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WAVE MOTION

 

The Doppler Effect

 

description

derivation of frequency change

applications to astronomy

speed traps

plasma temp

 

 

 

Description

The effect is caused by the relative motion of an observer and a source of waves.

The observed frequency(from the observer's viewpoint) is different from the actual frequency. The actual frequency is the frequency emitted from the source.

 

 

Derivation of frequency change

To aid understanding, the derivation is best broken down into a number of sections:

 

(derivation sub-menu)

 

observed wavelength - static observer infront of moving source

 

observed wavelength - static observer behind moving source

 

observed frequency - moving observer forward of source, moving towards it

 

observed frequency - moving observer behind source, moving towards it

 

 

original conditions - no observer

 

Consider a stationary source of waves S.

 

A wave is emitted and one period later another wave is just about to be produced.

 

The separation of successive crests is the wavelength λ .

 

 

From the Wave Equation,

 

 

doppler effect - equation #00

 

 

doppler effect - diagram #1

 

 

 

observed wavelength - static observer at B, infront of moving source

 

 

 

Doppler effect - diagram #2

 

 

The source moves towards B at velocity vs .

 

In the time span of one period(T secs.) the source moves towards B a distance vsT

(distance = velocity x time) to Sv.

 

In this time, the first wave has moved from S to B.

 

The source is considered to carry the second wave, which is on the point of being emitted.

 

The distance of the crest of the 1st wave at B, measured from S is dW1 .

 

The distance of the crest of the 2nd wave at Sv , measured from S is dW2 .

 

doppler effect derivation -equation #1

 

So the distance between crests is given by:

 

doppler effect derivation -equation #2          (i     

 

Remembering that,

 

doppler effect derivation -equation #3

 

we can substitute for T into (i

 

doppler effect derivation -equation #4

 

 

If λF is the wavelength forward of the crests, (the distance between the crests)

 

then,

doppler effect derivation -equation #5

 

return to 'derivation' sub menu

 

 

observed wavelength - static observer at A, behind moving source

 

 

 

Doppler effect - diagram #2

 

 

This is the same diagram as for the observer infront of the source.

 

But look again, this time from the left at A, looking towards the moving source.

 

This time we consider the distance between the wave at A and the wave about to be emitted at Sv .

Here the distance moved by the first wave from S and the distance moved by the source are added.

 

 

doppler effect derivation -equation #6

 

doppler effect derivation -equation #7

 

doppler effect derivation -equation #8

 

 

observed frequency - moving observer forward of source, moving towards it

 

 

Doppler effect - diagram #3

 

 

vO is the velocity of an observer moving towards the source.

 

This velocity is independent of the motion of the source.

 

Hence, the velocity of waves relative to the observer is c + vO .

 

The wavelength observed forward of the source is therefore given by:

 

doppler effect derivation -equation #9

 

Recalling the equation for 'a static observer infront of a moving source':

 

doppler effect derivation -equation #10

 

equating these two,

 

doppler effect derivation -equation #11

 

to make fF (frequency forward of the source) the subject,

 

doppler effect derivation -equation #12

 

 

return to 'derivation' sub menu

 

 

observed frequency - moving observer behind source, moving towards it

 

 

Doppler effect - diagram #4

 

 

Remembering that vO is independent of the motion of the source, the wavelength observed forward of the source is given by:

 

doppler effect derivation -equation #13

 

Recalling the equation for 'a static observer behind a moving source':

 

doppler effect derivation -equation #14

 

equating these two,

doppler effect derivation -equation #15

 

to make fB (frequency behind the source) the subject,

 

doppler effect derivation -equation #16

 

 

return to 'derivation' sub menu

 

 

 

important conclusions:

 

1. For an observer moving away from the source, the value of vs is negative.

 

2. The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time.

 

3. When the source is stationary(vs= 0), fA= fB . So it makes no difference whether the observer is infront or behind the source.

 

 

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Application in astronomy

 

A star or galaxy having a red shift means that the celestial object is moving away from us. Light rays become more spread out.

 

A blue shift indicates the opposite. The object is moving towards us. Light rays become bunched up.

 

 

Doppler effect - red & blue shifts

 

 

The effects are descriptions of how emission and absorption patterns in spectra are moved either towards the blue or the red end of the spectrum.

 

The movement is apparent when spectra are compared with spectra produced in the lab.

 

 

Doppler effect - red & blue shift  spectra

 

 

The rotation of celestial objects (eg the Sun, Saturn's rings) can be measured from the light moving towards and away from us at the limb.

 

 

Doppler effect - red & blue shift as a result of rotation

 

 

Similarly, double stars are detected by their star-light containing both red shifted and blue shifted lines.

 

This is a result of one star approaching the observer, while the other is receding

(a consequence of both stars revolving around a common centre).

 

 

Doppler effect - red & blue shift for a binary star

 

 

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Radar speed traps

 

The speed of an approaching car, or one speeding away, can be found from the change in frequency of microwaves reflected from it.

 

The following relation is used to calculate vcar the velocity of an approaching/receding car.

 

Doppler efect - equation #17

 

where,

 

c is the speed of light
f is the frequency of the microwaves
fbeat is the beat frequency produced by interference between the original and reflected waves

 

 

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Plasma Temperature

 

When the spectrum of a hot plasma is examined, the spectral lines are observed to broaden with increased temperature.

 

This is because atoms emitting light are moving away from an observer and at the same time coming towards him/her.

 

Plasmas are extremely hot gases with temperatures in excess of 106 deg. celsius.

 

The light emitted from an excited atom dropping to a particular energy state would normally be of one discrete wavelength.

 

However, the action of the Doppler effect means that wavelengths of slightly longer and shorter wavelengths are emitted. Hence the spectral line is broadened by the extra wavelengths being produced.

 

Spectral line broadening is proportional to the square root of the absolute temperature.

 

So by measuring the broadening of lines in a star's spectrum its surface temperature can be found.

 

 

 

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