The Doppler effect is the first wave phenomenon that we consider in this Additional Higher Level topic. It is a change in frequency in observed sound caused by relative motion between the source, the observer or the medium. Here we will learn how to calculate these changes in frequency according to the type of relative motion, explain the impact of a sonic boom and discuss the equivalent effect for light.
Key Concepts
The Doppler effect is the change in frequency in observed sound due to relative motion between source and observer.
This is often observed when a car drives past sounding its horn. The frequency is increased when the car approaches because the car catches up with the waves, causing them to be squashed. The result is a reduction in wavelength, which leads to a higher frequency as a result of the wave equation:
\(f={c\over \lambda}\)
This diagram shows how you might sketch the Doppler effect.
Doppler shift is also experienced when an observer moves towards or away from a source that is stationary relative to the medium. This effect is simply due to the relative velocity between the observer and source.
The equation to calculate the observed frequency is amended as follows:
\(f'=f({v\pm u_o \over v})\)
- \(f'\) is the observed frequency
- \(f_0\) is the frequency emitted by the source
- \(v\) is the speed of sound (ms-1)
- \(u_o\) is the speed of the observer (ms-1), where \(u_o>0\) for the source moving towards the observer
This equation comes from use of the relative velocity of the sound as it approaches the observer, now \(v+u_o\) and using \(\lambda={v\over f}\):
\(f'={v+u_o\over \lambda}\Rightarrow f'=f{(v+u_o)\over v}\)
Sound travels through media containing particles. The medium may too have a speed.
This diagram shows the effect when a steady drip of water falls from a bridge into the river below.
A sonic boom is generated when the speed of the source exceeds the speed of sound, creating a geometrical cone behind the object. This loud *BANG* from the shock waves can awaken sleeping people nearby and damage delicate objects.
Sonic booms can more easily be achieved by aircraft at height. The speed of sound decreases as the atmosphere becomes less dense, so the speed required to exceed the speed of sound also decreases.
Equation
We must use an approximate equation for all electromagnetic waves, in which we assume that the speed of the source does not approach the speed of light. Thus, there are no relativistic effects:
\({\Delta f \over f}={\Delta \lambda \over \lambda} \approx{v\over c}\)
- \(\Delta f\) is the change in observed frequency (Hz)
- \(f\) is the frequency emitted by the source (Hz)
- \(\Delta \lambda\) is the change in the oberved wavelength (m)
- \(\lambda\) is the wavelength emitted by the source (m)
- \(v\) is the speed of the source (ms-1)
- \(c\) is the speed of light (ms-1)
For a source moving away from the observer, the observed wavelength is higher than that at source, but the observed frequency is lower than that at source. A galaxy emitting visible light will have this red-shifted if it moves away from the observed. Red shift is evidence for the expansion of the universe, as the light from all distant galaxies is red-shifted. Thus, all galaxies are receding and so the universe must be expanding in all directions.
Uses
The Doppler effect in electromagnetic waves can also also be used for more 'earthly' means.
A Doppler radar produces velocity data about objects at a distance. Speed cameras are a good example. They emit a beam of radiation (microwaves or infrared) at an approaching vehicle where it is reflected. The Doppler effect occurs twice: once as the observer approaches the radiation and again as the reflection is emitted from the new moving source. The speed of the vehicle can be calculated from the change in frequency. Doppler radars can also be found in aviation, sounding satellites, Major League Baseball, meteorology, radar guns, and radiology and medicine.
In meteorology, the direction, speed and type of objects of precipitation may be detected. Storms may be analysed in this way to assess structure and likely severity.
In medicine, the direction and speed of blood flow in arteries and veins are determined. This technique is used in echocardiograms and medical ultrasonography and is an effective tool in diagnosis of vascular problems.
Use quizzes to practise application of theory.
START QUIZ!
The image is a still from a GeoGebra simulation.
The wavelength of the wave in front of the source is:
There are 16 waves produced in 4s. These fit in the space between the source and the first wavefront.
The wave has progressed 3 x 4 = 12 m
The source has moved 2 x 4 = 8 m
The difference between the source and wavefront distance = 4 m
\(\lambda'={vt-u_st \over ft}={4\over 16}=0.25\text{ m}\)
The image is a still from a GeoGebra simulation.
The wavelength of the wave behind the source is:
There are 16 waves produced in 4s. These fit in the space between the source and the first wavefront.
The wave has progressed -3 x 4 = -12 m
The source has moved 2 x 4 = 8 m
The difference between the source and wavefront distance = 20 m
\(\lambda'={vt-u_st \over ft}={20\over 16}=1.25\text{ m}\)
The image is a still from a GeoGebra simulation.
The frequency of the wave behind the source is:
For a stationary observer behind a moving source:
\(f' =f{v\over (v+u_s)} = 4 \times {3\over 5} =2.4 \text{ Hz} \)
The image is a still from a GeoGebra simulation.
The frequency of the wave in front of the source is:
For a stationary observer in front of a moving source:
\(f' =f{v\over (v-u_s)} = 4 \times {3\over 1} =12 \text{ Hz} \)
A student tries to fool his teacher by changing the pitch of his whistle as he cycles past. The trick worked and his teacher heard no Doppler shift.
The student whistled with...
The Doppler shift increases the pitch on approach and decreases the pitch on recession. The student compensated by whisting a lower pitch on approach and higher pitch on recession.
Two observers listen to the whistle of an approaching train.
Compared to A, the observer at B will hear a sound that is...
B is further away so the sound will be less loud as its energy is dissipated.
The component of the train's velocity towards B is less than towards A so the Doppler shift will have a reduced effect.
Two observers listen to the whistle of an approaching train.
As the train passes the pitch of the whistle drops. Compared to A the drop in pitch will... and the midpoint will take place...
The velocity at which the train approaches B decreases as the train gets nearer so the Doppler shift gradually reduces - this makes it last longer.
It takes more time for the sound to reach B as it is a greater distance from the track and so the drop in pitch will come later.
The image is a still from a GeoGebra simulation.
The frequency ahead of the source is 25 Hz. Calculate the velocity of the source.
\(f' =f{v\over (v-u_s)} \Rightarrow v-u_s={f\over f'}v \)
\(u_s=v(1-{f\over f'})=5(1-{10\over 25})=3 \text{ ms}^{-1}\)
How much of Doppler effect have you understood?
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