C.5 Doppler effect

The diagram shows the wavefronts emitted by a source of sound that is moving to the right, towards observer A and away from observer B.

Which row describes what observers A and B hear, compared with the sound emitted?

A distant star moves relative to the Earth with a speed much less than the speed of light c. Light from the star is observed on the Earth.

Which statement about the light received on the Earth is correct?

An ambulance with its siren sounding travels at constant speed along a straight road. A stationary pedestrian stands at the very edge of the road, so close that the ambulance moves effectively directly towards her, passes her and then moves directly away from her. The air is still. Which describes the frequency the pedestrian hears?

A line in the spectrum of a distant galaxy is observed at a longer wavelength than the same line measured in a laboratory on Earth. What can be deduced?

A drone carrying a buzzer flies in still air directly towards a stationary microphone. Compared with measurements made when the drone is stationary, which quantity measured at the microphone is unchanged?

A small loudspeaker moves at constant speed through still air while emitting sound of constant frequency. Viewed from above, the circular wavefronts are crowded close together on the north side of the loudspeaker and are widely spaced on the south side. Which statement is correct?

A hydrogen spectral line has a wavelength of 486 nm when measured in a laboratory. In the spectrum of a distant galaxy, the same line is observed at 492 nm. What can be concluded?

A cyclist rides directly towards a stationary siren that emits sound of constant frequency. The air is still. Which quantity, as measured by the cyclist, has the same value as it would if the cyclist were at rest?

The Sun rotates about an axis roughly perpendicular to the line of sight from Earth. Light is collected separately from its two opposite edges (limbs) on the solar equator. Which statement is correct?

Hydrogen atoms in a distant gas cloud emit radio waves of frequency 1420.4 MHz. The cloud is moving directly away from the Earth at 2.1 × 10⁵ m s⁻¹.

What is the frequency of these radio waves when they are received on the Earth?

A drone emitting a steady note flies at constant velocity in a straight line directly towards a stationary listener, passes just above her, and then flies directly away from her.

Ignoring the short interval as the drone passes overhead, which row describes the frequency she hears?

A loudspeaker S emits a note of constant frequency in still air. A cyclist O rides at constant speed directly towards S. The diagram shows some of the wavefronts emitted by S.

Which row describes the wavelength of the sound in the air between S and O, and the frequency heard by the cyclist, compared with those for a stationary listener?

A small source emits sound of constant frequency while moving along a straight line in still air. The diagram, drawn to scale, shows four wavefronts emitted at equal time intervals, seen from above at one instant. The small dots show where the source was when each wavefront was emitted and S is the position of the source at this instant.

Which row describes the motion of the source?

An observer on the Earth's equator measures the wavelength of an absorption line in the spectrum of sunlight. On the day of the measurements the Sun rises due east, is directly overhead at noon and sets due west. The Earth rotates from west to east. Ignore the orbital motion of the Earth.

Which row compares the wavelength measured at sunrise and at sunset with the wavelength measured at noon?

The hydrogen-alpha line has a laboratory wavelength of 656.28 nm. In the spectrum of a star it is observed at 656.06 nm. The speed of light is 3.00 × 10⁸ m s⁻¹. What is the velocity of the star along the line of sight?

A spiral galaxy is seen edge-on from Earth. Its centre recedes from Earth at 600 km s⁻¹. Stars at its two opposite edges orbit the centre at 200 km s⁻¹, moving along the line of sight. How do spectral lines from the two edges appear?

A star emits two spectral lines with laboratory wavelengths of 400 nm and 600 nm. The star recedes from Earth at a speed much less than c. How does the wavelength shift Δλ of the 600 nm line compare with that of the 400 nm line?

A traffic radar unit emits microwaves of frequency 24 GHz. The microwaves reflect from a car travelling directly towards the unit at 30 m s⁻¹. What is the difference between the emitted frequency and the frequency of the reflected microwaves received back at the unit?

A star moves in a circular orbit in a binary system whose orbital plane contains the line of sight from Earth. The centre of mass of the system is at rest relative to Earth. A spectral line with a laboratory wavelength of 600.0 nm is observed to vary steadily between 599.8 nm and 600.2 nm. What is the orbital speed of the star?

A medical Doppler ultrasound probe directs ultrasound along a blood vessel. The ultrasound reflects from blood cells moving away from the probe. The frequency of the reflected ultrasound differs from the emitted frequency by Δf. The probe frequency is changed from 4 MHz to 8 MHz, and the blood speed falls to half its original value. What is the new frequency difference?

A source moves at constant speed u through a medium in which the wave speed is v. Directly ahead of the source, the spacing of the wavefronts is half the spacing directly behind it. What is u?

Two absorption lines in the spectrum of a galaxy have laboratory wavelengths of 420 nm and 630 nm. The 420 nm line is observed to be red-shifted by Δλ.

What is (red shift of the 630 nm line)/Δλ?

A star moves in a circular orbit in a binary star system whose orbit is seen edge-on from the Earth. The binary system as a whole is not moving relative to the Earth. A line of laboratory wavelength λ₀ in the star's spectrum varies periodically between λ₀ − δ and λ₀ + δ.

What is the orbital speed of the star?

A motorboat whose engine emits a note of constant frequency f₀ travels at constant speed along a straight course across a lake. A stationary observer stands at the end of a jetty, 40 m from the course. The boat approaches from far away, passes the observer and moves far away again.

Which graph shows how the frequency f heard by the observer varies with time t?

Two stars X and Y move in circular orbits about their common centre of mass, which is at rest relative to the Earth. The Earth lies in the plane of the orbits. A spectral line of laboratory wavelength λ₀ appears in the spectra of both stars. The line from X varies between λ₀ − 0.030 nm and λ₀ + 0.030 nm, and the line from Y varies between λ₀ − 0.090 nm and λ₀ + 0.090 nm.

What is (mass of X)/(mass of Y)?

A published solar atlas gives the wavelengths of absorption lines in light from the east and west edges (limbs) of the Sun's equator. As seen from the Earth, the Sun's rotation carries the east limb towards the Earth and the west limb away from it. For six lines of wavelength λ, a student uses the atlas to find Δλ = λ_west − λ_east. Each value of Δλ has an uncertainty of ±0.3 pm (1 pm = 10⁻¹² m).

The graph shows Δλ against λ with error bars. The dashed lines are the lines of maximum and minimum gradient.

A student uses a small microwave radar module, of the type used in automatic doors, to test the Doppler effect for electromagnetic waves. The module transmits microwaves of frequency f = 24.125 GHz along a straight track that slopes down towards it. A trolley released from rest on the track rolls directly towards the module and reflects the microwaves back to it. The module outputs a signal whose frequency f_D is the difference between the transmitted and received frequencies. Theory predicts f_D = 2fv/c, where v is the speed of the trolley and c is the speed of light.

A light gate measures the time t for a card of length (0.1000 ± 0.0005) m on the trolley to pass through it, with an uncertainty of ±0.2 ms, and v is calculated from the card length and t. A data logger records f_D, with an uncertainty of ±4 Hz, as the card passes through the gate. The trolley is released from six different points. The graph shows f_D against v with error bars.

The Sun moves relative to the stars in its neighbourhood. A student uses a published catalogue of stellar spectra to determine the speed V of this motion. She selects seven groups of nearby stars in different directions in the sky. For each group the catalogue gives the mean shift Δλ = λ_observed − λ₀ of a calcium absorption line of laboratory wavelength λ₀ = 422.67 nm, after correction for the motion of the Earth. θ is the angle between the direction to the group and the direction in which the Sun is moving. The stars in each group move randomly, so that on average each group is at rest in the neighbourhood of the Sun. Each value of Δλ has an uncertainty of ±3 pm (1 pm = 10⁻¹² m).

The graph shows Δλ against cos θ, with error bars.

An online spectroscopic archive lists the measured wavelength λ of the hydrogen absorption line whose laboratory wavelength is 656.28 nm, in the spectrum of a star S that orbits a companion star. The wavelengths are quoted with an uncertainty of ±0.005 nm. A student calculates the velocity v of S along the line of sight (positive when S is moving away from the Earth) and plots v against time t.

A student uses archived spectra of sunlight reflected from Saturn's rings to test two models of the rings. In model 1 the rings are a solid disc rotating as a single body, so that the speed v of the ring material is proportional to its distance r from the centre of Saturn. In model 2 the rings consist of separate particles, each moving in its own circular orbit around Saturn, so that v² = GM/r, where M is the mass of Saturn.

At seven values of r, on the side of the rings where the ring material moves directly away from the Earth, the shift Δλ of a solar absorption line of laboratory wavelength 516.73 nm was measured, after correction for the motion of Saturn itself. Because the light is sunlight reflected by moving material, and the Sun and the Earth are in almost the same direction as seen from Saturn, Δλ/λ = 2v/c. Each Δλ has an uncertainty of ±0.3 pm. The uncertainty in r is negligible.

The graph shows lg(v/m s⁻¹) against lg(r/m) for six of the measurements.

A student whirls a small buzzer on a string in a horizontal circle of radius 1.20 m, at a constant rate of 2.0 revolutions per second. The buzzer emits sound of constant frequency 2.40 kHz. A listener L stands in the same horizontal plane, a large distance from the circle.

Figure 1 is a plan view. The buzzer moves anticlockwise through the points A, B, C and D.

The figure shows four absorption lines of hydrogen in a laboratory spectrum, and the same four lines in the spectrum of a distant galaxy. The laboratory wavelengths are 410.2 nm, 434.0 nm, 486.1 nm and 656.3 nm. The wavelengths measured in the galaxy spectrum are marked on the figure.

A Doppler ultrasound probe P is used to study the flow of blood in an artery in the arm. P is held against the skin and emits ultrasound of frequency 5.0 MHz. Red blood cells moving in the artery scatter some of the ultrasound back to P, which detects it. The speed of ultrasound in soft tissue and blood is 1.54 × 10³ m s⁻¹.

Astronomers detect an exoplanet by observing a periodic Doppler shift in the light from its host star. A sodium line with a laboratory wavelength of 589.0000 nm is observed to vary repeatedly between a minimum of 588.9996 nm and a maximum of 589.0004 nm. The speed of light is c = 3.00 × 10⁸ m s⁻¹.

A hydrogen line has a laboratory wavelength of 486.1 nm. The table gives the wavelength of this line observed in the spectra of three galaxies.

Galaxy | Observed wavelength / nm
A | 487.0
B | 485.6
C | 490.0

c = 3.00 × 10⁸ m s⁻¹.

A radar speed gun at a tennis tournament measures the speed of a serve. The gun emits microwaves of frequency 10.5 GHz towards a ball moving directly towards it. The reflected microwaves return to the gun.

The engine of a racing car emits a note of frequency 680 Hz. The car moves in a straight line at constant speed. The figure is a computer simulation, drawn to scale, of four sound wavefronts emitted by the car, seen from above at one instant. The small dots show the positions of the car when the wavefronts were emitted; dot 1 is where the largest wavefront was emitted. D is the position of the car at the instant shown. The speed of sound in air is 340 m s⁻¹.

A distant star lies in the plane of the Earth's orbit. The Earth moves around the Sun in a circular orbit of radius 1.50 × 10¹¹ m with a period of 365 days. An absorption line in the spectrum of the star has a laboratory wavelength of 589.000 nm. The table gives the wavelength of the line measured on two dates, six months apart, when the Earth was moving directly towards the star and directly away from it.

A student investigates the Doppler effect by releasing a small buzzer from rest at the top of a stairwell. The buzzer emits a note of constant frequency 3.20 kHz and falls 12.0 m vertically to a thick cushion on the floor, where it stops and continues to sound. Microphone M1 is at the point of release and microphone M2 is on the floor beside the cushion. Both microphones record the frequency of the sound they receive. Air resistance on the buzzer is negligible. The speed of sound in air is 340 m s⁻¹.

A deep-space probe transmits a radio signal of frequency 8.40 GHz. A ground station receives the signal with a frequency differing from the transmitted frequency by 280 kHz. The probe is moving directly away from Earth. c = 3.00 × 10⁸ m s⁻¹.

An astronomer measures three absorption lines in the spectrum of a galaxy. The table gives their laboratory and observed wavelengths.

Line | Laboratory wavelength / nm | Observed wavelength / nm
H-alpha | 656.3 | 669.4
H-beta | 486.1 | 495.8
Calcium K | 393.4 | 401.2

A space probe transmits a radio signal of frequency 8.400 000 GHz. A ground station receives the signal at 8.399 720 GHz. Assume the probe moves along the line joining it to the ground station.

A nova is an explosion on the surface of a white dwarf star. The explosion throws off a thin spherical shell of hot gas that expands in all directions. Assume that the speed of expansion is constant. Hydrogen in the shell emits light whose wavelength measured in the laboratory is 486.13 nm. Figure 1 shows how the intensity of this emission line, observed from the Earth, varies with wavelength. The white dwarf is at rest relative to the Earth.

A weather radar transmits pulses of microwaves of frequency 2.80 GHz and detects the microwaves reflected by raindrops. When the raindrops have a velocity component v along the radar beam, the frequency of the reflected microwaves received at the radar differs from the transmitted frequency f by Δf, where Δf = 2fv/c for v ≪ c.

A planet orbits a star of mass 2.2 × 10³⁰ kg. The gravitational pull of the planet makes the star move in a small circular orbit with the same period as the planet. Astronomers measure the wavelength of an absorption line in the spectrum of the star, whose laboratory wavelength is 500.00 nm. The graph shows how the change Δλ in the measured wavelength varies with time t. The Earth lies in the plane of the orbits.

A small satellite moves in a circular orbit at a height of 500 km above the surface of the Earth. It transmits a radio signal of frequency 437.100 MHz. A ground station directly below the path of the satellite records the difference Δf between the received frequency and 437.100 MHz as the satellite passes overhead. Figure 1 shows how Δf varies with time t while the satellite is above the horizon; the satellite is directly overhead at t = 0. Ignore the rotation of the Earth.

Mass of the Earth = 5.97 × 10²⁴ kg; radius of the Earth = 6.37 × 10⁶ m.

In a tokamak fusion reactor, a plasma of deuterium and tritium nuclei is held at a very high temperature by magnetic fields. The plasma also contains a small number of carbon ions, which emit light of laboratory wavelength 529.05 nm. A spectrometer views the plasma. Figure 1 shows how the intensity of this line, as observed, varies with wavelength.

Mass of a carbon ion = 1.99 × 10⁻²⁶ kg; mass of a deuterium ion = 3.34 × 10⁻²⁷ kg.

A Doppler weather radar emits microwaves of frequency 2.80 GHz towards a rotating thunderstorm. Two regions of rain, P and Q, lie on opposite sides of the storm's centre, each the same distance from it. They are 10.0 km apart, on a line through the centre that is perpendicular to the radar beam. The reflected signal from region P has a frequency 560 Hz higher than the emitted frequency. The reflected signal from region Q has a frequency 373 Hz lower than the emitted frequency.