∆t 2 .
If √ B 2 − AC ≥ 0 then solutions ∆t 1 and ∆t 2 have real values. We choose the solution for which the condition ( (26)) is fulfilled.
Generally we have that: Suppose that B 2 >> |AC|. We can write that From (34) and (36) it follows that Let make substitutions as it follows: Let define wavelength λ as follows λ = ∆τ * c We will assume that the inequality λ << d 1 holds.
And finally we have : We will consider following cases: There is only one solution ∆t 1 .
There are two solutions. This means that the signal sent from point Z(τ 2 ) will be registered twice by the receiver, at time t 1 + ∆t 1 and at time t 1 + ∆t 2 .
There are no solutions.
There is only one solution ∆t 2 .
This case is not possible.
There are no solutions.
This case is not possible.
There are two solutions. This means that the signal sent from point Z(τ 2 ) will be registered twice by the receiver, at time t 1 + ∆t 1 and at time t 1 + ∆t 2 .
Let us consider the case when u < c and v < c. We have the following implication: This means that ∆t 1 is the only solution of Equation (33).
Let define the frequencies f = 1 ∆τ and the f = 1 ∆t .
Now we can write that: In that way we prove that formula (56) is identical to formula (1).
After multiplying both sides of (55) by c we get the following expression where λ is wavelength of the emitted light and λ the wavelength we observe.
Doppler shift z f in frequency and Doppler shift z λ in wavelength are defined by the following formulas respectively: If | u r | << c then we have And if | v r | << c it follows that The frequency f can be determined in two ways, by direct measurement or by applying formula (56). While in the first case we have exactly one value in the second case that value depends on what is chosen as the reference for measuring the velocities u and v. Since f must have exactly one value it means that there is exactly one reference frame. The question is whether this frame is fixed or depends on how the sender and receiver of the signal are selected (for example the recipient is on Earth and the sender is somewhere in the Earth's orbit, the solar system, the Galaxy ...). At this point, we cannot give a precise answer to that question. If we accept the principle that measurements are made in relation to the receiver then it is obvious that this rule does not apply to sound waves. We can go so far as to claim that this principle has not been proven even in the case of electromagnetic waves, because there is no experiment to determine that the relative velocity at which the sender moves in relation to the observer is exactly that obtained by formula (3).
One of the ways in which it would be possible to choose a coordinate system (K) is already described in the paper in which we dealt with the problem of stellar aberration [3]. The ray connecting the sender and the receiver (telescope) represents the z − axis and the plane in which the objective lens lies is chosen as the (xy) plane.
The discussion on how to choose the reference coordinate systems (K) is not over yet and we will deal with this problem in the future.