Doppler effect

General

The relationship between observed frequency ${\displaystyle f}$ and emitting frequency ${\displaystyle f_{\text{0}}}$ of a wave propagating through a medium is given by: $${\displaystyle f=\left(\frac{v_{\text{m}}\pm v_{\text{r}}}{v_{\text{m}}\mp v_{\text{s}}}\right)f_0}$$ where

• ${\displaystyle v_{\text{m}}}$ is the propagation speed of the wave in the medium;
• ${\displaystyle v_{\text{r}}}$ is the speed of the wave receiver relative to the medium. In the formula, ${\displaystyle v_{\text{r}}}$ is added to ${\displaystyle v_{\text{m}}}$ if the receiver is moving towards the source, and subtracted if moving away;
• ${\displaystyle v_{\text{s}}}$ is the speed of the wave source relative to the medium. In the formula, ${\displaystyle v_{\text{s}}}$ is subtracted from ${\displaystyle v_{\text{m}}}$ if the source is moving towards the receiver, and added if moving away.

${\displaystyle v_{\text{m}}}$, ${\displaystyle v_{\text{r}}}$, and ${\displaystyle v_{\text{s}}}$ 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.

Consequences

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.

Applications

Sirens

A 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:

The reason the siren slides is because it doesn't hit you.

In other words, if the siren approached the observer directly, the pitch would remain constant, at a higher than stationary pitch, until the vehicle hit them, and then immediately jump to a new lower pitch. Because the vehicle passes by the observer, the radial speed does not remain constant, but instead varies as a function of the angle between his line of sight and the siren's velocity: $${\displaystyle v_{\text{radial}}=v_{\text{s}}\cos (\theta )}$$ where ${\displaystyle \theta }$ 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 widespread use in astronomy to measure the speed at which stars and galaxies are approaching or receding from us, resulting in so called blueshift or redshift, respectively. This may be used to detect if an apparently single star is, in reality, a close binary, to measure the rotational speed of stars and galaxies, or to detect exoplanets. This effect typically happens on a very small scale; there would not be a noticeable difference in visible light to the unaided eye. The use of the Doppler effect in astronomy depends on knowledge of precise frequencies of discrete lines in the spectra of stars.

Among the nearby stars, the largest radial velocities with respect to the Sun are +308 km/s (BD-15°4041, 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 speed means the star is receding from the Sun, negative that it is approaching.

The relationship between the expansion of the universe and the Doppler effect is not simply caused by the source moving away from the observer. In cosmology, the redshift of expansion is considered separate from redshifts due to gravity or Doppler motion.

Distant galaxies also exhibit peculiar motion distinct from their cosmological recession speeds. If redshifts are used to determine distances in accordance with Hubble's law, then these peculiar motions give rise to redshift-space distortions.

Radar

The Doppler effect is used in some types of radar to measure the velocity of detected objects. A radar beam is fired at a moving target – e.g. a motor car, as police use radar to detect speeding motorists – as it approaches to or recedes from the radar source. In the case of a car moving away from the source, each successive radar wave has to travel farther to reach the car, before being reflected and re-detected near the source. As each wave has to move farther, the gaps between the wave crests increase, increasing the wavelength of the radiation returning to the radar. In the opposite case, when the radar beam is fired at the moving car as it approaches, each successive wave travels a lesser distance, decreasing the wavelength. In either situation, calculations from the Doppler effect accurately determine the car's speed. For instance, a police radar operating at 24.15 GHz (K-band) detecting a vehicle traveling at 30 m/s (108 km/h) will measure a Doppler shift of approximately 4.83 kHz, a change easily detected by modern digital signal processing. Moreover, the proximity fuze, developed during World War II, relies upon Doppler radar to detonate explosives at the correct time, height, distance, etc.

Bats use echolocation in a similar way to locate moths. The Doppler shift affects the frequency of the wave incident upon the target (moth). When the wave is reflected back from the moth to the bat, the moth acts as the wave emitter and the bat as the wave receiver. The frequency of the reflected wave is again Doppler-shifted. A bat, emitting a wave at the frequency ${\displaystyle f}$ and flying at ${\displaystyle v_{\text{b}}}$ towards a moth flying at ${\displaystyle v_{\text{t}}}$ will detect a final reflected wave with a frequency: $${\displaystyle f_{\mathrm{b}\mathrm{d}}=f\left(\frac{v_{\text{m}}-v_{\text{t}}}{v_{\text{m}}-v_{\text{b}}}\right)\left(\frac{v_{\text{m}}+v_{\text{b}}}{v_{\text{m}}+v_{\text{t}}}\right)}$$

Medical

An echocardiogram can, within certain limits, produce an 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, abnormal communications between the left and right side of the heart, 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), Acoustic Doppler current profiler (ADCP), and acoustic Doppler velocimeter (ADV) have been developed to measure velocities in a fluid flow. The LDV emits a light beam, and the ADCP and 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 phase. 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 flow), 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.

Satellites

Satellite navigation

The Doppler shift can be exploited for satellite navigation such as in Transit and DORIS.

Satellite communication

Doppler also needs to be compensated in satellite communication. Fast moving satellites can have a Doppler shift of dozens of kilohertz relative to a ground station. The speed, thus magnitude of Doppler effect, changes due to earth curvature. Dynamic Doppler compensation, where the frequency of a signal is changed progressively during transmission, is used so the satellite receives a constant frequency signal. After realizing that the Doppler shift had not been considered before launch of the Huygens probe of the 2005 Cassini–Huygens mission, the probe trajectory was altered to approach Titan in such a way that its transmissions traveled perpendicular to its direction of motion relative to Cassini, greatly reducing the Doppler shift.

Doppler shift of the direct path can be estimated by the following formula: $${\displaystyle f_{\mathrm{D},\mathrm{d}\mathrm{i}\mathrm{r}}=\frac{v_{\mathrm{m}\mathrm{o}\mathrm{b}}}{{\lambda }_{\mathrm{c}}}\cos \varphi \cos \theta }$$ where ${\displaystyle v_{\text{mob}}}$ is the speed of the mobile station, ${\displaystyle {\lambda }_{\mathrm{c}}}$ is the wavelength of the carrier, ${\displaystyle \varphi }$ is the elevation angle of the satellite and ${\displaystyle \theta }$ is the driving direction with respect to the satellite.

The additional Doppler shift due to the satellite moving can be described as: $${\displaystyle f_{\mathrm{D},\mathrm{s}\mathrm{a}\mathrm{t}}=\frac{v_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{s}\mathrm{a}\mathrm{t}}}{{\lambda }_{\mathrm{c}}}}$$ where ${\displaystyle v_{\mathrm{r}\mathrm{e}\mathrm{l},\mathrm{s}\mathrm{a}\mathrm{t}}}$ is the relative speed of the satellite.

Audio

The Leslie speaker, most commonly associated with and predominantly used with the famous Hammond organ, takes advantage of the Doppler effect by using an electric motor to rotate an acoustic horn around a loudspeaker, sending its sound in a circle. This results at the listener's ear in rapidly fluctuating frequencies of a keyboard note.

Vibration measurement

A laser Doppler vibrometer (LDV) is a non-contact instrument for measuring vibration. The laser beam from the LDV is directed at the surface of interest, and the vibration amplitude and frequency are extracted from the Doppler shift of the laser beam frequency due to the motion of the surface.

Robotics

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. Such applications are specially used for competitive robotics where the environment is constantly changing, such as robosoccer.

Inverse Doppler effect

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. The first experiment that detected this effect was conducted by Nigel Seddon and Trevor Bearpark in Bristol, United Kingdom in 2003. Later, the inverse Doppler effect was observed in some inhomogeneous materials, and predicted inside a Vavilov–Cherenkov cone.

Primary sources