C.5Doppler effect + HL extra

An ambulance siren sounds higher as it races towards you and lower as it moves away. This change in frequency, caused by motion, is the Doppler effect. The same idea lets doctors measure blood flow, police measure your speed, and astronomers discover that the universe is expanding. Sections 1–6 are for everyone. Sections 7–8 (the equations for sound) are HL only and clearly marked.

Knowledge and science

Nature of science

HypothesesExperimentsEvidenceGlobal impact of science

Hypotheses. In 1842 Christian Doppler predicted that the motion of a source would change the frequency of its waves. He hoped this would explain the colours of double stars. That example turned out to be wrong, because stars don't move nearly fast enough, but the effect itself was real.

Experiments. In 1845 Christophorus Buys Ballot tested the idea for sound. He put trumpeters on an open train carriage near Utrecht and asked musicians with good pitch on the platform to name the notes they heard as the train went past.

Evidence. In the 1920s Edwin Hubble and others found that the light from almost every distant galaxy is redshifted, and more so for more distant galaxies. This is key evidence that the universe is expanding.

Global impact. Doppler radar tracks storms and aircraft, and Doppler ultrasound shows blood flow without surgery.

ToK: questions to think about

How far can we trust our senses? We can hear the Doppler effect for sound, but nobody can see a galaxy's redshift with their eyes. How does knowledge based on instruments compare with knowledge based on direct experience?
Can a theory be right for the wrong reasons? Doppler's example of coloured double stars was wrong, but his principle was correct. How should we judge a scientist whose idea is right but whose evidence is not?
What does "moving" mean? For sound, it matters whether the source or the observer moves through the air. For light, only the relative motion matters. Why should two kinds of wave behave so differently?
How do we know what we can't visit? The expansion of the universe is inferred from tiny shifts in spectral lines. How confident can we be in conclusions drawn about places we can never go?

How do physics, NoS and ToK fit together? →

1. What is the Doppler effect?

The Doppler effect is the change in the observed frequency (and wavelength) of a wave when the source and the observer are moving relative to each other.

When the source and the observer get closer together, the observed frequency is higher (and the wavelength shorter).
When they move apart, the observed frequency is lower (and the wavelength longer).
The source itself doesn't change: the siren still vibrates at the same frequency. Only what the observer receives changes.
Only motion along the line joining the source and observer matters. A source moving across your line of sight, at right angles, gives no shift at that moment.

The Doppler effect happens for all waves: sound, water waves and electromagnetic waves. But sound and light behave differently in one important way. Sound travels through a medium (air), so it matters whether the source or the observer is moving through the air. Light needs no medium, so only their relative motion matters.

2. Wavefronts: a moving source

A source emits a wavefront once every period $T$. Each wavefront spreads out from the point where it was emitted at the wave speed $v$. If the source is still, the wavefronts are concentric circles, equally spaced in every direction. If the source is moving, it emits each new wavefront from a slightly different place:

A source moving right at half the wave speed. The small dots show where it was when it emitted each wavefront. Ahead of it, the wavefronts are squashed together; behind it, they are stretched apart.

In front of the source (observer A), the wavefronts are closer together: a shorter wavelength. They still travel at the same speed $v$, so more of them arrive each second: a higher frequency.
Behind the source (observer B), the wavefronts are further apart: a longer wavelength and a lower frequency.
The wave speed doesn't change. Sound travels at the speed set by the air, however fast the source moves.

So an approaching siren sounds higher than its true pitch, and a receding one sounds lower. As it passes you, the pitch drops suddenly, from above the true frequency to below it.

3. Wavefronts: a moving observer

Now the source is still, so its wavefronts are evenly spaced circles. The wavelength in the air is the normal wavelength. But an observer moving towards the source runs into the wavefronts more often than an observer standing still:

The source is still, so the wavefronts are evenly spaced. The observer moving towards it meets them more often, so they hear a higher frequency.

Moving towards the source: the waves pass the observer faster, relative to the observer, so they hear a higher frequency.
Moving away: the waves pass more slowly, relative to the observer, so they hear a lower frequency.
The wavelength in the air is unchanged. Only the rate at which the observer meets the wavefronts changes. That's the key difference from a moving source.

4. The Doppler effect for light

Light from a moving source is also Doppler-shifted. When the relative speed $v$ between source and observer is much smaller than the speed of light ($v \ll c$), the fractional change in frequency or wavelength is:

$$\frac{\Delta f}{f} = \frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$$

$\Delta f$ and $\Delta\lambda$ are the changes (shifts); $f$ and $\lambda$ are the values emitted by the source; $v$ is the relative speed along the line of sight. Use the sizes of the changes, and decide the direction from the situation.

Source and observer moving apart: the wavelength increases, so the light is shifted towards the red end of the spectrum. This is a redshift.
Moving together: the wavelength decreases. This is a blueshift.
The speed of light doesn't change: it is $c$ for every observer. It is the frequency and wavelength that shift.
For everyday speeds, $\frac{v}{c}$ is tiny. A car at 30 $\text{m s}^{-1}$ shifts light by only one part in ten million. That's why we don't see colours change in traffic. Astronomers measure the much larger shifts from stars and galaxies.

Worked example: how fast is the galaxy moving?

Hydrogen in a laboratory emits a red spectral line of wavelength 656.3 nm. In the light from a distant galaxy, the same line is observed at 662.9 nm. Find the galaxy's speed and say whether it is moving towards or away from us.

$\Delta\lambda = 662.9 - 656.3 = 6.6$ nm. The wavelength has increased (redshift), so the galaxy is moving away from us.

$v = c\,\dfrac{\Delta\lambda}{\lambda} = 3.00\times10^{8} \times \dfrac{6.6}{656.3} = 3.0\times10^{6}\ \text{m s}^{-1}$. This is 1% of the speed of light, so $v \ll c$ and the formula is valid.

5. Redshift, blueshift and astronomy

Each element absorbs and emits light at its own set of exact wavelengths (E.1), so its spectral lines act like a fingerprint. Astronomers compare the lines in starlight with the same lines measured in a laboratory. If the whole pattern is shifted, the star or galaxy is moving.

The same pattern of absorption lines, shifted towards the red in light from a receding galaxy. Every line shifts by the same fraction of its wavelength.

The expanding universe: light from almost every distant galaxy is redshifted, and the further away the galaxy is, the bigger the redshift. So galaxies are moving apart. This is strong evidence that the universe is expanding.
Rotating stars and galaxies: as a star or galaxy spins, one edge moves towards us and the opposite edge moves away. Light from one edge is blueshifted and light from the other is redshifted. The difference tells astronomers how fast it rotates.
Binary stars and exoplanets: a star orbiting a companion, or being tugged by a planet, moves alternately towards and away from us. Its spectral lines shift back and forth regularly. Many planets around other stars were discovered this way.

6. Uses on Earth: medicine and radar

In these uses, a wave is sent out, reflects off a moving object, and comes back. The frequency of the reflected wave is compared with the frequency that was sent. The bigger the frequency shift, the faster the object is moving towards or away from the detector.

Doppler ultrasound: ultrasound reflects off red blood cells. The frequency shift shows how fast, and in which direction, blood flows through an artery or vein. Doctors use it to find narrowed arteries or blood clots, and to check a baby's heartbeat before birth.
Radar speed guns: police aim microwaves at a car. Waves reflected from an approaching car come back at a higher frequency, and the shift gives the car's speed.
Weather radar: microwaves reflect off raindrops. The shift shows how fast rain is moving towards or away from the radar, which reveals the winds and rotation inside storms.
Air traffic control and satellites: radar uses the Doppler shift to measure aircraft speeds, and the GPS system corrects for the Doppler shift of satellite signals.

For reflected waves, the shift is roughly twice as big as for a single trip. The moving object first receives a shifted wave, then sends it back as a moving source, which shifts it again.

7. Equation for a moving source HL

For sound and other mechanical waves, the wave speed $v$ is fixed relative to the medium. A source with frequency $f$ (period $T$) moves directly towards an observer at speed $u_s$:

In one period, wavefront 1 (emitted at $t = 0$) travels $vT$, but the source moves $u_sT$ after it. Wavefront 2 is emitted at $t = T$, so the gap between wavefronts ahead of the source is the shorter wavelength $\lambda' = (v - u_s)T$.

The observer receives waves of wavelength $\lambda' = (v - u_s)T$, still travelling at speed $v$. So the observed frequency is:

$$f' = \frac{v}{\lambda'} = \frac{v}{(v - u_s)T} = f\,\frac{v}{v - u_s}$$

For a source moving away, the gap is $(v + u_s)T$ instead. The data booklet combines both cases:

$$f' = f\,\frac{v}{v \pm u_s}$$

Moving source. Use $-$ when the source approaches (smaller denominator, so $f'$ is higher) and $+$ when it recedes. $v$ is the wave speed in the medium; $u_s$ is the speed of the source.

Worked example: an ambulance HL

An ambulance siren emits sound of frequency 700 Hz. The ambulance travels at 25 $\text{m s}^{-1}$ along a straight road past a pedestrian. The speed of sound is 340 $\text{m s}^{-1}$. Find the frequency the pedestrian hears (a) as it approaches and (b) after it has passed.

(a) Approaching, so use $-$: $f' = 700 \times \dfrac{340}{340 - 25} = 700 \times \dfrac{340}{315} = 756$ Hz.

(b) Receding, so use $+$: $f' = 700 \times \dfrac{340}{365} = 652$ Hz.

The pitch drops by about 100 Hz as the ambulance passes. Check: approaching should give a frequency above 700 Hz and receding one below, which it does.

Worked example: finding the speed of a train HL

A train's horn has a frequency of 400 Hz. A person on a platform hears 420 Hz as the train approaches. The speed of sound is 340 $\text{m s}^{-1}$. How fast is the train moving?

$420 = 400 \times \dfrac{340}{340 - u_s}$, so $340 - u_s = \dfrac{400 \times 340}{420} = 323.8$, giving $u_s = 16\ \text{m s}^{-1}$ (about 58 km/h).

8. Equation for a moving observer HL

Now the source is still and the observer moves directly towards it at speed $u_o$. The wavelength in the air is the normal $\lambda = \frac{v}{f}$, but the waves pass the observer at a relative speed of $v + u_o$. So the observer meets $\frac{v + u_o}{\lambda}$ wavefronts per second:

$$f' = \frac{v + u_o}{\lambda} = f\,\frac{v + u_o}{v}$$

$$f' = f\,\frac{v \pm u_o}{v}$$

Moving observer. Use $+$ when the observer approaches the source (so $f'$ is higher) and $-$ when they move away. $u_o$ is the speed of the observer.

Choosing the sign: don't memorise it. Ask yourself whether the source and observer are getting closer (frequency up) or further apart (frequency down), then choose the sign that makes $f'$ go the right way. IB questions never have the source and the observer moving at the same time.

Worked example: a cyclist and a siren HL

A stationary fire alarm emits a 500 Hz tone. A cyclist rides towards it at 8.0 $\text{m s}^{-1}$, then away from it at the same speed. The speed of sound is 340 $\text{m s}^{-1}$. What frequencies does the cyclist hear?

Towards: $f' = 500 \times \dfrac{340 + 8.0}{340} = 512$ Hz. Away: $f' = 500 \times \dfrac{340 - 8.0}{340} = 488$ Hz.

Why are the two equations different? For sound, the air is a special frame of reference: the waves always travel at $v$ relative to the air. A moving source changes the wavelength in the air; a moving observer changes the speed of the waves relative to them. At the same speed, these give slightly different answers. At 25 $\text{m s}^{-1}$ towards a 700 Hz source, a moving observer hears 751 Hz, while a moving source (above) gives 756 Hz. For light there is no medium, so only the relative speed matters, and the simple formula $\frac{\Delta f}{f} \approx \frac{v}{c}$ applies.

End of the HL-only content for C.5.

9. Common mistakes

Saying the source's frequency changes. The source emits the same frequency all the time. Only the observed frequency changes.
Saying the wave speed changes. Sound travels at the speed set by the air, and light at $c$, whatever the source does.
Thinking the pitch keeps rising as a source approaches. For a source moving at a steady speed straight towards you, the observed frequency is constant (and higher than normal). It drops suddenly as the source passes.
Mixing up redshift and blueshift. Moving apart means a longer wavelength: redshift.
Using $\frac{\Delta f}{f} \approx \frac{v}{c}$ for sound. That formula is for light (with $v \ll c$). HL For sound, use the moving-source or moving-observer equation.
HL Using the wrong equation. Ask who is moving through the air: the source ($u_s$ in the denominator) or the observer ($u_o$ in the numerator).
HL Choosing the sign by habit. Always check that approaching gives a higher frequency and receding a lower one.

10. Check your understanding

A car sounds its horn as it drives past you. Describe what you hear.

As it approaches, a constant note higher than the horn's true frequency. As it passes, the pitch drops quickly. As it drives away, a constant note lower than the true frequency.

Why don't we see the colour of a car change as it drives towards us?

The shift is $\frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$. For a car, $v/c$ is about $10^{-7}$, far too small for our eyes to notice.

A spectral line from a star appears at a shorter wavelength than in the laboratory. What does this tell you?

It is blueshifted, so the star is moving towards us. The size of the shift gives its speed along our line of sight: $v \approx c\,\frac{\Delta\lambda}{\lambda}$.

How can the Doppler effect show that a galaxy is rotating?

One side of the galaxy is moving towards us and the other side away. Light from one edge is blueshifted and light from the other is redshifted. The difference in the shifts gives the rotation speed.

HL A source and an observer approach each other at the same speed in two separate experiments: once with the source moving, once with the observer moving. Which gives the higher frequency for sound, and why are they different?

The moving source gives a slightly higher frequency ($\frac{v}{v - u}$ is bigger than $\frac{v + u}{v}$). They differ because sound travels at a fixed speed relative to the air: a moving source changes the wavelength in the air, while a moving observer changes the speed of the waves relative to themselves.

Practise C.5 questions