IB DP Physics · Theme C · Wave behavior

Doppler effect.

C.5 · Why motion changes the frequency you receive, for sound and for light
Standard level and higher level

SL + HL · 2 hours Additional HL · 2 hours

The journey

Three parts.

Part 01

The effect

Wavefront diagrams for a moving source and a moving observer

Part 02

Light

Red shift and blue shift, and what they reveal

Part 03 · HL only

Equations for sound

The observed frequency for a moving source and for a moving observer

C.5 DOPPLER EFFECT · SL + HL

Part 01

The effect.

The pitch of a siren drops as it passes. Nothing about the siren has changed

C.5 DOPPLER EFFECT · SL + HL

Relative motion

What the Doppler effect is.

The change in the observed frequency of a wave when the source and the observer move relative to each other

Coming together

Higher frequency

Sound: a higher pitch. Light: shifted towards the blue

Moving apart

Lower frequency

Sound: a lower pitch. Light: shifted towards the red

The source emits the same frequency throughout. Only what is received changes.

C.5 DOPPLER EFFECT · SL + HL

The siren

An ambulance drives past. As it approaches you hear a note higher than the one its driver hears; as it goes away you hear a lower one. The change happens as it passes, quite quickly, and it is a change of pitch, not of loudness. The driver, who is not moving relative to the siren, hears the same steady note the whole time.

Not the same as getting louder

The sound also grows louder as the ambulance approaches and fainter as it leaves, because it is nearer and then further. That is a separate effect. A source circling round you at a constant distance would stay equally loud and show no Doppler shift, because it is neither approaching nor receding.

Only motion along the line between them counts

A source moving at right angles to the line of sight gives no shift at that instant.

Saying that the frequency “of the source” changes. The source carries on exactly as before. It is the observed frequency that differs.

Wavefront diagram

A moving source.

Each wavefront is a circle centered on where the source was when it was emitted
Ahead: wavefronts closer together
Behind: further apart
The wave speed in the medium is unchanged

SOURCE MOVING TO THE RIGHT

C.5 DOPPLER EFFECT · SL + HL

Why the wavelength changes

Once a wavefront has left the source it spreads through the air at the speed of sound, and it does not care what the source does next. In the time before the next wavefront is sent, the source has moved forward a little. So in front of the source each new wavefront starts closer to the last one: the wavelength there is shorter. Behind, each starts further from the last, and the wavelength is longer.

From wavelength to frequency

All the wavefronts travel at the same speed \(v\) through the still air. An observer ahead receives a shorter wavelength at that speed, and \(f = \frac{v}{\lambda}\) gives a higher frequency. An observer behind receives a longer wavelength and a lower frequency.

Drawing it

Mark the positions of the source at equal time intervals. Draw the largest circle round the earliest position and the smallest round the latest. The circles must not cross, as long as the source is slower than the wave.

If the source moves as fast as the wave, the wavefronts ahead pile up on each other. For an aircraft that is the shock wave heard as a sonic boom.

Wavefront diagram

A moving observer.

The source is at rest: the wavefronts are evenly spaced circles
The wavelength in the medium is unchanged
Moving towards the source, the observer meets wavefronts more often
Moving away: less often

OBSERVER MOVING TOWARDS THE SOURCE

C.5 DOPPLER EFFECT · SL + HL

The same result for a different reason

With a moving source the wavelength in the air really is different in different directions. With a moving observer it is not: the pattern of wavefronts is perfectly symmetrical. The frequency changes because the observer is running into the wavefronts. Relative to the observer, the waves approach at the wave speed plus the observer’s own speed, so more of them arrive each second.

Wading into the sea

Stand still in shallow water and the waves pass you at a certain rate. Walk out towards them and they hit you more often, though the sea has not changed. Walk back to the beach and they catch you up less often.

Wavefront diagrams for both cases are required. The test of a correct diagram: moving source, circles with different centers; moving observer, circles with the same center.

The story · Prague, 1842 · the Netherlands, 1845

Background story · not assessed

An idea about stars, tested on a train.

Portrait of Christian Doppler, a man with dark hair and side whiskers in a dark coat — **Christian Doppler**

Printed portrait of C. H. D. Buys Ballot, an elderly man with white hair and a white beard — **C. H. D. Buys Ballot**, in 1887

1842: Doppler proposes that the color of a star’s light depends on its motion, in a paper on double stars
1845: Buys Ballot tests the idea with sound, on a Dutch railway
The pitch was higher as the source approached and lower as it moved away
1848: Fizeau works out the effect for light

The shift in a star’s light is far too small to see as a change of color. It is measured from its spectral lines.

Doppler: unknown author, public domain. Buys Ballot: Rijksmuseum, 1887, CC0.

C.5 DOPPLER EFFECT · SL + HL

Quick check

The effect.

01A source approaches an observer. The observed frequency is…?

Higher than the emitted frequency

02Does the frequency emitted by the source change?

No

03For a moving source, where are the wavefronts closest together?

Ahead of the source

04For a moving observer, does the wavelength in the medium change?

No

05A car circles you at a constant distance, sounding its horn. Is there a shift?

No: it is not approaching or receding

06In a moving-source diagram, where is the center of each circle?

Where the source was when it was emitted

PRESS → TO REVEAL EACH ANSWER

Part 02

Light.

The same effect for electromagnetic waves, and a way to measure the motion of stars

C.5 DOPPLER EFFECT · SL + HL

Electromagnetic waves

The Doppler effect for light.

\(\dfrac{|\Delta f|}{f} \approx \dfrac{|\Delta \lambda|}{\lambda} \approx \dfrac{v}{c}\)

\(v\): the relative speed of source and observer
Valid when \(v\) is much less than \(c\)

Moving apart

Red shift

Longer wavelength, lower frequency

Coming together

Blue shift

Shorter wavelength, higher frequency

C.5 DOPPLER EFFECT · SL + HL

A fractional change

The equation says that the shift, as a fraction of the original frequency or wavelength, equals the speed as a fraction of the speed of light. A source receding at 1% of the speed of light has every wavelength stretched by 1% and every frequency lowered by 1%.

Only relative motion

For sound, a moving source and a moving observer are different situations, because the air provides something to be moving relative to. Light needs no medium, so there is nothing to tell the two cases apart. Only the relative velocity of source and observer can matter, and one equation serves for both.

“Red” and “blue” are directions, not colors

A red shift means a move towards longer wavelengths, whether the light started in the visible, the ultraviolet or the radio. Red-shifted light does not usually look red.

The data booklet’s relation gives magnitudes. Here \(\Delta f = f_\mathrm{observed} - f_\mathrm{emitted}\) and \(\Delta\lambda = \lambda_\mathrm{observed} - \lambda_\mathrm{emitted}\). The signed shifts have opposite signs: \(\frac{\Delta f}{f} \approx -\frac{\Delta\lambda}{\lambda}\). For recession, \(\Delta\lambda > 0\) and \(\Delta f < 0\).

Reading motion from a spectrum

Shifted spectral lines.

THE SAME PATTERN, MOVED

Each element has lines at known wavelengths (E.1)
From a moving star, the whole pattern is shifted
The size of the shift gives the speed along the line of sight

C.5 DOPPLER EFFECT · SL + HL

Why lines are needed

A smooth, continuous spectrum shifted slightly would look the same as before: there would be nothing to measure. Spectral lines are sharp markers at wavelengths that are known precisely from laboratory measurements. The pattern of an element’s lines is recognizable even when all of them have moved, and the distance each has moved can be measured.

What has been learned this way

A star that wobbles to and fro, its lines shifting red and blue in turn, is being pulled by an unseen companion: this is how the first planet of a star like the Sun was found, in 1995. The two edges of a rotating galaxy are shifted in opposite directions, which gives its rate of rotation and so its mass. And almost every distant galaxy is red-shifted: the universe is expanding.

The shift gives only the part of the velocity along the line of sight. Motion across the sky does not shift the lines.

Worked example

The speed of a galaxy.

A hydrogen line: 656.3 nm in the laboratory The same line from a galaxy: 662.9 nm \(\Delta \lambda = 662.9 - 656.3 = 6.6 \,\mathrm{nm}\) \(\dfrac{\Delta \lambda}{\lambda} = \dfrac{v}{c}\) so \(v = c \dfrac{\Delta \lambda}{\lambda}\) \(v = 3.00 \times 10^{8} \times \dfrac{6.6}{656.3}\) \(v = \class{res}{3.0 \times 10^{6} \,\mathrm{m} \,\mathrm{s}^{-1}}\)

The wavelength is longer: a red shift, so the galaxy is moving away
1% of the speed of light: the approximation is good
\(\lambda\) on the bottom is the laboratory wavelength

C.5 DOPPLER EFFECT · SL + HL

Units cancel

\(\frac{\Delta \lambda}{\lambda}\) is a ratio, so both wavelengths can stay in nanometers. There is no need to convert them to meters, as long as both are in the same unit.

The same thing with frequency

The frequency of the line has fallen by the same fraction, 1.0%. A question may give frequencies instead of wavelengths; the method is identical.

How fast is that?

Three thousand kilometers every second, about a hundred times the speed of the Earth in its orbit. For galaxies it is modest: the most distant ones known are receding so fast that this simple equation no longer applies.

Dividing by the observed wavelength. The fractional change is measured against what the source emits: the laboratory value.

The story · Flagstaff, Arizona, 1912

Background story · not assessed

The nebulae are running away.

A brass spectrograph of tubes and a small telescope mounted on a frame, in a glass museum case — The **spectrograph** Vesto Slipher used at Lowell Observatory

In 1912 Slipher measured the shift of the lines of the Andromeda nebula: it is approaching at about 300 km s⁻¹
By 1917 he had measured 25 spiral nebulae. All but three were moving away
In 1929 Hubble and Humason set his speeds against distances: the further, the faster

The first evidence that the universe is expanding came from the Doppler shift of spectral lines.

Photograph: brewbooks, CC BY-SA 2.0.

C.5 DOPPLER EFFECT · SL + HL

Using the effect

Radar and medicine.

Radar: microwaves reflected from a moving car, aircraft or raindrop come back at a different frequency
Medicine: ultrasound reflected from moving blood gives the speed of the flow

A wave sent out, reflected from something moving, and compared with what was sent.

MEASURING A SPEED BY REFLECTION

C.5 DOPPLER EFFECT · SL + HL

A reflection doubles the shift

The moving car first receives the waves as a moving observer, at a shifted frequency. It then sends them back as a moving source, shifting them again. The frequency that returns to the radar differs from the one sent by about \(\frac{2v}{c} f\). A police radar at 24 GHz aimed at a car approaching at 30 m s⁻¹ sees a change of \(\frac{2 \times 30}{3.00 \times 10^{8}} \times 2.4 \times 10^{10} = 4.8\) kHz, which is easy to measure.

Blood flow

A probe on the skin sends ultrasound, typically a few megahertz, into an artery. The pulses reflect from the moving red blood cells, and the shift in frequency of the echo gives their speed. A narrowed artery shows up as a place where the blood speeds up. The same technique shows a heart valve leaking, or the heartbeat of an unborn child, without any surgery.

Weather radar measures the speed of raindrops towards or away from the station, and so the winds inside a storm.

The guide names medical physics and radar as the examples to know. Be able to say what the wave is, what it reflects from and what is found.

Quick check

Light.

01A star’s lines are shifted to shorter wavelengths. Which way is it moving?

Towards us: a blue shift

02When is \(\frac{\Delta \lambda}{\lambda} \approx \frac{v}{c}\) valid?

When \(v\) is much less than \(c\)

03A source recedes at 0.5% of \(c\). Change in a 600 nm line?

+3 nm

04A line moves from 500.0 nm to 501.0 nm. Speed of the source?

\(\frac{1.0}{500.0} \times 3.00 \times 10^{8} = 6.0 \times 10^{5} \,\mathrm{m} \,\mathrm{s}^{-1}\)

05What do the red shifts of distant galaxies show?

The universe is expanding

06Give one use of the Doppler effect in medicine.

Measuring the speed of blood flow

PRESS → TO REVEAL EACH ANSWER

Part 03 · Additional higher level

Equations for sound.

The observed frequency for a moving source and for a moving observer

C.5 DOPPLER EFFECT · ADDITIONAL HL

The wavelength changes

A moving source.

\(f' = f \left(\dfrac{v}{v \pm u_\mathrm{s}}\right)\)

\(v\): the speed of the wave · \(u_\mathrm{s}\): the speed of the source
Minus when the source approaches: \(f'\) is higher
Plus when it recedes: \(f'\) is lower

Source approaching In one period \(T\) the wave goes \(vT\), the source \(u_\mathrm{s} T\) \(\lambda' = vT - u_\mathrm{s} T = \dfrac{v - u_\mathrm{s}}{f}\) \(f' = \dfrac{v}{\lambda'} = f \dfrac{v}{v - u_\mathrm{s}}\)

C.5 DOPPLER EFFECT · ADDITIONAL HL

The derivation in words

The source emits a wavefront and, one period later, another. In that period the first wavefront has traveled a distance \(vT\) and the source has followed it for a distance \(u_\mathrm{s} T\). The second wavefront therefore starts only \(vT - u_\mathrm{s} T\) behind the first: that gap is the new wavelength. The wavefronts still travel at \(v\) through the air, so the observer receives them at a frequency \(\frac{v}{\lambda'}\).

Choosing the sign

Do not try to memorize which sign goes with which case. Decide first whether the observed frequency should go up or down, then pick the sign that does it. Approaching should raise the frequency, which needs a smaller number on the bottom: minus.

Confusing this \(v\) with the \(v\) in the equation for light. Here \(v\) is the speed of sound, about 340 m s⁻¹, and \(u_\mathrm{s}\) is the speed of the source.

Worked example

An ambulance passes.

Siren 500 Hz · \(u_\mathrm{s} = 30 \,\mathrm{m} \,\mathrm{s}^{-1}\) · \(v = 340 \,\mathrm{m} \,\mathrm{s}^{-1}\) Approaching \(f' = f \dfrac{v}{v - u_\mathrm{s}} = 500 \times \dfrac{340}{340 - 30}\) \(f' = \class{res}{548 \,\mathrm{Hz}}\)

Receding \(f' = f \dfrac{v}{v + u_\mathrm{s}} = 500 \times \dfrac{340}{340 + 30}\) \(f' = \class{res}{459 \,\mathrm{Hz}}\) A drop of 89 Hz as it goes by

C.5 DOPPLER EFFECT · ADDITIONAL HL

The shifts are not equal

Approaching, the note is 48 Hz above 500 Hz. Receding, it is 41 Hz below. The rise is larger than the fall, because the speed of the source appears on the bottom of the fraction. At low speeds the difference is slight; as the source nears the speed of sound, the approaching frequency grows without limit.

How the pitch changes as it passes

On a straight road the ambulance does not come directly at you unless you are standing in its path. What matters is the part of its velocity along the line joining it to you, which falls smoothly to zero as it draws level and then reverses. So the pitch slides down from 548 Hz to 459 Hz over a second or so; it does not step.

Check: approaching must give more than 500 Hz, receding less. If the answers are the other way round, the signs are swapped.

The wavelength does not change

A moving observer.

\(f' = f \left(\dfrac{v \pm u_\mathrm{o}}{v}\right)\)

\(u_\mathrm{o}\): the speed of the observer
Plus when the observer approaches: \(f'\) is higher
Minus when the observer recedes

Observer approaching a 500 Hz source at 30 m s⁻¹ \(f' = f \dfrac{v + u_\mathrm{o}}{v} = 500 \times \dfrac{340 + 30}{340}\) \(f' = \class{res}{544 \,\mathrm{Hz}}\) Moving away: \(500 \times \dfrac{310}{340} = 456 \,\mathrm{Hz}\)

C.5 DOPPLER EFFECT · ADDITIONAL HL

Where it comes from

The wavelength in the air is the ordinary one, \(\lambda = \frac{v}{f}\). An observer moving towards the source meets the wavefronts at a relative speed \(v + u_\mathrm{o}\), so the number met each second is \(\frac{v + u_\mathrm{o}}{\lambda} = f \frac{v + u_\mathrm{o}}{v}\).

Not the same as a moving source

The ambulance approaching a listener at 30 m s⁻¹ gave 548 Hz. A listener approaching the ambulance at 30 m s⁻¹ gets 544 Hz. The relative speed is the same in both cases, but the answers differ, because sound travels in air and it matters which of the two is moving through the air. The difference is small at low speeds and grows with speed.

The two together

In this course the source and the observer never both move in the same problem.

Speed on the top for the observer, speed on the bottom for the source. In both, choose the sign that moves the frequency the right way.

Worked example

Finding the speed of a train.

A horn of 400 Hz is heard at 425 Hz as the train approaches \(v = 340 \,\mathrm{m} \,\mathrm{s}^{-1}\) \(f' = f \dfrac{v}{v - u_\mathrm{s}}\) \(v - u_\mathrm{s} = \dfrac{f v}{f'}\) \(u_\mathrm{s} = v \left(1 - \dfrac{f}{f'}\right)\)

\(u_\mathrm{s} = 340 \times \left(1 - \dfrac{400}{425}\right)\) \(u_\mathrm{s} = \class{res}{20 \,\mathrm{m} \,\mathrm{s}^{-1}}\) 72 km h⁻¹

C.5 DOPPLER EFFECT · ADDITIONAL HL

Quick check · HL

Equations for sound.

01Source approaching: plus or minus on the bottom?

Minus

02Observer approaching: plus or minus on the top?

Plus

03For which case does the wavelength in the air change?

A moving source

04A 680 Hz source recedes at 34 m s⁻¹ (\(v\) = 340). Observed frequency?

\(680 \times \frac{340}{374} = 618 \,\mathrm{Hz}\)

05An observer moves at 17 m s⁻¹ towards a 680 Hz source. Observed frequency?

\(680 \times \frac{357}{340} = 714 \,\mathrm{Hz}\)

06Same relative speed: do a moving source and a moving observer give the same shift?

No, not for sound

PRESS → TO REVEAL EACH ANSWER

Summary

The Doppler effect on one slide.

The effect

Relative motion changes the observed frequency

Approaching: higher. Receding: lower

Wavefront diagrams

Source: bunched ahead

Observer: same wavelength, met more or less often

Light

\(\frac{|\Delta f|}{f} \approx \frac{|\Delta \lambda|}{\lambda} \approx \frac{v}{c}\)

Red shift: receding
Blue shift: approaching

Uses

Stars, radar, blood flow

Shifted spectral lines show how stars and galaxies move

Moving source

\(f' = f \left(\frac{v}{v \pm u_\mathrm{s}}\right)\)

Minus when approaching

Moving observer

\(f' = f \left(\frac{v \pm u_\mathrm{o}}{v}\right)\)

Plus when approaching

C.5 DOPPLER EFFECT · SL + HL

Summary

Key words.

Doppler effectChange in observed frequency due to relative motion

Observed frequencyWhat the observer receives

Emitted frequencyWhat the source sends: unchanged

Red shiftTowards longer wavelength: receding

Blue shiftTowards shorter wavelength: approaching

Spectral lineA marker at a known wavelength

Relative speedAlong the line joining source and observer

RadarSpeed from the shift of reflected microwaves

Doppler ultrasoundSpeed of blood from the shift of echoes

C.5 DOPPLER EFFECT · SL + HL