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Doppler Effect

DOPPLER EFFECT

CONCEPT

Almost everyone has experienced the Doppler effect, though perhaps without knowing what causes it. For example, if one is standing on a street corner and an ambulance approaches with its siren blaring, the sound of the siren steadily gains in pitch as it comes closer. Then, as it passes, the pitch suddenly lowers perceptibly. This is an example of the Doppler effect: the change in the observed frequency of a wave when the source of the wave is moving with respect to the observer. The Doppler effect, which occurs both in sound and electromagnetic wavesincluding light waveshas a number of applications. Astronomers use it, for instance, to gauge the movement of stars relative to Earth. Closer to home, principles relating to the Doppler effect find application in radar technology. Doppler radar provides information concerning weather patterns, but some people experience it in a less pleasant way: when a police officer uses it to measure their driving speed before writing a ticket.

HOW IT WORKS

Wave Motion and Its Properties

Sound and light are both examples of energy, and both are carried on waves. Wave motion is a type of harmonic motion that carries energy from one place to another without actually moving any matter. It is related to oscillation, a type of harmonic motion in one or more dimensions. Oscillation involves no net movement, only movement in place; yet individual points in the wave medium are oscillating even as the overall wave pattern moves.

The term periodic motion, or movement repeated at regular intervals called periods, describes the behavior of periodic waveswaves in which a uniform series of crests and troughs follow each other in regular succession. A period (represented by the symbol T ) is the amount of time required to complete one full cycle of the wave, from trough to crest and back to trough.

Period is mathematically related to several other aspects of wave motion, including wave speed, frequency, and wavelength. Frequency (abbreviated f ) is the number of waves passing through a given point during the interval of one second. It is measured in Hertz (Hz), named after nineteenth-century German physicist Heinrich Rudolf Hertz (1857-1894), and a Hertz is equal to one cycle of oscillation per second. Higher frequencies are expressed in terms of kilohertz (kHz; 103 or 1,000 cycles per second); megahertz (MHz; 106 or 1 million cycles per second); and gigahertz (GHz; 109 or 1 billion cycles per second.) Wavelength (represented by the symbol λ, the Greek letter lambda) is the distance between a crest and the adjacent crest, or a trough and an adjacent trough, of a wave. The higher the frequency, the shorter the wavelength.

Amplitude, though mathematically independent from the parameters discussed, is critical to the understanding of sound. Defined as the maximum displacement of a vibrating material, amplitude is the "size" of a wave. The greater the amplitude, the greater the energy the wave contains: amplitude indicates intensity, which, in the case of sound waves, is manifested as what people commonly call "volume." Similarly, the amplitude of a light wave determines the intensity of the light.

Frame of Reference

A knowledge of the fundamentals involved in wave motion is critical to understanding the Doppler effect; so, too, is an appreciation of another phenomenon, which is as much related to human psychology and perception as it is to physics. Frame of reference is the perspective of an observer with regard to an object or event. Things may look different for one person in one frame of reference than they do to someone in another.

For example, if you are sitting across the table from a friend at lunch, and you see that he has a spot of ketchup to the right of his mouth, the tendency is to say, "You have some ketchup right here"and then point to the left of your own mouth, since you are directly across the table from his right. Then he will rub the left side of his face with his napkin, missing the spot entirely, unless you say something like, "Nomirror image." The problem is that each of you has a different frame of reference, yet only your friend took this into account.

RELATIVE MOTION.

Physicists often speak of relative motion, or the motion of one object in relation to another. For instance, the molecules in the human body are in a constant state of motion, but they are not moving relative to the body itself: they are moving relative to one another.

On a larger scale, Earth is rotating at a rate of about 1,000 MPH (1,600 km/h), and orbiting the Sun at 67,000 MPH (107,826 km/h)almost three times as fast as humans have ever traveled in a powered vehicle. Yet no one senses the speed of Earth's movement in the way that one senses the movement of a caror, indeed, the way the astronauts aboard Apollo 11 in 1969 perceived that their spacecraft was moving at about 25,000 MPH (40,000 km/h). In the case of the car or the spacecraft, movement can be perceived in relation to other objects: road signs and buildings on the one hand, Earth and the Moon on the other. But humans have no frame of reference from which to perceive the movement of Earth itself.

If one were traveling in a train alongside another train at constant velocity, it would be impossible to perceive that either train was actually moving, unless one looked at a reference point, such as the trees or mountains in the background. Likewise, if two trains were sitting side by side, and one train started to move, the relative motion might cause a passenger in the unmoving train to believe that his or her train was the one moving. In fact, as Albert Einstein (1879-1955) demonstrated with his Theory of Relativity, all motion is relative: when we say that something is moving, we mean that it is moving in relation to something else.

Doppler's Discovery

Long before Einstein was born, Austrian physicist Christian Johann Doppler (1803-1853) made an important discovery regarding the relative motion of sound waves or light waves. While teaching in Prague, now the capital of the Czech Republic, but then a part of the Austro-Hungarian Empire, Doppler became fascinated with a common, but previously unexplained, phenomenon. When an observer is standing beside a railroad track and a train approaches, Doppler noticed, the train's whistle has a high pitch. As it passes by, however, the sound of the train whistle suddenly becomes much lower.

By Doppler's time, physicists had recognized the existence of sound waves, as well as the fact a sound's pitch is a function of frequencyin other words, the closer the waves are to one another, the higher the pitch. Taking this knowledge, he reasoned that if a source of sound is moving toward a listener, the waves in front of the source are compressed, thus creating a higher frequency. On the other hand, the waves behind the moving source are stretched out, resulting in a lower frequency.

After developing a mathematical formula to describe this effect, Doppler presented his findings in 1842. Three years later, he and Dutch meteorologist Christopher Heinrich Buys-Ballot (1817-1890) conducted a highly unusual experiment to demonstrate the theory. Buys-Ballot arranged for a band of trumpet players to perform on an open railroad flatcar, while riding past a platform on which a group of musicians with perfect pitch (that is, a finely tuned sense of hearing) sat listening.

The experiment went on for two days, the flatcar passing by again and again, while the horns blasted and the musicians on the platform recorded their observations. Though Doppler and Buys-Ballot must have seemed like crazy men to those who were not involved in the experiment, the resultas interpreted from the musicians' written impressions of the pitches they heardconfirmed Doppler's theory.

REAL-LIFE APPLICATIONS

Sound Compression and the Doppler Effect

As stated in the introduction, one can observe the Doppler effect in a number of settings. If a person is standing by the side of a road and a car approaches at a significant rate of speed, the frequency of the sound waves grows until the car passes the observer, then the frequency suddenly drops. But Doppler, of course, never heard the sound of an automobile, or the siren of a motorized ambulance or fire truck.

In his day, the horse-drawn carriage still constituted the principal means of transportation for short distances, and such vehicles did not attain the speeds necessary for the Doppler effect to become noticeable. Only one mode of transportation in the mid-nineteenth century made it possible to observe and record the effect: a steam-powered locomotive. Therefore, let us consider the Doppler effect as Doppler himself didin terms of a train passing through a s tation.

THE SOUND OF A TRAIN WHISTLE.

When a train is sitting in a station prior to leaving, it blows its whistle, but listeners standing nearby notice nothing unusual. There is no differenceexcept perhaps in degree of intensitybetween the sound heard by someone on the platform, and the sound of the train as heard by someone standing behind the caboose. This is because a stationary train is at the center of the sound waves it produces, which radiate in concentric circles (like a bulls-eye) around it.

As the train begins to move, however, it is no longer at the center of the sound waves emanating from it. Instead, the circle of waves is moving forward, along with the train itself, and, thus, the locomotive compresses waves toward the front. If someone is standing further ahead along the track, that person hears the compressed sound waves. Due to their compression, these have a much higher frequency than the waves produced by a stationary train.

At the same time, someone standing behind the traina listener on the platform at the station, watching the train recede into the distancehears the sound waves that emanate from behind the train. It is the same train making the same sound, but because the train has compressed the sound waves in front of it, the waves behind it are spread out, producing a sound of much lower frequency. Thus, the sound of the train, as perceived by two different listeners, varies with frame of reference.

THE SONIC BOOM: A RELATED EFFECT.

Some people today have had the experience of hearing a jet fly high overhead, producing a shock wave known as a sonic boom. A sonic boom, needless to say, is certainly not something of which Doppler would have had any knowledge, nor is it an illustration of the Doppler effect, per se. But it is an example of sound compression, and, therefore, it deserves attention here.

The speed of sound, unlike the speed of light, is dependant on the medium through which it travels. Hence, there is no such thing as a fixed "speed of sound"; rather, there is only a speed at which sound waves are transmitted through a given type of material. Its speed through a gas, such as air, is proportional to the square root of the pressure divided by the density. This, in turn, means that the higher the altitude, the slower the speed of sound: for the altitudes at which jets fly, it is about 660 MPH (1,622 km/h).

As a jet moves through the air, it too produces sound waves which compress toward the front, and widen toward the rear. Since sound waves themselves are really just fluctuations in pressure, this means that the faster a jet goes, the greater the pressure of the sound waves bunched up in front of it. Jet pilots speak of "breaking the sound barrier," which is more than just a figure of speech. As the craft approaches the speed of sound, the pilot becomes aware of a wall of high pressure to the front of the plane, and as a result of this high-pressure wall, the jet experiences enormous turbulence.

The speed of sound is referred to as Mach 1, and at a speed of between Mach 1.2 and Mach 1.4, even stranger things begin to happen. Now the jet is moving faster than the sound waves emanating from it, and, therefore, an observer on the ground sees the jet move by well before hearing the sound. Of course, this would happen to some extent anyway, since light travels so much faster than sound; but the difference between the arrival time of the light waves and the sound waves is even more noticeable in this situation.

Meanwhile, up in the air, every protruding surface of the aircraft experiences intense pressure: in particular, sound waves tend to become highly compressed along the aircraft's nose and tail. Eventually these compressed sound waves build up, resulting in a shock wave. Down on the ground, the shock wave manifests as a "sonic boom"or rather, two sonic boomsone from the nose of the craft, and one from the tail. People in the aircraft do not hear the boom, but the shock waves produced by the compressed sound can cause sudden changes in pressure, density, and temperature that can pose dangers to the operation of the airplane. To overcome this problem, designers of supersonic aircraft have developed planes with wings that are swept back, so they fit within the cone of pressure.

Doppler Radar and Other Sensing Technology

The Doppler effect has a number of applications relating to the sensing of movement. For instance, physicians and medical technicians apply it to measure the rate and direction of blood flow in a patient's body, along with ultra-sound. As blood moves through an artery, its top speed is 0.89 MPH (0.4 m/s)not very fast, yet fast enough, given the small area in which movement is taking place, for the Doppler effect to be observed. A beam of ultrasound is pointed toward an artery, and the reflected waves exhibit a shift in frequency, because the blood cells are acting as moving sources of sound wavesjust like the trains Doppler observed.

Not all applications of the Doppler effect fall under the heading of "technology": some can be found in nature. Bats use the Doppler effect to hunt for prey. As a bat flies, it navigates by emitting whistles and listening for the echoes. When it is chasing down food, its brain detects a change in pitch between the emitted whistle, and the echo it receives. This tells the bat the speed of its quarry, and the bat adjusts its own speed accordingly.

DOPPLER RADAR.

Police officers may not enjoy the comparisongiven the public's general impression of bats as evil, blood-thirsty creaturesbut in using radar as a basis to check for speeding violations, the police are applying a principle similar to that used by bats. Doppler radar, which uses the Doppler effect to calculate the speed of moving objects, is a form of technology used not only by law-enforcement officers, but also by meteorologists.

The change in frequency experienced as a result of the Doppler effect is exactly twice the ratio between the velocity of the target (for instance, a speeding car) and the speed with which the radar pulse is directed toward the target. From this formula, it is possible to determine the velocity of the target when the frequency change and speed of radar propagation are known. The police officer's Doppler radar performs these calculations; then all the officer has to do is pull over the speeder and write a ticket.

Meteorologists use Doppler radar to track the movement of storm systems. By detecting the direction and velocity of raindrops or hail, for instance, Doppler radar can be used to determine the motion of winds and, thus, to predict weather patterns that will follow in the next minutes or hours. But Doppler radar can do more than simply detect a storm in progress: Doppler technology also aids meteorologists by interpreting wind direction, as an indicator of coming storms.

The Doppler Effect in Light Waves

So far the Doppler effect has been discussed purely in terms of sound waves; but Doppler himself maintained that it could be applied to light waves as well, and experimentation conducted in 1901 proved him correct. This was far from an obvious point, since light is quite different from sound.

Not only does light travel much, much faster186,000 mi (299,339 km) a secondbut unlike sound, light does not need to travel through a medium. Whereas sound cannot be transmitted in outer space, light is transmitted by radiation, a form of energy transfer that can be directed as easily through a vacuum as through matter.

The Doppler effect in light can be demonstrated by using a device called a spectroscope, which measures the spectral lines from an object of known chemical composition. These spectral lines are produced either by the absorption or emission of specific frequencies of light by electrons in the source material. If the light waves appear at the blue, or high-frequency end of the visible light spectrum, this means that the object is moving toward the observer. If, on the other hand, the light waves appear at the red, or low-frequency end of the spectrum, the object is moving away.

HUBBLE AND THE RED SHIFT.

In 1923, American astronomer Edwin Hubble (1889-1953) observed that the light waves from distant galaxies were shifted so much to the red end of the light spectrum that they must be moving away from the Milky Way, the galaxy in which Earth is located, at a high rate. At the same time, nearer galaxies experienced much less of a red shift, as this phenomenon came to be known, meaning that they were moving away at relatively slower speeds.

Six years later, Hubble and another astronomer, Milton Humason, developed a mathematical formula whereby astronomers could determine the distance to another galaxy by measuring that galaxy's red shifts. The formula came to be known as Hubble's constant, and it established the relationship between red shift and the velocity at which a galaxy or object was receding from Earth. From Hubble's work, it became clear that the universe was expanding, and research by a number of physicists and astronomers led to the development of the "big bang" theorythe idea that the universe emerged almost instantaneously, in some sort of explosion, from a compressed state of matter.

WHERE TO LEARN MORE

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

Bryant-Mole, Karen. Sound and Light. Crystal Lake, IL: Rigby Interactive Library, 1997.

Challoner, Jack. Sound and Light. New York: Kingfisher, 2001.

Dispenzio, Michael A. Awesome Experiments in Light and Sound. Illustrated by Catherine Leary. New York: Sterling Publishing Company, 1999.

"The Doppler Effect." The Physics Classroom (Web site). <http://www.glenbrook.k12.il.us/gbssci/phys/Class/waves/u10l3d.html> (April 29, 2001).

Maton, Anthea. Exploring Physical Science. Upper Saddle River, N.J.: Prentice Hall, 1997.

Russell, David A. "The Doppler Effect and Sonic Booms" Kettering University (Web site). <http://www.kettering.edu/~drussell/Demos/doppler/doppler.html> (April 29, 2001).

Snedden, Robert. Light and Sound. Des Plaines, IL: Heinemann Library, 1999.

"Sound WaveDoppler Effect" (Web site). <http://csgrad.cs.vt.edu/~chin/doppler.html> (April 29, 2001).

"Wave MotionDoppler Effect" (Web site). <http://members.aol.com/cepeirce/b21.html> (April 29, 2001).

KEY TERMS

AMPLITUDE:

The maximum displacement of a vibrating material. In wave motion, amplitude is the "size" of a wave, an indicator of the energy and intensity of the wave.

CYCLE:

One complete oscillation. In wave motion, this is equivalent to the movement of a wave from trough to crest and back to trough. For a sound wave, in particular, a cycle is one complete vibration.

DOPPLER EFFECT:

The change in the observed frequency of a wave when the source of the wave is moving with respect to the observer. It is named after Austrian physicist Johann Christian Doppler (1803-1853), who discovered it.

FRAME OF REFERENCE:

The perspective an observer has with regard to an object or action. Frame of reference affect sperception of various physical properties, and plays a significant role in the Dopplereffect.

FREQUENCY:

In wave motion, frequency is the number of waves passing through a given point during the interval of one second. The higher the frequency, the shorter the wavelength. Measured in Hertz, frequency is mathematically related to wave speed, wavelength, and period.

HARMONIC MOTION:

The repeated movement of a particle about a position of equilibrium, or balance.

HERTZ:

A unit for measuring frequency, named after nineteenth-century German physicist Heinrich Rudolf Hertz (1857-1894). High frequencies are expressed in terms of kilohertz (kHz; 103 or 1,000 cycles per second); megahertz (MHz; 106 or 1 million cycles per second);and gigahertz (GHz; 109 or 1 billion cycles per second.)

INTENSITY:

Intensity is the rate at which a wave moves energy per unit of cross-sectional area. Where sound wavesare concerned, intensity is commonly known as "volume."

OSCILLATION:

A type of harmonic motion, typically periodic, in one or more dimensions.

PERIOD:

For wave motion, a period is the amount of time required to complete one full cycle. Period is mathematically related to frequency, wavelength, and wave speed.

PERIODIC MOTION:

Motion that is repeated at regular intervals. These intervals are known as periods.

PERIODIC WAVE:

A wave in which a uniform series of crests and troughs follow one after the other in regular succession.

RELATIVE MOTION:

The motion of one object in relation to another.

WAVELENGTH:

The distance between a crest and the adjacent crest, or the trough and an adjacent trough, of a wave. Wavelength, symbolized λ (the Greek letter lambda) is mathematically related to wave speed, period, and frequency.

WAVE MOTION:

A type of harmonic motion that carries energy from one place to another without actually moving anymatter.

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Doppler Effect

Doppler effect

The Doppler effect is an effect observed in light and sound waves as they move toward or away from an observer. One simple example of the Doppler effect is the sound of an automobile horn. Picture a person standing on a street corner. A car approaches, blowing its horn. As the car continues moving toward the person, the pitch of the horn appears to increase; its sound goes higher and higher. As the car passes the observer, however, the effect is reversed. The pitch of the car horn becomes lower and lower.

Explanation

All waves can be defined by two related properties: their wavelength and frequency. Wavelength is the distance between two adjacent (next to each other) and identical parts of the wave, such as between two wave crests (peaks). Frequency is the number of wave crests that pass a given point per second. For reference, the wavelength of visible light is about 400 to 700 nanometers (billionths of a meter), and its frequency is about 4.3 to 7.5 × 1014 hertz (cycles per second). The wavelength of sound waves is about 0.017 to 17 meters, and their frequency is about 20 to 20,000 hertz.

The car horn effect described above was first explained around 1842 by Austrian physicist Johann Christian Doppler (18031853). To describe his theory, Doppler used a diagram like the one shown in the accompanying figure of the Doppler effect. As a train approaches a railroad station, it sounds its whistle. The sound waves coming from the train travel outward in all directions. A person riding in the train would hear nothing unusual, just the steady pitch of the whistle's sound. But a person at the train station would hear something very different. As the train moves forward, the sound waves from its whistle move with it. The train is chasing or crowding the sound waves in front of it. An observer at the train station hears more waves per second than someone on the train. More waves per second means a higher frequency and, thus, a higher pitch.

An observer behind the train has just the opposite experience. Sound waves following the train spread out more easily. The second observer detects fewer waves per second, a lower frequency, and, therefore, a lower-pitched sound.

Words to Know

Hubble's law: The law that shows how the redshift of a galaxy can be used to determine its distance from Earth.

Redshift: The lengthening of the frequency of light waves as they travel away from an observer; most commonly used to describe movement of stars away from Earth.

It follows from this explanation that the sound heard by an observer depends on the speed with which the train is traveling. The faster the train is moving in the above example, the more its sound waves are bunched together or spread outthus, the higher or the lower the pitch observed.

Doppler effect in light waves

Doppler predicted that the effect in sound waves would also occur with light waves. That argument makes sense since sound and light are both transmitted by waves. But Doppler had no way to test his prediction experimentally. Doppler effects in light were not actually observed, in fact, until the late 1860s.

In sound, the Doppler effect is observed as a difference in the pitch of a sound. In light, differences in frequency appear as differences in color. For example, red light has a frequency of about 5 × 1014 hertz; green light, a frequency of about 6 × 1014 hertz; and blue light, a frequency of about 7 × 1014 hertz.

Suppose that a scientist looks at a lamp that produces a very pure green light. Then imagine that the lamp begins to move rapidly away from the observer. The Doppler effect states that the frequency of the light will decrease. Instead of appearing to be a pure green color, it will tend more toward the red end of the spectrum. The faster the lamp moves away from the observer, the more it will appear to be first yellow, then orange, then red. At very high speeds, the light coming from the lamp will no longer look green at all, but will have become red.

Applications

The green lamp example described above has been used to great advantage by astronomers when observing stars. The light of a star as seen

from Earth is always slightly different from its true color because all stars are in motion. When astronomers observe stars in our own Milky Way galaxy, for example, they find that the color of some stars is shifted toward the blue, while the color shift in other stars is toward the red. Blueshift stars are moving toward Earth, and redshift stars are moving away from Earth.

In 1923, American astronomer Edwin Hubble (18891953) made an interesting discovery. He found that all stars outside our own galaxy exhibit redshifts of light. That is, all stars outside our galaxy must be moving away from Earth. Furthermore, the farther away the stars are, the more their redshift and, thus, the faster they are moving away from us.

Hubble's discovery is one of the most important in all of modern astronomy. It tells us that the universe as a whole is expanding. Like dots on the surface of a balloon that's being blown up, galaxies throughout the universe are racing away from each other. One conclusion to be drawn from this discovery is thatat some time in the pastall galaxies must have been closer together at the center of the universe. Ever since that time, those galaxies have been moving away from each other. This conclusion is the basis for the currently popular theory about the creation of the universe, the big bang theory.

The Doppler effect has many other practical applications. Weather observers can bounce radar waves off storm clouds. By studying the frequency of the waves that return, they can determine the direction and speed with which the clouds are moving. Similarly, traffic police use radar guns to determine the speed of vehicles. The faster a car or truck is traveling, the greater the change in the frequency of the radar waves it reflects.

Sound waves are used for underwater observations. A submarine sends out sound waves that are reflected off other underwater objects, such as another submarine or a school of fish. The frequency of the reflected sound tells the direction and speed of the other object.

[See also Radar; Redshift; Sonar; Wave motion ]

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Doppler effect

Doppler effect, change in the wavelength (or frequency) of energy in the form of waves, e.g., sound or light, as a result of motion of either the source or the receiver of the waves; the effect is named for the Austrian scientist Christian Doppler, who demonstrated the effect for sound. If the source of the waves and the receiver are approaching each other (because of the motion of either or both), the frequency of the waves will increase and the wavelength will be shortened—sounds will become higher pitched and light will appear bluer. If the sender and receiver are moving apart, sounds will become lower pitched and light will appear redder. A common example is the sudden drop in the pitch of a train whistle as the train passes a stationary listener. The Doppler effect in reflected radio waves is employed in radar to sense the velocity of the object under surveillance. In astronomy, the Doppler effect for light is used to measure the velocity (and indirectly distance) and rotation of stars and galaxies along the direction of sight. In the spectrum of nearly every star there are wavelengths, characteristic of atoms, that lie near but not quite coincident to the same wavelengths as measured in the laboratory. The small deviations or shifts are generally due to the relative motion of the celestial object and the earth. Both blue shifts and red shifts are observed for various objects, indicating relative motion both toward and away from the earth. Such shifts have been used to measure the orbital velocity of the earth, to detect binary stars and variable stars, and to detect rotation of other galaxies. The Doppler effect is responsible for the red shifts of distant galaxies, and also of quasars, and thus provides the best evidence for the expansion of the universe, as described by Hubble's law. In addition to observations of visible light, the Doppler effect for radio waves is utilized by astronomers to determine the velocities of dust clouds in the spiral arms of the Milky Way galaxy. These observations provided the first direct proof that our own galaxy is rotating. The Doppler shift in radar pulses reflected from the surfaces of Venus and Mercury have been analyzed to obtain new values for their periods of rotation about their axes.

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Doppler effect

Doppler effect Change in frequency of a wave, when there is relative motion between the wave source and the observer. The amount of change depends on the velocities of the wave, source and observer. With a sound wave, the effect is demonstrated by the drop in pitch of a vehicle's siren as it passes an observer. With light, the velocity of the source or observer must be large for an appreciable effect to occur, such as the red shift of a rapidly receding galaxy. This effect was first described in 1842 by the Austrian physicist Christian Johann Doppler. See also navigation

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Doppler effect

Dop·pler ef·fect • n. Physics an increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move toward (or away from) each other. The effect causes the sudden change in pitch noticeable in a passing siren, as well as the redshift seen by astronomers.

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Doppler shift

Doppler shift The apparent change in frequency of waves whose source is moving towards or away from an observer. It was first described by the Austrian physicist Christian Doppler (1803–53).

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