A general derivation of the classical Doppler effect in 3D space

The classical Doppler effect has already been studied in many textbooks and papers. However, the formula currently used to describe the Doppler effect for a source and a receiver moving with constant velocities is not suitable for practical applications. Indeed, the formula makes use of two angles that depend on the state of the source-receiver system both at the emission instant and at the reception instant of sound, while in most real cases the state of the system is known only at one of the two instants. In this paper, two equivalent formulas are derived for the Doppler shift: one depending solely on the state of the system at the emission instant, and one depending solely on the state of the system at the reception instant. The paper also provides two more general expressions that describe the Doppler shift evolution over time. The expressions derived in this paper are compared with the existing ones in some special cases, obtaining consistent results.


Introduction
The classical Doppler effect is a purely kinematic phenomenon caused by the finite propagation speed of waves in media.Since waves do not propagate instantly, any signal emitted by a source reaches the receiver with some delay.If the source and/or the receiver are in motion with respect to the medium, their distance in general varies over time, and the delay varies accordingly.The variation of the delay provokes a deformation of the received wave compared to the emitted one along the time axis.In the case of periodic waves, for which a frequency can be defined, it is observed that the frequency n¢ perceived by the receiver differs from the frequency n emitted by the source.This phenomenon was first studied by Christian Andreas Doppler in 1842, and is therefore known as the Doppler effect, or Doppler shift [1].
The topic of this paper is the classical Doppler effect between a point-like source and a point-like receiver moving with constant velocities with respect to an inertial, non-dispersive medium.More precisely, we study what happens if the source emits a periodic, spherical wave, with the same intensity in all directions, propagating at a constant speed c.An example of such wave is a musical note, c being the speed of sound in air.More generally, the classical Doppler effect concerns any kind of waves which can be described in terms of classical mechanics.However, we will always refer to sound waves for convenience.This way, we can avoid sentences such as =the receiver moves at a lower speed than the propagation speed of waves in the medium?, stating, more briefly, =the receiver moves at subsonic speed?.
To begin with, let us define the main variables that describe the source-receiver system.We call ( ) r t S and ( ) r t R the positions of the source and receiver over time, v S and v R their respective velocities and v S and v R their speeds.We also call ( ) ( ) ( ) r r t t t R S r = the position of the receiver with respect to the source: its norm, ( ) t , r is the distance between the source and the receiver, while the unit vector ˆ( ) ( ) ( ) is the direction of the line linking the source to the receiver.We then define ( ) t S q the angle between ˆ( ) t r and v , S ( ) t R q the angle between ˆ( ) t r and v , R and a the constant angle between v S and v .R An example of a source-receiver system at a given instant t is shown in figure 1.Note that the vectors v , S v R and r do not necessarily lay on the same plane.
A simple case of the Doppler effect is the longitudinal case, which occurs when the source and the receiver move along the same line, namely when the angles S q and R q are equal to 0 or p in the four possible combinations.In this case, it can be shown that where the sign in the numerator is positive if the receiver moves towards the source (and negative otherwise), while the sign in the denominator is negative if the source moves towards the receiver (and positive otherwise).Equation (1) appears in several textbooks and papers [2][3][4][5], and it is known as the longitudinal Doppler effect formula.The Doppler effect in three dimensions is more difficult to describe.Due to its complexity, it is often studied in some special cases, such as the case of a resting receiver with respect to the medium.Suppose that the source emits a maximum at the instant t 0 0 = and that the receiver hears that maximum at a later instant t .0 ¢ Since sound takes some time to reach the receiver, the source moves in the meantime.Therefore, at t , 0 ¢ the direction ˆ( ) t 0 r ¢ in which an observer holding the receiver sees the source does not coincide with the direction ˆ( ) t 0 r from which the observer hears the sound coming, as shown in figure 2.
We will therefore call emission angle namely the angle between ˆ( ) t 0 r and v , S and reception angle ( ) t , S0 S 0 q q ¢ = ¢ namely the angle between ˆ( ) t 0 r ¢ and v S .The Doppler effect can be either expressed in terms of S0 q or .S0 q ¢ Textbooks usually report the formula in terms of S0 q [6-8]: ( ) S S / b = If both the source and the receiver move with respect to the medium, describing the Doppler effect is even more challenging.Since, in this case, both the source and the receiver are in motion, two emission angles and two reception angles can be defined, as shown in figure 3(a).More precisely, we call emission angles ( ) while we call reception angles ( ) Note that an observer operating the source can easily measure the emission angles S0 q and , R0 q as they depend on ˆ( ) t , 0 r which is the direction where the observer sees the receiver while he is emitting the sound.Similarly, an observer holding the receiver can easily measure the reception angles S0 q ¢ and : R0 q ¢ indeed, these angles depend on ˆ( ) t , 0 r ¢ which is the direction where the observer sees the source while he is hearing the sound.Therefore, one would expect a Doppler effect formula in terms of the emission angles or the reception angles, but such a formula has yet to be found.A few textbooks [9,10] report as a general formula the expression However, they fail to explain the meaning of S f and .R f In fact, S f and R f are neither the emission angles nor the reception angles, but the angles shown in figure 3(b), which we called hybrid angles.More precisely, if we call n the unit vector linking the position of the source at the emission instant t 0 to the position of the receiver at the reception instant t , 0 ¢ S f is the angle between n and v , S while R f is the angle between n and v .R This is explained in two papers [11,12], which also provide a derivation of equation (3).
Of course, equation (3) continues to hold in the case of a resting receiver or in the case of a resting source.If the receiver is at rest, 0 R b = and S f coincides with , S0 q so we get once again equation (2).If the source is at rest, q ¢ so we get a Doppler effect formula for a resting source in terms of the reception angle : The problem with equation ( 3) is that it is hard to apply in practice.For instance, if an observer operating the source wants to use equation (3) to get the frequency perceived by the receiver, he needs to know the direction where the receiver will be at the reception instant, namely n, to establish the angles S f and .R f However, the direction where the receiver will be at the reception instant is unknown to the observer.So, equation (3) expresses the Doppler shift in terms of two unknown angles, S f and , R f without explaining how to determine them.At this point, there are two possible ways to compute the Doppler shift: one can either find a mathematical way to establish the angles S f and R f in terms of physical quantities that are known to the observer, or approach the problem in a completely new way.In this paper, we choose the second path.
First, we set a rigorous definition of the Doppler shift, which offers a kinematical way to compute it.Then, we apply the definition to the case of a moving source and a moving receiver in 3D space.This way, we get two equivalent Doppler effect formulas: one in terms of the emission angles, which can be directly applied by an observer moving with the source, and one in terms of the reception angles, which can be directly applied by an observer moving with the receiver.We also obtain two more general expressions describing the Doppler shift evolution over time.Finally, to validate our results, we compare them with the known Doppler effect formulas in some special cases.

Definition of the doppler shift
The Doppler shift for a source-receiver system is usually defined as the ratio between the frequency n¢ perceived by the receiver and the frequency n emitted by the source.However, such a definition can only work once the definition of n¢ is set.Unfortunately, in most cases, even if the emitted wave is periodic, the received wave is not, so it is not possible to define its frequency in the classical sense of the term.In this paper, we define n¢ as follows.Suppose that the source emits a harmonic wave of period T. Suppose further that two consecutive maxima are emitted by the source at t t, ¢ -¢ between the reception of the two maxima.We then define the perceived frequency, , n¢ as T 1 .

/ ¢
At this point, it would seem natural to define the Doppler shift as with t t 1 = and t t T. 2 = + Equation (5) quantifies how shortened or stretched a single period of the emitted wave appears to the receiver.If we apply it to the longitudinal case, we get equation (1).Note that the right-hand side of equation (1) is constant.Thus,  DS in the longitudinal case does not depend on the frequency of the source.This statement is by no means obvious.In fact, the problem with equation ( 5) is that  DS depends on n if the definition is applied to the 3D case.The reason is explained below.
According to equation (5), to compute  DS we first need to establish the reception instants t 1 ¢ and t 2 ¢ of the two maxima.The reception instant t¢ of a maximum can be established by knowing its corresponding emission instant t, the state of the source-receiver system at t and the speed of sound.Let us assume that the state of the system at t 0 = is known, as well as the speed of sound.The state of the system at t can be determined from the state at t 0 = by simply knowing the elapsed time t.So, the reception instant of a maximum only depends on its emission instant: ( ) n In some special cases, such as the longitudinal case, n cancels out in computing the ratio.However, this does not happen in the 3D case.
The dependence of  DS on t is simply a consequence of the time evolution of the system: as the configuration of the system evolves, n¢ slowly changes over time, and  DS varies accordingly.The dependence on , n on the other hand, is more problematic.For example, by applying equation (5) in the case of a resting receiver, we do not get equation (2), but a more complex expression dependent on .
n More precisely, for a sound emitted at t 0, 0 = we get where 0 x is the initial distance between the source and the receiver divided by the speed of sound.
If we want the Doppler shift to be a physical quantity that does not depend on , n we need to choose the two emission instants t 1 and t 2 very close in time.More precisely, we need to compute the ratio ( ) ( )  This way, we get a physical quantity that describes how an infinitely small portion of the emitted wave is perceived by the receiver.In short, we define the Doppler shift for a sound emitted at t t 1 = and received at t t where the parenthesis  ( ) t t¢ highlights the emission and the reception instants of sound.If the state of system at t 0 = and the speed of sound are known, the reception instants t 1 ¢ and t 2 ¢ only depend on the emission instants t 1 and t , 2 and so does the ratio ( ) ( The limit t t  removes the dependence on t , 2 hence DS only depends on t , 1 or else t.At this point, some may argue that DS, despite being well defined, does not represent what is normally meant by the Doppler effect, i.e. the ratio between the frequency perceived by the receiver and the frequency emitted by the source.So, let us show why this is not true.Suppose that the source emits a periodic wave of period T. Computing with t t 1 = and t t T. 2 = + In other words, the Doppler shift is equal to the ratio / n n ¢ that one would observe over time if the source emitted a sound of a very high frequency.This means that, if the actual frequency emitted by the source is high enough, DS provides with good approximation the time deformation of the first period of the emitted wave, starting from time t. In the longitudinal case, the ratio ( ) ( ) = + in equation ( 7), the ratio ( ) ( ) / ¢ and assuming t d infinitely small is equivalent to computing the ratio in the limit t t.

Doppler effect in terms of the emission instant
Having set a rigorous definition of the Doppler shift, we are now ready to compute it.As we have already explained in section 2, the Doppler shift can be expressed in terms of the emission instant of sound, of the speed of sound and of the initial state of the source-receiver system.The state of the system at a generic instant t is given by the vectors ( ) r t , S ( ) r t , R v S and v , R namely a set of twelve scalars.However, due to the homogeneity and the isotropy of space, we know that DS cannot depend on the orientation of the axes nor on the origin of the reference frame.We therefore define the useful state of the system as the equivalence class of all those states that only differ for a roto-translation of the reference frame.Only six scalars are required to describe the useful state of the system: and , a which we will call essential variables.Thus, we expect DS to depend on t, c and on the initial values of the essential variables.
In order to compute the Doppler shift, we first need to establish the delay that occurs in the reception of a maximum.Suppose that the source emits a maximum at t 0 0 = and a maximum at t. Since v S and v R are constant, where r S0 and r R0 are the positions of the source and the receiver at the emission instant t 0. 0 = The position of the receiver with respect to the source at the instant t is therefore The fraction inside the norm is, by definition, the average velocity of the receiver in the time Finally, the time interval t t ¢is the delay that occurs in the reception of the maximum, which we will call .
t With these substitutions, equation (13) reads as By taking the square of equation ( 14) and bringing everything to the left-hand side, we get For convenience, we now introduce the dimensionless velocity and we also define ( ) Equation (15) can be written in terms of R b and ( ) t x by dividing both sides for c : 16) is a quadratic equation in , t so it can have up to two solutions.However, since a maximum cannot be received before it is emitted, equation (16) needs to satisfy the causality condition  ( ) t 0. t In the subsonic case ( 1 S b < and 1 R b < ) it is easy to show that equation (16) has only one acceptable solution: The dot products in equations (20) and (21) suggest introducing the angles and .a This way, By substituting equations (22) and (23) into equation (17), and by adding the like terms, we get an explicit expression for ( ) where we introduced the constants So, a general Doppler effect formula can be obtained by simply evaluating the derivative of ( ) t t from equation (24) and substituting it into equation (31): Equation (32) describes the Doppler shift for a sound emitted at t and received at t¢ as a function of the emission instant t.If we want to know the Doppler shift for a sound emitted at t 0 0 = and received at t , 0 ¢ we just need to set t 0 = in equation (32).If we also substitute equations (25) and (26) into equation (32), we get the explicit expression: Equation (33) expresses the Doppler shift for a sound emitted at t 0 in terms of the essential variables of the system computed at t .0 In particular, we note that DS depends on the angles S0 q and , R0 q namely the emission angles.So, equation (33) can be directly applied to compute the Doppler shift by an observer operating the source.
It is also important to point out that no assumptions were made on the state of the system at t , 0 which is simply what we chose as the origin of time.Due to the homogeneity of time, any instant can be chosen as the origin of time.Therefore, equation (33) holds for a sound emitted at any instant provided that the essential variables are computed at that precise instant.
The analysis that we have just conducted reveals two basic properties of the classical Doppler effect.First, we notice from equation (33) that, for c ,  ¥ S b and R b tend to 0 and DS tends to 1.In other words, no Doppler effect is observed if the wave propagates instantly.This result supports the very first claim of the Introduction: the Doppler effect is due to the finite propagation speed of waves in media.Second, we notice from equation (31) that, if the delay is constant, the derivative of ( ) t t is zero and DS 1. = This result supports another important claim of the Introduction: the Doppler effect is due to the variation of the delay in the reception of signals.

Example
To better understand the utility of equation (32), we now show how it can be applied in practice with an example.Suppose that the state of the system at the initial instant t 0 0 = is given by the vectors .Finally, DS can be evaluated as a function of t according to equation (32).The Doppler shift evolution over time is represented by the red curve in figure 5.
The blue curve in figure 5 represents instead the Doppler effect  DS on the single periods, defined in equation (5).The values of  DS were obtained through a computer simulation.The reception instants of the maxima were established by evaluating the position of the corresponding wavefronts, the position of the source and the position of the receiver every microsecond.
By definition,  DS tends to DS for , n  ¥ but, as we can see from figure 5, the two quantities are similar even at relatively low frequencies.More precisely, for the two quantities to be similar, the angles ( ) t S q and ( ) t R q must not vary significantly in between the emission of two consecutive maxima.In the subsonic case, a sufficient condition to fulfill this requirement is ( ) In our example, ( ) t 1/x roughly stays around 1Hz, so an emitted frequency of 20Hz is already high enough to show the similarity between  DS and DS.In most real cases, provided that the receiver stays far from the source, the condition ( )  t 1/ n x is largely satisfied.
4. Initial state of the source-receiver system.
from the first one by replacing the emission angles with the reception angles, by swapping all the physical quantities relative to the source with the corresponding physical quantities relative to the receiver and by inverting the result.Two additional expressions have also been derived, describing the Doppler shift evolution over time.

Figure 1 .
Figure 1.An example of a source-receiver system.

Figure 2 .
Figure 2. Doppler effect in the case of a resting receiver.

as shown in figure 4 .
Suppose further that the speed of sound is c 340m s. / = By definition, the essential variables of the system at t 0 are 0These can be used to compute the constants in equations (25), (26) and (27): is also a function of t and T , or else t and .n suppose that the source emits a harmonic wave of period T.