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An analysis of the classical Doppler effect
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2003 Eur. J. Phys. 24 497
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INSTITUTE OF PHYSICS PUBLISHING
EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 24 (2003) 497–505
PII: S0143-0807(03)58697-4
An analysis of the classical Doppler
effect
C Neipp, A Hernández, J J Rodes, A Márquez,
T Beléndez and A Beléndez
Departamento de Fı́sica, Ingenierı́a de Sistemas y Teorı́a de la Señal, Universidad de Alicante,
Apartado 99, E-03080 Alicante, Spain
E-mail: [email protected]
Received 23 January 2003
Published 22 July 2003
Online at stacks.iop.org/EJP/24/497
Abstract
The Doppler effect is a phenomenon which relates the frequency of the harmonic
waves generated by a moving source with the frequency measured by an
observer moving with a different velocity from that of the source. The classical
Doppler effect has usually been taught by using a diagram of moving spheres
(surfaces with constant phase) centred at the source. This method permits an
easy and graphical interpretation of the physics involved for the case in which
the source moves with a constant velocity and the observer is at rest, or the
reciprocal problem (the source is at rest and the observer moves). Nevertheless
it is more difficult to demonstrate, by this method, the relation of the frequencies
for a moving source and observer. We present an easy treatment where the
Doppler formulae are obtained in a simple way. Different particular cases will
be discussed by using this treatment.
1. Introduction
The Doppler effect is a phenomenon of intrinsic kinematic character. This allowed
Kapoulitsas [1] to treat in a simple way a generalized Doppler effect by supposing that the
positions of both the source and the observer are arbitrary functions of time. By doing this,
Kapoulitsas analysed different particular cases, showing the validity of this method. There are
also other treatments of the Doppler effect, for instance Saknidy [2] commented on a unified
treatment of both the classical and relativistic Doppler effect by using four-dimensional space.
This well-known effect (the classical and relativistic Doppler effect) has, then, even recently
attracted the attention of a great number of authors [1–7] who in some way have tried to
clarify different aspects of this phenomenon. Recently, a very easy derivation of two Doppler
formulae was given by Donges [7]. These formulae were obtained for the case in which the
observer is at rest and the source is moving with a constant velocity, and also for the case in
which the source is at rest and the observer is moving with a constant velocity. The study was
limited to one-dimensional motion. We want to generalize the treatment made by Donges for
0143-0807/03/050497+09$30.00
© 2003 IOP Publishing Ltd
Printed in the UK
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C Neipp et al
positions of the source and observer that are arbitrary functions of time in three-dimensional
space. By doing this the Doppler formula for both a moving source and an observer with
constant velocity will also be obtained.
In this work we will first, in section 2, establish a relation between the phase of a harmonic
wave measured by the observer and the phase of the wave emitted by a source. This relation
depends on the propagation time, i.e. the time it takes the wave to travel from the source to
the observer. In addition this propagation time depends on the functions which describe the
variation with time of the positions of the source and the observer. Thus, we can study different
cases for different kinds of motion. In section 3 the Doppler formula in the general case is
derived. In section 4 the basic Doppler formula for the case in which the observer and the
source move with constant velocities in the same direction is analysed. In section 5 we consider
the case in which the observer is moving and the source is at rest, whereas in section 6 the
source will be supposed to be at rest and the observer to be moving. In both sections 5 and 6
we allow for the source and the observer to move in different directions.
2. Relation between the phases for observer and source
First we will consider the case when the source (S) and the observer (O) are at rest. The
temporal oscillation, yS , of a monochromatic wave with angular frequency, ω, at the location
of the source can be expressed as:
yS = A sin ϕS ;
ϕS = ωt
(1)
where ϕS is the phase of the wave emitted by the source.
If the locations of the observer and the source are described by the position vectors, rO
and rS , respectively, the time, τ , which the wave takes to propagate from S to O is
|rO − rS |
(2)
τ=
c
where c is the propagation velocity of the wave.
The temporal oscillation at the location, yO , of the observer can then be expressed by:
yO = A sin ϕO ;
ϕO = ω(t − τ )
(3)
where ϕO is the phase measured by the observer.
Equation (3) shows that there is a phase shift between the oscillations at S and O, although
the angular frequency measured by the observer and that of the wave emitted by the source are
the same.
We can generalize equations (1)–(3) to the case where the observer and the source move
along rO (t) and rS (t). Figure 1 describes this situation. Now the propagation time, τ , is given
by:
|rO (t) − rS (t − τ )|
.
(4)
τ=
c
This equation describes the situation for a wavefront arriving at O at time t, which was emitted
from the source at a time t − τ and which travels the distance from S to O in a time interval τ .
By using equations (3) and (4) the temporal evolution of the phase at O can be described,
and the frequency of the wave at O can also be obtained for different situations.
3. Doppler formula for the general case
In this section we will derive the Doppler formula for the general case. For two waves emitted
by the source with a time difference dt, there will be a phase difference ω dt if the observer and
the source are stationary. However, by (3) and (4), if the observer and the source are moving,
there will be a phase difference:
dϕO = ω(dt − dτ ).
(5)
An analysis of the classical Doppler effect
499
Figure 1. Vector positions of the source and the observer, which are arbitrary functions of time.
This phase difference is due to the relative motion of the source to the observer with
velocity vrel and can be calculated as follows: the distances these waves have to cover differ by
vrel · n dt, where n is the outgoing normal to the wavefront, along the line joining the observer
and the source (always pointing to the observer). Therefore, they reach the observer not with
a time difference dt, but with
vrel · n dt
dt +
crel · n
where crel is the velocity of the wave relative to the moving source. Consequently, from (5)
we get
vrel · n dt
dτ = −
.
(6)
crel · n
If vS is the velocity of the source and vO the velocity of the observer, vrel = vS − vO and
crel = c − vS . Therefore we get for the corresponding phase difference seen by the observer:
(vO − vS ) · n
c − vO · n
dϕO = ω 1 −
dt = ω
dt.
(7)
c − vS · n
c − vS · n
From (7) the angular frequency, ω , measured by the observer can be identified as:
c − vO · n
.
(8)
ω = ω
c − vS · n
Equation (8) gives the frequency measured by the observer provided the velocities of the source
and observer are known. In some cases, where the motion of the source and the observer can be
easily described, equation (8) could be used to obtain the temporal evolution of the frequency
measured by the observer. Nonetheless, in general situations it is more useful to use an
alternative expression.
From equation (5)
dτ
dϕO = ω 1 −
dt.
(9)
dt
So the angular frequency measured by the observer can also be expressed as:
dτ
.
(10)
ω = ω 1 −
dt
C Neipp et al
500
Equation (10) permits us to obtain the temporal evolution of the angular frequency
measured by the observer provided the temporal dependence τ (t) is known for each kind
of motion. The aim of this work is to calculate τ (t) for different particular cases by using the
time-integrated motion of the observer and the source (equation (4)) and express the general
Doppler formula (10) as a function of t. A similar approach can be found in [1], although
Kapoulitsas treated the problem in terms of t (t), where t corresponds to the time in which
the wave was emitted from the source.
4. The source and the observer move along the same direction with constant
velocities
For simplicity, it will be supposed that motion takes place along the x direction. In this case
the position of the observer can be described by:
rO (t) = (x O0 + vO t)ux
(11)
and the position of the source by:
rS (t) = (x S0 + vS t)ux
(12)
where ux is a unit vector in the direction of the x axis.
Substituting (11) and (12) into equation (4) we obtain:
cτ = ±[(x O0 − x S0 ) + (vO − vS )t + vS τ ].
(13)
The (+) sign corresponds to the case rO (t) > rS (t) (case (i)) whereas the (−) sign corresponds
to the case: rS (t) > rO (t) (case (ii)). For simplicity, we will study case (i), although a similar
treatment will give the solution for case (ii).
From equation (13), τ can be obtained as a function of time, t, as follows:
τ=
(x O0 − x S0 ) + (vO − vS )t
.
(c − vS )
(14)
From equation (14) and by using (10) it is easy to obtain the angular frequency measured by
the observer as:
c − vO
vO − vS
=ω
ω = ω 1 −
.
(15)
c − vS
c − vS
This relation is valid provided
rO (t) − rS (t)
> 0.
c − vS
(16)
In a similar way (taking the minus sign in equation (13)) it can be demonstrated that:
ω = ω
c + vO
.
c + vS
(17)
which is valid, provided that:
rO (t) − rS (t)
< 0.
c + vS
(18)
These are the classical formulae for the Doppler effect when the source and the observer
move with constant velocity in the same direction; they can be found in many introductory
physics texts, e.g. [8–10].
An analysis of the classical Doppler effect
501
Figure 2. Motion of the observer with constant velocity.
5. The observer moves and the source is at rest
In this case we will suppose that the observer moves in a non-radial direction. For simplicity
the source will be supposed to be located at the origin of the reference system adopted, whereas
the observer moves along rO (t). In this case equation (4) becomes
|rO (t)|
.
(19)
c
If ur is a unit vector in the radial direction from the origin pointing to the location of the
observer expression (19) becomes
τ=
rO (t) · ur
.
c
By differentiating equation (20) with respect to time:
dτ
1 drO (t)
dur
=
· ur + rO (t) ·
dt
c
dt
dt
τ=
(20)
(21)
and by using equation (10) the angular frequency measured by the observer can be obtained
as:
vO · ur
ω = ω 1 −
(22)
c
where it has been taking into account that drO (t)/dt is vO and the second term in the bracket
of (21) is zero since ur · (dur /dt) = 0.
Finally equation (22) can be expressed as:
vO
cos β
(23)
ω = ω 1 −
c
where β is the angle between ur and vO .
5.1. The observer moves with a constant velocity
In the case in which the observer moves with a constant velocity, the vector position of the
observer can be expressed as:
rO (t) = rO0 + vO t.
(24)
C Neipp et al
502
Figure 3. Frequency as a function of time in the case in which the observer moves with a constant
velocity and the source is at rest.
For simplicity, we will study the case in which the observer moves in a perpendicular
direction to the x-axis (figure 2). The initial position vector of the observer is assumed to be
rO0 = rO0 ux
(25)
whereas the velocity of the observer will be
vO = vO u y
(26)
where ux is a unit vector in the direction of x axis and u y a unit vector in the direction of the
y-axis.
From figure 2 it is easy to calculate the value of cos β as:
vO t
cos β = .
(27)
2
rO0 + (vO t)2
By substituting equation (27) into (23) it can be checked that the angular frequency measured
by the observer is
vO2 t
.
(28)
ω =ω 1− 2
c rO0
+ (vO t)2
Figure 3 shows the angular frequency measured by the observer as a function of time
obtained from equation (28). It is interesting to note that the asymptotes of the curves shown
in figure 3 are the values of the Doppler angular frequencies for the case in which the observer
moves in a radial direction (direction of the source). This can be easily checked by equation (22)
for high values of time, t. The physical meaning is easy to understand, because the observer
is seen to move from the source in a radial direction provided that the time, t, is large enough.
5.2. The observer moves with constant acceleration in the direction of the source
Another interesting example is the case in which the observer moves in a radial direction
accelerated with respect to the source.
The position of the observer can be expressed as follows:
rO (t) = rO0 + 12 at 2
(29)
An analysis of the classical Doppler effect
503
where a is the acceleration of the observer and the initial velocity is supposed to be zero for
simplicity. Since the observer moves in the direction of the source cos β = 1, so from (23):
a
(30)
ω =ω 1− t
c
as vO = at.
Therefore, in the case in which the observer is accelerated with respect to the source
the frequency varies linearly with time t. It can also be seen that there is a limiting time
(tlim = c/a) corresponding to ω = 0. If t > tlim the frequency measured by the observer
is negative, which is not physically possible. It can be easily checked that when t = tlim the
velocity of the observer is c, thus when t > tlim the observer moves with velocity vO > c.
6. The source moves and the observer is at rest
In this case the observer will be supposed to be at rest whereas the source moves through rS (t).
In this case equation (6) becomes
τ=
|rS (t − τ )|
.
c
(31)
6.1. The source moves with constant velocity
The vector position of the source can be described by the following equation:
rS (t) = rS0 + vS t.
(32)
We will assume that:
rS0 = rS0 ux
vS = vS u y .
(33)
(34)
Equation (31) becomes:
τ=
|rS0 ux + vS (t − τ )u y |
=
c
2
rS0
+ vS2 (t − τ )2
c
.
(35)
Hence:
2
τ 2 (vS2 − c2 ) − 2vS2 tτ + (rS0
+ vS2 t 2 ) = 0.
(36)
For vS < c this has a real, positive solution (by (4), τ must be non-negative):
2
+ vS2 t 2 )
2vS2 t − 4vS4 t 2 − 4(vS2 − c2 )(rS0
τ=
.
2(vS2 − c2 )
(37)
From equation (37) and by using (10) the angular frequency measured by the observer
can be obtained as:
c2
vS2 t
.
(38)
ω =ω 2
1− c − vS2
v 4 t 2 − (v 2 − c2 )(r 2 + v 2 t 2 )
S
S
S0
S
This expression was derived by Kapoulitsas [1], although as commented in section 3 he treated
the problem in terms of t − t .
It is easy to see that (38) is equivalent to (8) for vO = 0 and vS · n = −vS cos θ , where θ
is the angle between vS and −n. In this case (8) gives
c
(39)
ω = ω
c + vS cos θ
C Neipp et al
504
Figure 4. Frequency as a function of time in the case in which the source moves with a constant
velocity and the observer is at rest.
and from the geometry in this case
vS (t − τ )
vS (t − τ )
=
cos θ = cτ
2
2
rS0 + vS (t − τ )2
(40)
by (35). Therefore we readily get
ω = ω
c2 τ
.
c2 τ + (t − τ )vS2
(41)
But this expression is just (38), as it can be seen by solving from (37) for the square root there
and substituting into (38).
Figure 4 shows the general form of the frequency measured by the observer as a function
of time obtained from equation (38). Again, it can be observed that the asymptotes of the
curves shown in figure 4 are the values of the Doppler frequencies for the case in which the
source moves in a radial direction (direction of the observer).
7. Conclusions
The classical Doppler effect is analysed using a simple treatment. The model proposed is based
on the derivation made by Donges of the Doppler formulae for the case in which the observer
moves towards the source or the source moves towards the observer. A generalization to the
case in which the source and the observer move with positions which are arbitrary functions
of time is made.
By using this theoretical treatment the basic formulae for the case in which the observer and
the source move in the same direction with constant velocity are obtained in a very simple way.
In addition other cases are analysed and discussed, showing that this method can be generalized
to the case in which the vector positions of the source and the observer are arbitrary functions
of time.
An analysis of the classical Doppler effect
505
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