Sunday, December 5, 2021

#### 1. Introduction

Have you noticed how the pitch of a police car or ambulance siren changes as it passes where you are standing, or how an approaching car or train sounds different to when it is travelling away from you? If you haven’t, try to do an experiment by paying extra careful attention the next time it happens to see if you can notice a difference in pitch.This doesn’t apply to just vehicles and trains but anything that emits waves, be those sound waves or any other electromagnetic (EM) waves.
The effect actually occurs if you move towards or away from the source of the sound as well. This effect is known as the Doppler effect and will be studied in this chapter.
Investigation: Creating the Doppler effect in class
You can create the Doppler effect in class. One way of doing this is to get:

• string, and
• a tuning fork.

Tie the string to the base of the tuning fork. Strike the tuning fork to create a note and then hold the other end of the string and swing the tuning fork in circles in the air in a horizontal plane.
WARNING!
The string needs to be very securely tied to the tuning fork to ensure that it does not come loose during the demonstration. The class should be able to hear that the frequency heard when the tuning fork is moving is different to the frequency heard when it is stationary.

#### 2. The Doppler Effect With Sound

You, the person hearing the sounds, are called the observer or listener and the thing emitting the sound is called the source. As mentioned in the introduction, there are two situations which lead to the Doppler effect:

1. When the source moves relative to a stationary observer.
2. When the observer moves relative to a stationary source.

In points 1 and 2 above there is relative motion between the source and the observer. Both the source and the observer can be moving at the same time but we won’t deal with that case in this chapter.
DEFINITION: Doppler effect
The Doppler effect is the change in the observed frequency of a wave when the source or the detector moves relative to the transmitting medium. The Doppler effect occurs when a source of waves and/or observer move relative to each other, resulting in the observer measuring a different frequency of the waves than the frequency that the source is emitting. The medium that the waves are travelling through, the transmitting medium, is also stationary in the cases we will study. The question that probably comes to mind is: ”How does the Doppler effect come about?”. We can understand what is happening by thinking through the situation in detail.
Case 1: Moving source, stationary observer
Let us consider a source of sound waves with a constant frequency and amplitude. The sound waves can be represented as concentric circles where each circle represents a crest or peak as the wavefronts radiate away from the source. This is because the waves travel away from the source in all directions and the distance between consecutive crests or consecutive troughs in a wave is constant (the wavelength as we learnt in Grade 10). In this ﬁgure the crests are represented by by the black lines and the troughs by the orange lines. The sound source is the police car in the middle and is stationary. For the Doppler effect to take place (manifest), the source must be moving relative to the observer. Let’s consider the following situation: The source (represented by the black dot) emits one wave (the black circles represent the crests of the sound wave) that moves away from the source at the same rate in all directions. The distance between the crests represents the wavelength (λ) of the sound. The closer together the crests, the higher the frequency (or pitch) of the sound according to f=​$$\frac{v}{λ}$$, where v (speed of sound) is constant.

As this crest moves away, the source also moves
and then emits more crests. Now the two circles are not concentric any more, but on the one side they are closer together and on the other side they are further apart. This is shown in the next diagram. If the source continues moving at the same speed in the same direction, then the distance between crests on the right of the source is constant. The distance between crests on the left is also constant. The distance between successive crests on the left is constant but larger than the distance between successive crests on the right. When a car approaches you, the sound waves that reach you have a shorter wavelength and a higher frequency. You hear a sound with a higher pitch. When the car moves away from you, the sound waves
that reach you have a longer wavelength and lower frequency. You hear a sound with a lower pitch.

Case 2: Moving observer, stationary source
Just as we did before, let us consider a source (a police car) of sound waves with a constant frequency and amplitude. There are two observers, one on the left that will move away from the source and one on the right that will move towards the source. We have three diagrams:

1. shows the overall situation with the siren starting at time t1;
2. shows the situation at time t when the observers are moving; and
3. shows the situation at t after the observers have been moving for a time interval,
Δt = t3-t2

The crests and troughs are numbered so you can see how they move further away and so that we can track which ones an observer has measured. The observers can hear the sound waves emitted by the police car and they start to move (we ignore the time it takes them to accelerate). The frequency of the wave that an observer measures is the number of complete waves cycles per unit time. By numbering the crests and troughs we can see which complete wave cycles have been measured by each of the observers in time, Δt . To ﬁnd the frequency we divide the number of wave cycles by Δt . In the time interval that passed, the observer moving towards the police car observed the crests and troughs numbered 1 through 5 (the portion of the wave is highlighted below). The observer moving away encountered a smaller portion of the wavefront, crest 3 and trough 4. The time interval for each of them is the same. To the observers this will mean that the frequency they measured is different. The motion of the observer will alter the frequency of the measured sound from a stationary source:

• An observer moving towards the source measures a higher frequency.
• An observer moving away from the source measures a lower frequency.

It is important to note that we have only looked at the cases where the source and observer are moving directly towards or away from each other and these are the only cases we will consider.
NOTE:
We didn’t actually need to analyse both cases. We could have used either explanation because of relative motion. The case of a stationary source with moving observer is the same as the case of the stationary observer and the moving source because the relative motion is the same. Do you agree? Discuss with your friends and try to convince yourselves that this is the case. Being able to explain work to each other will help you understand it better. If you don’t understand it, you won’t be able to explain it convincingly. For a real conceptual test, discuss what you think will happen if the source and the observer are both moving, in the same direction and at the same speed.
The formula that provides the relationship between the frequency emitted by the source and the frequency measured by the listener is: where fL is the frequency perceived by the observer (listener),

• fs is the frequency of the source,
• v is the speed of the waves,
• vL the speed of the listener and
• vs the speed of the source.

Note: The signs show whether or not the relative motion of the source and observer is towards each other or away from each other: Worked example 1: Ambulance siren
QUESTION
The siren of an ambulance emits sound with a frequency of 700 Hz. You are standing on the pavement. If the ambulance drives past you at a speed of 20 m.s-1 , what frequency will you hear, when
1. a) the ambulance is approaching you
2. b) the ambulance is driving away from you
Take the speed of sound in air to be 340 m.s-1

SOLUTION
Step 1: Analyse the question
The question explicitly asks what frequency you will hear when the source is moving a certain speed. This tells you immediately that the question is related to the Doppler effect. The values given in the question are all in S.I. units so no conversions are required.
Step 2: Determine how to approach the problem based on what is given
We know that we are looking for the observed frequency with a moving source. The change in frequency can be calculated using: To correctly apply this we need to conﬁrm that it is valid and determine what signs we need to use for the various speeds. You (the listener) are not moving but we have to consider two different cases, when the ambulance is moving towards you (a) and away from you (b). We have been told that if the source is moving towards the observer then mwe will use subtraction in the denominator and if it is moving away, addition. This means:

fs= 700 Hz
v = 340 m.s-1
vL= 0 because you, the observer, are not moving
vS= -20 m.s-1 for (a) and
vS= +20 m.s-1 for (b) Step 5: Quote the ﬁnal answer
When the ambulance is approaching you, you hear a frequency of 743,75 Hz and when it is going away you hear a frequency of 661,11 Hz
Worked example 2: Moving observer
QUESTION
What is the frequency heard by a person driving at 15 m.s1 toward a factory whistle that is blowing at a frequency of 800 Hz. Assume that the speed of sound in air is 340 m.s1
SOLUTION
Step 1: Analyse the question
The question explicitly asks what frequency you will hear when the observer is moving at a certain speed. This tells you immediately that the question is related to the Doppler shift. The values given in the question are all in S.I. units so no conversions are required. Worked example 3: Doppler effect
QUESTION
A train approaches a station at a constant speed of 20 m.s1 with its whistle blowing at a frequency of 458 Hz. An observer, standing on the platform, hears a change in pitch as the train approaches him, passes him and moves away from him.

1. Name the phenomenon that explains the change in pitch heard by the observer. (1 mark)
2. Calculate the frequency of the sound that the observer hears while the train is approaching him. Use the speed of sound in air as 340 m.s1 . (4 marks)
3. How will the observed frequency change as the train passes and moves away from the observer? Write down only INCREASES, DECREASES or REMAINS THE SAME. (1 mark)
4. How will the frequency observed by the train driver compare to that of the sound waves emitted by the whistle? Write down only GREATER THAN, EQUAL TO or LESS THAN. Give a reason for the answer. (2 marks )

[TOTAL: 8 marks]
SOLUTION Question 4
Equal to, because …

• the velocity of train driver relative to the whistle is zero. OR
• the train driver has same velocity as the whistle. OR
• there is no relative motion between source and observer.

(2 marks)
[TOTAL: 11 marks]
Ultrasound and the Doppler effect
Ultrasonic waves (ultrasound) are sound waves with a frequency greater than 20 000 Hz (the upper limit of human hearing). These waves can be used in medicine to determine the direction of blood ﬂow. The device, called a Doppler ﬂow meter, sends out sound waves. The sound waves can travel through skin and tissue and will be reﬂected by moving objects in the body (like blood). The reﬂected waves return to the ﬂow meter where its frequency (received frequency) is compared to the transmitted frequency. Because of the Doppler effect, blood that is moving towards the ﬂow meter will change the sound to a higher frequency and blood that is moving away from the ﬂow meter will cause a lower frequency. Ultrasound can be used to determine whether blood is ﬂowing in the right direction in the circulation
system of unborn babies, or identify areas in the body where blood ﬂow is restricted due to narrow
veins. The use of ultrasound equipment in medicine is called sonography or ultrasonography.
Figure 6.3: Colour Doppler imaging of cervicocephalic
ﬁbromuscular dysplasia

Exercise 6 – 1: The Doppler effect with sound

1. Suppose a train is approaching you as you stand on the platform at the station. As the train approaches the station, it slows down. All the while, the engineer is sounding the hooter at a constant frequency of 400 Hz. Describe the pitch of the hooter and the changes in pitch of the hooter that you hear as the train approaches you. Assume the speed of sound in air is 340 m.s-1.
2. Passengers on a train hear its whistle at a frequency of 740 Hz. Anja is standing next to the train tracks. What frequency does Anja hear as the train moves directly toward her at a speed of 25 m.s-1? Assume the speed of sound in air is 340 m.s-1.
3. A small plane is taxiing directly away from you down a runway. The noise of the engine, as the pilot hears it, has a frequency 1,15 times the frequency that you hear. What is the speed of the plane? Assume the speed of sound in air is 340 m.s-1.
4. In places like Canada during winter temperatures can get as low as 35 C . This affects the speed of sound in air and you can use the Doppler effect to determine what the speed of sound is. On a winter’s day in Canada with a temperature of 35 C , a source emits sound at a frequency of 1050 Hz and moves away from an observer at 25 m.s-1. The frequency that the observer measures is 971,41 Hz, what is the speed of sound?Cecil approaches a source emitting sound with a frequency of 437,1 Hz.
1. How fast does Cecil need to move to observe a frequency that is 20 percent higher?
2. If he passes the source at this speed, what frequency will he measure when he is moving away?
3. What is a practical means of achieving this speed? Assume the speed of sound in air is 340 m.s-1

#### 3. The Doppler Effect With Light

The Doppler effect with light
Light is a wave and earlier you learnt how you can study the properties of one wave and apply the same ideas to another wave. The same applies to sound and light. We know the Doppler effect is relevant in the context of sound waves when the source is moving. Therefore, in the context of light (EM waves), the frequency of observed light
should be different to the emitted frequency when the source of the light is moving relative to the observer.
A frequency shift of light in the visible spectrum could result in a change of colour which could be observable with the naked eye. There will still be a frequency shift for frequencies of EM radiation we cannot see.
We can apply all the ideas that we learnt about the Doppler effect to light. When talking about light we use slightly different terminology to describe what happens. If you look at the colour spectrum (more details in Chapter 12) then you will see that blue light has a shorter wavelength than red light. Since for light, c = f λ, shorter wavelength equals higher frequency. Relative to the middle of the visible spectrum (approximately green light) longer wavelengths (or lower frequencies) are redder and shorter wavelengths (or higher frequencies) are bluer. So we call shifts towards longer wavelengths ”redshifts” and shifts towards shorter wavelengths ”blueshifts”. A shift in wavelength implies that there is also a shift in frequency. Longer wavelengths of light have lower frequencies and shorter wavelengths have higher frequencies. From the Doppler effect we know that when the source moves towards the observer any waves they emit that you measure are shifted to shorter wavelengths (blueshifted). If the source moves away from the observer, the shift is to longer wavelengths (redshifted).
The expanding universe
Stars emit light, which is why we can see them at night. Galaxies are huge collections of stars. An example is our own Galaxy, the Milky Way, of which our sun is only one of the billions of stars!
Using large telescopes like the Southern African Large Telescope (SALT) in the Karoo, astronomers can measure the light from distant galaxies. The spectrum of light can tell us what elements are in the stars in the galaxies because each element has unique energy levels and therefore emits or absorbs light at particular wavelengths.
These characteristic wavelengths are called spectral lines because the lines show up as discrete frequencies in the spectrum of light from the star. If these lines are observed to be shifted from their usual wavelengths to shorter wavelengths, then the light from the galaxy is said to be blueshifted. If the spectral lines are shifted to longer wavelengths, then the light from the galaxy is said to be redshifted. If we think of the blueshift and redshift in Doppler effect terms, then a blueshifted galaxy
would appear to be moving towards us (the observers) and a redshifted galaxy would appear to be moving away from us.
Edwin Hubble (20 November 1889 – 28 September 1953) measured the Doppler shift of a large sample of galaxies. He found that the light from distant galaxies is redshifted and he discovered that there is a proportionality relationship between the redshift and the distance to the galaxy. Galaxies that are further away always appear more
redshifted than nearby galaxies. Remember that a redshift in Doppler terms means a velocity of the light source directed away from the observer. So why do all distant galaxies appear to be moving away from our Galaxy? None of them seem to be moving towards us. The reason is that the universe is expanding! Some of the galaxies will be moving in our direction but more slowly than the space between us and them is expanding. The expansion is so large that it is the primary effect that we observe. The primary reason the light is redshifted isn’t actually because all of the Doppler effect, it is redshifted because the space is expanding, the waves are being stretched out. If the Doppler effect were a larger effect then some of the galaxies would still be blueshifted (just less than if space were not expanding).
IMPORTANT!
You might think that this means we are at the centre of the universe. This isn’t correct, the situation will look the same from every galaxy because space is expanding in all directions. There are two things you can do to help you visualize this a little better. One thing to try is to get a balloon and draw some dots on it with a marker. As you blow the balloon up all the dots get further away from all the other dots. The dots represent galaxies in a two-dimensional, expanding universe (the balloon surface). Another thing to imagine is baking raisin bread. As the bread rises, the distance between all the raisins gets larger. Every raisin thinks that all the other raisins are moving away from it. In this picture the bottom vertex represents the beginning of time, the
ﬂat surface represents space. As you move up through the panels you are moving later in time and the expansion of the the ﬂat surface shows the expansion of the universe. The galaxies shown on the surface get further away from each other just because of the expansion of space.

#### 4. Chapter Summary

• The Doppler effect is a change in observed frequency due to the relative motion of a source and an observer.
• The following equation can be used to calculate the frequency of the wave ac- cording to the observer or listener: • If the direction of the wave from the listener to the source is chosen as positive, the velocities have the following signs: • The Doppler effect can be observed in all types of waves, including ultrasound, light and radiowaves.
• Sonography makes use of ultrasound and the Doppler effect to determine the direction of blood flow.
• Light is emitted by stars. Due to the Doppler effect, the frequency of this light decreases and the stars appear redder than if they were stationary. This is called a red shift and means that the stars are moving away from the Earth. This is one of the reasons we can conclude that the Universe is expanding. Table 6.1: Units used in Doppler  effect.
Exercise 6 – 2:

1. Write a definition for each of the following terms.
1. Doppler effect
2. Redshift
3. Ultrasound

2. The hooter of an approaching taxi has a frequency of 500 Hz. If the taxi is travelling at 30 m·s−1 and the speed of sound is 340 m·s−1 , calculate the frequency of sound that you hear when

1. the taxi is approaching you.
2. the taxi passed you and is driving away.

Draw a sketch of the observed frequency as a function of time.
3. A truck approaches you at an unknown speed. The sound of the truck’s engine has a frequency of 210 Hz, however you hear a frequency of 220 Hz. The speed of sound is 340 m·s−1 .
a) Calculate the speed of the truck.
b) How will the sound change as the truck passes you? Explain this phenomenon in terms of the wavelength and frequency of the sound.
4. [Extension question] A police car is driving towards a fleeing suspect at v/35 m·s−1 , where v is the speed of sound. The frequency of the police car’s siren is 400 Hz. The suspect is running away at v/68 . What frequency does the suspect hear?
5. Explain how the Doppler effect is used to determine the direction of flow of blood in veins.
6. An person in a car travelling at 22,2 m·s−1 approaches a source emitting sound waves with a frequency of 410 Hz. What is the frequency of the sound waves observed by the person? Assume the speed of sound in air is 340 m·s−1 .

#### Examples

Rational and Irrational Numbers
Rational Numbers Irrational Numbers #### Problems

Rational and Irrational Numbers