C.5 Doppler effect

An ambulance siren sounds higher in pitch as the ambulance approaches and lower as it moves away, even though the siren itself emits a constant note. This change in the observed frequency caused by motion is the Doppler effect. It happens for every kind of wave, and because the size of the shift depends on speed, it gives a way of measuring how fast something is moving towards or away from an observer.

Police speed guns, weather radar, ultrasound scans of blood flow and the measurement of the motion of stars and galaxies all rely on it. For light, Doppler shifts gave the first detection of a planet orbiting a Sun-like star.

What the Doppler effect is

The Doppler effect has a simple cause. The frequency an observer measures is the number of wavefronts that reach the observer per second. If the source or the observer moves, that number changes, even though the source still emits the same number of waves each second.

• When the source and the observer approach each other, the observed frequency is higher than the emitted frequency.

• When they move apart, the observed frequency is lower than the emitted frequency.

• When there is no relative motion along the line joining them, there is no shift.

For sound, a higher frequency is heard as a higher pitch. For light, a higher frequency means a shorter wavelength, a shift towards the blue end of the visible spectrum; a lower frequency means a shift towards the red end.

The effect was first predicted by Christian Doppler in 1842. Three years later it was tested with sound: trumpeters playing a steady note on a moving train were heard by musicians beside the track, who recorded the change in pitch as the train approached and receded. Doppler had predicted the shift for starlight too, and astronomers confirmed this later in the nineteenth century.

Wavefront diagrams for a moving source

A wavefront is a line joining points on a wave that are in phase, such as the crests. Wavefronts drawn one period apart are one wavelength apart. A stationary source in still air emits circular wavefronts centred on the source, evenly spaced in every direction, so every stationary observer receives the emitted frequency $f$.

When the source moves, each wavefront still spreads out at the wave speed $v$ from the point where it was emitted, because the speed of sound depends only on the medium, not on the motion of the source. But each new wavefront is emitted from a point further along the source's path. The wavefronts are therefore crowded together ahead of the source and spread apart behind it.

• Observer A, ahead of the source. The wavelength is shorter. The waves still travel at $v$, so more wavefronts arrive per second: the observed frequency is higher, $f' = v/\lambda' > f$.

• Observer B, behind the source. The wavelength is longer, fewer wavefronts arrive per second, and the observed frequency is lower.

• An observer to the side, at right angles to the motion at the moment the sound is emitted, receives approximately the emitted frequency, because the source is neither approaching nor receding at that moment.

The frequency heard as a source passes

A siren driving past at constant speed gives a characteristic sound: a steady high note while it is still some distance away and approaching, a rapid drop as it passes, then a steady low note as it moves away. The observed frequency depends on the component of the source's velocity along the line to the observer. Far away, almost all of the velocity is along that line; as the source passes, the component falls to zero and then reverses.

The closer the observer is to the source's path, the faster the drop. An observer standing directly in the path would hear the frequency fall instantly from the high value to the low value.

A moving observer

When the source is stationary and the observer moves, the wavefronts are evenly spaced circles, so the wavelength in the medium is unchanged. An observer moving towards the source meets the wavefronts more often than a stationary observer would, because the wavefronts approach at $v + u_\mathrm{o}$ relative to the observer, where $u_\mathrm{o}$ is the observer's speed. The observed frequency is higher. An observer moving away meets fewer wavefronts per second and measures a lower frequency.

For sound the two cases are physically different, because the medium (the air) provides a reference against which both the source and the observer move. A source moving at a certain speed towards a stationary observer gives a slightly different shift from an observer moving at the same speed towards a stationary source. For low speeds the two shifts are almost equal.

The Doppler effect for light

Electromagnetic waves need no medium, and the speed of light is the same for every observer. There is therefore no distinction between a moving source and a moving observer for light: only the relative velocity of the source and the observer along the line joining them matters.

When the relative speed $v$ is much smaller than the speed of light $c$, the fractional change in frequency and the fractional change in wavelength have the same size, given by

Here $f$ and $\lambda$ are the emitted frequency and wavelength (measured in the source's own frame, for example in a laboratory), $\Delta f$ and $\Delta\lambda$ are the sizes of the shifts, $v$ is the relative speed of the source and the observer along the line of sight, and $c = 3.00\times10^{8}\ \text{m s}^{-1}$. The approximation is accurate to within a few per cent up to about a tenth of $c$, and far better at the speeds of stars in the Galaxy, which are well below 1% of $c$.

Redshift and blueshift

The words come from the visible spectrum, where red has the longest wavelength and blue and violet the shortest. They are used for any electromagnetic wave: a radio signal from a receding spacecraft is "redshifted" even though it is nowhere near red. The equation gives the size of the shift; the direction comes from whether the wavelength has increased (redshift) or decreased (blueshift).

Measuring the motion of stars and galaxies

A star's light passes through its cooler outer layers, where atoms absorb particular wavelengths. Its spectrum is therefore an absorption spectrum, a continuous spectrum crossed by dark spectral lines whose pattern identifies the elements present: hydrogen, for example, absorbs at 410 nm, 434 nm, 486 nm and 656 nm. If the star is moving along the line of sight, every line is shifted by the same fraction of its wavelength, so the pattern is still recognisable and the shift can be measured.

Comparing the measured wavelength of a line with its laboratory wavelength gives $\Delta\lambda$, and then $v = c\,\Delta\lambda/\lambda$ gives the speed of the star towards or away from Earth. This speed along the line of sight is called the radial velocity. Motion across the sky, at right angles to the line of sight, produces no shift.

The Andromeda galaxy, the nearest large galaxy to the Milky Way, shows a blueshift: it is approaching the Sun at about $300\ \text{km s}^{-1}$. Much of this is the Sun's own orbital motion around the Galaxy, which carries it roughly towards Andromeda. Almost all more distant galaxies, by contrast, show redshifts, so they are moving away from the Milky Way. This pattern was one of the key observations that led to the idea of an expanding universe.

Finding planets around other stars

A planet and its star orbit their common centre of mass, so the star moves in a small circle and its radial velocity rises and falls with the orbital period of the planet. The star's spectral lines therefore shift periodically, alternately towards red and towards blue. In 1995 Michel Mayor and Didier Queloz detected this wobble in the star 51 Pegasi: its radial velocity varies by about $56\ \text{m s}^{-1}$ either side of its average, with a period of 4.23 days. It was the first exoplanet (a planet outside the Solar System) found orbiting a Sun-like star, and the discovery earned the 2019 Nobel Prize in Physics.

Rotating bodies

When a star, a planet or a galaxy rotates, one edge moves towards the observer while the opposite edge moves away. Light from the approaching edge is blueshifted and light from the receding edge is redshifted. Comparing the two edges gives the speed of rotation. For a star too distant to resolve, light from both edges arrives together, so each spectral line is spread over a range of wavelengths and looks broader than it would for a non-rotating star.

Radar: reflection from a moving target

A radar speed gun sends a beam of microwaves at a moving vehicle and detects the waves reflected back. The vehicle acts first as a moving observer, receiving a shifted frequency, and then as a moving source, re-emitting that shifted frequency back towards the gun. The frequency is therefore shifted twice, and for $v \ll c$ the total shift is double the one-way shift:

The gun mixes the reflected signal with the emitted one and measures the difference frequency $\Delta f$, which is easy to measure accurately because it is only a few kilohertz. Weather radar uses the same principle to measure the speed of raindrops, and so the wind speed inside a storm.

Doppler ultrasound and blood flow

In a Doppler ultrasound scan, a probe placed on the skin sends ultrasound (typically a few megahertz) into the body. Red blood cells moving in an artery or vein reflect some of it back. Cells moving towards the probe return a slightly higher frequency and cells moving away a lower one. The shift gives the speed and direction of the blood flow, which shows narrowed arteries, blood clots and the blood flow in a fetus.

The shifts are small fractions of the emitted frequency because blood moves at less than about $1\ \text{m s}^{-1}$, much slower than ultrasound travels in tissue (about $1500\ \text{m s}^{-1}$). The shift is usually a few kilohertz, in the audible range, so scanners often play it through a loudspeaker as well as showing it as a colour map, with flow towards the probe in one colour and away in another. Because sound is a mechanical wave, the precise equations for this case are those for sound.

Doppler equations for sound (HL)

In the equations, $f$ is the frequency emitted by the source, $f'$ the observed frequency, $v$ the speed of the wave in the medium, $u_\mathrm{s}$ the speed of the source and $u_\mathrm{o}$ the speed of the observer, all measured relative to the medium. Only one of the source and the observer moves at a time, and the motion is along the line joining them.

A moving source

A source of frequency $f$ and period $T = 1/f$ moves at speed $u_\mathrm{s}$ towards a stationary observer. It emits a wavefront at A. One period later that wavefront has travelled a distance $vT$, while the source has moved $u_\mathrm{s}T$ to B, where it emits the next wavefront.

The distance between the two wavefronts, the wavelength ahead of the source, is

The waves travel towards the stationary observer at $v$, so the observed frequency is $f' = v/\lambda'$:

The minus sign applies when the source moves towards the observer (a smaller denominator, so $f' > f$) and the plus sign when it moves away ($f' < f$). As $u_\mathrm{s}$ approaches $v$ the wavefronts ahead of the source pile up on each other; a source faster than sound produces a shock wave, the sonic boom of a supersonic aircraft.

A moving observer

For a stationary source the wavelength is unchanged, $\lambda = v/f$. An observer moving towards the source at speed $u_\mathrm{o}$ meets the wavefronts at a relative speed $v + u_\mathrm{o}$, so the number met per second is $(v + u_\mathrm{o})/\lambda$:

The plus sign applies when the observer moves towards the source and the minus sign when it moves away.

For a moving observer the equation is linear in $u_\mathrm{o}$, so equal speeds towards and away from the source give equal and opposite changes in frequency.

Finding the speed of a moving source

Problems often give two observed frequencies, one as a source approaches and one as it recedes, without the emitted frequency. Dividing one equation by the other removes $f$ and leaves an equation for the speed.

Doppler ultrasound calculations

Blood cells reflecting ultrasound are first moving observers and then moving sources, so, as for radar, the shift is doubled. Because the blood speed $u$ is tiny compared with the wave speed $v$ in tissue, the two sound equations give almost the same shift, and the result simplifies to $\Delta f/f \approx 2u/v$ for blood moving directly towards or away from the probe.

Comparing a moving source and a moving observer

The graph of frequency ratio against speed shows the two equations side by side. At speeds much less than the wave speed, a source and an observer moving at the same speed give nearly the same shift, $\Delta f/f \approx u/v$, which has the same form as the equation for light. At higher speeds the two cases differ, because a moving source changes the wavelength in the medium while a moving observer does not.

Applying the ideas in exam questions

Calculation questions at both levels usually ask for a speed from a measured shift, or for an observed frequency or wavelength from a given speed. State the direction of motion whenever the question asks whether the source is approaching or receding.

Paper 1A: wavelength and frequency for a moving source

Paper 1A: speed of a star

Paper 1B: the spectrum of a binary star

An astronomer measures the wavelength $\lambda$ of a hydrogen line in the spectrum of a star that orbits a companion. The laboratory wavelength of the line is 656.280 nm. Each measurement has an uncertainty of $\pm 0.002\ \text{nm}$.

Paper 2: a microwave speed trap

Paper 2 (HL): a passing train

Paper 1A (HL): a moving observer

Self-check

• To do:

  I can define the Doppler effect and state when the observed frequency rises or falls.

• To do:

  I can draw and interpret wavefront diagrams for a moving source.

• To do:

  I can explain the difference between a moving source and a moving observer for sound.

• To do:

  I can describe how the pitch of a passing source changes, and why.

• To do:

  I can use $\Delta f/f = \Delta\lambda/\lambda \approx v/c$ for light and say when it is valid.

• To do:

  I can define redshift and blueshift and use spectral line shifts to find the speed and direction of a star or galaxy.

• To do:

  I can explain how Doppler shifts show planets around other stars and the rotation of a body.

• To do:

  I can explain why the shift is doubled for reflection from a moving target and use this for radar.

• To do:

  I can describe how Doppler ultrasound measures blood flow.

• To do:

  I can use the moving-source and moving-observer equations for sound, choosing the correct sign (HL).

• To do:

  I can find the speed of a source from the frequencies heard as it approaches and recedes (HL).

The answer is D.

B has the direction reversed. A halves the answer and D doubles it, the factor of 2 that belongs only to reflected waves.

D comes from the moving-source equation, $f/(1 - 0.10) = 1.11f$. A and B are the corresponding answers for motion away from the source.