Doppler effect

 The Doppler effect is a phenomenon observed in waves, such as sound waves and light waves, where the frequency or wavelength of the wave appears to change when the source of the wave and the observer are in relative motion. This effect is named after the Austrian physicist Christian Doppler, who first described it 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]

The reason for the Doppler effect is that when the source of the waves is moving towards the observer, each successive wave crest is emitted from a position closer to the observer than the crest of the previous wave.^[4][5] Therefore, each wave takes slightly less time to reach the observer than the previous wave. Hence, the time between the arrivals of successive wave crests at the observer is reduced, causing an increase in the frequency. While they are traveling, the distance between successive wave fronts is reduced, so the waves "bunch together". Conversely, if the source of waves is moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the frequency. The distance between successive wave fronts is then increased, so the waves "spread out".

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 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, such as electromagnetic waves or gravitational waves, only the difference in velocity between the observer and the source needs to be considered. If this relative speed is not negligible compared to the speed of light, a more complicated relativistic Doppler effect arises.

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 ${\displaystyle �}$ and emitted frequency ${\displaystyle {�}_{\text{0}}}$ is given by:^[8]

$${\displaystyle �=\left(\frac{�\pm {�}_{\text{r}}}{�\pm {�}_{\text{s}}}\right){�}_0}$$

where

• ${\displaystyle �}$ is the propagation speed of waves in the medium;
• ${\displaystyle {�}_{\text{r}}}$ is the speed of the receiver relative to the medium, added to ${\displaystyle �}$ if the receiver is moving towards the source, subtracted if the receiver is moving away from the source;
• ${\displaystyle {�}_{\text{s}}}$ is the speed of the source relative to the medium, added to ${\displaystyle �}$ if the source is moving away from the receiver, subtracted if the source is moving towards the receiver.

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:

$${\displaystyle \frac{�}{{�}_{��}}=\frac{{�}_0}{{�}_{��}}=\frac{1}{�}}$$

where

• ${\displaystyle {�}_{��}}$ is the wave's speed relative to the receiver;
• ${\displaystyle {�}_{��}}$ is the wave's speed relative to the source;
• ${\displaystyle �}$ is the wavelength.

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 ${\displaystyle {�}_{\text{s}}}$ and ${\displaystyle {�}_{\text{r}}}$ are small compared to the speed of the wave, the relationship between observed frequency ${\displaystyle �}$ and emitted frequency ${\displaystyle {�}_{\text{0}}}$ is approximately^[8]

Observed frequency	Change in frequency
${\displaystyle �=\left(1+\frac{\mathrm{\Delta }�}{�}\right){�}_0}$	${\displaystyle \mathrm{\Delta }�=\frac{\mathrm{\Delta }�}{�}{�}_0}$

where

• ${\displaystyle \mathrm{\Delta }�=�-{�}_0}$
• ${\displaystyle \mathrm{\Delta }�=-({�}_{\text{r}}-{�}_{\text{s}})}$ is the opposite of the relative speed of the receiver with respect to the source: it is positive when the source and the receiver are moving towards each other.

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