Doppler effect
The Doppler effect, 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. 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 analysed 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.

# Development
Doppler first proposed the effect in 1842 in the monograph Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels - Versuch einer das Bradley'sche Aberrations-Theorem als integrierenden Theil in sich schliessenden allgemeineren Theorie (On the coloured light of the binary refracted stars and other celestial bodies - Attempt of a more general theory including Bradley's theorem as an integral part). The hypothesis was tested for sound waves by the Dutch scientist Christoph Hendrik Diederik Buys Ballot in 1845. He confirmed that the sound's pitch was higher when the sound source approached him, and lower 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"). It is often overlooked that in Doppler's publications (and also Einstein's in his discussion of the Doppler effect) he explicitly acknowledges that his formulae are only approximate because he made several mathematical approximations in his derivation. Doppler's derivation is repeated more or less verbatim in most modern textbooks but often without the warning that the formulas are only valid in some (experimentally often seen) limits. In Britain, John Scott Russell made an experimental study of the Doppler effect. In 1848, Russell reported his study of the Doppler effect. (J.S. Russell, "On certain effects produced on sound by the rapid motion of the observer", Brit. Assn. Rep., vol. 18, p. 37 (1848).)
An English translation of Doppler's 1842 monograph can be found in the book by Alec Eden, "The search for Christian Doppler", Springer-Verlag 1992. In this book, Eden felt doubtful regarding Doppler's conclusions on the colour of double stars, but he was convinced regarding Doppler's conclusions on sound.

# General
The relationship between observed frequency f' and emitted frequency f is given by:
Because we are using an inertial reference system, the velocity of an object moving towards the observer is considered as negative, so the observed frequency is higher than its emitted frequency (this is because the source's velocity is in the denominator). Conversely, the velocity of an object moving away from the observer is considered as positive, so the observed frequency is lower than its emitted frequency. When the object is at the same position as the observer, the observed frequency is briefly equal to its emitted frequency.
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 increase 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 f is given by:
As mentioned previously, 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.

# Analysis
It is important to realize that 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 perceived 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
where v is the speed of the waves in the medium and vs, r is the speed of the source with respect to the medium (positive if moving away from the observer, negative if moving towards the observer), radial to the observer.
With a relatively slow moving source, vs, r is small in comparison to v and the equation approximates to
A similar analysis for a moving observer and a stationary source yields the observed frequency (the observer's velocity being represented as vo):
where the same convention applies : vo 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. 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 classicali.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.

# Common error
Craig Bohren pointed out that many 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. 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.

# Applications

## Everyday
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:
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:
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
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-154041, 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 measurement
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.

## Radar
The Doppler effect is also used in some forms of radar to measure the velocity of detected objects. A radar beam is fired at a moving targeta car, for example, as radar is often used by police to detect speeding motoristsas it approaches or recedes from the radar source. Each successive wave has to travel further to reach the car, before being reflected and re-detected near the source. As each wave has to move further, 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.
The proximity fuze which was developed during World War II also relies on Doppler radar.

## Medical imaging and blood flow measurement
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.

## Flow measurement
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 measurement
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 acoustics
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 an another submarine, then the relative velocity can be calculated.

## Audio
The Leslie speaker, associated with and predominantly used with the Hammond B-3 Organ, takes advantage of the Doppler Effect by using an electrically driven motor to constantly rotate the speaker 360 degrees, rapidly alternating the perceived frequency output of the keyboard note.