Doppler Effect

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수업 노트

The Doppler Effect

The Doppler effect is the change in observed frequency (and wavelength) of a wave due to relative motion between the source and the observer.
When a source moves towards a stationary observer, the observed frequency increases (higher pitch) and the wavelength decreases.
When a source moves away from a stationary observer, the observed frequency decreases (lower pitch) and the wavelength increases.
The source itself continues to emit sound at a constant frequency; only the observed frequency changes.
The Doppler effect applies to all waves, including sound and electromagnetic waves (light).
For light, motion away from the observer causes redshift (longer wavelength, lower frequency), while motion towards causes blueshift (shorter wavelength, higher frequency).

Representing the Doppler Effect

Wavefront diagrams show how the spacing of wavefronts changes when the source moves.
For a moving source, wavefronts are squashed in the direction of motion and stretched behind.
The wavelength in front of the source becomes shorter (λ – Δλ) and the frequency increases.
The wavelength behind the source becomes longer (λ + Δλ) and the frequency decreases.
The change in wavelength, Δλ, determines the size of the Doppler shift: a bigger Δλ means a bigger shift.

The Doppler Effect of Light

For a non-relativistic light source (v << c), the Doppler shift is given by: Δf/f = Δλ/λ ≈ Δv/c.
Δf is the change in frequency, f the original frequency, Δλ the change in wavelength, λ the original wavelength, Δv the relative velocity, and c the speed of light.
The change in wavelength is Δλ = λ₀ − λ, where λ₀ is the observed wavelength and λ the reference wavelength.
The relative velocity along the line joining source and observer is Δv = v_s − v_o; if the observer is stationary, Δv = v_s.
The equation can be rearranged as Δλ/λ = (λ₀ − λ)/λ ≈ v/c or Δf/f = (f₀ − f)/f ≈ v/c.
The Doppler shift has no units because it is a ratio of like quantities.

Spectral Lines and Redshift

Doppler shift is observed in atomic spectral lines from stars and galaxies.
Each spectral line corresponds to an element in the source's composition.
When lines are shifted towards the red end (longer wavelengths), the source is moving away from Earth.
When lines are shifted towards the blue end (shorter wavelengths), the source is moving towards Earth.
Redshift is defined as the fractional increase in wavelength (or decrease in frequency) due to the source and observer receding from each other.
Blueshift is the fractional decrease in wavelength (or increase in frequency) due to the source and observer approaching each other.

Galactic Redshift and the Expanding Universe

Almost all galaxies show redshift, meaning they are receding from Earth.
This led to the idea that space itself is expanding, stretching light waves as they travel.
The expansion can be compared to dots on an inflating balloon: as the balloon expands, the dots move apart, but they do not move through the rubber themselves.
The greater the distance to a galaxy, the greater its redshift and the faster it is receding.
The furthest galaxies appear the most redshifted and are receding the fastest.
Redshift provided evidence for the Big Bang and the expansion of the universe.

Positive and Negative Velocities

If the calculated velocity of a galaxy relative to Earth is positive, the galaxy is moving towards Earth (observed frequency > reference frequency).
If the calculated velocity is negative, the galaxy is moving away from Earth (observed frequency < reference frequency).
Keeping track of the minus sign in calculations tells you the direction of motion.
The speed of light c is given in the data booklet and does not need to be memorised.

Equations for the Doppler Effect of Sound

For a moving source and stationary observer: f' = f (v / (v ± u_s)).
Use v − u_s in the denominator when the source moves towards the observer (frequency increases).
Use v + u_s in the denominator when the source moves away from the observer (frequency decreases).
For a moving observer and stationary source: f' = f ((v ± u_o) / v).
Use v + u_o in the numerator when the observer moves towards the source (frequency increases).
Use v − u_o in the numerator when the observer moves away from the source (frequency decreases).
The speed of sound in air is typically 340 m s⁻¹.
The equations can also be written in terms of wavelength: for a moving source, λ' = λ (1 ± u_s/v).

Applications and Problem-Solving

The Doppler effect is used to measure blood flow using ultrasound.
It is used to find planetary orbits around distant stars and to map the expansion of the universe.
When a wave reflects off a moving object, the change in frequency is doubled compared to a stationary observer detecting the wave directly.
In calculations, clearly label the source and observer to choose the correct signs in the equations.
Remember that the speed of the wave (e.g., speed of sound) does not change due to the Doppler effect; only frequency and wavelength change.

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연습 문제

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1.A train is moving towards a stationary observer. The train sounds its horn. What are the correct changes in frequency, wavelength and pitch of the horn as heard by the observer?
Easy
- AFrequency: higher, Wavelength: shorter, Pitch: higher
- BFrequency: higher, Wavelength: shorter, Pitch: lower
- CFrequency: lower, Wavelength: shorter, Pitch: lower
- DFrequency: lower, Wavelength: longer, Pitch: higher
2.A team of naturalists are researching the movement of whales in the ocean. They plan to calculate the velocity of a whale using the Doppler effect. The whale pod is moving towards the research team who are in a stationary boat. Which equation will allow the researchers to investigate the velocity of the whales using the frequency of sound in water?
Easy
- Af' = f (v / (v + u_s))
- Bf' = f ((v + u_o) / v)
- Cf' = f (v / (v - u_s))
- Dλ' = λ (1 + u_s / v)
3.Which diagram correctly represents redshift?
Medium
- ASpectral lines shifted towards the red end of the spectrum
- BSpectral lines shifted towards the blue end of the spectrum
- CSpectral lines unchanged from the reference spectrum
- DSpectral lines split into two sets, one red and one blue
4.What did the discovery of the Doppler redshift give scientists evidence for?
Easy
- ANewton's Third Law
- BThe formation of solar systems
- CThe Big Bang
- DBlueshift
5.Which equation can be used to calculate the Doppler shift for the sound of a person running away from an observer whilst blowing a whistle?
Easy
- Af' = f ((v + u_o) / v)
- Bf' = f (v / (v - u_s))
- Cf' = f ((v - u_o) / v)
- Df' = f (v / (v + u_s))
6.Which of the following are uses or applications of the Doppler effect? (Select all that apply.)
Medium
- AMapping the expansion of the universe
- BMeasuring the rate of blood flow in patients
- CFinding planetary orbits around distant stars
- DRecording the speed of sound
- EMeasuring the speed of a passing car with radar
7.Which statement about redshift is true?
Easy
- ARedshift shows all galaxies are moving towards the Earth
- BRedshift shows that the space between galaxies is expanding
- CRedshift is the change in pitch of the sound waves emitted from galaxies
- DRedshift is the expansion of stars
8.Which of the following equations will give an increase in the frequency of the observed wave according to the Doppler Effect?
Medium
- Af' = f ((v - u_o) / v)
- Bλ' = λ (1 + u_s / v)
- Cf' = f ((v + u_o) / v)
- Df' = f (v / (v + u_s))

더 많은 문제

9.Which of the following statements about the Doppler effect is correct?
Medium
- AWhen a galaxy moves towards the Earth the spectral absorption lines observed will have a longer wavelength
- BWhen a car starts to move away from an observer, the sound from the exhaust appears at a lower frequency to the observer than the driver
- CWhen a train moves towards a station, its horn will sound lower to the passengers on the platform compared to the passengers in the train
- DLight from a star moving towards the Earth is redshifted
10.Which of the following is an equation to calculate the Doppler shift of light?
Medium
- AΔf / f ≈ c / v
- BΔf / f ≈ v / c
- Cf / Δf ≈ c / v
- DΔf / f ≈ v / c²
11.A train is moving away from an observer on a platform with a constant speed v/17, where v is the speed of sound. The train emits sound of frequency f. What is the observed speed of sound and the observed frequency as the train recedes?
Medium
- AObserved speed of sound: v, Observed frequency: 18f/17
- BObserved speed of sound: v, Observed frequency: 17f/18
- CObserved speed of sound: v/17, Observed frequency: 17f/18
- DObserved speed of sound: 17v, Observed frequency: 17f/18
12.In the laboratory on Earth, hydrogen emits a spectral line that is measured by an observer to have a wavelength of 400 nm. The same spectral line is emitted by a galactic source that is moving away from Earth at speed of 0.2c, where c is the speed of light. What is the wavelength of the same spectral line from the galaxy as measured by the observer on Earth?
Medium
- A480 nm
- B320 nm
- C80 nm
- D2000 nm
13.A car travels in a straight line towards a stationary observer at a speed of 0.2v, where v is the speed of sound. The car's horn emits a pure sound of frequency f. What is the frequency of the sound of the horn as heard by the observer?
Medium
- A0.80f
- B1.25f
- C0.83f
- D1.20f
14.An observer approaches a stationary source of sound. The speed of the emitted sound and its wavelength, measured at the source, are v and λ respectively. Which of the following is the observed wave speed and the observed wavelength, as measured by the moving observer?
Medium
- AObserved wave speed: > v, Observed wavelength: < λ
- BObserved wave speed: = v, Observed wavelength: > λ
- CObserved wave speed: < v, Observed wavelength: > λ
- DObserved wave speed: = v, Observed wavelength: < λ
15.A spectral line is observed at 530 nm in the laboratory. The same spectral line is observed in a galaxy which is receding from Earth at a velocity of 3.00 × 10⁶ m s^−1. What is the wavelength of the spectral line from the galaxy?
Medium
- A534.7 nm
- B5.3 nm
- C535.3 nm
- D583 nm
16.A star emits light of frequency f. This light is observed on Earth to have a frequency f' which is of a lower frequency than f. Which of the following correctly describes the speed and direction of motion of the star?
Medium
- ASpeed: ((f − f') / f) c, Direction: Towards Earth
- BSpeed: ((f − f') / f') c, Direction: Towards Earth
- CSpeed: ((f − f') / f) c, Direction: Away from Earth
- DSpeed: ((f − f') / f') c, Direction: Away from Earth
17.Two lines P and Q in the emission spectrum of a gas are measured on the Earth by a stationary observer. The observer then moves towards the gas sample, and the emission spectrum is recorded as they are moving. What is the correct pattern for the shifted spectral lines, P' and Q'?
Medium
- AP' and Q' are both shifted towards shorter wavelengths, with the shift proportional to their original wavelengths
- BP' and Q' are both shifted towards longer wavelengths, with the shift proportional to their original wavelengths
- CP' is shifted towards shorter wavelengths and Q' towards longer wavelengths
- DP' and Q' are both shifted towards shorter wavelengths by the same absolute amount
18.Which of the following statements is true about redshift?
Medium
- ARedshift led to the idea that galaxies are expanding
- BA greater redshift means a faster speed of recession
- CRedshift shows a few of the galaxies in the universe are moving away from a single point
- DRedshift causes electromagnetic waves to change frequency and speed
19.A car of terrified observers drives away from a stationary, but hungry T. rex, at a constant speed of 10 m s^−1. They hear the roar of the T. rex at a frequency of 170 Hz. The speed of sound in air is 340 m s^−1. What is the frequency of the sound as emitted by the T. rex?
Medium
- A165 Hz
- B175 Hz
- C180 Hz
- D220 Hz
20.A stationary source emits sound wave with frequency f. An observer moving towards the source measures the sound as having frequency f'. The speed of the observer is 20% of the speed of sound. What is the value of f' when the emitted frequency is 10 Hz?
Hard
- A0.8 Hz
- B1.2 Hz
- C10.2 Hz
- D12 Hz
21.The flow of blood can be monitored using a machine which emits ultrasound of 10 MHz and is then able to measure the change in frequency when the pulse returns having reflected off a red blood cell. When a wave is reflected from a moving object, the change in frequency is doubled compared to a situation where a sound is detected by a moving observer. Assume that the speed of sound in the body is the same as that of water, 1500 m s^−1. What is the frequency detected if blood is flowing away at 0.15 m s^−1?
Hard
- A10.002 MHz
- B2000 Hz
- C9.998 MHz
- D−2000 Hz
22.A boat sits on a lake, and is hit by 4 waves every 10 s. As the boat starts to move in the same direction as the waves, exactly 2 waves from behind hit the boat every 10 s. The speed of the boat is 1.5 m s^−1. What is the speed of the waves?
Hard
- A1.5 m s^−1
- B3 m s^−1
- C1500 m s^−1
- D340 m s^−1
23.In a binary star system, two stars orbiting each other can be observed using a spectrometer. The star system as a whole is moving away from the Earth at a constant velocity. The spectral lines appear to fluctuate. A graph can be plotted of the relative velocity of each star compared to the observer. At which point on the graph will the stars be in the following position?
Hard
- AWhen the stars are side-by-side (perpendicular to the line of sight)
- BWhen one star is moving directly towards Earth and the other directly away
- CWhen both stars are moving directly away from Earth at the same speed
- DWhen both stars are moving directly towards Earth at the same speed
24.Photographs of lines in the spectrum of the Sun show changes in wavelength. The atmosphere of the Sun rotates with the Sun. The lines from the east limb are shifted towards the red end of the spectrum, while those from the west limb are shifted towards the blue end. What does this indicate?
Medium
- AThe east limb is moving towards Earth and the west limb away
- BThe east limb is moving away from Earth and the west limb towards Earth
- CThe Sun is moving away from Earth as a whole
- DThe Sun is moving towards Earth as a whole
25.The Doppler effect is the change in observed frequency due to relative motion between a source and an observer.
Easy
True or false?
26.When a source of sound moves away from a stationary observer, the observed frequency is higher than the emitted frequency.
Easy
True or false?
27.Redshift is observed when a light source is moving away from the observer.
Easy
True or false?
28.The speed of sound in air changes when the source of the sound moves.
Easy
True or false?
29.The Doppler shift equation for light, Δf/f ≈ v/c, is valid only when the relative speed v is much less than the speed of light c.
Easy
True or false?
30.If a galaxy's observed frequency is greater than the reference frequency, the galaxy is moving away from Earth.
Medium
True or false?
31.Match each term with its correct definition.
Medium
- Redshift
- Blueshift
- Doppler effect
- Wavelength increases, frequency decreases
- Wavelength decreases, frequency increases
- Change in observed frequency due to relative motion
32.Place the steps of the Doppler shift calculation for a receding galaxy in the correct order.
Medium
- Identify the reference wavelength λ and observed wavelength λ0
- Calculate the change in wavelength Δλ = λ0 − λ
- Use the equation Δλ / λ = v / c
- Rearrange to find v = c Δλ / λ
- Substitute values and calculate v
33.Match each Doppler effect term with its correct description.
Medium
- Redshift
- Blueshift
- Doppler effect
- Fractional increase in wavelength due to source and observer receding
- Fractional decrease in wavelength due to source and observer approaching
- Change in observed frequency due to relative motion between source and observer
34.Place the steps for calculating the speed of a receding galaxy using the Doppler shift equation in the correct order. Use the equation Δλ/λ = v/c.
Medium
- Identify the reference wavelength λ and observed wavelength λ0 from the spectra
- Calculate the change in wavelength using Δλ = λ0 − λ
- Rearrange the Doppler shift equation to make v the subject: v = cΔλ/λ
- Substitute the values into the equation and calculate v
- State whether the galaxy is moving towards or away from Earth based on the sign of v
35.Which of the following are correct statements about the Doppler effect for sound? (Select all that apply.)
Medium
- AWhen a source moves towards a stationary observer, the observed frequency is higher than the emitted frequency.
- BWhen a source moves away from a stationary observer, the observed wavelength is shorter than the emitted wavelength.
- CThe speed of sound in air is independent of the motion of the source.
- DWhen an observer moves towards a stationary source, the observed frequency is higher than the source frequency.
- EThe Doppler effect causes a permanent change in the frequency emitted by the source.
36.Which of the following statements about redshift are true? (Select all that apply.)
Medium
- ARedshift is observed when a light source moves away from the observer.
- BA greater redshift indicates a faster speed of recession.
- CRedshift provides evidence that the universe is expanding.
- DRedshift causes the speed of light to decrease.
- ERedshift is observed when a light source moves towards the observer.
37.Which equation represents the Doppler shift for light when the source is moving away from the observer?
Medium
- AΔf/f ≈ v/c
- BΔf/f ≈ −v/c
- CΔλ/λ ≈ −v/c
- Df' = f(v/(v + us))
38.A stationary source emits sound of frequency 500 Hz. An observer moves away from the source at 20 m s⁻¹. The speed of sound in air is 340 m s⁻¹. What is the observed frequency?
Medium
- A471 Hz
- B531 Hz
- C500 Hz
- D529 Hz
39.A source of sound moves towards a stationary observer at one-third the speed of sound. The emitted frequency is f. What is the observed frequency?
Medium
- A0.75f
- B1.33f
- C1.50f
- D0.67f
40.A distant galaxy is observed to have a spectral line at 650 nm, whereas the same line in the laboratory is at 600 nm. What is the speed of the galaxy relative to Earth? (c = 3.0 × 10⁸ m s⁻¹)
Hard
- A2.5 × 10⁷ m s⁻¹, moving away
- B2.5 × 10⁷ m s⁻¹, moving towards
- C3.0 × 10⁷ m s⁻¹, moving away
- D1.5 × 10⁷ m s⁻¹, moving away
41.What term describes the change in observed frequency due to relative motion between a source and an observer?
Easy
42.What is the name for the shift in wavelength towards the red end of the spectrum when a light source moves away from an observer?
Easy
43.What is the name for the shift in wavelength towards the blue end of the spectrum when a light source moves towards an observer?
Easy
44.A stationary source emits sound of frequency 500 Hz. An observer moves towards the source at 34 m s^−1. The speed of sound in air is 340 m s^−1. Calculate the observed frequency in Hz.
Medium
45.A galaxy is receding from Earth at 1.5 × 10⁶ m s^−1. A spectral line in the laboratory has a wavelength of 500 nm. Calculate the observed wavelength from the galaxy in nm. (Speed of light c = 3.0 × 10⁸ m s^−1.)
Medium
46.Complete the sentence about the Doppler effect for a moving source.
Easy
When a source of sound moves towards a stationary observer, the observed frequency is ____ and the observed wavelength is ____.
47.Complete the equation for the Doppler shift of light.
Medium
Δf / f ≈ ____ / ____
48.A galaxy is receding from Earth at 1.5 × 10⁶ m s⁻¹. A spectral line in the laboratory has a wavelength of 500 nm. Calculate the observed wavelength from the galaxy in nm. (c = 3.0 × 10⁸ m s⁻¹)
Medium
49.Complete the sentence about the Doppler effect for a moving source of sound.
Medium
When a source of sound moves towards a stationary observer, the observed frequency is ____ than the emitted frequency, and the observed wavelength is ____ than the emitted wavelength.
50.Explain the difference between redshift and blueshift in terms of the motion of a light source relative to an observer.
Hard
51.Explain how the Doppler effect provides evidence that the universe is expanding.
Hard
52.A stationary source emits sound of frequency 500 Hz. An observer moves towards the source at 34 m s^−1. The speed of sound in air is 340 m s^−1. Calculate the observed frequency in Hz.
Medium
- A550 Hz
- B605 Hz
- C495 Hz
- D660 Hz
53.A galaxy is receding from Earth at 1.5 × 10⁶ m s^−1. A spectral line in the laboratory has a wavelength of 500 nm. Calculate the observed wavelength from the galaxy in nm. (Speed of light c = 3.0 × 10⁸ m s^−1.)
Medium
- A502.5 nm
- B452.2 nm
- C603.1 nm
- D552.8 nm
54.A galaxy is receding from Earth at 1.5 × 10⁶ m s⁻¹. A spectral line in the laboratory has a wavelength of 500 nm. Calculate the observed wavelength from the galaxy in nm. (c = 3.0 × 10⁸ m s⁻¹)
Medium
- A552.8 nm
- B452.2 nm
- C603.1 nm
- D502.5 nm

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