– the most important phenomenon in wave physics. Before going straight to the heart of the matter, a little introductory theory.
Hesitation– to one degree or another, a repeating process of changing the state of a system around an equilibrium position. Wave- this is an oscillation that can move away from the place of its origin, spreading in the medium. The waves are characterized amplitude, length And frequency. The sound we hear is a wave, i.e. mechanical vibrations of air particles propagating from a sound source.
Armed with information about waves, let's move on to the Doppler effect. And if you want to learn more about vibrations, waves and resonance, welcome to our blog.
The essence of the Doppler effect
The most popular and simple example that explains the essence of the Doppler effect is a stationary observer and a car with a siren. Let's say you are standing at a bus stop. An ambulance with a siren on is heading down the street towards you. The frequency of sound you will hear as the car approaches is not the same.
The sound will initially be of a higher frequency as the car comes to a stop. You will hear the true frequency of the siren sound, and the frequency of the sound will decrease as you move away. This is it Doppler effect.
The frequency and wavelength of the radiation perceived by the observer changes due to the movement of the radiation source.
If Cap is asked who discovered the Doppler effect, he will answer without hesitation that Doppler did it. And he will be right. This phenomenon, theoretically substantiated in 1842 year by Austrian physicist Christian Doppler, was subsequently named after him. Doppler himself derived his theory by observing ripples on water and suggesting that the observations could be generalized to all waves. It was later possible to experimentally confirm the Doppler effect for sound and light.
Above we looked at an example of the Doppler effect for sound waves. However, the Doppler effect is not only true for sound. There are:
- Acoustic Doppler effect;
- Optical Doppler effect;
- Doppler effect for electromagnetic waves;
- Relativistic Doppler effect.
It was experiments with sound waves that helped provide the first experimental confirmation of this effect.
Experimental confirmation of the Doppler effect
Confirmation of the correctness of Christian Doppler's reasoning is associated with one of the interesting and unusual physical experiments. IN 1845 meteorologist from Holland Christian Ballot took a powerful locomotive and an orchestra consisting of musicians with perfect pitch. Some of the musicians - these were trumpeters - rode on the open area of the train and constantly played the same note. Let's say it was A of the second octave.
Other musicians were at the station listening to what their colleagues were playing. Absolute hearing of all participants in the experiment reduced the likelihood of error to a minimum. The experiment lasted two days, everyone was tired, a lot of coal was burned, but the results were worth it. It turned out that the pitch of sound really depends on the relative speed of the source or observer (listener).
Application of the Doppler effect
One of the most widely known applications is determining the speed of moving objects using speed sensors. Radio signals sent by radar are reflected from cars and returned back. In this case, the frequency offset at which the signals are returned is directly related to the speed of the machine. By comparing speed and frequency change, speed can be calculated.
The Doppler effect is widely used in medicine. The operation of ultrasound diagnostic devices is based on it. There is a separate technique in ultrasound called Dopplerography.
The Doppler effect is also used in optics, acoustics, radio electronics, astronomy, radar.
By the way! For our readers there is now a 10% discount on any type of work
The discovery of the Doppler effect played an important role in the development of modern physics. One of the confirmations big bang theory is based on this effect. How are the Doppler effect and the Big Bang related? According to the Big Bang theory, the Universe is expanding.
When observing distant galaxies, a red shift is observed - a shift of spectral lines to the red side of the spectrum. Explaining the red shift using the Doppler effect, we can draw a conclusion consistent with the theory: galaxies are moving away from each other, the Universe is expanding.
Formula for the Doppler effect
When the theory of the Doppler effect was criticized, one of the arguments of the scientist’s opponents was the fact that the theory was contained on only eight pages, and the derivation of the Doppler effect formula did not contain cumbersome mathematical calculations. In our opinion, this is only a plus!
Let u – speed of the receiver relative to the medium, v – speed of the wave source relative to the medium, With - speed of propagation of waves in the medium, w0 - frequency of source waves. Then the formula for the Doppler effect in the most general case will look like this:
Here w – frequency that the receiver will record.
Relativistic Doppler effect
In contrast to the classical Doppler effect, when electromagnetic waves propagate in a vacuum, to calculate the Doppler effect, SRT should be used and relativistic time dilation should be taken into account. Let the light - With , v – speed of the source relative to the receiver, theta – the angle between the direction to the source and the velocity vector associated with the receiver’s reference system. Then the formula for the relativistic Doppler effect will look like:
Today we talked about the most important effect of our world - the Doppler effect. Do you want to learn how to solve Doppler effect problems quickly and easily? Ask student service specialists and they will be happy to share their experience! And at the end - a little more about the Big Bang theory and the Doppler effect.
The Doppler effect is a physical phenomenon consisting of a change in the frequency of waves depending on the movement of the source of these waves relative to the observer. As the source approaches, the frequency of the waves it emits increases and the length decreases. As the source of waves moves away from the observer, their frequency decreases and the wavelength increases.
For example, in the case of sound waves, as the source moves away, the pitch of the sound will decrease, and as the source approaches, the pitch of the sound will become higher. Thus, by changing the pitch, you can determine whether a train, a car with a special sound signal, etc. is approaching or moving away. Electromagnetic waves also exhibit the Doppler effect. If the source is removed, the observer will notice a shift in the spectrum to the “red” side, i.e. towards longer waves, and when approaching - to the “violet”, i.e. towards shorter waves.
The Doppler effect turned out to be an extremely useful discovery. Thanks to him, the expansion of the Universe was discovered (the spectra of galaxies are red-shifted, therefore, they are moving away from us); a method for diagnosing the cardiovascular system by determining the speed of blood flow has been developed; Various radars have been created, including those used by the traffic police.
The most popular example of the Doppler effect propagation: a car with a siren. When she drives towards you or away from you, you hear one sound, and when she passes by, you hear a completely different one - a lower one. The Doppler effect is associated not only with sound waves, but also with any others. Using the Doppler effect, we can determine the speed of something, be it a car or celestial bodies, provided that we know the parameters (frequency and wavelength). Everything related to telephone networks, Wi-Fi, security alarms - the Doppler effect can be observed everywhere.
Or take a traffic light - it has red, yellow and green colors. Depending on how fast we move, these colors can change, but not among themselves, but shift towards violet: yellow will go into green, and green into blue.
Well why? If we move away from the light source and look behind us (or the traffic light moves away from us), the colors will shift towards red.
And it’s probably worth clarifying that the speed at which red can be confused with green is much higher than the speed at which you can drive on the roads.
Answer
Comment
The essence of the Doppler effect is that if a sound source approaches or moves away from the observer, then the frequency of the sound emitted by it changes from the observer's point of view. For example, the sound of the engine of a car that passes by changes. It is higher as it approaches you and suddenly becomes lower as it flies past you and begins to move away. The higher the speed of the sound source, the greater the frequency change.
By the way, this effect is true not only for sound, but also, say, for light. It’s just more obvious for sound - it can be observed at relatively low speeds. Visible light has such a high frequency that small changes due to the Doppler effect are invisible to the naked eye. However, in some cases the Doppler effect should be taken into account even in radio communications.
If you don’t delve into strict definitions and try to explain the effect, as they say, on your fingers, then everything is quite simple. Sound (like light or a radio signal) is a wave. For clarity, let's assume that the frequency of the received wave depends on how often we receive the "crests" of the schematic wave (dropboxusercontent.com). If the source and receiver are stationary (yes, relative to each other), then we will receive “ridges” with the same frequency with which the receiver emits them. If the source and receiver begin to approach each other, then we will begin to receive more often, the higher the speed of approach - the speeds will add up. As a result, the sound frequency at the receiver will be higher. If the source begins to move away from the receiver, then each next “ridge” will take a little more time to reach the receiver - we will begin to receive “ridges” a little less frequently than the source emits them. The sound frequency at the receiver will be lower.
This explanation is somewhat schematic, but it reflects the general principle.
In short, the change in observed frequency and wavelength when the source and receiver move relative to each other. Associated with the finiteness of the speed of wave propagation. If the source and receiver get closer, the frequency increases (the wave peak is recorded more often); move away from each other - the frequency drops (the peak of the wave is recorded less frequently). A common illustration of the effect is the siren of the special services. If an ambulance approaches you, the siren squeals; when it drives away, it buzzes loudly. A separate case is the propagation of an electromagnetic wave in a vacuum - a relativistic component is added there and the Doppler effect also manifests itself in the case when the receiver and source are motionless relative to each other, which is explained by the properties of time.
The essence of the Doppler effect is the dependence of the oscillation frequency on the speed of the oscillation source relative to the receiver. For example, if you throw a sounding tuning fork away from you, the sound will seem lower (the oscillation frequency will decrease), and if a tuning fork is thrown at you, the sound will seem higher to you (the oscillation frequency will increase). This also applies to vibrations of a different nature - light and radio waves. Famous examples. 1) Due to the shift of radiation from distant stars down the spectrum, towards the red color, the hypothesis of an “expanding universe” arose. 2) Homing missiles, aimed at high-speed targets (enemy aircraft and missiles) by the radio wave reflected from the targets, receive oscillations of a changed frequency, this change is called “Doppler shift”, and radio heads are sometimes called “Doppler”.
Registered by the receiver, caused by the movement of their source and/or the movement of the receiver. It is easy to observe in practice when a car with a siren on drives past the observer. Suppose the siren produces a certain tone, and it does not change. When the car is not moving relative to the observer, then he hears exactly the tone that the siren makes. But if the car moves closer to the observer, the frequency of the sound waves will increase (and the length will decrease), and the observer will hear a higher pitch than the siren actually emits. At the moment when the car passes by the observer, he will hear the very tone that the siren actually makes. And when the car drives further and moves away rather than closer, the observer will hear a lower tone due to the lower frequency (and, accordingly, longer length) of the sound waves.
For waves propagating in any medium (for example, sound), it is necessary to take into account the movement of both the source and the receiver of the waves relative to this medium. For electromagnetic waves (such as light), which do not require any medium to propagate, all that matters is the relative motion of the source and receiver.
Also important is the case when a charged particle moves in a medium with a relativistic speed. In this case, Cherenkov radiation, which is directly related to the Doppler effect, is recorded in the laboratory system.
Where f 0 is the frequency with which the source emits waves, c- speed of propagation of waves in the medium, v- the speed of the wave source relative to the medium (positive if the source approaches the receiver and negative if it moves away).
Frequency recorded by a fixed receiver
u- the speed of the receiver relative to the medium (positive if it moves towards the source).
Substituting the frequency value from formula (1) into formula (2), we obtain the formula for the general case.
 |
(3) |
Relativistic Doppler effect
In the case of electromagnetic waves, the formula for frequency is derived from the equations of special relativity. Since electromagnetic waves do not require a material medium to propagate, only the relative speed of the source and the observer can be considered.
 |
Where With- speed of light, v- relative speed of the receiver and source (positive if they move away from each other).
How to observe the Doppler effect
Since the phenomenon is characteristic of any oscillatory processes, it is very easy to observe for sound. The frequency of sound vibrations is perceived by ear as pitch. You need to wait for a situation when a fast-moving car passes by you, making a sound, for example, a siren or just a beep. You will hear that when the car approaches you, the pitch of the sound will be higher, then, when the car reaches you, it will drop sharply and then, as it moves away, the car will honk at a lower note.
Application
Doppler radar
Links
- Using the Doppler effect to measure ocean currents
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See what “Doppler shift” is in other dictionaries:
Doppler shift- Doplerio poslinkis statusas T sritis fizika atitikmenys: engl. Doppler displacement; Doppler shift vok. Doppler Verschiebung, f rus. Doppler shift, m; Doppler shift, n pranc. déplacement Doppler, m; déviation Doppler, f … Fizikos terminų žodynas
Doppler frequency shift- Doplerio dažnio poslinkis statusas T sritis radioelektronika atitikmenys: engl. Doppler frequency displacement; Doppler frequency shift vok. Doppler Frequenzverschiebung, f rus. Doppler frequency shift, m; Doppler frequency shift, n… … Radioelektronikos terminų žodynas
Red shift is a shift in the spectral lines of chemical elements to the red (long wavelength) side. This phenomenon may be an expression of the Doppler effect or gravitational redshift, or a combination of both. Spectrum shift... Wikipedia
Increasing wavelengths (l) of lines in electricity. mag. source spectrum (shift of lines towards the red part of the spectrum) compared to the lines of the reference spectra. Quantitatively K. s. characterized by the value z=(lprin lsp)/lsp, where lsp and lprin... ... Physical encyclopedia
The gravitational blue shift of a quantum (photon) or other elementary particle (such as an electron or proton) when it falls into a gravitational field (created by a yellow star at the bottom ... Wikipedia
A decrease in the frequencies of electromagnetic radiation is one of the manifestations of the Doppler effect. The name "K. With." due to the fact that in the visible part of the spectrum, as a result of this phenomenon, the lines are shifted towards its red end; K. s. observed... ... Great Soviet Encyclopedia
The change in the oscillation frequency w or wavelength l perceived by the observer when the source of oscillations and the observer move relative to each other. The emergence of D. e. The easiest way to explain is by following. example. Let a motionless source emits... Physical encyclopedia
The theories of relativity form an essential part of the theoretical basis of modern physics. There are two main theories: particular (special) and general. Both were created by A. Einstein, particular in 1905, general in 1915. In modern physics, particular... ... Collier's Encyclopedia
A branch of astronomy that studies space objects by analyzing the radio emission coming from them. Many cosmic bodies emit radio waves that reach the Earth: these are, in particular, the outer layers of the Sun and planetary atmospheres, clouds of interstellar gas.… … Collier's Encyclopedia
Hot glowing celestial bodies like the Sun. Stars vary in size, temperature and brightness. In many respects, the Sun is a typical star, although it seems much brighter and larger than all other stars, since it is located much closer to... ... Collier's Encyclopedia
Yushkevich R.S., Degtyareva E.R.
The article provides a derivation of formulas for the Doppler effect without using the law of addition of velocities, but using the principle of constancy of the speed of light only relative to the light source. The spatial limit of the possibility of receiving electromagnetic waves has been determined. The dependence of the speed of light on distance is considered. The coefficient for calculating the speed of light is determined.
To explain the effect, we assume that the light coming from the light source is connected to the source and propagates from it at a speed s = 3 10 8 m/s relative to the source. For the receiver, the speed of light relative to the source will be added to the speed of the source v.
To determine the frequency dependence of light ν from speed v, consider the propagation of light from two sources, one of which Ѕ moves away from the receiver at a speed v, and the other S 0 rests.
Rice. 1.
Identical sources emit light of the same frequency ν 0 . Light travels at the same speed relative to its sources With, therefore the emitted wavelength λ 0 will be the same. Light will approach the receiver from a moving source at a speed With-v and wavelength λ 0 will be accepted in time T =(period), and from a stationary source - in time T 0 =. Periods are the reciprocal quantities of the oscillation frequencies and . Let's substitute the values T And T 0 into the resulting equalities
dividing them term by term, we get
,
we get [p. 181].
(1)
In the case when the source and receiver are approaching, you need a sign v replace with the opposite, we get 
. Note that With-v And c are the speeds of light relative to the receiver and the light source, respectively.
Now consider the case when the light source moves perpendicular to the direction of the receiver. Considering that light is associated with a source, it propagates relative to it at a speed With and carries him with speed v in order for it to hit the receiver it must be directed at a certain angle α So sinα= . In this case, the component of the speed of light coinciding with the direction to the receiver A will be , the component v in this direction is equal to 0. In order not to repeat the previous reasoning, we use formula (1), With-v we replace it with , and the speed c relative to the source remains unchanged. As a result we get:
which corresponds to the result obtained in the experiments of Ives [p. 181].
Rice. 2.
When light passes from a source to a receiver, its frequency changes from ν 0
to ν.
From the formula с=λν it follows that the wavelength must also change. If a wave of length came from a light source λ 0
, then the receiver will receive it differently, say λ
. Get value λ
it is possible by taking advantage of the fact that λ
And ν
quantities are inversely proportional 
. Substituting the value ν
from the formula (1),
we get
To be more confident, we obtain this formula in a different way.
Any light receiver can also be an emitter, which means that it has the same light-carrying medium as the source, and light propagates in it at speed With. Light, passing from the source medium to the receiver medium, gains speed With relative to the receiver.
Wave length λ 0 from the source to the interface between the source and receiver media approaches at a speed With -v and the boundary will pass in time C from the very beginning of the wave’s entry into the sphere of the receiver’s medium, its beginning acquires speed c relative to the receiver and in time T it will travel the distance λ = cT. Substituting the value T, we get:
Rice. 3.
In the first half of the twentieth century. The American scientist Hubble discovered in the spectra of distant stars a shift of spectral lines towards the red part of the spectrum compared to laboratory spectra - “red shift”. This meant that the received wavelength λ is greater than λ 0 and the farther the star is, the greater the “red shift”.
Into the formula (2) includes four quantities λ, λ 0 , s And v. By the time the “red shift” was discovered, the speed of light with Einstein’s postulate was fixed constant relative to any frame of reference, which means λ 0 , associated with the speed of light c and the source of radiation, turned out to be constant. In the formula (2) variable quantity λ , turned out to be related to the speed of the source v. Increase λ causes an increase v.
“Redshift” is observed in stars located in all directions, so the expansion of the Universe was recognized.
In astronomy, the connection between λ And v determined by another formula
(3)
for a receding radiation source.
For the same phenomenon and the same quantities, two formulas establish different dependences! To understand this, let’s compare the results that these formulas give for different v. Restrictions on the speed value v no formula required. For convenience, we denote the wavelengths λ 3 And λ 2 according to the designation of the formulas (3) And ( 2 ) in which they are included. At v=0 :
At 0< v< с compare by division:
If v"With, then λ 3 ≈ λ 2. Under these two conditions, the results are practically consistent with each other.
When v = c; 
λ 2 turns to infinity, while formula (1) gives 
. It turns out that the light wave does not reach the receiver from the source, it travels at a speed With will move from the source to the receiver and, together with the source, will move away from it at the same speed c - c = 0.
The third comparison requires concluding which formula correctly reflects reality. Origin of the formula (2) discussed at the beginning of the article. Now let's look at how the formula is obtained (3).
Rice. 4.
Let's imagine that the light source is surrounded by a medium in which the light propagates to the receiver at a speed With. Light source at a point A began to emit a wave. Let us denote the emission time of one wave T(period). From the moment the wave begins to appear, it begins to move towards the receiver in the environment at a speed With and for the time T will move away from the point A to a distance st. But during this same time, the source, moving from the receiver, will end up at the point WITH, having covered the distance AC =vT, where the end of the wave will be. Distance from WITH to B and will be the wavelength λ = сТ +vT = (c +v)T
If the source is not moving, then v = 0 and the wavelength will be λ 0 = cT. By dividing λ on λ 0, we get:
At the beginning of the article, we looked at the medium that provides the speed of light c; it is either connected to the source or to the receiver of light. The first one gives formulas (1) and (2). The probability that the second one, from a distant light receiver, influenced the speed of light more than the environment of the light source is negligible. A medium remains, not connected with either the source or the receiver of light, which acts like air (matter) on the propagation of sound. But the negative result of Michelson’s experiments to detect the “ethereal wind” proved that such a medium does not exist in nature. It remains to give preference to formula (2). It was previously noted that when the light source moves away at a speed v = c, the wave will not reach the receiver and the signal will not be received.
Hubble introduced the law that bears his name [p. 120]
v= HD,
where v is the speed of removal of the light source, D is the distance between the source and the receiver, H is the coefficient of proportionality, called the Hubble constant.
.
1 Mpc = 10 6 pc; 1pc (parsec) = 3.26 light years = 3. 10 13 km.
Let's find the distance at which v = c: 
;
D- this is the radius of the sphere that limits the reception of direct electromagnetic radiation from the vastness of the Universe. From the zones adjacent to this sphere in its inner part, electromagnetic radiation can only come in the form of radio waves. In nature, there is no priority direction in the distribution of stars, so radio emission must come from all directions evenly.
Let's consider the option when v>s. In this case, formulas (1) and (2) give: 
And .
This means that the wave must come from the direction opposite to where the emitter is located.
At v= 2s we have
.
The wave will arrive without a “red shift”. The limit of possible reception of electromagnetic radiation defined in the article will be correct if Hubble’s law is true and the “red shift” is caused solely by the removal of the emitter. If other factors are discovered that reduce the speed of light relative to the receiver (and they may exist), then the wave reception limit can be approached.
Let us now turn to the formulas (1) And (2). In them c-v is the speed of light relative to the receiver, let's denote it c 1 =c-v where v=c-c 1.In formulas v represents the difference in the speed of light, regardless of the nature of its occurrence. It is generally accepted that this is the result of removing the light source. But this speed difference can also arise due to the decrease in the speed of light with increasing distance. Light is a flow of energy quanta and it is possible that their speed may decrease.
Let us assume that the speed of light decreases with increasing distance from the light source, figuratively speaking, “light ages.”
It is known that the speed of light decreases when passing from an optically less dense medium to a more dense one. This is due to the fact that the conditions for the passage of light change. The decrease in speed is characterized by the refractive index n;, Where With- speed of light in vacuum a from 1- speed in another environment.
If, by assumption, the speed of light decreases with increasing distance from the light source, then the conditions for its passage also change, which can also be characterized by the refractive index n. We find that the reduced speed of light will be .
In the article “Fizeau's Experience” (journal “Modern Science-Intensive Technologies” No. 2, 2007) to determine the speed of light in a moving medium, the refractive index n was used in the form 
, where part of the indicator is determined by the emitting atom, and is determined by the conditions for the passage of light in the medium.
Let us apply this representation of the refractive index for vacuum. If we accepted the assumption that the speed of light decreases in a vacuum, and vacuum is a homogeneous medium, then the decrease in the speed of light should depend only on the distance and is proportional to it. Therefore we can write where D- distance to light source, μ - proportionality coefficient is a constant value. The speed of received light will be
The difference between the initial and reduced speeds of light will be
This expresses the relationship between the decrease in the speed of light and the distance D. The connection between these same quantities is expressed by Hubble’s law where v- the speed of removal of the star, which for the light receiver is the difference c-c 1.
Let's compare the values v, which these two equations give for the distance limits D.
If , then from the first equation we obtain: , n=1 (for short distances) and . From Hubble's law we also get .
If this coincidence is not accidental, it can be assumed that the quanta of light energy are associated with the emitter, and this is also indicated by the connection of the light-carrying medium with the light source.
To determine the speed from 1, we need to decide regarding n equation:
and find the speed through n from 1.
For small values of D, Hubble's law can be used.
There is a clear contradiction in the article. Based on the concept of the expansion of the Universe, it was concluded that there is a limit to the possible reception of electromagnetic waves, but based on the natural decrease in the speed of light, there is no such limit. It turns out that the discovery of such a boundary will be evidence of the expansion of the Universe.
The article also assumes, without convincing grounds, that the speed of light depends on distances. The basis for this assumption will be discovered when considering the process of emission of light quanta by an atom.
REFERENCES:
- Zisman G.A., Todes O.M., Course of general physics vol.3. - M.: “Science”, 1972.
- Vorontsov - Velyaminov B.A. Astronomy 10. - M.: “Enlightenment”, 1983.
Bibliographic link
Yushkevich R.S., Degtyareva E.R. THE DOPPLER EFFECT AND THE SPEED OF LIGHT // Fundamental Research. – 2008. – No. 3. – P. 17-24;URL: http://fundamental-research.ru/ru/article/view?id=2764 (access date: 02/16/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"
It is known that when a fast moving electric train approaches a stationary observer, its sound signal seems higher, and when moving away from the observer, it appears lower than the signal of the same electric train, but stationary.
Doppler effect call the change in the frequency of waves recorded by the receiver, which occurs due to the movement of the source of these waves and the receiver.
The source, moving towards the receiver, seems to compress a spring - a wave (Fig. 5.6).
This effect is observed during the propagation of sound waves (acoustic effect) and electromagnetic waves (optical effect).
Let's consider several cases of manifestation acoustic Doppler effect .
Let the receiver of sound waves P in a gaseous (or liquid) medium be motionless relative to it, and the source I move away from the receiver with a speed along the straight line connecting them (Fig. 5.7, A).
The source is displaced in the medium in a time equal to the period of its oscillations, by a distance , where is the oscillation frequency of the source.
Therefore, when the source moves, the wavelength in the medium is different from its value with a stationary source:
,
where is the phase velocity of the wave in the medium.
The wave frequency recorded by the receiver is
 | (5.7.1) |
If the source velocity vector is directed at an arbitrary angle to the radius vector connecting the stationary receiver with the source (Fig. 5.7, b), That
 | (5.7.2) |
If the source is stationary and the receiver approaches it at a speed along the straight line connecting them (Fig. 5.7, V), then the wavelength in the medium is . However, the speed of propagation of the wave relative to the receiver is equal to , so the frequency of the wave recorded by the receiver
| (5.7.3) |
In the case when the speed is directed at an arbitrary angle to the radius vector connecting the moving receiver with a stationary source (Fig. 5.7, G), we have:
This formula can also be represented as (if)
| , | (5.7.6) |
where is the speed of the wave source relative to the receiver, and is the angle between the vectors and . The quantity equal to the projection onto the direction is called radial velocity of the source.
Optical Doppler effect
When the source and receiver of electromagnetic waves move relative to each other, it is also observed Doppler effect , i.e. wave frequency change, registered by the receiver. In contrast to the Doppler effect we considered in acoustics, the laws of this phenomenon for electromagnetic waves can only be established on the basis of the special theory of relativity.
Relationship describing Doppler effect For electromagnetic waves in vacuum, taking into account Lorentz transformations, has the form:
 .
| (5.7.7) |
At low speeds of movement of the wave source relative to the receiver, the relativistic formula for the Doppler effect (5.7.7) coincides with the classical formula (5.7.2).
If the source moves relative to the receiver along the straight line connecting them, then we observe longitudinal Doppler effect .
In case of approaching the source and receiver ()
 ,
| (5.7.8) |
and in case of their mutual removal ()
 .
| (5.7.9) |
In addition, from the relativistic theory of the Doppler effect it follows the existence transverse Doppler effect , observed at and , i.e. in cases where the source moves perpendicular to the observation line (for example, the source moves in a circle, the receiver is in the center):
 .
| (5.7.10) |
The transverse Doppler effect is inexplicable in classical physics. It represents a purely relativistic effect.
As can be seen from formula (5.7.10), the transverse effect is proportional to the ratio, therefore it is much weaker than the longitudinal one, which is proportional to (5.7.9).
In the general case, the relative velocity vector can be decomposed into components: one provides a longitudinal effect, the other provides a transverse effect.
The existence of the transverse Doppler effect follows directly from time dilation in moving reference frames.
The first experimental verification of the existence of the Doppler effect and the correctness of the relativistic formula (5.7.7) was carried out by American physicists G. Ives and D. Stilwell in the 30s. Using a spectrograph, they studied the radiation of hydrogen atoms accelerated to speeds of m/s. In 1938 the results were published. Summary: the transverse Doppler effect was observed in full accordance with relativistic frequency transformations (the emission spectrum of atoms turned out to be shifted to the low-frequency region); the conclusion about time dilation in moving inertial frames of reference has been confirmed.
The Doppler effect has found wide application in science and technology. This phenomenon plays a particularly important role in astrophysics. Based on the Doppler shift of absorption lines in the spectra of stars and nebulae, it is possible to determine the radial velocities of these objects relative to the Earth: at using formula (5.7.6)
 .
| (5.7.11) |
American astronomer E. Hubble discovered in 1929 a phenomenon called cosmological redshift and consisting in the fact that the lines in the emission spectra of extragalactic objects are shifted towards lower frequencies (longer wavelengths). It turned out that for each object the relative frequency shift ( is the frequency of the line in the spectrum of a stationary source, is the observed frequency) is exactly the same for all frequencies. Cosmological redshift is nothing more than the Doppler effect. It indicates that the Metagalaxy is expanding, so that extragalactic objects are moving away from our Galaxy.
The Metagalaxy is understood as the totality of all star systems. With modern telescopes you can observe a part of the Metagalaxy, the optical radius of which is equal to 
. The existence of this phenomenon was theoretically predicted back in 1922 by the Soviet scientist A.A. Friedman based on the development of the general theory of relativity.
Hubble established a law according to which the relative redshift of galaxies increases in proportion to their distance .
Hubble's Law can be written in the form
 ,
| (5.7.12) |
Where H– Hubble constant. According to the most recent estimates, conducted in 2003, . (1 pc (parsec) is the distance that light travels in a vacuum in 3.27 years ( 
)).
In 1990, the Hubble Space Telescope was launched into orbit aboard the Discovery shuttle (Fig. 5.8).
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| Rice. 5.8 | Rice. 5.9 |
Astronomers have long dreamed of a telescope that would operate in the visible range, but would be located outside the Earth's atmosphere, which greatly interferes with observations. Hubble not only did not disappoint the hopes placed on it, but even exceeded almost all expectations. He fantastically expanded the “field of vision” of humanity, looking into the unimaginable depths of the Universe. During its operation, the space telescope transmitted 700 thousand magnificent photographs to earth (Fig. 5.9). In particular, he helped astronomers determine the exact age of our Universe - 13.7 billion years; helped confirm the existence of a strange but powerful form of energy in the Universe - dark energy; proved the existence of supermassive black holes; amazingly clearly captured the fall of a comet on Jupiter; showed that the process of formation of planetary systems is widespread in our Galaxy; discovered small protogalaxies by detecting the radiation emitted by them when the age of the Universe was less than 1 billion years.
Radar laser methods for measuring the speeds of various objects on Earth (for example, a car, an airplane, etc.) are based on the Doppler effect. Laser anemometry is an indispensable method for studying the flow of liquid or gas. The chaotic thermal motion of the atoms of a luminous body also causes a broadening of lines in its spectrum, which increases with increasing speed of thermal motion, i.e. with increasing gas temperature. This phenomenon can be used to determine the temperature of hot gases.
