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16.1The Doppler effect at low speeds

If you stand by the edge of a road and a car moving at high speed approaches you will hear a high-pitched sound. The instant the car moves past you, the frequency of the sound will drop abruptly and will stay low as the car moves away from you: this is the Doppler effect the change in observed frequency when there is relative motion between the source and the observer. It is more pronounced if the car going by is a Formula 1 racing car!

The Doppler effect is the change in the observed frequency of a wave when there is relative motion between the source and the observer.

We can explain most aspects of the Doppler effect using wavefront diagrams. Consider first a source of sound, S, that is stationary in still air. The source emits spherical wavefronts of frequency, f. (See Figure 16.1).

Figure 16.1: The wavefronts emitted by a stationary source are concentric. The common centre is the position of the source.

This means that f wavefronts are emitted per second. An observer who moves towards the stationary source will meet one wavefront after the other more frequently and so will measure a higher frequency of sound than f. The distance between the wavefronts does not change and so the moving observer will measure the same wavelength of sound as the source. Similarly, if the observer moves away from the source, then he or she will meet wavefronts less frequently and so will measure a frequency lower than f. The wavelength will be the same as that measured by the source.

If it is the source that is moving, then the wavefronts will look like those in Figure 16.2.

Figure 16.2: A source is moving away from a stationary observer A and is approaching the stationary observer B.

How do we understand this diagram? The large wavefront was emitted first. The position of the source when this wavefront was emitted was at the centre of this wavefront. The second wavefront was emitted later, and in this time the position of the source moved to the right. So this wavefront has its centre shifted to the right, and similarly for the third wavefront.

The diagram shows that the wavefronts between the source and observer B are getting closer to each other. This means that observer B will meet them more frequently, i.e. he or she will measure a frequency higher than f. Because of the bunching of the wavefronts the wavelength measured will be less than the wavelength measured by the source. For observer A, the wavefronts are further apart, and the frequency measured by this observer will be less than f. The wavelength measured is larger than that at the source because the wavefronts are further apart. These are the main features of the Doppler effect for sound.

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Exam tip

A common mistake is to confuse frequency with intensity (or loudness) of the sound. As a source approaches at constant speed, the intensity keeps increasing because the source is getting closer. The frequency observed has a constant value, higher than the frequency emitted.

The Doppler effect for light

The Doppler effect also applies to light. In the case in which the speed of the source or the observer is small compared to the speed of light, the approximate Doppler formula is:

Δ f f≈v c

In this formula v is the speed of the source or the observer, c is the speed of light and f is the emitted frequency. Then Δf gives the change in the observed frequency.

(Note that this approximate formula may also be used for sound provided the speed of the source or the observer is small compared to the speed of sound.)

Since c = f λ it also follows that:

Δ λ λ≈v c

In this formula v is the speed of the source or the observer, c is the speed of light and λ is the emitted wavelength. Then Δλ gives the change in the observed wavelength.

(This may also be used for sound but only for the case of a moving source. As we saw earlier, a moving observer will measure the same wavelength as that emitted.)

Remember that if the source of light approaches, then the frequency increases and the wavelength decreases. When the wavelength decreases we say we have a blue-shift the shift in wavelength towards smaller wavelengths for an approaching source. If the source of light moves away then the frequency decreases and the wavelength increases. In this case we speak of a red-shift the shift in wavelength towards larger wavelengths for a source getting further away.

Light from distant galaxies measured on earth shows a red-shift, i.e. it is longer than the wavelength emitted. This means that the galaxies are moving away from us. This great discovery in 1929 is convincing evidence for an expanding universe, a universe where galaxies get further apart from each other.

CHECK YOURSELF 1

A source emitting sound at 500 Hz and an observer are both moving towards the right as shown in Figure 16.3. Both have speed 10 m s−1. The speed of sound is 340 m s−1.

What is the frequency heard by the observer?

Figure 16.3: For check yourself question 1.

Worked Example 16.1

Hydrogen atoms in a distant galaxy emit light of wavelength 656 nm. The light received on earth is measured to have a wavelength of 689 nm.

State whether the galaxy is approaching the earth or moving away, and calculate the speed of the galaxy.

Answer

The received wavelength is longer than that emitted, and so the galaxy is moving away from earth (we have a red-shift). Using Δ λ≈ν c λ we get:

v=c Δ λ λ

v=3.00×10 8×(689−656)656=1.5×10 7 m s−1

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Science in context

The Doppler effect was first proposed to explain changes in the wavelength of light from binary stars as they rotate around each other. The effect also explains the change in pitch that occurs when a fast-moving source of sound passes by. Applying the theory to different types of waves in different areas of physics has led to hand-held radar guns to check for speeding vehicles and improvements in weather forecasting using the Doppler shift in radio waves reflected from moving cloud systems. But the striking application is in medicine, where Doppler imaging techniques are now routinely used to measure the speed of flow of blood cells in an artery, to monitor heart functions and examine the development of a fetus.

NATURE OF SCIENCE

Christian Doppler (1803–1853) proposed the effect that now bears his name in 1842. It was not immediately accepted. An “experiment” of musicians playing horns on a moving train platform in 1845 provided some support for his theory for sound, but the effect for light was severely attacked by a colleague of Doppler at the Austrian Academy of Sciences. Dejected and sick with tuberculosis, Doppler resigned his position and left for Venice where he died a few months later. Definitive verification of Doppler’s ideas came in 1868 and 1872, long after Doppler’s death, when astronomers measured the shift of spectral lines from stars.

Doppler would have greatly enjoyed knowing that the expansion of the universe, discovered by Edwin Hubble (1889–1953), was made by applying Doppler’s red-shift formula. Observers on earth who measure the light emitted by galaxies find that the wavelength is longer than that emitted. The galaxies must be moving away.

But there would be another twist in the story. The modern view based on Einstein’s general theory of relativity is that space in between galaxies is being stretched. This stretching makes all distances, including wavelengths, get larger. Yet the Doppler effect remains as a simple, intuitive if not completely correct description of what is actually going on, which gives the right answer for galaxies that are not too far away.

Links

The formula for the Doppler shift for light given previously is only approximate and is valid if the speed of the source or receiver is small compared to the speed of light. Things change drastically when this condition is not satisfied. In that case special relativity has to be used because clocks ‘in motion’ tick at a different rate compared to those ‘at rest’ (Chapter 6). This results in a more complicated formula for the Doppler shift. In the nonrelativistic Doppler effect it is the velocity along the line of sight between source and observer that is important. In relativity there is a new effect involving the component of velocity at right angles to the line of sight. This transverse Doppler effect, predicted by relativity, was one of the early tests of Einstein’s theory.

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TEST YOUR UNDERSTANDING

1Explain the Doppler effect by drawing appropriate wavefront diagrams for:

aa stationary source and a moving observer

ba moving source and a stationary observer.

2Calculate the speed of a galaxy emitting light of wavelength 5.48 × 10−7 m which when received on earth is measured to have a wavelength of 5.65 × 10−7 m.

3A story told by Feynman tells of a man who appeared in front of a judge, accused of running a red traffic light. The man’s defense was that due to the Doppler effect the red light appeared green as he was approaching it. The judge, who knew physics, accepted the man’s defense but fined him very heavily anyway. Why?

4Light from a nearby galaxy is emitted at a wavelength of 657 nm and is observed on earth at a wavelength of 654 nm.

aDeduce the speed of this galaxy.

bState what, precisely, can be deduced about the direction of the velocity of this galaxy.

5The sun rotates about its axis with a period of 24.5 days at the equator. The radius of the sun is 7.00 × 108 m. Discuss the shifts in wavelength of light emitted from the sun’s equator and received on earth. Assume that the sun emits monochromatic light of wavelength 5.00 × 10−7 m.

6In a binary star system, two stars orbit a common point and move so that they are always in diametrically opposite positions. Light from both stars reaches an observer on earth. Assume that both stars emit light of wavelength 6.58 × 10−7 m.

Figure 16.4: For question 6.

aWhen the stars are in the position shown in Figure 16.4a, the observer on earth measures a wavelength of light of 6.58 × 10−7 m from both stars. Explain why there is no Doppler shift in this case. (We ignore effects of special relativity.)

bWhen the stars are in the position shown in Figure 16.4b, the observer on earth measures two wavelengths in the received light, 6.50 × 10−7 m and 6.70 × 10−7 m. Determine the speed of each of the stars.

7Ultrasound of frequency 5.000 MHz reflected from red blood cells moving in an artery is found to show a frequency shift of 2.4 kHz. The speed of ultrasound in blood is v = 1500 m s–1.

aExplain why the appropriate formula for the frequency shift is Δ f f=2 u v, where u is the speed of the blood cells.

bEstimate the speed of the blood cells.

cIn practice, a range of frequency shifts is observed. Explain this observation.