# Doppler effect

The Doppler effect ( the Doppler Fizeau effect rarely ) the temporal compression and expansion of a signal with variations in the distance between transmitter and receiver for the duration of the signal. The cause is the change of the running time. This purely kinematic effect occurs for all signals with a certain speed, usually the speed of light or speed of sound spread. Spreads the signal in a medium of, its state of motion is taken into account.

For periodic signals increases or decreases the observed frequency. With the passing of an ambulance that affects both the pitch ( " iiiiuuuu ") and the alternating frequency of the siren. At low speeds relative to the speed of propagation of this ratio are also on the relative frequency change. In reflected signal, as in the radar Doppler and Doppler ultrasound, doubled with the running time and the Doppler shift.

- 3.1 Longitudinal Doppler Effect
- 3.2 Transverse Doppler effect
- 3.3 Doppler effect at any angle
- 3.4 Doppler effect and astronomical redshift

- 4.1 Radar Technology
- 4.2 Medical Diagnostics
- 4.3 Laser Doppler
- 4.4 Other Applications

## History

The Doppler effect has been known by Christian Doppler, who tried to convince them in the year 1842 astronomers that this effect is the cause for the more pronounced in double stars color differences ( in fact it is to temperature differences ). His thought experiment for the quantitative derivation of the term of generated by the minute water waves to a boat in variable distance is similar to the situation had the Doppler effect actually observed 180 years ago in the Ole Rømer and correctly interpreted. Rømer was interested in the suitability of the rapidly rotating moons of Jupiter as a clock signal to solve the longitude problem. He found that the observed time -shifted by up to about ± 10 minutes, depending on the position of the Earth to Jupiter. Merit of Doppler is the transmission of the effect on the wavelength of light. In the French -speaking world, this is often attributed to Armand Fizeau (1848 ). Both, however, did not provide experiments on it.

A quantitative experiment on the Doppler effect with sound waves was carried out in 1845 by the physicist Christoph Buys- Ballot. He posted to several trumpeter both on a moving railroad train and off the track. On passing each one of them should play a G and determine the other belonged pitch. There was a shift of one semitone, corresponding to a speed of 70 km / h

Twenty years later, William Huggins found the predicted spectroscopic Doppler shift in the light from stars. He showed that Sirius steadily away from us.

Another century later, the accuracy of the astronomical unit of 10-4 ( from the horizontal parallax of Eros ) was by radar measurements between Earth and Venus improved to 10-6 initially based on distance measurements in the lower conjunctions of 1959 and 1961 (eg at JPL by amplitude modulation with up to 32 Hz ), then 10-8 by Doppler measurements on the carrier frequencies for several months before and after the lower conjunctions of 1964 and 1966. results were as 300 years previously specified as runtime, since the value of the speed of light was then known only to six digits.

For proof of the perihelion of Mercury ranged Doppler measurements of the years 1964-1966 - with optical methods were needed and a half centuries.

## Details of acoustic Doppler effect

In the explanation of the acoustic Doppler effect is to distinguish whether the sound source, the observer, or both relative to the medium ( the still air ) move.

### Moving observer at rest, signal source

As an example, assume that the siren of the ambulance emits sound waves with a frequency of 1000 Hz. This means that exactly 1/1000 of a second after the first wave crest follows a second wave crest. The waves propagate at the speed of sound at 20 ° C.

As long as the ambulance is, the wavelength of the sound, ie the distance of the wave peaks:

For an observer on the street these peaks are indeed depending on the distance to a slight delay. However, the time between two wave peaks does not change. The fundamental frequency of the perceived sound is the same for each distance from the observer and ambulance.

The situation changes when the ambulance zufährt with the speed to the observer. As the car moved in the time between the two wave peaks, the distance between them shortened a bit. They shortened by the way traveled by the car during the period of 1 /1000 second:

The indices and refer to the sender or observer of the shaft. Since both the wave crests move at the same speed of sound to the observer, the shortened distance between them is preserved, and the second wave crest comes not only 1/1000 second to after the first, but a little earlier. This gives the observer the frequency (the pitch) of sirene higher () appears.

Quantitatively simply obtains the frequency change by substituting the relationship into the above formula for. For the perceived by the observer frequency, this results in:

The frequency of the sound source, the propagation velocity of the sound and the speed of the sound source (ie the ambulance ) mean.

When the ambulance drove past the observer, it behaves in reverse order: the distance between the wave crests (wavelength ) increases, and the observer hears a lower tone. Mathematically, the above formula applies as well, you just have to work for a negative velocity.

The operations of the signal source directly to the observer or directly away from him are special cases. The signal source is moving anywhere in space at the speed so the Doppler shift for a stationary receiver

Be specified. is the time-dependent unit vector of the direction of the signal source S describes the observer B.

### Observer moves, the signal source at rest

Even with static sound source S and the moving observer B occurs a Doppler effect, but here is the cause of another: When the car rests, also does not change the distance between the peaks, ie, the wavelength remains the same. However, the peaks seem to come quickly one after the other at the observer when this moves towards the ambulance:

Again, there is again the case of a receding observer by inserting a negative speed.

For an arbitrary motion of the observer B with the velocity vector of the Doppler effect results in a fixed position transmitter S to

Wherein again the unit vector describing the direction of the source S is the observer B, which in the general case, as the velocity vector can be time-dependent.

As you can see, the equations (1) and (2) are not identical (only in the limit they approach each other ). Obviously, this is the extreme case: the observer moves with the speed of sound to the signal source to reach it, the wave crests twice as fast, and he hears a tone double frequency. Moves, however, the signal source with the speed of sound is the distance between the wave crests virtually zero, they overlap and there is an extreme compression of the air (see sound barrier breakdown ). Since then all wave crests arrive simultaneously at the observer, the above formula would be theoretically an infinite frequency - virtually any sound you can hear a certain frequency, but the sonic boom.

### Moving observer and source

By combining the equations (1) and (2) can be derived an equation which describes the perceived frequency of the observer, when the transmitter and the receiver are in motion.

Transmitter and receiver are approaching each other:

Transmitter and receiver are moving away from each other:

Here, the speed of the observer and the speed of the sound waves of the transmitter relative to the medium.

### General Doppler law for sound sources

In general, the frequency difference can be written as:

Here, the speed of the observer and the sender of the sound waves, respectively relative to the medium (eg air). The upper sign is valid for operation approach ( moving in the direction of the transmitter or receiver). That is, both speeds are measured positive in the direction of the observer or transmitter. With or the above-mentioned special cases arise. Next we see that the effect is canceled ( ie there are no pitch change ), though. This corresponds to the case when the transmitter and receiver both move in the same direction at the same speed relative to the medium. Usually occurs as a case when the medium is moving even while the transmitter and receiver rest (wind). Therefore, it is independent of the wind speed at no Doppler effect.

One of the formulas is not to say that they were derived under the assumption that source and observer move directly to each other. In real cases, the ambulance drives past in a certain minimum distance to the observer. This has the consequence that the distance between source and observer does not change uniformly, and therefore - is a continuous transition in pitch from higher to lower to hear - especially immediately before and after the passing.

## Doppler effect without medium

Electromagnetic waves propagate in a vacuum, without the media. When the waves of the transmitter is moved relative to the receiver, in this case a displacement of the frequency occurs. This relativistic Doppler effect is due to the fact that the waves propagate with finite velocity, namely the speed of light. One can interpret it as a geometric effect of spacetime.

### Longitudinal Doppler effect

Wherein electromagnetic waves in vacuum ( optical Doppler effect ), there is no medium, so the observed frequency change depends only on the relative velocity of the source and observer; whether fueling the source, the observer, or both, has no influence on the amount of frequency change.

Due to the principle of relativity, each observer should be regarded as dormant. However, it must be in the calculation of the Doppler effect, in addition to the above considerations also dilation of the observer moving relative to the source account. Thus we obtain for the relativistic Doppler effect:

V> 0 for reducing the distance between source and observer.

### Transverse Doppler effect

If an object moves at a certain time transversely to the observer, it is possible to neglect the variation in the distance at that time; Accordingly, one would expect here also no Doppler effect. However, the theory of relativity states that any object due to its motion is subject to a time dilation, because of the frequency is also reduced. This effect is known as the transverse Doppler effect. The formula for this is

In which case the vacuum speed of light and the speed of the signal source respectively.

The transverse Doppler effect can in non- relativistic speeds ( speeds so far below the speed of light), however, be disregarded.

### Doppler effect at any angle

The Doppler effect is generally dependent on the angle of the direction of movement to the axis of source - recipient. The frequency change for any angle α is given by

When used for the angle α of 0 °, 90 ° or 180 °, then the above equations are obtained for the longitudinal and transverse Doppler effect. It can be seen also that the angle at which the Doppler effect disappears is dependent on the relative velocity, unlike the Doppler effect for sound, where he is always 90 °.

### Doppler effect and astronomical redshift

Even if the observed effects of the Doppler effect and are identical astronomical red shift ( reduction in the observed frequency of the electromagnetic radiation of a star or a galaxy ), so both can still not be confused, since they have completely different causes. The relativistic Doppler effect is only main cause of the change in frequency when the transmitter and receiver as described above to move through space-time and their distance is so small that the expansion of the space lying between them in the ratio is low. From a certain distance far outweighs that part that is caused by the expansion of space-time itself, so that the proportion of the discussion here Doppler effect can be neglected altogether.

## Applications

Doppler effect occurs in the echo of the transmitted acoustic and electromagnetic signals.

### Radar technology

When Doppler radar to calculate the speed of approach of an object from the measured change in frequency between the transmitted and reflected signal. The special feature of an active radar apparatus is that the Doppler effect occurs twice. The first time in analogy to the case of signal source at rest observer moves on the transmit path from the radar to the moving object: a radar warning receiver in this object would measure a simple Doppler frequency proportional to the radial velocity. Reflecting the transmitted frequency plus the frequency Doppler, the reflecting object can be considered as a signal source for a frequency equal to the original transmitted frequency plus the Doppler frequency. This signal is now subject to the terms analogous to the case of signal source is moving, observers at rest. In this way, the Doppler shift occurs again, the receiver in the radar device registers twice the Doppler frequency.

- In meteorology, the Doppler radar for the determination of rotational motion in supercells ( tornadoes ) will be used.
- The military and air traffic control use of the Doppler effect, among others, the passive radar and the fixed target suppression.
- Also for speed detection in so-called speed traps on the road, a Doppler radar is used.
- A synthetic aperture radar is mainly based on the assignment of signals through the course of changing its Doppler shift.

### Medical Diagnostics

In the medicine of the acoustic Doppler effect is used with ultrasound imaging to display the blood flow velocity and to measure. He has proven to be extremely useful. There is this one:

- Color Doppler: Red: river to the sound probe to
- Blue: flow from the sonic probe away

### Laser Doppler

For non-contact measurement of the velocity distribution of fluids (liquids and gases) the Laser Doppler Anemometry (LDA) is used. Another application, the laser Doppler vibrometer (LDV ), relates to the measurement of the vibration velocity of surfaces. Here caused by the surface movement frequency shift of a laser beam reflected at the measuring point for determining the oscillation speed is used in this measuring point.

### Other Applications

- For water waves ( gravity waves ), the carrier medium a constant flow rate is subject, see wave transformation.
- The now disconnected satellite navigation system Transit used the Doppler effect for position determination. It is used actively in Argos, a satellite-based positioning system. In modern GNSS satellites, the Doppler effect is first of disturbing. He forces the receiver to scan a wider frequency range. On the other hand, can be obtained from the frequency shift additional information and so accelerate the coarse positioning. This procedure is called Doppler Aiding. See also: Doppler satellite.
- In music, the Doppler effect is used to produce sound effects, such as the rotating speakers of a Leslie cabinet.
- In the Mössbauer spectroscopy of the Doppler effect of moving gamma-ray source is used to change the energy of the photons of the source is minimal. Thereby able to interact with the nuclear hyperfine levels of a corresponding absorber these photons.

## Example

A stationary observer hears a sound source is moving exactly on him f'zu with the frequency (v / c), see equation ( 1) when it moves away from him, with the frequency f'weg (v / c ), see equation (2). With sound sources of the relativistic transverse Doppler effect does not matter. The further the observer is away from the linear trajectory, the slower the radial velocity component will change during approach. The speed of the frequency change depends on the shortest distance between the observer and the source. The chart on the right shows the frequency dependence relative to a stationary observer at the origin. The red line corresponds to the frequency he hears when it happens, the signal source at a large distance, the blue at close range. Maximum and minimum frequencies are not symmetrical to the natural frequency, because the velocity V is not very much smaller than the speed of sound c. We have the relations (1) and (2).

If the coordinates of the moving signal source is known, can be seen from the frequency response of their own location can be derived ( see, eg, Transit ( satellite system ) ).

The sound samples give the pitches that hears a resting observer when a source flies past him. They neglect the effect that the receding source is heard longer than the Approaching to:

Increases the relative speed, move the frequencies: