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SL IB Physics Your notes

Doppler Effect
Contents
The Doppler Effect
The Doppler Effect of Light
Galactic Redshift

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The Doppler Effect

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The Doppler Effect
When a source of sound, such as the whistle of a train or the siren of an ambulance, moves away from
an observer:
It appears to decrease in frequency, i.e. it sounds lower in pitch
The source of the sound however, remains at a constant frequency
This frequency change due to the relative motion between a source of sound or light and an observer is
known as the Doppler effect (or Doppler shift)
When the observer and the source of sound (e.g. ambulance siren) are both stationary:
The waves appear to remain at the same frequency for both the observer and the source
When the observer and the source of sound (e.g. ambulance siren) are moving relative to each other
The waves appear to have a different frequency for both the observer and the source

Stationary source and observer. The wavelength of the waves are the same for both observers
When the source starts to move towards the observer, the wavelength of the waves is shortened
The sound, therefore, appears at a higher frequency to the observer

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Your notes

Moving source at speed vs and stationary observer. The waves are closer together closer near the
stationary observer
Notice how the waves are closer together between the source and the observer compared to point P
and the source
This also works if the source is moving away from the observer
If the observer was at point P instead, they would hear the sound at a lower frequency due to the
wavelength of the waves broadening
The frequency is increased when the source is moving towards the observer
The frequency is decreased when the source is moving away from the observer
The same phenomena occurs for electromagnetic waves, such as light
Waves moving away from the observer are red-shifted
Their wavelengths shift to the red end of the electromagnetic spectrum
This is equivalent to sound waves appearing at a lower frequency to the observer
Waves moving towards the observer are blue-shifted
Their wavelengths shift to the blue end of the electromagnetic spectrum
This is equivalent to sound waves appearing at a higher frequency to the observer
This is because red light has a longer wavelength than blue light

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Your notes

Red shift and blue shift for electromagnetic waves

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Worked example
Your notes
A cyclist rides a bike ringing their bell past a stationary observer.
Which row correctly describes the Doppler shift caused by the sound of the bell?

Answer: D
If the cyclist is riding past the observer, the wavelength of sound waves are going to become
longer
This rules out options A and C
A longer wavelength means a lower frequency (from v = fλ )
Lower frequency creates a lower sound pitch
Therefore, the answer is row D

Examiner Tip
The relationship between frequency and wavelength is determined by the wave equation, which is
given in your data booklet. The speed v of the wave does not change.

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Representing The Doppler Effect

Wavefront diagrams help visualise the Doppler effect for moving wave sources and stationary Your notes
observers

Wavefronts are even in a stationary object but are squashed in the direction of the moving wave source
Δλ is the change in wavelength
The bigger the change, the bigger the doppler shift
A moving object will cause the wavelength, λ, (and frequency) of the waves to change:
The wavelength of the waves in front of the source decreases (λ – Δλ) and the frequency increases
The wavelength behind the source increases (λ + Δλ) and the frequency decreases
The Doppler shift is observed by all waves including sound and light

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The Doppler Effect of Light

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The Doppler Effect of Light
The Doppler shift for a light-emitting non-relativistic source can be described using the equation:
∆f ∆λ ∆v
= ≈
f λ c
Where:
Δf = change in frequency (Hz)
f = reference (original) frequency (Hz)
Δλ = change in wavelength (m)
λ = reference (original) wavelength (m)
Δv = relative velocity of the source and observer (m s–1)
c = the speed of light (m s–1)
The sign ≈ means 'approximately equals to'
This equation only works if v << c
The change in wavelength Δλ is equal to:
∆λ = λ − λ
0

Where:
λ0 = observed wavelength of the source (m)
Since the fractions have the same units on the numerator (top number) and denominator (bottom
number), the Doppler shift has no units
The relative speed between the source and observer along the line joining them is given by:
∆v = v − v
s o

Where:
vs = velocity of the source of the light (m s–1)
vo = velocity of the observer (m s–1)
Usually, we calculate the speed of the source of electromagnetic waves relative to an observer which
we assume to be stationary
Therefore vo = 0, hence ∆v = vs = v
Where v is the velocity at which the source of the electromagnetic waves is moving from the
observer
Hence, the Doppler shift equation can be written in terms of wavelength:
∆λ λ0 − λ v
= ≈
λ λ c
It can also be written in terms of frequency:

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∆f f0 − f v
= ≈
f f c Your notes
Spectral Lines
Doppler shift can easily be seen in atomic spectral lines from planets and stars

Spectral lines showing red shift

Each line represents an element making up the composition of the galaxy
The lines are identical to those measured in the lab and the light measured from the distant galaxy
Since the lines all move to the left (the red end of the spectrum) this means the galaxy is travelling away
from Earth

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Worked example
Your notes
A stationary source of light is found to have a spectral line of wavelength 438 nm. The same line from a
distant star that is moving away from us has a wavelength of 608 nm.
Calculate the speed at which the star is travelling away from Earth.
Answer:
Step 1: List the known quantities
Unshifted wavelength, λ = 438 nm
Shifted wavelength, λ0 = 608 nm
Change in wavelength, Δλ = (608 – 438) nm = 170 nm
Speed of light, c = 3.0 × 108 m s–1
Step 2: Write down the Doppler equation and rearrange for velocity v

∆λ v
=
λ c
c ∆λ
v=
λ
Step 3: Substitute values to calculate v

(3.0 × 108) × 170

v= = 1.16 × 108 m s–1
438

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Worked example
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The stars in a distant galaxy can be seen to orbit about a galactic centre. The galaxy can be
observed 'edge-on' from the Earth.
Light emitted from a star on the left-hand side of the galaxy is measured to have a wavelength of
656.44 nm. The same spectral line from a star on the right-hand side is measured to have a wavelength
of 656.12 nm.
The wavelength of the same spectral line measured on Earth is 656.28 nm.
(a) State and explain which side of the galaxy is moving towards the Earth.
(b) Calculate the rotational speed of the galaxy.

Answer:
(a)
The light from the right-hand side (656.12 nm) is observed to be at a shorter wavelength than the
reference line (656.28 nm)
Therefore, the right-hand side has been blue-shifted and must be moving towards the Earth
(b)
Step 1: List the known quantities

Observed wavelength on LHS, λ = 656.44 nm

LHS
Observed wavelength on RHS, λ = 656.12 nm
RHS
Reference wavelength, λ = 656.28 nm
Speed of light, c = 3.0 × 108 m s−1
Step 2: Calculate the average change in wavelength

λ LHS − λ RHS 656 . 44 − 656 . 12

∆λ = =
2 2
∆ λ = 0.32 nm
Step 3: Write down the Doppler equation and rearrange for velocity v

∆λ v
=
λ c
c ∆λ
v=
λ
Step 4: Substitute values into the velocity equation

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(3 × 108) × 0 . 32
v=
656 . 28 Your notes
Rotational speed: v = 1 . 46 × 105 m s −1 = 146 km s −1

Examiner Tip
You need to know that in the visible light spectrum red light has the longest wavelength and
the smallest frequency compared to blue light which has a shorter wavelength and higher
frequency.
The second worked example didn't change the wavelengths from nm into m, since it doesn't matter in
the equation as the units will cancel out.

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Galactic Redshift
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Galactic Redshift
In space, the Doppler effect of light can be observed when spectra of distant stars and galaxies are
observed, this is known as:
Redshift if the object is moving away from the Earth
The wavelength is increasing but the frequency is decreasing
Blueshift if the object is moving towards the Earth
The wavelength is decreasing but the frequency is increasing

Stars and galaxies can be red or blueshifted to an observer on Earth

Redshift is defined as:
The fractional increase in wavelength (or decrease in frequency) due to the source and
observer receding from each other
Redshift can be observed by comparing the light spectrum produced from a close object, such as our
Sun, with that of a distant galaxy
The light from the distant galaxy is shifted towards the red end of the spectrum (compared to the
Sun's spectra)
This provides evidence that the universe is expanding

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Your notes

Comparing the light spectrum produced by the Sun with light from a distant galaxy

Positive and Negative Velocities

If the speed of the galaxy relative to Earth is positive then the galaxy is moving towards the Earth
This is the case when the observed frequency f0 is greater than the reference frequency f
If the speed of the galaxy relative to Earth is negative then the galaxy is moving away from the Earth
This is the case when the observed frequency f0 is less than the reference frequency f

An Expanding Universe
After the discovery of Doppler redshift, astronomers began to realize that almost all the galaxies in the
universe are receding
This led to the idea that the space between the Earth and the galaxies must be expanding
This expansion stretches out the light waves as they travel through space, shifting them towards the
red end of the spectrum

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The expansion of the universe can be compared to dots on an inflating balloon

As the balloon is inflated, the dots all move away from each other
In the same way, as the rubber stretches when the balloon is inflated, space itself is stretching out Your notes
between galaxies
Just like the dots, the galaxies move away from each other, however, they themselves do not move
Another observation from looking at the light spectra produced by distant galaxies is that the greater
the distance to the galaxy, the greater the redshift
This means that the greater the degree of redshift, the faster the galaxy is moving away from Earth

The further a galaxy is from Earth, the greater its redshift tends to be

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The furthest galaxies appear to be redshifted the most and are receding the fastest

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Worked example
Your notes
The spectra below show dark absorption lines against a continuous visible spectrum.

A particle line in the spectrum of light from a source in the laboratory has a frequency of 4.570 × 1014
Hz. The same line in the spectrum of light from a distant galaxy has a frequency of 4.547 × 1014 Hz.
Calculate the speed of the distant galaxy in relation to the Earth. State whether it is moving towards or
away from the Earth.
Answer:
Step 1: Write down the known quantities
Observed frequency, f0 = 4.547 × 1014 Hz
Reference (source) frequency, f = 4.570 × 1014 Hz
Shift in frequency, Δf = observed − source = (4.547 – 4.570) × 1014 = –2.3 × 1012 Hz
Speed of light, c = 3.0 × 108 m s–1
Step 2: Write down the Doppler redshift equation

∆f f0 − f v
= ≈
f f c
Step 3: Rearrange for speed v, and calculate

c Δf (3 . 0 × 108) × ( − 2 . 3 × 1012 )
v= = = –1.5 × 106 m s–1
f 4 . 570 × 1014
Step 4: Write a concluding sentence

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The velocity is negative, so the source is moving away from the Earth
OR
The observed frequency is less than the source frequency, therefore, there is a decrease in Your notes
frequency, so the source is receding, or moving away, from the Earth at 1.5 × 106 m s–1

Examiner Tip
Keep travel of the minus signs in your calculation, as this gives you information about whether the
object is moving away or towards the observer.
The speed of light is given in your data booklet, you will not need to memorise this value.

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