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The Doppler effect (or Doppler shift), named after Christian Doppler, is the change in frequency and wavelength of a wave for an observer moving relative to the source of the waves. It is commonly heard when a vehicle sounding a siren approaches, passes and recedes from an observer. The received frequency is increased (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is decreased during the recession.

For waves that propagate in a medium, such as sound waves, the velocity of the observer and of the source are relative to the medium in which the waves are transmitted. The total Doppler effect may therefore result from motion of the source, motion of the observer, or motion of the medium. Each of these effects is analyzed separately. For waves which do not require a medium, such as light or gravity in special relativity, only the relative difference in velocity between the observer and the source needs to be considered.

## DevelopmentEdit

Doppler first proposed the effect in 1842 in his treatise "Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels" (On the coloured light of the binary stars and some other stars of the heavens).[1] The hypothesis was tested for sound waves by Buys Ballot in 1845. He confirmed that the sound's pitch was higher than the emitted frequency when the sound source approached him, and lower than the emitted frequency when the sound source receded from him. Hippolyte Fizeau discovered independently the same phenomenon on electromagnetic waves in 1848 (in France, the effect is sometimes called "effet Doppler-Fizeau"). In Britain, John Scott Russell made an experimental study of the Doppler effect (1848).[2]

An English translation of Doppler's 1842 treatise can be found in the book The Search for Christian Doppler by Alec Eden.[1]

## GeneralEdit

The relationship between observed frequency f and emitted frequency f0 is given by:

$f = \left( \frac{v}{v + v_{s,r}} \right) f_0 \,$
where
$v \;$ is the velocity of waves in the medium
$v_{s,r} \,$ is the radial component of the velocity of the source (the object emitting the wave) along a line from the source to the observer; positive if moving away from the observer.

The observed frequency is increased if the source is moving towards the observer. It is decreased if the source is moving away.

For all paths of an approaching object, the observed frequency that is first heard is higher than the object's emitted frequency. Thereafter there is a monotonic decrease in the observed frequency as it gets closer to the observer, through equality when it is level with the observer, and a continued monotonic decrease as it recedes from the observer. When the observer is very close to the path of the object, the transition from high to low frequency is very abrupt. When the observer is far from the path of the object, the transition from high to low frequency is gradual.

In the limit where the speed of the wave is much greater than the relative speed of the source and observer (this is often the case with electromagnetic waves, e.g. light), the relationship between observed frequency f; and emitted frequency f0 is given by:

Change in frequencyObserved frequency
$\Delta f=-\frac{f_0v_{s,r}}{c}=-\frac{v_{s,r}}{\lambda}$
$f=\left(1-\frac{v_{s,r}}{c}\right)f_0$
where
$f_{0} \,$ is the transmitted frequency
$v_{s,r} \,$ is the velocity of the transmitter relative to the receiver in meters per second: negative when moving towards one another, positive when moving away
$c \,$ is the speed of wave (e.g. 3×108 m/s for electromagnetic waves travelling in a vacuum)
$\lambda \,$ is the wavelength of the transmitted wave subject to change.

These two equations are only accurate to a first order approximation. However, they work reasonably well in the case considered by Doppler: when the speed between the source and receiver is slow relative to the speed of the waves involved and the distance between the source and receiver is large relative to the wavelength of the waves. If either of these two approximations are violated, the formulae are no longer accurate.

## AnalysisEdit

The frequency of the sounds that the source emits does not actually change. To understand what happens, consider the following analogy. Someone throws one ball every second in a man's direction. Assume that balls travel with constant velocity. If the thrower is stationary, the man will receive one ball every second. However, if the thrower is moving towards the man, he will receive balls more frequently because the balls will be less spaced out. The inverse is true if the thrower is moving away from the man. So it is actually the wavelength which is affected; as a consequence, the received frequency is also affected. It may also be said that the velocity of the wave remains constant whereas wavelength changes; hence frequency also changes.

If the moving source is emitting waves through a medium with an actual frequency f0, then an observer stationary relative to the medium detects waves with a frequency f given by

$f = f_0 \left ( \frac {v}{v + v_{s}} \right )$

where v is the speed of the waves in the medium and vs is the speed of the source with respect to the medium (positive if moving away from the observer, negative if moving towards the observer).

A similar analysis for a moving observer and a stationary source yields the observed frequency (the receiver's velocity being represented as vr):

$f = f_0 \left ( \frac {v}{v + v_{r}} \right )$

where the same convention applies: vr is positive if the observer is moving away from the source, and negative if the observer is moving towards the source.

These can be generalized into a single equation with both the source and receiver moving.

$f = f_0 \left ( \frac {v}{v + v_{s,r}} \right )$
which can be written as: $f = f_0 \left (1 - \frac {v_{s,r}}{v+v_{s,r}} \right )$

Where vs,r is the source to receiver velocity radial component.
With a relatively slow moving source, vs,r is small in comparison to v and the equation approximates to

$f = f_0 \left (1 - \frac {v_{s,r}}{v} \right )$.

However the limitations mentioned above still apply. When the more complicated exact equation is derived without using any approximations (just assuming that everything: source, receiver, and wave or signal are moving linearly) several interesting and perhaps surprising results are found. For example, as Lord Rayleigh noted in his classic book on sound, by properly moving it would be possible to hear a symphony being played backwards. This is the so-called "time reversal effect" of the Doppler effect. Other interesting cases are that the Doppler effect is time dependent in general (thus we need to know not only the source and receivers' velocities, but also their positions at a given time) and also in some circumstances it is possible to receive two signals or waves from a source (or no signal at all). In addition there are more possibilities than just the receiver approaching the signal and the receiver receding from the signal.

All these additional complications are for the classical—i.e., nonrelativistic Doppler effect. However, all these results also hold for the relativistic Doppler effect as well.

The first attempt to extend Doppler's analysis to light waves was soon made by Fizeau. In fact, light waves do not require a medium to propagate and the correct understanding of the Doppler effect for light requires the use of the Special Theory of Relativity. See relativistic Doppler effect.

## Sloppy languageEdit

Craig Bohren pointed out that some physics textbooks erroneously state that the observed frequency increases as the object approaches an observer and then decreases only as the object passes the observer.[3] In fact, the observed frequency of an approaching object declines monotonically from a value above the emitted frequency, through a value equal to the emitted frequency when the object is closest to the observer, and to values increasingly below the emitted frequency as the object recedes from the observer. Bohren proposed that this common misconception might occur because the intensity of the sound increases as an object approaches an observer and decreases once it passes and recedes from the observer. Or it is just sloppy language (in/decreases instead of is in/decreased)

## ApplicationsEdit

### Sirens Edit

The siren on a passing emergency vehicle will start out higher than its stationary pitch, slide down as it passes, and continue lower than its stationary pitch as it recedes from the observer. Astronomer John Dobson explained the effect thus:

"The reason the siren slides is because it doesn't hit you."

In other words, if the siren approached the observer directly, the pitch would remain constant (as vs, r is only the radial component) until the vehicle hit him, and then immediately jump to a new lower pitch. Because the vehicle passes by the observer, the radial velocity does not remain constant, but instead varies as a function of the angle between his line of sight and the siren's velocity:

$v_{s, r}=v_s\cdot \cos{\theta}$

where vs is the velocity of the object (source of waves) with respect to the medium, and $\theta$ is the angle between the object's forward velocity and the line of sight from the object to the observer.

### Astronomy Edit

The Doppler effect for electromagnetic waves such as light is of great use in astronomy and results in either a so-called redshift or blueshift. It has been used to measure the speed at which stars and galaxies are approaching or receding from us, that is, the radial velocity. This is used to detect if an apparently single star is, in reality, a close binary and even to measure the rotational speed of stars and galaxies.

The use of the Doppler effect for light in astronomy depends on our knowledge that the spectra of stars are not continuous. They exhibit absorption lines at well defined frequencies that are correlated with the energies required to excite electrons in various elements from one level to another. The Doppler effect is recognizable in the fact that the absorption lines are not always at the frequencies that are obtained from the spectrum of a stationary light source. Since blue light has a higher frequency than red light, the spectral lines of an approaching astronomical light source exhibit a blueshift and those of a receding astronomical light source exhibit a redshift.

Among the nearby stars, the largest radial velocities with respect to the Sun are +308 km/s (BD-15°4041, also known as LHS 52, 81.7 light-years away) and -260 km/s (Woolley 9722, also known as Wolf 1106 and LHS 64, 78.2 light-years away). Positive radial velocity means the star is receding from the Sun, negative that it is approaching.

### Temperature measurementEdit

Another use of the Doppler effect, which is found mostly in astronomy, is the estimation of the temperature of a gas which is emitting a spectral line. Due to the thermal motion of the gas, each emitter can be slightly red or blue shifted, and the net effect is a broadening of the line. This line shape is called a Doppler profile and the width of the line is proportional to the square root of the temperature of the gas, allowing the Doppler-broadened line to be used to measure the temperature of the emitting gas.

### RadarEdit

Main article: Doppler radar

The Doppler effect is used in some types of radar, to measure the velocity of detected objects. A radar beam is fired at a moving target — e.g. a motor car, as police use radar to detect speeding motorists — as it approaches or recedes from the radar source. Each successive radar wave has to travel farther to reach the car, before being reflected and re-detected near the source. As each wave has to move farther, the gap between each wave increases, increasing the wavelength. In some situations, the radar beam is fired at the moving car as it approaches, in which case each successive wave travels a lesser distance, decreasing the wavelength. In either situation, calculations from the Doppler effect accurately determine the car's velocity. Moreover, the proximity fuze, developed during World War II, relies upon Doppler radar to explode at the correct time, height, distance, et cetera.

### Medical imaging and blood flow measurementEdit

An echocardiogram can, within certain limits, produce accurate assessment of the direction of blood flow and the velocity of blood and cardiac tissue at any arbitrary point using the Doppler effect. One of the limitations is that the ultrasound beam should be as parallel to the blood flow as possible. Velocity measurements allow assessment of cardiac valve areas and function, any abnormal communications between the left and right side of the heart, any leaking of blood through the valves (valvular regurgitation), and calculation of the cardiac output. Contrast-enhanced ultrasound using gas-filled microbubble contrast media can be used to improve velocity or other flow-related medical measurements.

Although "Doppler" has become synonymous with "velocity measurement" in medical imaging, in many cases it is not the frequency shift (Doppler shift) of the received signal that is measured, but the phase shift (when the received signal arrives).

Velocity measurements of blood flow are also used in other fields of medical ultrasonography, such as obstetric ultrasonography and neurology. Velocity measurement of blood flow in arteries and veins based on Doppler effect is an effective tool for diagnosis of vascular problems like stenosis.[4]

### Flow measurementEdit

Instruments such as the laser Doppler velocimeter (LDV), and Acoustic Doppler Velocimeter (ADV) have been developed to measure velocities in a fluid flow. The LDV emits a light beam and the ADV emits an ultrasonic acoustic burst, and measure the Doppler shift in wavelengths of reflections from particles moving with the flow. The actual flow is computed as a function of the water velocity and face. This technique allows non-intrusive flow measurements, at high precision and high frequency.

### Velocity profile measurementEdit

Developed originally for velocity measurements in medical applications (blood flows), Ultrasonic Doppler Velocimetry (UDV) can measure in real time complete velocity profile in almost any liquids containing particles in suspension such as dust, gas bubbles, emulsions. Flows can be pulsating, oscillating, laminar or turbulent, stationary or transient. This technique is fully non-invasive.

### Underwater acousticsEdit

In military applications the Doppler shift of a target is used to ascertain the speed of a submarine using both passive and active sonar systems. As a submarine passes by a passive sonobuoy, the stable frequencies undergo a Doppler shift, and the speed and range from the sonobuoy can be calculated. If the sonar system is mounted on a moving ship or another submarine, then the relative velocity can be calculated.

### AudioEdit

The Leslie speaker, associated with and predominantly used with the Hammond B-3 Organ, takes advantage of the Doppler Effect by using an electric motor to rotate a speaker continuously, rapidly alternating the received frequency of a keyboard note.

## Further readingEdit

• Doppler and the Doppler effect, E. N. da C. Andrade, Endeavour Vol. XVIII No. 69, January 1959 (published by ICI London). Historical account of Doppler's original paper and subsequent developments.
• Adrian, Eleni (24 June, 1995). "Doppler Effect". NCSA. Retrieved on 2008-07-13.

## ReferencesEdit

1. 1.0 1.1 Alec Eden The search for Christian Doppler, Springer-Verlag, Wien 1992. Contains a facsimile edition with an English translation
2. Scott Russell, John (1848). "On certain effects produced on sound by the rapid motion of the observer". Report of the Eighteen Meeting of the British Association for the Advancement of Science (John Murray, London in 1849) 18 (7): 37–38. Retrieved on 8 July 2008.
3. Bohren, C. F. (1991). What light through yonder window breaks? More experiments in atmospheric physics. New York: J. Wiley.
4. D. H. Evans and W. N. McDicken, Doppler Ultrasound, Second Edition, John Wiley and Sons, 2000