The Doppler effect is the shift of Fréquence of a acoustic Onde or electromagnetic between measurement with the emission and measurement with the reception when the transmitter and the receiver are in movement one compared to the other. If one indicates in a general way this physical phenomenon under the name of Doppler effect one reserves the d&rsquo term; effect Doppler with the light waves.

This effect was presented by Christian Doppler in 1842 in the article Über das farbige Licht der Doppelsterne und einige andere Gestirne of Himmels , was confirmed on the sounds by the researcher Dutch Christoph Hendrik Diederik Buys Ballot (by using musicians playing a note gauged on a train of the line Utrecht - Amsterdam), and was also proposed by Hippolyte Fizeau for the electromagnetic waves in 1848.

The Doppler effect appears for example for the sound waves in the perception height of the sound of an engine of car, or siren of an emergency vehicle. The sound is different according to whether one is in the vehicle (the transmitter is motionless compared to the receiver), that the vehicle approaches the receiver (the sound becomes acuter) or that it moves away (the sound becomes more serious).

This effect is used to measure a Speed, for example that of a car, or that of the Sang when medical examinations are carried out (in particular the echography S in Obstétrique or Cardiologie). It is of great importance in Astronomie because it makes it possible to directly determine the speed of approach or distancing of the celestial objects (stars, galaxies, clouds of gas, etc). Let us state however that the shift towards the cosmological red, which represents the apparent escape of the galaxies and constitutes a proof of the expansion of space, is of another nature : it is not justiciable to a Doppler treatment because it is due (in a picturesque way) to a stretching of space producing itself a stretching wavelengths (the wavelength of a radiation following the size of the Universe accurately).

Physical explanation

A person is upright in water, at the edge of the shore. Vaguenesses arrive to him on the feet every ten seconds. The person walks, then runs in direction the broad one: she goes to the meeting of the waves, those reach it with a higher Fréquence (for example every eight seconds, then every five seconds). The person then makes half-turn and walk then runs in direction of the beach; the waves reach it with less low frequency, for example all the twelve, then fifteen seconds.

The frequency of the waves does not depend on the movement of the person compared to water (it is in particular independent of the presence or not of a current), but of the movement of the person compared to the transmitter of the waves (in fact a place with broad where the current is opposed to the wind).

In an opposite way, one can imagine a mobile source of waves, for example a Aéroglisseur whose air blast would generate waves at a regular frequency. If the hovercraft moves in a direction, then the waves are tightened forwards movement and more spaced towards the back of the movement; a closed lake, the waves will strike the bank at different frequencies.

Mathematical formulation

Effect Doppler galiléen

Let us suppose that the transmitter and the receiver move on a line. There are three referential galiléens to consider:

• (1) the reference frame of the medium in which the wave is propagated (for example atmosphere for a sound wave). One notes C the speed of the wave in this reference frame (it is not inevitably the Speed of light).
• (2) the reference frame related to the transmitter (source): let us call v_em the algebraic speed of the transmitter (source) compared to the reference frame (1).
• (3) the reference frame related to the receiver: let us call v_rec the speed of the receiver compared to the reference frame (1).

To convention speeds will be counted like positive in the direction of propagation of the signal (of the transmitter towards the receiver). Thus a positive speed v_em will correspond to a bringing together between source and receiver while a positive speed v_rec will correspond to a distance.

If ƒ_em is the Fréquence of the Onde in the Référentiel of the source, then the receiver will receive a wave of frequency ƒ _rec

$f\_\; \{rec\}\; =\; \backslash \; frac\; \{c-v\_\; \{rec\}\}\; \{c-v\_\; \{EM\}\}\; \backslash \; cdot\; f\_\; \{EM\}\; =\; \backslash \; frac\; \{1\; (v\_\; \{rec\}\; /c)\}\; \{1\; (v\_\; \{EM\}\; /c)\}\; \backslash \; cdot\; f\_\; \{EM\}$

Indeed, let us suppose that the source emits beeps at a frequency ƒ_em and that the relative movement between transmitter and receiver is done according to the line uniting them. When the second beep is produced, the first beep traversed a distance

D ₀ = C · T _em

in the reference frame (1), with T _em = 1/ƒ_em. The source being moved of v _EM · T _em during time T _em, the distance separating two beeps is

D ₁ = ( C - v _EM )· T _em.

Let us calculate time T_rec separating detection from both beeps by the receiver. This last receives the first beep . At the end of this time T_rec , he traversed the distance v _rec · T_rec at the time when it receives the second beep . During this amount of time T_rec the second beep will thus have traversed the distance

D ₂ = D ₁ + v _rec · T_rec = C · T_rec ,

what gives well:

$f\_\; \{rec\}\; =\; \{1\; \backslash \; over\; T\_\; \{rec\}\}\; =\; \{C\; -\; v\_\; \{rec\}\; \backslash \; over\; d\_1\}\; =\; \{C\; -\; v\_\; \{rec\}\; \backslash \; over\; C\; -\; v\_\; \{EM\}\}\; \backslash \; cdot\; \{1\; \backslash \; over\; T\_\; \{EM\}\}\; =\; \{C\; -\; v\_\; \{rec\}\; \backslash \; over\; C\; -\; v\_\; \{EM\}\}\; \backslash \; cdot\; f\_\; \{EM\}$

If only the source is mobile compared to the reference frame ( v_rec = 0), one has then:

$f\_\; \{rec\}\; =\; \backslash \; frac\; \{C\}\; \{c-v\_\; \{EM\}\}\; \backslash \; cdot\; f\_\; \{EM\}\; =\; \backslash \; frac\; \{1\}\; \{1\; (v\_\; \{EM\}\; /c)\}\; \backslash \; cdot\; f\_\; \{EM\}$

and so only the receiver is mobile compared to the reference frame ( v_em = 0), one a:

$f\_\; \{rec\}\; =\; \backslash \; frac\; \{c-v\_\; \{rec\}\}\; \{C\}\; \backslash \; cdot\; f\_\; \{EM\}\; =\; (1\; -\; \backslash \; frac\; \{v\_\; \{rec\}\}\; \{C\})\; \backslash \; cdot\; f\_\; \{EM\}$

It is seen clearly that the two situations are not symmetrical: indeed, if the receiver “flees” the transmitter at an high speed with C , it will never receive wave, whereas if the transmitter flees a motionless receiver, this one will always receive a wave. One cannot reverse the role of the transmitter and the receiver. In the traditional case, there is dissymmetry in the frequential shift according to whether the transmitter or the receiver is moving (the received frequencies differ by the terms from the second order for the same frequency from emission). This dissymmetry is due to the presence of the medium in which the waves are propagated, it is justified for the sound waves.

For the electromagnetic waves, the propagation being able to be done in the vacuum, this dissymmetry is unfounded. One must then deal with the problem within the framework of restricted relativity and one then expects to find an effect perfectly symmetrical since one cannot distinguish between speed of the transmitter and speed of the receiver, the only cash relative speed between the two. It is what us will show.

In the case of electromagnetic waves, the speed of the wave is the Speed of light which depends on the nature of the medium (and in particular of sound Index of refraction), but not of the reference frame.

Fast relativistic calculation

Before giving the formula of the relativistic Doppler effect in the general case here initially a fast simplified demonstration of the relativistic formula if all the movements are done along the same axis, that the length whose the signal is propagated. The principle of calculation consists in taking account of the effect of dilation of the time which accompanies the passage by a reference mark at rest with a reference mark moving.

We change notation to prepare us with a symmetrization of the problem. Speed between the transmitter and the receiver will be noted v and will be counted as positive if it corresponds at a speed of distancing. It is the convention generally adopted in Astronomie for the radial Speed. Consequently if the source only moves, its speed of the former formulas is v_em = - v and if it is the receiver which only moves, its speed is v_rec = +v .

• Considérons initially that it is the source which moves. If one calculated it by the traditional formula preceding the frequency of the signal to the reception would be

$f\_\; \{rec\}\; =\; \backslash \; frac\; \{f\_\; \{EM\}\}\; \{1\; +\; (v/c)\}\; =\; \backslash \; frac\; \{f\_\; \{EM\}\}\; \{1\; +\; \backslash \; beta\}\; \backslash$   with   $\backslash \; \backslash \; beta=v/c\; \backslash .$

If one holds account maintaining of the factor of dilation of time of restricted relativity

$\backslash \; gamma\; =\; 1\; \backslash \; sqrt\; \{1\; -\; (v^2/c^2)\}\; =\; (1\; -\; \backslash \; beta^2)\; ^\; \{-\; 1/2\}\; \backslash$

who increases the durations measured by the fixed receiver, the frequency observed will decrease by the factor reverses $-\; (v^2/c^2)\; ^\; \{1/2\}$ so that the frequency f_rec becomes

$f\_\; \{rec\}\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{(1\; -\; \backslash \; beta^2)\}\}\; \{1\; +\; \backslash \; beta\}\; f\_\; \{EM\}\; =\; \backslash \; frac\; \{1\; \backslash \; beta\}\; \{\backslash \; sqrt\; \{(1\; -\; \backslash \; beta^2)\}\}\; f\_\; \{EM\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{1\; -\; \backslash \; beta\}\; \{1\; +\; \backslash \; beta\}\}\; f\_\; \{EM\}\; \backslash .$

• Considérons now that it is the receiver which moves. With the formula galiléenne we would have

$f\_\; \{rec\}\; =\; (1\; -\; \backslash \; beta)\; f\_\; \{EM\}\; \backslash .$

As previously it is necessary to take account of the relativistic factor γ  . But as here it is the receiver which is moving and the source which is fixed, it is the expression of $f\_\; \{EM\}\; =\; f\_\; \{rec\}/(1\; -\; \backslash \; beta)$ which must be multiplied by $-\; (v^2/c^2)\; ^\; \{1/2\}$. We thus obtain the same formula as previously

$f\_\; \{rec\}\; =\; \backslash \; frac\; \{1\; -\; \backslash \; beta\}\; \{\backslash \; sqrt\; \{(1\; -\; \backslash \; beta^2)\}\}\; f\_\; \{EM\}\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{(1\; -\; \backslash \; beta^2)\}\}\; \{1\; +\; \backslash \; beta\}\; f\_\; \{EM\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{1\; -\; \backslash \; beta\}\; \{1\; +\; \backslash \; beta\}\}\; f\_\; \{EM\}\; \backslash ,$

what shows well that the Doppler effect is perfectly symmetrical and depends only on the relative speed between the transmitter and the receiver.

Let us notice that the relativistic Doppler effect combines two effects, the effect galiléen and the effect of deceleration of the clocks. First the radial Speed utilizes between source and observer, the second the value total speed.

Relativistic Doppler effect

Let us deal with the problem now in a complete way.

In restricted relativity a photon is entirely characterized by its quadrivector energy-impulse P . This quantity is defined independently of all Frame of reference but it is useful when one wants to make measurements or calculi to specify the value of the components of this quadrivector. So in a frame of reference the frequency of the photon is $\backslash \; nu$ and the unit vector along the way of the photon is the vector with 3 dimensions $\backslash \; vec\; \{N\}$, the quadrivector P is

$\backslash \; mathbf\; \{P\}\; =\; \backslash \; left\; (\backslash \; frac\; \{H\; \backslash \; naked\}\; \{C\},\; \backslash ,\; \backslash \; frac\; \{H\; \backslash \; naked\}\; \{C\}\; \backslash ,\; \backslash \; vec\; \{N\}\; \backslash \; right)\; =\; (p\_0,\; \backslash ,\; p\_1,\; \backslash ,\; p\_2,\; \backslash ,\; p\_3)$

where H is the Constante of Planck.

Let us consider a star of which we receive the photons on Earth. Let us choose a terrestrial reference mark Oxyz such that the axis OX is directed along the speed v of star. Restricted relativity teaches us whereas the components $(p\text{'}\_0,\; \backslash ,\; p\text{'}\_x,\; \backslash ,\; p\text{'}\_y,\; \backslash ,\; p\text{'}\_z)$ of a quadrivector P in the reference mark moving of star change in the components $(p\_0,\; \backslash ,\; p\_x,\; \backslash ,\; p\_y,\; \backslash ,\; p\_z)$ in the terrestrial reference mark according to the following formulas of Lorentz

$\backslash \; begin\; \{boxes\}$

p_0 = \ gamma (p'_0 + \ beta p'_x) \ \ p_x = \ gamma (\ beta p'_0 + p'_x) \ \ p_y = p'_y \ \ p_z = p'_z \ end {boxes}

with always

$\backslash \; beta\; =\; v/c\; \backslash ,$   and   $\backslash \; gamma\; =\; 1\; \backslash \; sqrt\; \{1\; -\; \backslash \; beta^2\}$

By using the notations of the preceding paragraphs the frequencies of the photon are $\backslash \; naked\; \backslash ,\; =\; \backslash ,\; f\_\; \{rec\}$ in the terrestrial reference mark and $\backslash \; nu\text{'}\; \backslash ,\; =\; \backslash ,\; f\_\; \{EM\}$ in the reference mark of transmitting star. The equations of Lorentz give then (the components of the quadrivector are proportional to the frequency and the common factor of proportionality h/c disappears)

$f\_\; \{rec\}\; =\; \backslash \; gamma\; (1\; +\; \backslash \; beta\; \backslash \; cos\; \backslash \; theta\text{'})\; f\_\; \{EM\}\; \backslash \; \backslash ,$

where $\backslash \; theta\text{'}$ is the angle which forms the photon with the axis OX in the reference mark of the star . If the quantity $\backslash \; beta\text{'}\_\; \{rad\}$ corresponds to the radial component relative speed between transmitter and receiver in the reference mark of star, i.e.

$\backslash \; beta\text{'}\_\; \{rad\}\; =\; v\; \backslash \; cos\; \backslash \; theta\text{'}/c\; \backslash ,$

one can write the relativistic Doppler formula in the form

$f\_\; \{rec\}\; =\; \backslash \; frac\; \{1\; +\; \backslash \; beta\text{'}\_\; \{rad\}\}\; \{\backslash \; sqrt\; \{1\; -\; \backslash \; beta^2\}\}\; \backslash ,\; f\_\; \{EM\}\; \backslash ,$

who gives again the formulas presented well above when one takes $\backslash \; cos\; \backslash \; theta\text{'}=-1$.

We find the fact that the relativistic effect is to some extent the combination of the traditional Doppler effect due to the radial Speed and of the phenomenon of deceleration of the clocks inherent in the restricted Relativité.

It is easy to find the angle $\backslash \; theta$ which the luminous ray with the axis OX in the terrestrial reference mark forms . The difference between the directions of the photon in the terrestrial reference mark and the reference mark of star constitutes the phenomenon of Aberration of the light. According to the equations of Lorentz written above one has

$\backslash \; begin\; \{boxes\}\; \backslash \; cos\; \backslash \; theta\; =\; p\_x/p\_0\; =\; (\backslash \; beta\; +\; \backslash \; cos\; \backslash \; theta\text{'})/(1\; +\; \backslash \; beta\; \backslash \; cos\; \backslash \; theta\text{'})\; \backslash \; \backslash$

\ sin \ theta = p_y/p_0 = \ gamma^ {- 1} \ sin \ theta'/(1+ \ beta \ cos \ theta') \ end {boxes}

These formulas give a complete relativistic description of the Doppler effect.

Let us note a subtlety in the phenomenon of aberration. If the photon is radially propagated in a reference mark it will also make it in the other. In other words if $\backslash \; cos\; \backslash \; theta\text{'}\; \backslash ,\; =\; \backslash ,\; -\; 1$ then $\backslash \; cos\; \backslash \; theta\; \backslash ,\; =\; \backslash ,\; -\; 1$. On the other hand if speed is perpendicular to the direction of the photon in a reference mark it will not be it in any rigor in the other. Indeed if $\backslash \; cos\; \backslash \; theta\text{'}\; \backslash ,\; =\; \backslash ,\; 0$ then $\backslash \; cos\; \backslash \; theta\; \backslash ,\; =\; \backslash ,\; \backslash \; beta$. And if $\backslash \; cos\; \backslash \; theta\; \backslash ,\; =\; \backslash ,\; 0$ then $\backslash \; cos\; \backslash \; theta\text{'}\; \backslash ,\; =\; \backslash ,\; -\; \backslash \; beta$.

Applications

In Medicine

In 1958, continuous Doppler (which is a Cristal emitting and receiving uninterrupted ultrasounds) allowed the study of blood circulation in the vessels ( Rushmer ). First pulsated Doppler (emission of the ultrasound into discontinuous and fenestrates of fixed temporal listening, making it possible to analyze the speed of blood to a definite depth) was introduced by Baker in 1970.

• Doppler, coupled or not with an echographic examination , makes it possible to analyze the speed of the Sang. One can thus quantify flows, escapes or contractings.

Indeed, the échodoppler is used in medicine to measure the speed of red blood corpuscles and to calculate the diameter of a blood-vessel (aorta…).

• In Cardiology, one can analyze the speed of the cardiac walls using tissue Doppler, it is the Doppler imagery of fabrics, or TDI ( tissular dopplar imaging )

See also: Echography Doppler

In Meteorology

The Doppler effect is used by the laser Vibromètre for measurement of Vibration S in Mécanique. It is also used by the anemometers laser (LDV) for the velocity measurement of flow of the Fluide S.

Road traffic

The Doppler effect allows the police force and with the Gendarmerie to determine the speed of the Automobile S. For that they use a radar whose frequency is perfectly known. The measurement of the frequency of the echo gives the speed of the vehicle. Modern technology makes it possible today to have automatic radars and binoculars radar.

See too

• relativistic Calculations/the voyage in the future of the others, where the Doppler effect is used to analyze the Paradoxe of the twins within the framework of restricted relativity.

• a way much simpler and elegant to solve the paradox of the twins but with a transverse Doppler effect.
• intuitive Illustration

Simple: Doppler effect

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