One mountain, one water, one city , One person, one pen, one world . Hello! , I am a Herbie, Welcome to my official account ！

Doppler effect （ Doppler shift ）

• Introduction to the people
• The phenomenon
• history
• The formula
• prove
• reference

Introduction to the people

Christian · Andreas · Doppler （Christian Andreas Doppler,1803 year 11 month 29 Japan －1853 year 3 month 17 Japan ） Doppler effect discoverer , Austrian mathematician 、 physical scientist . Born in a stonemason family in Salzburg, Austria , Studied at Vienna Institute of Technology .1841 He became a professor of mathematics at Prague Institute of Technology ,1850 year , Doppler was the first Dean of the school of physics at the University of Vienna ,1853 Died in Venice, Italy , At the age of 49 year .

The phenomenon

Doppler effect （Doppler effect） It is when the wave source has relative motion with the observer , The phenomenon that the frequency of the wave received by the observer is different from that of the wave source . A common example in life is that the whistle of a speeding train in the distance becomes shrill （ That is, the frequency becomes higher , The wavelength becomes shorter ）, And the whistle of the train leaving us became low （ That is, the frequency becomes lower , Longer wavelength ）, This is the phenomenon of Doppler effect , The same phenomenon also occurs in the ringing of private cars and the ringing of trains , Audio see Doppler effect of sound . And intuitive feelings about frequency and wavelength, such as The figure below Shown （ come from wiki）.

history

Doppler effect was created by Austrian physicists in 1842 First proposed in , Followed by Buys Ballot Verify in sound waves , Then 1848 year Hippolyte Fizeau Independent found the same phenomenon in electromagnetic waves , Same year John Scott Russell The Doppler effect is experimentally studied . At present, this effect is widely used in astronomy 、 Radar and medical fields .

The formula

The frequency relationship between the observer and the emitter is ：

among , Is the observed frequency ; Is the original emission frequency from the medium ; Is the traveling speed of the wave in the medium ; Is the moving speed of the observer relative to the medium , If it is close to the emission source, the forward operation symbol is Number , Instead of Number ; Is the moving speed of the emission source relative to the medium , If you are close to the observer, the forward operation symbol is Number , Instead of Number .

prove

This article only introduces in traditional physics （ That is, the effect of relativity is not considered ） The basic principle of radar detection is used to prove the above Doppler formula , See here Radar Basics — Doppler frequency of moving target . The working principle of radar is to send some radio pulses to detect targets , And calculate the distance of the target by measuring the signal delay of the received echo , That is, the delay time is the two-way distance traveled by the pulse divided by its propagation speed . In fact, it can go further , Radar can also capture the Doppler frequency of moving targets through Doppler effect , So as to accurately reflect its movement speed , So for radar , How does the Doppler frequency of a moving target come into being , And what is the mathematical relationship between it and target speed and radar operating frequency ？

Suppose the radar transmits pulses at a specific frequency , If this pulse encounters a moving target and reflects back , Then the pulse waveform will be compressed more tightly . This means that the echo signal has time delay and strength weakening relative to the transmitted signal , Its carrier frequency will also have to rise . This shows that we can complete the speed extraction of moving targets by capturing the weak change of carrier frequency .

For the sake of simplicity , Suppose the target is at speed Radial motion relative to radar , The radar transmission width is Electromagnetic pulse of , It is in free space according to the speed of light To spread , Suppose that The front edge of the radar pulse is just the target , Because the radar pulse width is Therefore, at this time, the distance of the trailing edge of the radar pulse lagging behind the leading edge in space is , Then the front edge of the radar pulse will start back propagation after being reflected by the target , And the trailing edge of radar pulse and moving target , Then they continue to move forward at their own set speed and experience a short period After time , They will be in Time meets in space . During this time , The front edge of the reflected pulse will travel the same distance as the back edge , Together, they will form the time boundary of the transmitted pulse And it is obvious from the figure that the transmitted pulse is indeed narrower compressed by the moving target . We assume that the new pulse width is .

The following two equations are obtained from the simple distance relationship through the graph ：

Divide the two equations to get the ratio of pulse width compressed as

From this formula, the pulse compression ratio is greater than 1, And the faster the moving target is , Pulse compression is becoming more and more intense . For general discussion , We assume that the original pulse width just includes Integer oscillation period . Therefore, according to the definition of frequency, that is, the number of cycles of oscillation per unit time , We can get that the carrier frequency is

The number of oscillation periods of the carrier in the pulse will never change because the pulse width is compressed , Therefore, it can be obtained that the signal frequency of the reflected pulse is

take （3） Medium use 、 as well as To express, you get

So the proof is over .

However, this shows that the frequency of the reflected signal will increase according to the same pulse width compression ratio , Thus, the difference between it and the frequency of the transmitted signal can be predicted , It is the Doppler frequency of the moving target that

Three conclusions can be drawn from this ：
1、 The greater the radial velocity of the target , The higher the Doppler frequency .
2、 The shorter the working wavelength of radar , The higher the Doppler frequency .
3、 The velocity of the target can be calculated by measuring the Doppler frequency ：.

It should be noted that 2 Because of the round trip of the beam , Different from the maximum Doppler frequency shift commonly used in communication .

reference

[1] https://en.wikipedia.org/wiki/Doppler_effect

[2] https://www.bilibili.com/video/BV1634y1o7EZ?spm_id_from=333.337.search-card.all.click