Doppler effect

The Doppler effect (also Doppler shift) is the change in the frequency of a wave in relation to an observer who is moving relative to the source of the wave.^[1]^[2]^[3] The Doppler effect is named after the physicist Christian Doppler, who described the phenomenon in 1842. A common example of Doppler shift is the change of pitch heard when a vehicle sounding a horn approaches and recedes from an observer. Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing by, and lower during the recession.^[4]

"Doppler" redirects here. For other uses, see Doppler (disambiguation).

When the source of the sound wave is moving towards the observer, each successive cycle of the wave is emitted from a position closer to the observer than the previous cycle.^[4]^[5] Hence, from the observer's perspective, the time between cycles is reduced, meaning the frequency is increased. Conversely, if the source of the sound wave is moving away from the observer, each cycle of the wave is emitted from a position farther from the observer than the previous cycle, so the arrival time between successive cycles is increased, thus reducing the frequency.

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.^[3] The total Doppler effect in such cases may therefore result from motion of the source, motion of the observer, motion of the medium, or any combination thereof. For waves propagating in vacuum, as is possible for electromagnetic waves or gravitational waves, only the difference in velocity between the observer and the source needs to be considered.

$c$ is the propagation speed of waves in the medium;

$v_{\text{r}}$ is the speed of the receiver relative to the medium, added to $c$ if the receiver is moving towards the source, subtracted if the receiver is moving away from the source;

$v_{\text{s}}$ is the speed of the source relative to the medium, added to $c$ if the source is moving away from the receiver, subtracted if the source is moving towards the receiver.

In classical physics, where the speeds of source and the receiver relative to the medium are lower than the speed of waves in the medium, the relationship between observed frequency $f$ and emitted frequency $f_{\text{0}}$ is given by:^[8] $f=\left({\frac {c\pm v_{\text{r}}}{c\pm v_{\text{s}}}}\right)f_{0}$ where

Note this relationship predicts that the frequency will decrease if either source or receiver is moving away from the other.

Equivalently, under the assumption that the source is either directly approaching or receding from the observer: ${\frac {f}{v_{wr}}}={\frac {f_{0}}{v_{ws}}}={\frac {1}{\lambda }}$ where

If the source approaches the observer at an angle (but still with a constant speed), 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 coming from a direction perpendicular to the relative motion (and was emitted at the point of closest approach; but when the wave is received, the source and observer will no longer be at their closest), 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.

If the speeds $v_{\text{s}}$ and $v_{\text{r}}\,$ are small compared to the speed of the wave, the relationship between observed frequency $f$ and emitted frequency $f_{\text{0}}$ is approximately^[8]

where

Consequences[edit]

With an observer stationary relative to the medium, if a moving source is emitting waves with an actual frequency $f_{0}$ (in this case, the wavelength is changed, the transmission velocity of the wave keeps constant; note that the transmission velocity of the wave does not depend on the velocity of the source), then the observer detects waves with a frequency $f$ given by $f=\left({\frac {c}{c\pm v_{\text{s}}}}\right)f_{0}$

A similar analysis for a moving observer and a stationary source (in this case, the wavelength keeps constant, but due to the motion, the rate at which the observer receives waves and hence the transmission velocity of the wave [with respect to the observer] is changed) yields the observed frequency: $f=\left({\frac {c\pm v_{\text{r}}}{c}}\right)f_{0}$

Assuming a stationary observer and a wave source moving towards the observer at (or exceeding) the speed of the wave, the Doppler equation predicts an infinite (or negative) frequency as from the observer's perspective. Thus, the Doppler equation is inapplicable for such cases. If the wave is a sound wave and the sound source is moving faster than the speed of sound, the resulting shock wave creates a sonic boom.

Lord Rayleigh predicted the following effect in his classic book on sound: if the observer were moving from the (stationary) source at twice the speed of sound, a musical piece previously emitted by that source would be heard in correct tempo and pitch, but as if played backwards.^[9]

Applications[edit]

Acoustic Doppler current profiler[edit]

An acoustic Doppler current profiler (ADCP) is a hydroacoustic current meter similar to a sonar, used to measure water current velocities over a depth range using the Doppler effect of sound waves scattered back from particles within the water column. The term ADCP is a generic term for all acoustic current profilers, although the abbreviation originates from an instrument series introduced by RD Instruments in the 1980s. The working frequencies range of ADCPs range from 38 kHz to several Megahertz. The device used in the air for wind speed profiling using sound is known as SODAR and works with the same underlying principles.

Robotics[edit]

Dynamic real-time path planning in robotics to aid the movement of robots in a sophisticated environment with moving obstacles often take help of Doppler effect.^[10] Such applications are specially used for competitive robotics where the environment is constantly changing, such as robosoccer.

Inverse Doppler effect[edit]

Since 1968 scientists such as Victor Veselago have speculated about the possibility of an inverse Doppler effect. The size of the Doppler shift depends on the refractive index of the medium a wave is traveling through. Some materials are capable of negative refraction, which should lead to a Doppler shift that works in a direction opposite that of a conventional Doppler shift.^[24] The first experiment that detected this effect was conducted by Nigel Seddon and Trevor Bearpark in Bristol, United Kingdom in 2003.^{[p 6]} Later, the inverse Doppler effect was observed in some inhomogeneous materials, and predicted inside a Vavilov–Cherenkov cone.^[25]

Doppler, C. (1842). . Publisher: Abhandlungen der Königl. Böhm. Gesellschaft der Wissenschaften (V. Folge, Bd. 2, S. 465–482) [Proceedings of the Royal Bohemian Society of Sciences (Part V, Vol 2)]; Prague: 1842 (Reissued 1903). Some sources mention 1843 as year of publication because in that year the article was published in the Proceedings of the Bohemian Society of Sciences. Doppler himself referred to the publication as "Prag 1842 bei Borrosch und André", because in 1842 he had a preliminary edition printed that he distributed independently.

Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (About the coloured light of the binary stars and some other stars of the heavens)

"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.

David Nolte (2020). The fall and rise of the Doppler effect. Physics Today, v. 73, pgs. 31 - 35.

DOI: 10.1063/PT.3.4429

Adrian, Eleni (24 June 1995). . NCSA. Archived from the original on 12 May 2009. Retrieved 2008-07-13.