The Doppler effect (also Doppler shift) is the change in the frequency or, equivalently, the period of a wave in relation to an observer who is moving relative to the source of the wave. It 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 approaches and recedes from an observer. Compared to the emitted sound, the received sound has a higher pitch during the approach, identical at the instant of passing by, and lower pitch during the recession. 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. Hence, from the observer's perspective, the period or 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 period or time between successive cycles is increased, thus reducing the frequency. For waves propagating in vacuum, as is possible for electromagnetic waves or gravitational waves, only the relative velocity between the observer and the source needs to be considered. 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 in such cases may therefore result from motion of the source, motion of the observer, motion of the medium, or any combination thereof.
History
Doppler first proposed this 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). 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 independently discovered the same phenomenon on electromagnetic waves in 1848. In France, the effect is sometimes called "effet Doppler-Fizeau" but that name was not adopted by the rest of the world as Fizeau's discovery was six years after Doppler's proposal. In Britain, John Scott Russell made an experimental study of the Doppler effect (1848).
General The relationship between observed frequency f {\displaystyle f} and emitting frequency f 0 {\displaystyle f_{\text{0}}} of a wave propagating through a medium is given by:
f = ( v m ± v r v m ∓ v s ) f 0 {\displaystyle f=\left({\frac {v_{\text{m}}\pm v_{\text{r}}}{v_{\text{m}}\mp v_{\text{s}}}}\right)f_{0}}
where
v m {\displaystyle v_{\text{m}}} is the propagation speed of the wave in the medium;
v r {\displaystyle v_{\text{r}}} is the speed of the wave receiver relative to the medium. In the formula, v r {\displaystyle v_{\text{r}}} is added to v m {\displaystyle v_{\text{m}}} if the receiver is moving towards the source, and subtracted if moving away;
v s {\displaystyle v_{\text{s}}} is the speed of the wave source relative to the medium. In the formula, v s {\displaystyle v_{\text{s}}} is subtracted from v m {\displaystyle v_{\text{m}}} if the source is moving towards the receiver, and added if moving away.
v m {\displaystyle v_{\text{m}}} , v r {\displaystyle v_{\text{r}}} , and v s {\displaystyle v_{\text{s}}} here are not vectors as velocities, but their magnitudes as speeds. This relationship predicts that the observed frequency by the receiver will decrease if the distance between the source and receiver is increasing. Note that the speed of the wave is determined by the medium, not by the speed of the source. 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.
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