Sound and light Doppler effects

Denis Michel

Universite de Rennes1, Irset, Rennes, France. E-mail: denis.michel@live.fr

Although electromagnetic and acoustic waves pro-foundly differ in their nature, comparing theirDoppler effects is instructive and reveals persistentconceptual traps. The principle of the Doppler ef-fect was presented by Christian Doppler in 1842long before the advent of relativity theory, butwhile the relativistic Doppler effect is now wellestablished, its non-relativistic version retrospec-tively called classical, suffers from misleading intu-itions such as: (1) there is no transverse classicalDoppler effect; (2) the Galilean Doppler effect cor-responds to the relativistic one without Lorentzdilation factor. Additional pitfalls concern thesound Doppler effect which, in addition to be-ing Galilean, results from asymmetric contribu-tions of the sources and receptors and dependson a material medium for its propagation, possi-bly modifying the effective velocity of the wave.Moreover, contrary to light, the information onthe sound Doppler effect and the location of thesource are transmitted through different channels,sound and light respectively, thereby complicat-ing data interpretation and increasing confusion.A thorough revision is proposed here addressingthese issues and providing a complete set of newcandidate Doppler formulas, non-collinear, two-and three-dimensional and in their angular andlinear versions.

Keywords:Sound Doppler effect; Transverse Doppler effect.

Highlights:• Two- and three-dimensional angular Doppler formulas are

reconstructed.

• The Galilean transverse Doppler effect exists.

• It is technically unfeasible to verify the relativistic transverse

Doppler effect of Einstein.

• The traditional formula for non-collinear sound Doppler ef-

fect is disqualified.

• A general candidate Doppler formula is proposed for the

sound Doppler effect in the presence of wind.

• Longitudinal non-collinear and non-angular Doppler formu-

las are presented.

1 Introduction

One of the major results of the theory of the special rela-tivity theory was the establishment of the laws of lightaberration and Doppler effect, but the Doppler effectnamed for its inventor, extends beyond electromagneticwaves and also exists for waves obeying Galilean rules. Al-though relativistic transformations are more subtle thanGalilean transformations, Galilean Doppler effects appearmore complicated and, as a matter of fact, their currentformulations are misled. According to the historical re-minders of [1], one of the difficulties encountered by Chris-tian Doppler was that his theory seemed too simple math-ematically to describe physics, at the time when the mostcelebrated tool was the differential equations. However, ifsome differential systems present technical subtleties, theyremain globally intuitive while conversely, the conceptionof mathematically simple processes may be tricky, as istypically illustrated by the classical Doppler effect. Thefirst sections of this study will focus on the Doppler ef-fect understood as Galilean in the mathematical sense, byconsidering the receiver as immobile. Then the additionalparticularities of the sound Doppler will be introduced,including the source-receiver asymmetry and the displace-ment of the propagation medium.

2 Problems with the angular ver-sion of the classical Doppler ef-fect

2.1 The currently accepted formula

The current Doppler formula describing the change in fre-quency of a moving source perceived by a static observeris

fmov

f=

1

1− β cos θ(1)

where β is the ratio of the source velocity to the wavevelocity (β = v/c). A first ambiguity in this formula con-cerns the nature of the angle θ between the trajectory ofthe source and the direction of the receiver. The phe-nomenon of aberration is not specific to relativity but itis strangely neglected in the so called classical Doppler ef-fect. The origin of the angle considered is generally thesource but in addition it is necessary to specify whether

1

arX

iv:2

112.

1366

1v2

[ph

ysic

s.cl

ass-

ph]

22

Feb

2022

the position of the source is to be considered when thewave is emitted or received. As the Doppler effect is nat-urally carried and detected through the same wave, thefirst possibility appears reasonable and it would thereforebe the angle θ′ of Fig.1,

fmov

f=

1

1− β cos θ′(2)

even if this angle is considered as apparent by the observerwho knows that it no longer corresponds to the true posi-tion of the source when the Doppler effect is received.

Figure 1. Parameters used to represent the sound Doppler

effect. (A) Angular representation. As the speed of light can

be considered as infinite compared to that of sound, the sound

wave carrying the Doppler effect emitted by the source S un-

der an angle θ′ towards the receiver R, and the light carrying

the image of the source, emitted under an angle θ towards

the receiver, arrive together at the receiver. On the whole

trajectory, D is the shortest distance separating the source

and the receiver. (B) In this linear representation, the X axis

corresponds to the trajectory of the source. The unit of length

chosen here is the shortest distance D, which allows to define

the normalized distance x = X/D.

Eq.(2) is given in textbooks [2] and online courses [3], butdisagrees with the hypothesis of a spherical propagationof the wave around its point of emission, which predicts aDoppler formula of the form [4]

fmov

f=

1√1 + β2 − 2β cos θ′

(3)

In an attempt to reconcile the traditional (Eq.(2)) andexact (Eq.(3)) formulas, the former could be rewritten asa square root of a square,

fmov

f=

1√1 + β2 cos2 θ′ − 2β cos θ′

(4)

But Eq.(3) and Eq.(4) are still different and in addi-tion one cannot invoke the approximation of a very smallangle θ′ (which would make cos2 θ′ ≈ 1) because of the

alleged prediction that the Doppler effect must vanish(λmov/λ = 1) when θ′ = π/2 and cos θ′ = 0. So we seethat the classical Doppler formula is based on several lay-ers of misleading intuitions. To make matters worse, theseerrors are likely to have been supported by a resemblancewith the relativistic Doppler effect.

2.2 How the relativistic Doppler effectcould have reinforced a flawed classi-cal approach

The formula of the classical Doppler effect (Eq.(2)), haslikely been consolidated by the advent of the relativisticDoppler effect, whose form can suggest the confusing ideathat it is simply the classical Doppler effect corrected bythe relativistic dilation factor.

2.2.1 The puzzling idea that the classical Dopplereffect is the primary effect of the relativisticone

According to approximate relativistic theories, the classi-cal formula would be a ”primary” Doppler effect of purelykinetic nature, which must be completed with a so-called”secondary” effect of time dilation by the Lorentz factor(multiplication of the periods by 1/

√1− β2) to obtain the

relativistic Doppler effect. In fact, in relativity the kineticand temporal effects cannot be dissociated and the Lorentzfactor itself includes the change of kinetic energy. Never-theless, this questionable principle was accepted because itapparently works. In the famous longitudinal relativisticDoppler formulas, for the approach,

fmov

f=

√1− β2

1− β=

√1 + β

1− β(5a)

and for the recession,

fmov

f=

√1− β2

1 + β=

√1− β1 + β

(5b)

It can be extended to all intermediate points, not collinearto the source path, by applying the same correction to thegeneral equation Eq.(2).

fmov

f=

√1− β2

1− β cos θ′(5c)

It happens that this latter equation is indeed Einstein’srelativistic Doppler formula where θ′ is the reception an-gle. This unfortunate identity has logically reinforced thesupposed validity of Eq.(2) as the classical Doppler ef-fect formula for generations of researchers and teachers,and convinced them that the relativistic Doppler effect ofEq.(5c) is simply the classical Doppler effect of Eq.(2) cor-rected by the dilatation factor of special relativity. Theconsensus generated by this apparent evidence likely in-hibited naive questions such as, for instance, if the onlydifference between the classical and relativistic Doppler ef-fects is the dilation of the periods for the latter, then why

2

would the classical and relativistic aberration rules be dif-ferent? Indeed, the generalized correction by the dilationfactor has a homothetic effect unable in itself to modifythe angles. But a majority consensus in the scientific com-munity naturally tends to inhibit legitimate questions.In addition, the secondary Doppler effect supposedly spe-cific to relativity, was further consolidated by the inap-propriate use of the arithmetic mean in the most famousvalidation tests of the relativistic Doppler effect, eitherlongitudinal [5] or transversal [6] (appendix B). As we willsee in section 6, the idea that this relativistic secondaryDoppler effect is the only one perceptible in the transversesituation, was based on a misconception of the GalileanDoppler effect which presents in fact a very similar trans-verse Doppler effect.

3 Galilean and relativistic wavebubble equations arising fromgalilean and relativistic transfor-mations

Galilean Doppler and aberration formulas can be deducedfrom the rays connecting the source to the wavefront,taking into account that the wavefront of a classical waveis the surface of a sphere and that of a relativistic waveis the surface of an ellipsoid with rotational symmetryaround the axis of the trajectory [7].

3.1 The Galilean wave bubble

The Galilean transformations are simple: x′ = x+vt, y′ =y, z′ = z and t′ = t. Hence, the surface of the Galileanwavefront emitted from a source moving at speed v in thex direction, is a sphere of Cartesian equation

(x+ vt)2 + y2 + z2 = (ct)2 (6a)

For a single period wave bubble (t = T = 1) and using theparameter β = v/c

(x+ β)2 + y2 + z2 = 1 (6b)

The wavelengths are directly obtained by convert-ing this Cartesian equation in polar equation [4], as theradii ρ using polar and deviation angles such that x =ρ sin θ cosϕ, y = ρ sin θ sinϕ and z = ρ cos θ. The radiusof the Galilean wave bubble derived from the completeCartesian equation Eq.(6b) reads

ρ =

√1− β2(1− sin2 θ cos2 ϕ)− β sin θ cosϕ (7)

shown in Fig.2A and which reduces in the two spatial di-mensions x = ρ cos θ and y = ρ sin θ, to the Galilean circle

ρ =λmov

λ=

√1− β2 sin2 θ − β cos θ (8)

which is equivalent to a Doppler effect formula ex-pressed using wavelentghs. However, this Doppler effect isnot valid in optics since the Doppler effect of electromag-netic waves, called relativistic, should take into accountthe relativistic transformations.

3.2 The relativistic wave bubble

Lorentz transformations rewritten by Poincare [8] assum-

ing c = 1, are: x′ =x+ βt√1− β2

, y′ = y, z′ = z and

t′ =t+ βx√1− β2

. They transform the coordinate x into

x′ = x√

1− β2 + βt′, so that the relativistic wavefrontis, for one period t′ = T = 1 and the wavelength cT = 1,the surface of the single-period ellipsoid:

(x√

1− β2 + β)2 + y2 + z2 = 1 (9)

which gives upon conversion in polar coordinates theremarkably elegant equation

ρ =

√1− β2

1 + β sin θ cosϕ(10)

shown in Fig.2B and reducing in 2D polar coordinates toan ellipse, better known in normal coordinates as the rel-ativistic Doppler effect

ρ =λmov

λ=

√1− β2

1 + β cos θ(11)

Figure 2. Perspective view of the shapes of the Galilean (A)

and relativistic (B) single period wavefront surfaces, drawn to

Eq.(7) and Eq.(10) respectively, for β = 0.9. The source is

located at the intersection of the axes.

4 Galilean and relativistic Dopplerand aberration formulas

The above equations of wave bubbles are also those ofDoppler effects expressed in wavelengths. They can bemore appropriately expressed with frequencies, in theGalilean case for an immobile receptor in a stationarypropagation medium. The angle θ used in these equa-tions takes into account the position of the source whenthe Doppler effect is received, but as this position is not

3

visible by the observer in the case of light, it is useful todefine another angle, θ′ taking into account the formerposition of the source precisely when it emitted the wavewhose Doppler effect is detected.

4.1 Two usable angles and two sets of for-mulas

Two angles, linked together by aberration relations, canbe used to describe Doppler effects:• the angle θ, defined between the source path and the di-rection of the source-receiver line when the Doppler effectis received, and• the angle θ′, defined between the trajectory of the sourceand the line connecting the point of emission to the re-ceiver (Fig.1A).

4.2 The Galilean formulas

The reciprocal relations of aberration linking these twoangles are

cos θ =cos θ′ − β√

1 + β2 − 2β cos θ′(12a)

and

cos θ′ = cos θ

√1− β2 sin2 θ + β sin2 θ (12b)

and using the tangent function often employed to describethe aberration,

tan θ′ =sin θ

(√1− β2 sin2 θ − β cos θ

)cos θ

√1− β2 sin2 θ + β sin2 θ

(12c)

Accordingly, two Doppler formulas related through theaberration relations, can be established depending on theangle considered. The Doppler effect is, in function of θ,(

fmov

f

)θ

=1√

1− β2 sin2 θ − β cos θ(13a)

and in function of θ′(fmov

f

)θ′

=1√

1 + β2 − 2β cos θ′(13b)

corresponding to Eq.(3).

4.3 The relativistic formulas

The above formulas are Galilean and do not apply to elec-tromagnetic waves whose wavefronts are ellipsoidal sur-faces [7, 9]. The aberration relations are in this case

cos θ =cos θ′ − β

1− β cos θ′(14a)

cos θ′ =cos θ + β

1 + β cos θ(14b)

tan θ′ =sin θ

√1− β2

β + cos θ(14c)

and the Doppler effect can also be written in two ways(fmov

f

)θ

=1 + β cos θ√

1− β2(15a)

(fmov

f

)θ′

=

√1− β2

1− β cos θ′(15b)

The relativistic formulas are demonstrated by the stan-dard relativistic approach in [9] further detailed in [10].For alternative geometric demonstrations of all these for-mulas, Galilean and relativistic, see [4]. Although theGalilean transformations are much simpler than the rel-ativistic (or Lorentz) ones, paradoxically the relativisticDoppler and aberration formulas appear simpler and moreelegant than their Galilean counterparts, even before ap-plying them additional complications detailed later, in-cluding the propagation medium and the source-receptorasymmetry. The Doppler profiles described by these equa-tions are shown in Fig.3 and Fig.4 as functions of θ andθ′.

Figure 3. Angular representation of the Doppler effect of a

Galilean wave emitted towards a fixed receiver by a moving

source.

θ′ refers to the real point of emission of the wave andθ refers to the real position of the source, the differencebetween the two being due to the delay in the travel ofthe wave whose speed is finite. This comparison of theGalilean (Fig.3) and relativistic (Fig.4) profiles clearlyshows that: (i) The only common point between them,giving the same Doppler effect, is (θ, θ′) = (π/2, cos−1 β).(ii) The difference between the Galilean and relativisticDoppler effects is not due to the absence of a transverseeffect for the first one contrary to the second one, becausethe ratio fmov/f = 1 is never obtained for an angle of π/2.(iii) Contrary to popular belief, the relativistic Dopplereffect is not merely the Galilean Doppler effect correctedby the Lorentz expansion factor, as erroneously suggested

4

by the usual Eq.(2) supposed to be the relativistic formulaEq.(15b) lacking the Lorentz factor. As a matter of fact,the polar curve described by Eq.(2) is an ellipse but not acircle, which definitely disqualifies it as a Galilean Dopplerformula.

Figure 4. Angular representation of the Doppler effect of

electromagnetic waves, called relativistic.

4.4 Dual information on the Doppler ef-fect and the position of the source

Since no information can exceed the speed of light, rel-ativistic formulas do not suffer from any ambiguity intheir measurement, because the information on the posi-tion of the source and on the Doppler effect necessarilypass through the same channel. This is not the case forthe Doppler effect of low speed waves because the ac-tual position of the source can be determined visually,i.e. transmitted by light [11]. Therefore, the informa-tion on the sound and the position of the source, arrivingthrough different channels (Fig.1A) can be combined (asillustrated in the appendix D).

5 Doppler effects as functions ofdistance

5.1 Converting angles to distances

The representations of the Doppler effect as a function ofthe angle of the source varying from 0 to π shown in Figs.3and 4, amplify the central part of the trajectory aroundπ/2, and compress the more distant locations until infin-ity. Hence, they are poorly appropriate to ordinary mea-surements for a source evolving at constant speed alonga linear coordinate X, as that shown in the appendix D.To superimpose these records on a theoretical curve, it

is first necessary to establish the correspondence betweenangles and distances. Several units can be chosen for theX axis, one of the simplest consists in giving to the incre-ment of X the value of the minimum distance D betweenthe source a the receiver. We can then define a relativedistance x = X/D (Fig.1B) which allows to standardizeall the data for sources passing more or less far away. Thecorrespondence of any angle ϑ with the distance is

D = −X tanϑ (16)

X

D= x = − cosϑ√

1− cos2 ϑ(17)

cosϑ = − x√1 + x2

(18)

By applying this relation to the angles θ and θ′ pre-sented previously, we obtain, for the classical Doppler for-mula (

fmov

f

)classical

=1

1 +βx′√

1 + x′2

(19)

and for the formulas deduced from the spherical wave-front, the Doppler effects are described as functions of thecoordinates of the position of the source (P ) and of theemission point (E), by setting ϑ = θ or θ′ respectively.(

fmov

f

)P

=

√1 + x2

βx+√

1− β2 + x2(20)

and (fmov

f

)E

=1√

1 + β2 + 2βx′√

1 + x′2

(21)

5.2 Linear Galilean aberration

The light Doppler effect of a star is naturally measured bypointing the telescope at this star, but we are aware thatit has changed location while the light was flying towardsthe telescope, so that its true position is invisible. By con-trast for the sound, the informations on the Doppler effectand on the location of the source are carried by differentchannels and can be recovered simultaneously. Indeed,the Doppler effect of the sound is naturally carried bythe acoustic wave but the information on the position ofthe source is generally visual, i.e. carried by a light wave(Fig.1A). When the wave emitted in X arrives at the levelof the receiver, the source will have continued to progressover a distance depending on the duration ∆t of the flightof the wave from the source to the receiver. This pathof length c∆t is the hypotenuse of a right triangle whoseother two sides are the shortest distance D, and the dis-tance X separating the source from the nearest point. SoPythagoras says

(c∆t)2 = D2 +X2 (22a)

5

from which

∆t =

√D2 +X2

c= D

√1 + x2

c(22b)

During this time, the source will have traveled

∆X = v∆t = βD√

1 + x2 (22c)

or in normalized distance

∆x = β√

1 + x2 (22d)

The point of emission can be calculated from the actualposition of the source when the Doppler effect is detected.

The angle θ′ = cos−1(− x√

1 + x2

)whose origin is the

point of emission, is expected to become θ when replacingx by x+ ∆x:

θ = cos−1

−(x+ β√

1 + x2)√1 + (x+ β

√1 + x2)2

which can be rewritten

= cos−1

(− x√

1 + x2

)− β√

1 + β2 − 2β

(− x√

1 + x2

)

(23)

In this form, Eq.(23) is clearly analogous to the aber-ration formula Eq.(12a), recovered here from a linear rep-resentation, as illustrated in Fig.5.

Figure 5. Representation of the Galilean angular aberration

as a function of distances. Dashed curve: angle θ between

the trajectory of the source and the direction source-receiver

according to Eq.(23). Solid line curve: angle θ′ between the

trajectory of the source and the line connecting the emission

point to the receiver.

The introduction of Eq.(23) in the Doppler for-mula Eq.(13a), indeed gives back the curve of Eq.(21).Conversely, the introduction in the Doppler formula

Eq.(13b) of the angle θ obtained by conversion of

cos−1(− x√

1 + x2

)by the aberration formula Eq.(12b),

gives the curve of Eq.(20). The Doppler functions derivedfrom this approach are represented in Fig.6.

6 The Galilean transverse Dopplereffect

The acceptance by the scientific community of the classi-cal Doppler effect (dotted line in Fig.6), could have beenfavored by the mistaken intuition that the Doppler ef-fect cancels (fmov/f = 1) when the source is the clos-est (x = 0). The absence of Doppler effect predicted bythe currently admitted Eq.(2) seems very reasonable [3].It can indeed seem reasonable but it is nevertheless erro-neous, as revealed by a rigorous analysis of the sphericalwave (Table 1). This is typically a misleading intuition.For a moving source and static receiver, the transverseDoppler effect of sound, obtained for a reception angleθ′ = π/2, is (

fmov

f

)transverse

=1√

1 + β2(24)

When this effect is heard, the source is at the distance βDfrom the nearest point (Table 1). By comparison, the fa-mous relativistic transverse effect envisioned by Einsteinas a possible confirmation of special relativity theory [12],is (

fmov

f

)transverse

=√

1− β2 (25)

For small values of β, these effects are almost identical

since both are 1 − β2

2 + O(β4) and differ only by β4/4.The difference between the transverse Doppler effects cal-culated in the Galilean and relativistic ways, would beonly 1.5% for a speed as phenomenal as half the speedof light, making the discrimination proposed by Einsteintechnically very delicate. In spite of the importance of theverification of the transverse relativistic effect suggestedby Einstein, only one study has confirmed this prediction[6]. In addition, a retrospective analysis of this work raisessome concerns: (i) The uncertainty margin on the resultof Eq.(25) exceeds the value of Eq.(24), especially as theauthors explained that they had to widen their angle to91◦ to measure this effect. (ii) The experimental setupof these authors (see the Fig.1 of [6]) is intriguing sincethe rays at 90◦ to the trajectory are canalized while thereception angle is diluted in a cascade of mirrors, whenit is the reception angle that should be π/2 for the trans-verse effect (Fig.4 and Table 2). (iii) Finally these authorsused the arithmetic mean to average wavelengths and de-fine the so-called secondary Doppler effect (appendix B).The Galilean Doppler effect studied above for a movingsource must now be completed with a series of additionalsubtleties for the sound.

6

Figure 6. Doppler effect of the sound as a function of the relative position of a moving source on its path, for a stationary

receiver in the absence of wind, expressed as a function of either the coordinate of the emission point (lower red curve drawn

to Eq.(21)), or of the position of the source detected visually (upper blue curve drawn to Eq.(20)). The dashed curve is that

of the classical Doppler formula drawn to Eq.(19), shown for comparison. The increment of the x coordinate is the minimum

distance between the source and the receiver. Some remarkable points from these curves are listed in the appendix A (Table 1)

in comparison with those of the relativistic Doppler effect (Table 2).

7 The steady wind

An additional difficulty for the sound is that its medium ofpropagation can itself move. The sound wave has in factno intrinsic material existence other than alternating com-pressions and depressions in the medium. A general move-ment of this medium, banally known as wind, is thereforeexpected to move the waves it contains and can of coursefacilitate or hinder their propagation. A first overview canbe obtained for a mobile source and immobile receiver. Inthe same way that β is the ratio between the velocity ofthe source v and that of the wave c, the ratio betweenthe wind velocity vw and that of the wave will be noted ω.The wave bubbles decentered by the velocity of the source,are now translated non-symmetrically apart from its tra-jectory. However, this displacement of wavefronts by thewind between stationary sources and receivers, does notinduce any change in the perceived frequency, by compen-sation between wave speed and wavelength. Indeed, inthe direction of the wind, the wave fronts progress morerapidly but their spacing lengthens in the same ratio be-cause the wavefronts are emitted at the same rate f0 bythe source whether there is wind or not. The successivewave crests are separated by the same time interval butby a larger spatial interval:

λw =f0

c+ vw

Hence, these wavefronts which are more spaced but travel

faster arrive at the same frequency.

Figure 7. Influence of wind on the wave bubble offset illus-

trated by a few pairs of values (ψ, φ), drawn to Eq.(31). All

angular combinations maintain the spherical surface.

Inversely, in the direction opposite to the wind, thewave fronts progress more slowly but their spacing is re-duced in the same ratio. These wavefronts that are denserbut travel more slowly also arrive at the same frequency.So in all cases for static and comobile S and R,

7

c

λ=cwind

λwind= f

This reasoning holds for all directions, that is to sayfor any arbitrary angle θ in the Doppler formula. Butcontrary to the velocity of the wave, that of the sourceis considered independent of the wind, so if the sourcemoves in the wind, the basal reference condition shouldbe the presence of wind only. For each angle θ, the wave-crest spacing for the combined motion of the wind and thesource ρω+β should be weighted by that caused by wind

only ρω, to give the Doppler effect(fmov

f

)wind

=ρωρω+β

(26)

The distances between the source and the wavefrontsurface depend not only on the velocity of the source com-pared to that of the sound (β), but also on the velocityof the wind compared to the sound (ω), according to theGalilean law of velocity addition β + ω. These values of ρcan be calculated by repeating the approach of Cartesianto polar conversion (section 3, [4]).

Using three-dimensional polar coordinates, the sound wave bubble is shifted by the wind blowing with zenithaland asimuthal angles such that

(x+ β + ω sinψ cosφ)2 + (y + ω sinψ sinφ)2 + (z + ω cosψ)2 (27)

It is converted in polar coordinates by replacing x by ρ sin θ cosϕ, y by sin θ sinϕ and y by ρ cos θ, yielding the equationfor ρ

ρ2 + 2ρ (β sin θ cosϕ+ ω(cos θ cosψ + sin θ sinψ cos(ϕ− φ))) + ω2 + β2 + 2βω sinψ cosφ− 1 = 0 (28)

NotingA = sin θ cosϕ

B = cos θ cosψ + sin θ sinψ cos(ϕ− φ)

C = sinψ cosφ

the above quadratic equation reads

ρ2 + 2ρ (βA+ ωB) + ω2 + β2 + 2βωC − 1 = 0 (29)

and its solution isρβ+ω =

√1− β2(1−A2)− ω2(1−B2)− 2βω(C −AB)− βA− ωB (30)

or, retranslated in elementary components,

ρβ+ω =√

1− β2 [1− (sin θ cosϕ)2]− ω2 [1− (cos θ cosψ + sin θ sinψ cos(ϕ− φ))2]

−2βω [sinψ cosφ− sin θ cosϕ (cos θ cosψ + sin θ sinψ cos(ϕ− φ))]

− β sin θ cosϕ− ω (cos θ cosψ + sin θ sinψ cos(ϕ− φ))

(31)

The three-dimensional representation of this equation in Fig.7 for the angles θ and ϕ shows that it still yields spheres.This equation in absence and presence of source motion, gives the full Doppler formula for an immobile receiver.

ρωρβ+ω

=

√1− ω2(1−B2)− ωB√

1− β2(1−A2)− ω2(1−B2)− 2βω(C −AB)− βA− ωB(32)

which is developed in the appendix Eq.(C.4). In the plane (~β, ~ω), this formula reduces to

(fmov

f

)wind

=

√1− ω2 sin2(θ − ψ)− ω cos(θ − ψ)√

1− [ω sin(θ − ψ) + β sin θ]2 − ω cos(θ − ψ)− β cos θ

(33)

of which a direct step-by-step demonstration is proposed in the appendix C, and which is illustrated in Fig.8 for theset of values: β = 0.5, ω = 0.3 and ψ = π/4 or 3π/4.

8

Figure 8. Effect of an homogenous wind on the Doppler effect perceived by a static receiver. (A) Doppler effects for a mobile

source and immobile detector in absence of wind (red surface) or in presence of wind blowing under the angle ψ shown in the

bottom inset and ranging from 0 to π. (B) Examples of Doppler profiles extracted from the panel A in the absence of wind

(red continous curve), in the presence of wind blowing with the sound (ψ = 3π/4, dashed blue curve) and against the sound

(ψ = π/4, dotted blue curve). Curves drawn to Eq.(33) with β = 0.5 and ω = 0.3.

When the wind is collinear to the trajectory (ψ = θ =0), Eq.(33) reduces to

(fmov

f

)wind

=1± ω

1± ω ± β=

c± vwc± vw ± vS

(34)

which depends on the relative velocity signs (otherwisegiven by the cosines). A headwind (ω negative) naturallyhinders the progression of the wavefront in the x directionwhile a downwind (ω positive) favors it.

8 Asymmetric roles of the sourceand receiver

When a source S and a receiver R move one with respect tothe other with a relative velocity v, for the electromagneticwaves one cannot attribute the movement specifically to Sor R and therefore define anything other than the relativevelocity. By contrast for the sound wave, it is possible tospecify the precise contributions of S and R in their rel-ative motion, thanks to an additional point of referenceoffered by a sort of substratum grid. In the simple dia-grams of Fig.9, this grid can be compared to the fibers ofthe paper of the printed article or to the pixel coordinateson the screen, which make it possible to distinguish theabsolute movements of S and R. Note that this spatial

reference grid is not the propagation medium, because incase of wind one can also define an absolute velocity ofthe medium itself, relatively to some absolutely static ref-erence grid. In ordinary experiments, the fixed spatial ref-erence chosen is concretely the terrestrial ground assumedfixed. This major difference with light explains why forthe sound, for the same relative velocity v, the Dopplereffect can take different values depending on whether thedisplacement on the grid is made by S or R.

Figure 9. A sound pulse is emitted at t0 by the source S

when spaced from the receiver R by D0. The sound travels

through a static medium relatively to which either R (middle

line) or S (bottom line), is immobile.

9

The resolution of this asymmetry, which is simple but sub-tle, is an additional illustration of the fact underlined inthe introduction that sophisticated mathematics are chal-lenged in this field by simple algebra associated to gooddiagrams. A convenient approach in case of dual motionof the source and receiver with and without wind, is basedon a principle of conservation of the wave phase [13]. Avisualization is proposed below in the case of a distancing(because it is easier to draw) between S and R (Fig.9). Sand R recede form each other at relative velocity v and inaddition, move relative to the static medium supportingsound propagation at speed c. At time t0, when spacedfrom the receiver by D0, the source emits a light beamtowards the receiver. Different results are expected de-pending on whether this is the source or the receiver whichmoves relatively to the static background.

8.1 The receiver is immobile

When only the source moves, as shown in the middlescheme of Fig.9, its movement no longer affects the dis-tance to travel by the sound already emitted at t0. Hence,the sound reaches the receiver at time tR after crossingthe initial distance D0. The duration of the sound travelis therefore

tR − t0 = D0/c (35a)

At tR, the new spacing between the source and the receiverhas become

DR = D0 + v(tR − t0) (35b)

Replacing the duration (tR − t0) in this equation by thevalue given by Eq.(35a), yields a distance ratio

DR

D0= 1 +

v

c(35c)

For a single period, as the source moves back, the nextwavefront following that sent at t0 will be necessarily de-layed such that

DR

D0=cTRcT0

=f0fR

and the Doppler effect is

fRf0

=1

1 +v

c

(35d)

8.2 The source is immobile

In this case (bottom situation of Fig.9), the sound travelis longer because the wavefront must reach the receiverwhich is concommitantly escaping.

tR − t0 = DR/c (36a)

and

DR = D0 + v(tR − t0) (36b)

Replacing the duration in Eq.(36b) by the value given byEq.(36a), yields the Doppler effect

fRf0

= 1− v

c(36c)

Since the relative velocity v can be precisely attributedto S and R, it is split into vS and vR components. Thegeneral formula for collinear recession Doppler effect is,using velocity magnitudes,

(fRf0

)r

=1− vR

c

1 +vSc

(37)

This simple reasoning can be repeated in the case ofcollinear approach

(fRf0

)a

=1 +

vRc

1− vSc

(38)

and generalized to non-collinear movements using angles.The asymmetry of the sound Doppler effect has strangeconsequences, long established in [10]. For example let usconsider three cases of approach where the relative veloc-ity between S and R is the speed of sound.

c = vS + vR

If only the source moves, the Doppler effect will befR/f0 = ∞ (sound barrier). If only the receiver ad-vances towards the immobile source, the Doppler effectis fR/f0 = 2, and if both the source and the receiver ad-vance one towards the other at c/2, the resulting Dopplereffect is fR/f0 = 3. Such subtleties do not exist for light.

8.3 No background medium for light

In absence of substratum grid, it is impossible to assignthe relative movement to either the source or the observer.As the contributions of S and R in the relative velocityare undistinguishable, it seems natural to average the twoextreme situations described above (only S moves or onlyR moves). As discussed in the appendix B, the type ofmathematical mean appropriate for averaging frequenciesis the geometric mean. Interestingly, the geometric meanof Eq.(35d) and Eq.(36c) is

⟨fRf0

⟩=

√√√√√1− v

c

1 +v

c

(39)

In the special relativity theory, uniform motion cannotbe attributed specifically to one of the relatively movingobjects, so that the total distance crossed by light DL, isnot DR nor D0 as above, but

DL = c(tR − t0) =c

v(DR −D0) (40)

10

This should be the fundamental principle of distancecalculation in astronomy [14]. As an exercise, the rela-tionships between DR, D0, DL and the resulting Dopplereffect have been calculated for different theoretical modesof space expansion in [15]. But the differences betweenthe sound and the light do not stop there.

8.4 Triple collinear motion of the source,receiver and medium

A wind blowing in the direction of the sound (called down-wind, from left to right in Fig.9), is expected to decreasefrequency whereas a headwind blowing against the sound(from right to left in Fig.9), is expected to increase fre-quency. The calculations can be summarized in the ex-treme cases of recession illustrated in Fig.9, in which ei-ther the receiver or the source remains completely immo-bile. The duration tR−t0 of the sound travel, noted below∆t, depends on the velocity of the wind vw.

8.4.1 Immobile receiver

When only the source moves, as shown in the middle sit-uation of Fig.9,

DR = D0 + vS∆t (41)

• Downwind. A wind blowing as the sound path increasesit speed and push the wavefront forward,

∆t =D0

c+ vw(42a)

Replacing ∆t by this value in Eq.(41) gives

DR = D0

(1 +

vSc+ vw

)(42b)

so thatfRf0

=1

1 +vS

c+ vw

(42c)

• Headwind

∆t =D0

c− vw(43a)

Replacing ∆t by this value in Eq.(41) gives

DR = D0

(1 +

vSc− vw

)(43b)

so thatfRf0

=1

1 +vS

c− vw

(43c)

8.4.2 Immobile source

When only the source moves as in the bottom of Fig.9,

DR = D0 + vR∆t (44)

• Downwind

∆t =DR

c+ vw(45a)

Replacing ∆t by this value in Eq.(44) gives

DR

(1− vR

c+ vw

)= D0 (45b)

so thatfRf0

= 1− vRc+ vw

(45c)

• Headwind

∆t =DR

c− vw(46a)

Replacing ∆t by this value in Eq.(44) gives

DR

(1− vR

c− vw

)= D0 (46b)

so thatfRf0

= 1− vRc− vw

(46c)

When both the source and the receiver are moving,the complete formulas for collinear recession are, usingvelocity magnitudes,

• Downwind

fRf0

=1− vR

c+ vw

1 +vS

c+ vw

(47)

• Headwind

fRf0

=1− vR

c− vw1 +

vSc− vw

(48)

In summary, the simplest writing, easiest to rememberfor all the collinear situations using velocity magnitudes, is

• For the recession

fRf0

=cw − vRcw + vS

(49)

with cw = c + vw when the wind blows with the soundand cw = c− vw when it blows against the sound.

• For the approach

fRf0

=cw + vRcw − vS

(50)

with cw = c + vw when the wind blows in the same di-rection as the sound and cw = c − vw when the wind isopposite to the direction of the sound.

The extension of these collinear equations to a generalformula holding for all the combinations of relative veloci-ties: of the source (β = vs/c), the receiver (γ = vR/c) andthe medium (ω = vw/c) and for arbitrary angles, wouldrequire splitting the velocity components and repeatingthe polar conversion approach in three-dimensional coor-dinates. A reduced version is obtained by confining the

11

receiver velocity vector in the plane (x, y) = (~β, ~ω). In-stead of staying static in the wind as in section 6, thedetector moves with a velocity γ = vR/c, making an an-gle α with the source trajectory. While the propagationof the successive wavefronts results from a combination ofthe source and wind velocity vectors, the motion of the re-ceiver is assumed to be autonomous and to follow a linearpath whose polar equation is

ρ =ρ0

sin(α− θ)

crossing the successive spherical wavecrests. Depending onits direction, the receiver either comes in front of the wave-crests increasing their perceived frequency or moves awayreducing the frequency. As the magnitude of the receivervelocity component towards the source is γ cos(α− θ), itscontribution to the Doppler effect according to [11], com-bined to those of the wind and the source defined here totake into account the aberration, gives

fmov

f=

√1− ω2 sin2(θ − ψ)− ω cos(θ − ψ) + γ cos(α− θ)√

1− [ω sin(θ − ψ) + β sin θ]2 − ω cos(θ − ψ)− β cos θ

(51)

of which a numerical example is shown in Fig.10.

Figure 10. Doppler effects drawn to Eq.(51) for the dual

motion of the source (β = 0.5) and of the detector (γ = 0.4),

without wind (bottom surface) or in presence of wind (ω = 0.3)

blowing with an angle ψ of π/4 (top surface).

This reduced formula differs in several aspects fromthose proposed in [16, 13, 11] but is not yet satisfactoryand usable in practice because θ also evolves with themovement of the receiver. As illustrated in Fig.11 drawnto Eq.(51) simplified in the absence of wind

fmov

f=

1 + γ cos(α− θ)√1− (β sin θ)2 − β cos θ

the frequency perceived by the detector results from thecombination of two main parameters: (i) the magnitudeof its angular velocity towards the source, itself depending

on the relative values of α and θ, and (ii) the spacing be-tween the successive wave crests it crosses. As these twoparameters are not parallel functions of θ, this can gener-ate internal peaks in the Doppler profiles, as illustrated inFig.10 and Fig.11.

Figure 11. Snapshots of source-receiver configurations (A)

The doppler effect depends on the density of wavecrests along

the line connecting R to S, and on the magnitude of the re-

vocity of R toward S, which depends itself on the combination

of θ and α. (B) Doppler profiles drawn to Eq.(51) in absence

of wind and for different values of the angle α).

In the examples shown in Fig.11A, on the one hand,an increase in θ reduces the angle α− θ and therefore in-creases the magnitude of γ cos(α − θ), but on the other

12

hand the increase of θ also goes with a spacing of thewavecrests with inverse effects on the frequency, makingEq.(51) poorly useful in practice.

9 Can wavelengths and frequen-cies be used indifferently to de-scribe Doppler effects?

Doppler effects are often described in terms of wavelengthwhen the effects generated by waves such as the Dopplereffect, sound and colors, are not determined by wave-lengths λ but by frequencies f . Frequencies and wave-lengths are mutually constrained by the velocity of thewave

c = fλ

So as long as c is constant, f and λ can be used indif-ferently to describe the colors, the Doppler effect and thesound, but one should keep in mind that the real determi-nant is the frequency, itself proportional for electromag-netic waves to the energy through E = hf where h is aconstant (of Planck). In this case, according to the fun-damental law of nature of the conservation of energy, thefrequency is an invariant, so that in the absence of energyvariation between reference points, any modification of cnecessarily leads to a joint modification of λ leaving f un-changed, as are the Doppler effect and colors. However thecolors are commonly associated with wavelengths (520nmfor green, 630nm for red, etc), but this relation is validonly under the implicit assumption that c is constant. Ifthe speed of light is more or less decreased, for example bypassing through materials with a higher refractive indexn, which changes the speed of light to V = c/n, where cis its speed in vacuum, the wavelengths are lengthened inthe same proportion but the frequencies do not change.We have all noticed that our bathing suit does not changecolor when seen through water, although its wavelengthhas changed. The color is unchanged because it dependson the frequency only. Variations in wave velocity are evenmore critical for sound due in particular to the presence ofwind, real or apparent, in the propagation medium (air inordinary conditions). A particularly instructive situationto illustrate this question is shown in Fig.12.

9.1 Application of Fig.12 to the sound

Two cars are driving at the same speed and in parallel,with the siren of the one shown below emitting a con-tinuous sound. The relative speed of the two vehicles iszero, yet the wavelength received by the other vehicle isshortened by a factor of

√1− β2 in the direction orthogo-

nal to the travel, because the wavefronts are shifted back-ward by the ”apparent wind” of the propagation medium.However, the driver of the upper car does not perceiveany Doppler effect owing to the principle explained in sec-tion 7. The wavelength decreases by the same factor asthe wave velocity, so that the frequency is unchanged, in

agreement with the absence of relative motion of the twocars. We know that one swims more slowly to cross ariver with a current and that the crossing period is there-fore longer. It is the same in the case of Fig.12 where therole of the current is played by the apparent wind. Thewavelength is the adjustment variable that guarantees theinvariance of the frequency. The joint decrease of the wavespeed is not only valid in the direction orthogonal to thetrajectory but in all directions. Accordingly, the modifi-cation by the wind of the velocity of sound found in [13]is

cw =

√c2 − v2w sin2 ψ + vw cosψ (52)

which is equivalent to the Doppler effect of Eq.(8) ifreplacing vw/c by −β.

Figure 12. Two cars, one of which is equipped with a siren

of known frequency, drive in parallel at the same speed. The

relative velocity of the two vehicles is zero and yet the wave-

lengths are shortened on the axis connecting them, because of

the apparent wind due to the speed of the cars, which shifts

the wavefronts backwards. This shortening of the wavelengths

(by a factor of√

1 − β2) does not generate a Doppler effect

because the velocity of sound is reduced in the same propor-

tion, which keeps the frequency unchanged.

9.2 Treatment of this situation for thelight

The situation in Fig.12 is also worth considering for a lightwave (by increasing the speed of luminous cars). In thiscase, there is no apparent wind since there is no propa-gation medium. For an observer considered immobile andexternal to the system of Fig.12, the speed of the wave c isconstant but the Lorentz factor involved in time dilationwill play its role by expanding the wavelengths preciselyby the factor 1/

√1− β2, thus restoring the original wave-

length and maintaining unchanged both the frequency andthe wavelength received by the upper driver.

9.3 A counter-example invalidating thecurrent sound Doppler effect

In addition to demonstrate new formulas for the Galileanand sound Doppler effects, another approach to con-

13

vince readers reluctant to question a pre-existing formulas,would be to provide a proof that this pre-existing formulais necessarily wrong as it leads to absurd results. As ex-plained above, the driver at the top of Fig.12 does not per-ceive any sound Doppler effect from the siren of the bot-tom car; this fact is naturally expected from the absenceof relative velocity between the source and the receiver.Now let us consider that the top car is immobile (in thiscase, of course, the diagram of Fig.12 would be a snap-shot and no longer a stationary state). Which Dopplereffect is then predicted by the traditional formula? (i) Ifthe aberration effect is neglected, as it is the case in mostdescriptions of the classical Doppler effect, then the cosinebeing zero and according to the absence of transverse effectfor the sound, the Doppler effect is still 1; so whether theupper car is moving or immobile, the perceived Dopplereffect would be the same, which would seem very puz-zling. (ii) Now if admitting that aberration also exists fora classical wave, the transverse Doppler effect would exist(fmov/f = 1/

√1− β2) contrary to dogma. This simple

example is sufficient to reveal an internal contradictionin the usual formulation of the so-called classical Dopplereffect.

10 Conclusions

The Doppler effect is generally envisioned as a long estab-lished phenomenon corresponding to a dead branch of ba-sic physics now confined to general education. However,a critical look reveals that the research on the acousticDoppler effect is currently incomplete and paved with falseideas and invalid formulas. This joint study of the lightand sound Doppler effects allows to perceive the speci-ficities of each type of wave and shows that the classicalDoppler effect has been strangely shaped from the knowl-edge of the relativistic one, while the two Doppler effectsobey radically different laws. The most frequent sourcesof error concerning the sound Doppler effect are reviewedhere. In particular, the relativistic Doppler effect is notmerely a classical effect corrected by the relativistic dilata-tion factor. Besides, from a pedagogical point of view, weshould in general be wary of the famous representationsof wave crests as shifted nested circles to predict Dopplereffects. Indeed, on the one hand, light wavefronts are notperceived as circles but as ellipses [4], and on the otherhand even for sound, as explained in sections 7 and 9,the nested spheres sometimes describe accurately wave-length distortions without affecting frequencies and there-fore the Doppler effects either. Many more subtleties couldbe added to the complexity of the sound. For instance, notonly does the wind modify the sounds, but it also adds itsown sound. Conversely, a sound emitted by a fast sourcemay neither be the sound of the wind nor correspond toany sound of the source at rest. The number of parame-ters to be taken into account to predict the Doppler effectof sound as well as the complexity of its three-dimensionalformulation, show how elegant the Doppler effect of lightis comparatively. For light, the absence of a propagation

medium, formerly postulated under the name of ether,means that the only absolute velocity is the relative veloc-ity [7]. The absence of the absolute substratum simplifiesthings considerably and is much more comfortable con-ceptually than the strange immaterial Galilean referenceframe, in respect to which one can define absolute veloc-ities for everything, in the case of sound for the air, thesource and the detector.

References

[1] Nolte D.D. The fall and rise of the Doppler effect. Phys.Today 73 (2020) 30-35.

[2] Halliday D., Resnick R., Walker J. Fundamentals ofPhysics. John Wiley & Sons Inc. 10th ed. 2013.

[3] Fowler M. The Doppler effect.https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/DopplerEffect.htm

[4] Michel D. Galilean and relativistic Doppler/aberration ef-fects deduced from spherical and ellipsoidal wavefronts re-spectively. Optik 250 (2022) 168242.

[5] Ives H.E., Stillwell G.R., An experimental study of the rateof a moving atomic clock. J. Opt. Soc. Am. 28 (1938) 215-226.

[6] Hasselkamp D., Mondry E., Scharmann A. Direct observa-tion of the transversal Doppler-shift. Z. Phys. A 289 (1979)151-155.

[7] Poincare H. Le principe de Relativite. La dynamique del’electron. Revue Generale des Sciences Pures et Appliquees.19 (1918) 386-402.

[8] Poincare H. Sur la dynamique de l’electron. C.R. Acad. Sci.140 (1905) 1504-1508.

[9] Einstein A. Zur Elektrodynamik bewegter Korper (On theelectrodynamics of moving bodies), Annal. Phys. 17 (1905)891-921.

[10] Joos G., Freeman, I.M. Theoretical Physics. Hafner pub-lishing company NY. 1958.

[11] Mangiarotty R.A., Turner B.A. Wave radiation dopplereffect correction for motion of a source, observer and thesurrounding medium. J. Sound Vib. 6 (1967) 110-116.

[12] Einstein A. Possibility of a new examination of the rela-tivity principle, Annal. Phys. 23 (1907) 197-198.

[13] Spees A.H. Acoustic doppler effect and phase invariance.Am. J. Phys. 24 (1956) 7-10.

[14] Harrison E. The redshift-distance and velocity-distancelaws. Astrophys. J. 403 (1993) 28-31.

[15] Michel D. Analytical relationships between source-receiverdistances, redshifts and luminosity distances under puremodes of expansion. Adv. Astrophys. 2 (2017) 217-230.

[16] Young R.W. The Doppler effect for sound in a moving

medium. J. Acoust. Soc. Am. 6 (1934) 112-114.

14

Appendices

A Key points to compare the Galilean and relativistic Doppler effects

Table 1: Some Galilean correspondences between angles, relative distances and Doppler effects. The angle unit is the radianand the distance unit is the minimum distance between the source and the receiver. These values hold for a stationary receiverand correspond to the Doppler profiles of Fig.6. The line highlighted in gray corresponds to the transverse Doppler effect andthat highlighted in yellow color is the only common point with the relativistic Doppler effect (Table 2).

Origin of the angle Distance to the nearest point Doppler effect

source point of emission of the source (image) of the emission point (sound)

θ θ′ x x′ fmov/f

0 0 −∞ −∞ 1

1− β

π

2cos−1 β 0 − β√

1− β2

1√1− β2

cos−1−β2

cos−1β

2

β√4− β2

− β√4− β2

1

cos−1− β√1 + β2

π

2β 0

1√1 + β2

π π +∞ +∞ 1

1 + β

15

Table 2: Some relativistic correspondences between angles, distances and Doppler effects. The line highlighted in graycorresponds to the transverse Doppler effect and that highlighted in yellow color is the only common point with the GalileanDoppler effect of Table 1.

Origin of the angle Distance to the nearest point Doppler effect

source emission point of the source of the emission point

θ θ′ x x′ fmov/f

0 0 −∞ −∞√

1 + β

1− β

π

2cos−1 β 0 − β√

1− β2

1√1− β2

cos−1−1−√

1− β2

βcos−1

1−√

1− β2

β

1√2

√1√

1− β2− 1 − 1√

2

√1√

1− β2− 1 1

cos−1−β π

2

β√1− β2

0√

1− β2

π π +∞ +∞√

1− β1 + β

16

B Average Doppler effects

The concept of secondary Doppler effect supposedlyspecific to relativity, was further consolidated by the inap-propriate use of the arithmetic mean for averaging Dopplereffects. In the articles validating the relativistic Dopplereffect, longitudinal [5] and transverse [6], it is explainedthat the relativistic Doppler effects include, contrary tothe classical one, a secondary transverse effect through acurious reasoning. Ives and Stilwell simultaneously mea-sured the longitudinal wavelengths of approach (λa) andrecession (λr), with and against the motion of the parti-cles. They then compared the wavelength shifts to theirso-called ”center of gravity” conceived as an arithmeticmean [6]. Knowing the relativistic longitudinal effects tobe demonstrated, they calculated

λmean =λa + λr

2

=1

2

(λ0

√1− v

c

1 + vc

+ λ0

√1 + v

c

1− vc

)

=λ0√1− v2

c2

∼ λ0 +λ02

v2

c2

(B.1)

They concluded that λmean 6= λ0 due to transverseDoppler shift

∆λ

λ0=λmean − λ0

λ0∼ 1

2

v2

c2

The conclusion of these authors comes from an erro-neous use of the arithmetic mean. Perhaps judging theappearance of Eq.(B.1) satisfactory, they did not look atwhat is going on for the frequency fmean correspondingto this λmean. Yet, since the Doppler effect during theapproach for wavelengths corresponds to Doppler effectduring the recession for frequencies and vice-versa, theywould have found that the result is the same

fmean =f0√

1− v2

c2

(B.2)

But for any photon, the product: frequency × wavelengthis a well known constant

fλ = c (B.3)

and therefore the above approach is obviously wrong aswe would have

fmean λmean =ν0 λ0

1− v2

c2

6= c (B.4)

In fact, the arithmetic mean used in [5, 6] is inappropriatefor averaging Doppler effects because it cannot work forboth frequencies and wavelengths. Mathematically, thereare several modes of averaging which apply differently tothe specific situations. These different types of averagesinclude, when applied to two Doppler effects,

• The arithmetic mean:1

2

(fmov1

f+fmov2

f

)

• The geometric mean:

(fmov1

f

fmov2

f

) 12

• The harmonic mean:2f

fmov1 + fmov

2

The appropriate one is necessarily the geometric mean,because it is the only one holding for both periods andfrequencies, such that

〈f1, f2〉 =1

〈T1, T2〉(B.5)

Besides, the use of geometric averages for wavelengthshas already been applied empirically and satisfies the ruleof color reflectance fusion. Therefore, let us apply thegeometric mean to the Doppler effects obtained in front ofand behind the closest point. We can use the linear coor-dinate for which the closest point corresponds to x = 0, orthe angular coordinate for which the closest point corre-sponds to θ = π/2, or ξ = 0 defined such that sin ξ = cos θ.

For the Gallilean effect:

Using x and Eq.(20), the geometric mean of theGallilean Doppler effect is

∀x,⟨f(−x)

f0,f(+x)

f0

⟩=

1√1− β2

(B.6)

The same result is naturally obtained using angles andEq.(13a)

∀ξ,⟨f(π2 − ξ)

f0,f(π2 + ξ)

f0

⟩=

1√1− β2

(B.7)

Strikingly, this result is independant of the distance.

For the relativistic effect:

Using the linear coordinate

∀x,⟨f(−x)

f0,f(+x)

f0

⟩=

√1 + x2(1− β2)

(1 + x2)(1− β2)(B.8)

and using the angular coordinate,

∀ξ,⟨f(π2 − ξ)

f0,f(π2 + ξ)

f0

⟩=

√1− β2 sin2 ξ

1− β2(B.9)

which correspond to the Galilean result only for x = 0 orξ = 0.

17

C Step-by-step establishment of the 2D combination of speed and wind

The sphere Cartesian equation

(ρ cos θ + β + ω cosψ)2 + (ρ sin θ + ω sinψ)2 = 1 (C.1a)

developed in polar coordinates is

ρ2 + ω2 + β2 + 2ωρ sin θ sinψ + 2ωρ cos θ cosψ + 2βρ cos θ + 2βω cosψ = 1 (C.1b)

and since2ωρ sin θ sinψ + 2ωρ cos θ cosψ = 2ωρ cos(θ − ψ)

the quadratic equation for ρ is:

ρ2 + 2ρ(ω cos(θ − ψ) + β cos θ) + (β2 + ω2 + 2βω cosψ − 1) = 0 (C.1c)

Its solution is

ρ =√

(ω cos(θ − ψ) + β cos θ)2 − ω2 − β2 − 2βω cosψ + 1− ω cos(θ − ψ)− β cos θ (C.2a)

ρ =√

1 + ω2 cos2(θ − ψ) + β2 cos2 θ + 2βω cos(θ − ψ) cos θ − ω2 − β2 − 2βω cosψ − ω cos(θ − ψ)− β cos θ (C.2b)

ρ =√

1− ω2 + ω2 cos2(θ − ψ)− β2 + β2 cos2 θ − 2βω cosψ + 2βω cos(θ − ψ) cos θ − ω cos(θ − ψ)− β cos θ (C.2c)

ρ =√

1− ω2(1− cos2(θ − ψ))− β2(1− cos2 θ)− 2βω(cosψ − cos(θ − ψ) cos θ)− ω cos(θ − ψ)− β cos θ (C.2d)

and given that: cosψ − cos(θ − ψ) cos θ = sin θ sin(θ − ψ),

ρ =

√1− ω2 sin2(θ − ψ)− β2 sin2 θ − 2βω sin θ sin(θ − ψ)− ω cos(θ − ψ)− β cos θ (C.2e)

and finally

ρ =

√1− [ω sin(θ − ψ) + β sin θ]

2 − ω cos(θ − ψ)− β cos θ (C.2f)

This wavefront bubble shifted by the wind is the denominator of the Doppler effect, whose numerator is, in theabsence of movement of the detector, the same equation reduced to the wind only (β = 0)

fmov

f=

√1− ω2 sin2(θ − ψ)− ω cos(θ − ψ)√

1− [ω sin(θ − ψ) + β sin θ]2 − ω cos(θ − ψ)− β cos θ

(C.3)

With an additional dimension, the same approach gives the three-dimensional Doppler formula:√1− ω2 [1− (cos θ cosψ + sin θ sinψ cos(ϕ− φ))2]− ω (cos θ cosψ + sin θ sinψ cos(ϕ− φ))

fmov

f= (C.4)√

1− β2 [1− (sin θ cosϕ)2]− ω2 [1− (cos θ cosψ + sin θ sinψ cos(ϕ− φ))2]

−2βω [sinψ cosφ− sin θ cosϕ (cos θ cosψ + sin θ sinψ cos(ϕ− φ))]

−β sin θ cosϕ− ω (cos θ cosψ + sin θ sinψ cos(ϕ− φ))

18

D Classical Doppler measurement

This section aims at: (1) validating the linear Dopplerapproach described in this study; and (2) addressing thequestion of the classical transverse effect currently con-sidered inexistent. Any ordinary film is a co-recordingof image and sound, but since these two types of wavesreached the camera and microphone at different speeds,

they describe in fact separate moments in the recent past.To illustrate concretely this subtlety, let us analyse theshift in sound frequency during the passage of an aircraft,through the parallel analysis of image and sound.

Figure D1. Doppler effect illustrated by a dominant frequency recorded during the passage of an aircraft at low altitude. The

blue lines connecting the images of the planes to the spectrogram indicate the actual concomitance of the sound and images on

the film, while the red lines connect the recorded frequencies to their real points of emission.

D.1 Determination of the aircraft speedand rest frequency

The asympotic values of the apparent frequencies heardwhen the source arrives, written fa and that measuredwhen the source goes away, written fr, are sufficient todetermine the source velocity, even in absence of knowl-edge of the source frequency f0. Indeed, fa and fr arerelated through

f0 = fa (1− β) = fr (1 + β) (D.1a)

from which

β =fa − frfa + fr

(D.1b)

The frequencies given by the spectrogram fa=6750 Hzand fr=4338 Hz, give β = 0.2175 (at 15◦C, 74 m/s or 266

km/h). Once β is known, the equalities of Eq.(D.1a) thenallow to find the rest frequency: f0 = 5282 Hz. Note thatalthough it is called rest frequency, f0 may not exist whenthe aircraft is stopped with the engines on, for instance ifthis sound is generated by the flux of apparent wind.

D.2 Curve fitting and conclusions

The theoretical equation combining the image and soundrecorded simultaneously, is Eq.(20):

fmov

f0=

√1 + x2

βx+√

1− β2 + x2

where the ordinate is the sound Doppler effect and the ab-scissa x is the spatial coordinate of the source determinedvisually. Introducing the value of β measured previously

19

in this equation gives the horizontal increment x = 1. Atx = 0 (5156 Hz, Doppler effect of 1.025) the line of sightof the observer is perpendicular to the plane trajectory.The Doppler effect for x = 0 is expected to be

fmov

f0 orthogonal

=1√

1− β2

As explained in the main text, this is not the transverseDoppler effect which is

fmov

f0 transverse

=1√

1 + β2

This effect (5156 Hz, Doppler effect of 0.977) is re-ceived only when the aircraft has moved away from thetransverse position. Upon reception of the transverseDoppler effect, the plane it located at a distance βD fromthe closest point corresponding, given the delay of 0.323seconds measured from the video, to 110 m from thetransverse position. In summary, the accuracy of curvefitting shown in Fig.D1 can be checked by verifying thefrequencies for the following two points:

• x = 0→ fmov = f0/√

1− β2

• x = β → fmov = f0/√

1 + β2

• In addition, for the agreement between the image andsound, x = 0 must coincide with the most transverseposition of the source. This can be appreciated here forinstance through the apparent orientation of the wingsand the alignment of the side windows of the cockpit.

Once these three criteria are fulfilled, the rest of thecurve fits remarkably well (Fig.D1). On Fig.D1, the bluelines join the images and the sounds which are superim-posed on the video. But this apparent simultaneity is onlyan illusion of reception, as shown by the red lines whichlink the sound to the position of the plane where they wereactually emitted. This offset is naturally due to the dif-ference in speed between light and sound to get from theplane to the camera [11]. The sound received when theairplane is seen perfectly in profile was sent at the posi-tion x = −β/

√1 + β2, which would belong to the curve

drawn to Eq.(21) if added on the same diagram.

20