## Summary

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.

## Get the basics

#### Definition

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.

#### Equation

For a moving source and stationary observer, the observed frequency can be calculated using the following equation:

\(f'=f({v\over v\pm u_s})\)

- \(f'\) is the observed frequency
- \(f\) is the frequency emitted by the source
- \(v\) is the speed of sound (ms
^{-1}) - \(u_s\) is the speed of the source (ms
^{-1}), where \(u_s>0\) for the source moving towards the observer

This equation emerges from the wave equation. Ahead of the source, the waves have been squashed into the distance \(vt-u_st\) (equal to the distance traveled by the sound less the distance caught up by the source):\(\lambda'={vt-u_st \over ft}={v-u_s\over f}\)

Substituting into the wave equation for the observed wave: \(f'={c\over \lambda'}=f({v\over v-u_s})\)

The converse is true for an observer behind the source.

Notice that, the greater the speed of the source, the more the observed frequency is changed. However, the speed of the source relative to an observer within the source itself is constant.

### Using the equation

#### Calculating velocity

We can use a determined values of the frequency of the sound emitted by a source car in the video above as it approaches and recedes and passes the observer to find the velocity of the source.

#### Measuring the Doppler effect

- Determine the frequency of the source by measuring the time for the formation of, for example, 10 waves: \(f={1\over T}\)
- Determine the frequency ahead of the source by measuring the time for 10 waves to pass a chosen point or line.
- Using the “set scale” tool set the width of the video window to 1m.
- Measure the velocity of the source and waves from the gradient of displacement-time graphs.
- Use the Doppler equation to calculate the change in frequency.

## Stretch for 7

### Moving observer

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 \(\lambda={v\over f}\):

\(f'={v+u_o\over \lambda}\Rightarrow f'=f{(v+u_o)\over v}\)

### Moving medium

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.

### Sonic boom

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.

### Electromagnetic waves

#### 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.