Exercises: Changes due to displacement of emitter and receiver (Doppler effect)

An ambulance with its siren on approaches a pedestrian, passes them and moves away.

1. a) Describe how the pitch perceived by the pedestrian changes as the ambulance approaches and as it moves away (higher or lower pitch).
2. b) Explain the change using the compression and spreading out of the wavefronts, using a diagram in your notebook to help you.
3. c) Does the frequency actually emitted by the siren change? Distinguish between emitted frequency and perceived frequency.

When a source approaches a stationary observer, the observer perceives the frequency $f' = f \cdot \frac{v}{v - v_f}$ (formula given; speed of sound in air: $v = 340\ \text{m/s}$).

An ambulance emits sound at $500\ \text{Hz}$ and approaches a pedestrian at $20\ \text{m/s}$. Calculate the frequency perceived by the pedestrian.

When the source moves away from a stationary observer, the formula is $f' = f \cdot \frac{v}{v + v_f}$ (formula given; $v = 340\ \text{m/s}$).

1. a) Using the same ambulance as before ($f = 500\ \text{Hz}$, $v_f = 20\ \text{m/s}$), calculate the frequency perceived as it moves away.
2. b) Verify that it is lower than the emitted frequency and explain it qualitatively using the spacing of the wavefronts.

When the observer moves towards a stationary source, the formula is $f' = f \cdot \frac{v + v_0}{v}$ (formula given; $v = 340\ \text{m/s}$).

A cyclist rides at $10\ \text{m/s}$ towards a loudspeaker emitting a $600\ \text{Hz}$ tone. The figure shows the situation.

Calculate the frequency perceived by the cyclist.

In the spectra of distant galaxies, the absorption lines of hydrogen appear shifted towards longer wavelengths than those measured in the laboratory.

1. a) What is this shift called, and what does it indicate about the movement of the galaxy relative to Earth?
2. b) Relate the widespread observation of this phenomenon to the expansion of the universe.
3. c) Reason why the optical Doppler effect does not require a material medium, unlike the acoustic Doppler effect.

The H$\alpha$ line of hydrogen is measured in the laboratory at $\lambda_0 = 656,3\ \text{nm}$ and appears in the spectrum of a galaxy at $\lambda' = 662,0\ \text{nm}$. The figure shows both measurements.

Redshift is defined as $z = \frac{\lambda' - \lambda_0}{\lambda_0}$ (formula given).

Calculate $z$ and interpret its sign in terms of approach or recession.

Without making calculations, state in each case whether the perceived frequency is greater than, less than, or equal to the emitted frequency, and justify your answer:

1. a) A train approaching the station sounding its whistle.
2. b) The same train moving away from the station.
3. c) A pedestrian running towards a stationary siren.
4. d) A galaxy moving away from Earth.
5. e) Source and observer in relative rest.

A traffic radar emits microwaves that reflect off vehicles and return to the device. The diagram shows the emission and the echo.

1. a) Explain the two successive Doppler effects experienced by the wave: the car as an 'observer' receiving the signal and as a 'source' re-emitting the echo.
2. b) Reason why the echo returns with a higher frequency if the car is approaching the radar.
3. c) What wave phenomenon makes it possible for the signal to return to the radar?

An ambulance emits sound at $700\ \text{Hz}$ and travels at $90\ \text{km/h}$ (speed of sound: $v = 340\ \text{m/s}$).

1. a) Express the speed of the ambulance in $\text{m/s}$.
2. b) Calculate the frequency perceived by a stationary pedestrian as it approaches, using $f' = f \cdot \frac{v}{v - v_f}$.
3. c) Calculate it as it moves away, using $f' = f \cdot \frac{v}{v + v_f}$.

A traffic radar emits microwaves at $f = 2,4 \cdot 10^{10}\ \text{Hz}$. For speeds much smaller than the speed of light, the frequency shift of the echo is $\Delta f \approx \frac{2 \cdot f \cdot v}{c}$ (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$). An approaching car returns an echo with $\Delta f = 4 \cdot 10^{3}\ \text{Hz}$.

1. a) Rearrange the formula for the speed of the car and calculate it in $\text{m/s}$.
2. b) Express it in $\text{km/h}$.

For speeds small compared to the speed of light, the redshift satisfies $z \approx \frac{v}{c}$ (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$). A galaxy exhibits $z = 0,02$.

1. a) Calculate its speed in $\text{m/s}$ and in $\text{km/s}$.
2. b) State whether the galaxy is approaching or moving away from us, justifying your answer using the sign of $z$.

If both the source and the observer move towards each other at the same time, the perceived frequency is $f' = f \cdot \frac{v + v_0}{v - v_f}$ (formula given; $v = 340\ \text{m/s}$).

A train sounds a $400\ \text{Hz}$ whistle as it approaches at $30\ \text{m/s}$, while a cyclist pedals at $10\ \text{m/s}$ towards it. The figure shows the situation.

Calculate the frequency perceived by the cyclist.

In a Doppler ultrasound scan of a blood vessel, the frequency shift of the echo is $\Delta f \approx \frac{2 \cdot f \cdot v_{\text{sangre}}}{v}$ (formula given; speed of sound in tissue: $v = 1540\ \text{m/s}$). The probe emits at $5\ \text{MHz}$ and the blood flows at $0,3\ \text{m/s}$.

1. a) Calculate $\Delta f$.
2. b) Compare the result with the range of human hearing ($20\ \text{Hz}$ to $20\ \text{kHz}$) and reason why the equipment converts this shift into an audible tone for the doctor.

A car with its alarm sounding at $800\ \text{Hz}$ passes a stationary person while travelling at $20\ \text{m/s}$ ($v = 340\ \text{m/s}$).

1. a) Calculate the frequency perceived as it approaches.
2. b) Calculate the frequency perceived as it moves away.
3. c) Determine the sudden drop in frequency perceived at the exact moment the car passes.

The siren of an emergency motorbike emits sound at $850\ \text{Hz}$ and, as it approaches, a pedestrian perceives $920\ \text{Hz}$ ($v = 340\ \text{m/s}$).

1. a) Rearrange $f' = f \cdot \frac{v}{v - v_f}$ for $v_f$.
2. b) Calculate the speed of the motorbike in $\text{m/s}$ and express it in $\text{km/h}$.

The Sun rotates on its axis: its eastern limb approaches us at $2\ \text{km/s}$ and its western limb moves away at the same speed. The figure outlines solar rotation.

For small speeds, $\frac{\Delta \lambda}{\lambda} = \frac{v}{c}$ holds true (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the shift $\Delta \lambda$ of the $\lambda = 589\ \text{nm}$ line in the light coming from each limb, in nanometres.
2. b) Determine the wavelength difference between the lines from both limbs, which is what spectrographs measure to study solar rotation.

An ambulance emits sound at $600\ \text{Hz}$ and travels at $72\ \text{km/h}$ ($v = 340\ \text{m/s}$).

1. a) Express the speed in $\text{m/s}$.
2. b) Calculate the frequencies perceived by a stationary pedestrian as it approaches and as it moves away.
3. c) Determine the drop in frequency perceived just as it passes, and use it to explain the characteristic change in pitch of sirens when they overtake us.
4. d) Draw in your notebook the wavefronts of a moving source and indicate where they bunch up and where they spread apart.

You can do it on paper and send a photo with your answer.

A traffic radar emits at $1,0 \cdot 10^{10}\ \text{Hz}$ and measures the echo from vehicles using $\Delta f \approx \frac{2 \cdot f \cdot v}{c}$ (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the $\Delta f$ produced by a car approaching at $144\ \text{km/h}$ ($40\ \text{m/s}$).
2. b) On a road section with a speed limit of $120\ \text{km/h}$, reason whether this vehicle commits a speed offence.
3. c) Explain the origin of the factor $2$ in the formula (double Doppler effect on the outbound and return path of the echo) and why the radar must measure frequencies with extraordinary precision.

The H$\alpha$ line of hydrogen ($\lambda_0 = 656,3\ \text{nm}$ in the laboratory) is observed in the spectrum of a galaxy at $\lambda' = 678,9\ \text{nm}$ (data: $z = \frac{\Delta \lambda}{\lambda_0}$; $v \approx z \cdot c$; Hubble's law $v = H_0 \cdot d$, with $H_0 = 70\ \text{km/s}$ per $\text{Mpc}$; $c = 3 \cdot 10^{5}\ \text{km/s}$).

1. a) Calculate the redshift $z$ of the galaxy.
2. b) Calculate its speed in $\text{km/s}$.
3. c) Calculate its distance in $\text{Mpc}$.
4. d) Explain what cosmological consequence is deduced from the fact that distant galaxies systematically show $z > 0$.

In front of a moving source, the perceived frequency is $f' = f \cdot \frac{v}{v - v_f}$ (formula given; $v = 340\ \text{m/s}$). The Mach number is the ratio of an aircraft's speed to the speed of sound. The figure shows the wavefronts of a source moving at nearly the speed of sound.

1. a) Calculate $f'$ for an aircraft at Mach $0,9$ ($v_f = 306\ \text{m/s}$) whose engine emits a noise of $1000\ \text{Hz}$.
2. b) Analyse what happens to the denominator of the formula and to $f'$ as $v_f$ approaches $v$ (Mach $1$).
3. c) Relate this to the shock wave and sonic boom: explain why an observer on the ground does not hear a supersonic aircraft approaching until after it has already passed them.

The radial velocity method detects exoplanets from the wobble they induce in their star: spectral lines oscillate periodically with $\frac{\Delta \lambda}{\lambda} = \frac{v}{c}$ (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$). In the $\lambda = 656,3\ \text{nm}$ line of a star, a maximum oscillation of $\Delta \lambda = 1,3 \cdot 10^{-4}\ \text{nm}$ is measured. The diagram shows the wobble of the star.

1. a) Express both lengths in metres and calculate the maximum speed of the star.
2. b) Explain why this periodic oscillation reveals an invisible planet that forces the star to orbit the common centre of mass of the system.
3. c) Reason what additional information about the planet the period of oscillation would provide.

A weather radar emits at $5,6 \cdot 10^{9}\ \text{Hz}$ and measures the echo from raindrops using $\Delta f \approx \frac{2 \cdot f \cdot v}{c}$ (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate $\Delta f$ for raindrops moving away from the radar at $15\ \text{m/s}$.
2. b) State whether the echo returns with a higher or lower frequency than the emitted frequency and justify your answer.
3. c) Explain how, by sweeping the beam in different directions, the radar reconstructs a wind map inside a storm.

A low Earth orbit satellite transmits at $f = 2,2\ \text{GHz}$ and moves at $7,6\ \text{km/s}$ relative to the ground station. For $v \ll c$, $\frac{\Delta f}{f} = \frac{v}{c}$ holds true (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the maximum shift $\Delta f$, expressed in $\text{kHz}$.
2. b) Determine the frequency range that the station must sweep through during a full pass, with the satellite first approaching and then moving away.
3. c) Reason why satellite receivers must continuously retune during each pass.

A bat flies at $6\ \text{m/s}$ towards a stationary moth while emitting ultrasound at $80\ \text{kHz}$ ($v = 340\ \text{m/s}$). The diagram shows the emission and the echo.

1. a) Calculate the frequency 'received' by the moth, using $f_1 = f \cdot \frac{v}{v - v_f}$.
2. b) The reflected echo behaves as if the moth were a stationary source of frequency $f_1$ and the bat an observer approaching at $6\ \text{m/s}$: calculate the final frequency heard by the bat, $f_2 = f_1 \cdot \frac{v + v_0}{v}$.
3. c) Compare $f_2$ with the emitted frequency $f$ and reason why this double Doppler effect facilitates prey detection.
4. d) Predict qualitatively how $f_2$ would change if the moth were also moving, fleeing from the bat or flying towards it.

Unit challenge. How does wave physics condition the acoustics of your environment? Design a home sound audit following these stages:

1. Check the ambient temperature ($T$) of your locality on AEMET to calculate the theoretical speed of sound: $v = 331,5 + 0,6 \cdot T\text{ m/s}$. Measure the echo delay facing a wall at a known distance ($d \pm \Delta d$) with your smartphone by taking five readings. Tabulate the data ($t \pm \Delta t$) and obtain $v_{\text{exp}} \pm \Delta v$.

2. Using a free audio app (such as Phyphox), measure the fundamental frequency and intensity level in $\text{dB}$ of a constant emitter. Analyse the sound characteristics and their attenuation after interposing various household materials (absorption and reflection).

3. Record the passage of a vehicle or a bicycle with a continuous bell at constant speed emitting a tone $f_0$. Extract the received frequencies upon approaching and receding from the spectrogram. Calculate the source speed using the Doppler effect equation and plot the theoretical curves of $f'$ versus source speed (approaching and receding), placing your measured values on them.

4. Write a short report integrating your graphs, physical conclusions, and a justified proposal to mitigate urban noise in your area.

Assessment criteria:

• Rigour in experimental data processing, uncertainty calculation, and quality of the table and graph.

• Correct application of theoretical models of sound characteristics, propagation, and the Doppler effect.

• Scientific coherence of the report and feasibility of the proposed acoustic solutions.