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Waves Wrap Up and Review Doppler Effect Optics Nature of Light Lana Sheridan De Anza College June 5, 2017

Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

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Page 1: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Waves Wrap Up and ReviewDoppler Effect

OpticsNature of Light

Lana Sheridan

De Anza College

June 5, 2017

Page 2: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Last time

• standing waves and sound

• musical instruments

• standing waves in rods and membranes

• beats

• nonsinusoidal waves

• intensity of a wave

Page 3: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Overview

• sound level

• the Doppler effect

• bow and shock waves

• talk about waves

• nature of light

• speed of light

Page 4: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Intensity of a Wave

Intensity

the average power of a wave per unit area

I =Pavg

A

Intensity is used for waves that move on 3 dimensional media, suchas sound or light.

The waves travel in one direction, and the area A is arrangedperpendicular to the direction of the wave travel.

Page 5: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Power and Intensity of Sound Waves

Power of a sound wave:

Poweravg =1

2ρvω2A s2max

Dividing this by the area gives the intensity of a sound arriving onthat area:

I =1

2ρv(ωsmax)

2

This can be written in terms of the pressure variation amplitude,∆Pmax = ρvωsmax:

I =(∆Pmax)

2

2ρv

Page 6: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Decibels: Scale for Sound Level

The ear can detect very quiet sounds, but also can hear very loudsounds without damage.

(Very, very loud sounds do damage ears.)

As sound that has twice the intensity does not “sound like” it istwice as loud.

Many human senses register to us on a logarithmic scale.

Roughly, noise A sounds twice as loud noise B if noise A is 10times as loud as noise B.

Page 7: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Decibels: Scale for Sound Level

Decibels is the scale unit we use to measure loudness, because itbetter represents our perception than intensity.

The sound level β is defined as

β = 10 log10

(I

I0

)

• I is the intensity

• I0 = 1.00× 10−12 W/m2 is a reference intensity at thethreshold of human hearing

The units of β are decibels, written dB.

Page 8: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Question

Quick Quiz 17.31 Increasing the intensity of a sound by a factorof 100 causes the sound level to increase by what amount?

(A) 100 dB

(B) 20 dB

(C) 10 dB

(D) 2 dB

1Serway & Jewett, page 515.

Page 9: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Perception of Loudness and Frequency

Human hearing also depends on frequency.

Humans can only hear sound in the range 20-20,000 Hz. 17.4 The Doppler Effect 517

old of pain. Here the boundary of the white area appears straight because the psy-chological response is relatively independent of frequency at this high sound level. The most dramatic change with frequency is in the lower left region of the white area, for low frequencies and low intensity levels. Our ears are particularly insen-sitive in this region. If you are listening to your home entertainment system and the bass (low frequencies) and treble (high frequencies) sound balanced at a high volume, try turning the volume down and listening again. You will probably notice that the bass seems weak, which is due to the insensitivity of the ear to low frequen-cies at low sound levels as shown in Figure 17.7.

17.4 The Doppler EffectPerhaps you have noticed how the sound of a vehicle’s horn changes as the vehicle moves past you. The frequency of the sound you hear as the vehicle approaches you is higher than the frequency you hear as it moves away from you. This experience is one example of the Doppler effect.3

To see what causes this apparent frequency change, imagine you are in a boat that is lying at anchor on a gentle sea where the waves have a period of T 5 3.0 s. Hence, every 3.0 s a crest hits your boat. Figure 17.8a shows this situation, with the water waves moving toward the left. If you set your watch to t 5 0 just as one crest hits, the watch reads 3.0 s when the next crest hits, 6.0 s when the third crest

Infrasonicfrequencies

Sonicfrequencies

Ultrasonicfrequencies

Large rocket engine

Jet engine (10 m away) Rifle

Thunderoverhead

Rock concert

Underwater communication

(Sonar)

Car hornMotorcycle School cafeteria

Urban traffic Shout

ConversationBirds

BatsWhispered speechThreshold of

hearing

Sound level (dB)

1 10 100 1 000 10 000 100 000Frequency f (Hz)

220

200

180

160

140

120

100

80

60

40

20

0

Threshold ofpain

bFigure 17.7 Approximate ranges of frequency and sound level of various sources and that of normal human hearing, shown by the white area. (From R. L. Reese, University Physics, Pacific Grove, Brooks/Cole, 2000.)

3Named after Austrian physicist Christian Johann Doppler (1803–1853), who in 1842 predicted the effect for both sound waves and light waves.

c

vwavesS

vboatS

In all frames, the waves travel to the left, and their source is far to the right of the boat, out of the frame of the figure.

a

vwavesS

b

vwavesS

vboatS

Figure 17.8 (a) Waves moving toward a stationary boat. (b) The boat moving toward the wave source. (c) The boat moving away from the wave source.

Low frequency sounds need to be louder to be heard.

1Figure from R. L. Reese, University Physics, via Serway & Jewett.

Page 10: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Perception of Loudness and Frequency

Human hearing also depends on frequency.

Humans can only hear sound in the range 20-20,000 Hz. 17.4 The Doppler Effect 517

old of pain. Here the boundary of the white area appears straight because the psy-chological response is relatively independent of frequency at this high sound level. The most dramatic change with frequency is in the lower left region of the white area, for low frequencies and low intensity levels. Our ears are particularly insen-sitive in this region. If you are listening to your home entertainment system and the bass (low frequencies) and treble (high frequencies) sound balanced at a high volume, try turning the volume down and listening again. You will probably notice that the bass seems weak, which is due to the insensitivity of the ear to low frequen-cies at low sound levels as shown in Figure 17.7.

17.4 The Doppler EffectPerhaps you have noticed how the sound of a vehicle’s horn changes as the vehicle moves past you. The frequency of the sound you hear as the vehicle approaches you is higher than the frequency you hear as it moves away from you. This experience is one example of the Doppler effect.3

To see what causes this apparent frequency change, imagine you are in a boat that is lying at anchor on a gentle sea where the waves have a period of T 5 3.0 s. Hence, every 3.0 s a crest hits your boat. Figure 17.8a shows this situation, with the water waves moving toward the left. If you set your watch to t 5 0 just as one crest hits, the watch reads 3.0 s when the next crest hits, 6.0 s when the third crest

Infrasonicfrequencies

Sonicfrequencies

Ultrasonicfrequencies

Large rocket engine

Jet engine (10 m away) Rifle

Thunderoverhead

Rock concert

Underwater communication

(Sonar)

Car hornMotorcycle School cafeteria

Urban traffic Shout

ConversationBirds

BatsWhispered speechThreshold of

hearing

Sound level (dB)

1 10 100 1 000 10 000 100 000Frequency f (Hz)

220

200

180

160

140

120

100

80

60

40

20

0

Threshold ofpain

bFigure 17.7 Approximate ranges of frequency and sound level of various sources and that of normal human hearing, shown by the white area. (From R. L. Reese, University Physics, Pacific Grove, Brooks/Cole, 2000.)

3Named after Austrian physicist Christian Johann Doppler (1803–1853), who in 1842 predicted the effect for both sound waves and light waves.

c

vwavesS

vboatS

In all frames, the waves travel to the left, and their source is far to the right of the boat, out of the frame of the figure.

a

vwavesS

b

vwavesS

vboatS

Figure 17.8 (a) Waves moving toward a stationary boat. (b) The boat moving toward the wave source. (c) The boat moving away from the wave source.

Low frequency sounds need to be louder to be heard.1Figure from R. L. Reese, University Physics, via Serway & Jewett.

Page 11: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

The Doppler Effect

The frequency of a sound counts how many wavefronts (pressurepeaks) arrive per second.

If you are moving towards a source of sound, you encounter morewavefronts per second → the frequency you detect is higher!

518 Chapter 17 Sound Waves

hits, and so on. From these observations, you conclude that the wave frequency is f 5 1/T 5 1/(3.0 s) 5 0.33 Hz. Now suppose you start your motor and head directly into the oncoming waves as in Figure 17.8b. Again you set your watch to t 5 0 as a crest hits the front (the bow) of your boat. Now, however, because you are moving toward the next wave crest as it moves toward you, it hits you less than 3.0 s after the first hit. In other words, the period you observe is shorter than the 3.0-s period you observed when you were stationary. Because f 5 1/T, you observe a higher wave frequency than when you were at rest. If you turn around and move in the same direction as the waves (Fig. 17.8c), you observe the opposite effect. You set your watch to t 5 0 as a crest hits the back (the stern) of the boat. Because you are now moving away from the next crest, more than 3.0 s has elapsed on your watch by the time that crest catches you. Therefore, you observe a lower frequency than when you were at rest. These effects occur because the relative speed between your boat and the waves depends on the direction of travel and on the speed of your boat. (See Section 4.6.) When you are moving toward the right in Figure 17.8b, this relative speed is higher than that of the wave speed, which leads to the observation of an increased fre-quency. When you turn around and move to the left, the relative speed is lower, as is the observed frequency of the water waves. Let’s now examine an analogous situation with sound waves in which the water waves become sound waves, the water becomes the air, and the person on the boat becomes an observer listening to the sound. In this case, an observer O is moving and a sound source S is stationary. For simplicity, we assume the air is also station-ary and the observer moves directly toward the source (Fig. 17.9). The observer moves with a speed vO toward a stationary point source (vS 5 0), where stationary means at rest with respect to the medium, air. If a point source emits sound waves and the medium is uniform, the waves move at the same speed in all directions radially away from the source; the result is a spherical wave as mentioned in Section 17.3. The distance between adjacent wave fronts equals the wavelength l. In Figure 17.9, the circles are the intersections of these three-dimensional wave fronts with the two-dimensional paper. We take the frequency of the source in Figure 17.9 to be f, the wavelength to be l, and the speed of sound to be v. If the observer were also stationary, he would detect wave fronts at a frequency f. (That is, when vO 5 0 and vS 5 0, the observed frequency equals the source frequency.) When the observer moves toward the source, the speed of the waves relative to the observer is v9 5 v 1 vO , as in the case of the boat in Figure 17.8, but the wavelength l is unchanged. Hence, using Equation 16.12, v 5 lf, we can say that the frequency f 9 heard by the observer is increased and is given by

f r 5v rl

5v 1 vO

l

Because l 5 v/f , we can express f 9 as

f r 5 av 1 vO

v b f 1observer moving toward source 2 (17.15)

If the observer is moving away from the source, the speed of the wave relative to the observer is v9 5 v 2 vO . The frequency heard by the observer in this case is decreased and is given by

f r 5 av 2 vO

v b f ( observer moving away from source) (17.16)

These last two equations can be reduced to a single equation by adopting a sign convention. Whenever an observer moves with a speed vO relative to a stationary source, the frequency heard by the observer is given by Equation 17.15, with vO interpreted as follows: a positive value is substituted for vO when the observer moves

Figure 17.9 An observer O (the cyclist) moves with a speed vO toward a stationary point source S, the horn of a parked truck. The observer hears a fre-quency f 9 that is greater than the source frequency.

O

O

S

vS

Page 12: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

The Doppler Effect

518 Chapter 17 Sound Waves

hits, and so on. From these observations, you conclude that the wave frequency is f 5 1/T 5 1/(3.0 s) 5 0.33 Hz. Now suppose you start your motor and head directly into the oncoming waves as in Figure 17.8b. Again you set your watch to t 5 0 as a crest hits the front (the bow) of your boat. Now, however, because you are moving toward the next wave crest as it moves toward you, it hits you less than 3.0 s after the first hit. In other words, the period you observe is shorter than the 3.0-s period you observed when you were stationary. Because f 5 1/T, you observe a higher wave frequency than when you were at rest. If you turn around and move in the same direction as the waves (Fig. 17.8c), you observe the opposite effect. You set your watch to t 5 0 as a crest hits the back (the stern) of the boat. Because you are now moving away from the next crest, more than 3.0 s has elapsed on your watch by the time that crest catches you. Therefore, you observe a lower frequency than when you were at rest. These effects occur because the relative speed between your boat and the waves depends on the direction of travel and on the speed of your boat. (See Section 4.6.) When you are moving toward the right in Figure 17.8b, this relative speed is higher than that of the wave speed, which leads to the observation of an increased fre-quency. When you turn around and move to the left, the relative speed is lower, as is the observed frequency of the water waves. Let’s now examine an analogous situation with sound waves in which the water waves become sound waves, the water becomes the air, and the person on the boat becomes an observer listening to the sound. In this case, an observer O is moving and a sound source S is stationary. For simplicity, we assume the air is also station-ary and the observer moves directly toward the source (Fig. 17.9). The observer moves with a speed vO toward a stationary point source (vS 5 0), where stationary means at rest with respect to the medium, air. If a point source emits sound waves and the medium is uniform, the waves move at the same speed in all directions radially away from the source; the result is a spherical wave as mentioned in Section 17.3. The distance between adjacent wave fronts equals the wavelength l. In Figure 17.9, the circles are the intersections of these three-dimensional wave fronts with the two-dimensional paper. We take the frequency of the source in Figure 17.9 to be f, the wavelength to be l, and the speed of sound to be v. If the observer were also stationary, he would detect wave fronts at a frequency f. (That is, when vO 5 0 and vS 5 0, the observed frequency equals the source frequency.) When the observer moves toward the source, the speed of the waves relative to the observer is v9 5 v 1 vO , as in the case of the boat in Figure 17.8, but the wavelength l is unchanged. Hence, using Equation 16.12, v 5 lf, we can say that the frequency f 9 heard by the observer is increased and is given by

f r 5v rl

5v 1 vO

l

Because l 5 v/f , we can express f 9 as

f r 5 av 1 vO

v b f 1observer moving toward source 2 (17.15)

If the observer is moving away from the source, the speed of the wave relative to the observer is v9 5 v 2 vO . The frequency heard by the observer in this case is decreased and is given by

f r 5 av 2 vO

v b f ( observer moving away from source) (17.16)

These last two equations can be reduced to a single equation by adopting a sign convention. Whenever an observer moves with a speed vO relative to a stationary source, the frequency heard by the observer is given by Equation 17.15, with vO interpreted as follows: a positive value is substituted for vO when the observer moves

Figure 17.9 An observer O (the cyclist) moves with a speed vO toward a stationary point source S, the horn of a parked truck. The observer hears a fre-quency f 9 that is greater than the source frequency.

O

O

S

vS

The speed you see the waves traveling relative to you isv ′ = v + v0, while relative to the source the speed is v .

f ′ =v ′

λ=

(v + v0

v

)f

(v and v0 are positive numbers.)

Page 13: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

The Doppler Effect

The speed you see the waves traveling relative to you isv ′ = v + v0, while relative to the source the speed is v .

f ′ =v ′

λ=

(v + v0

v

)f

(v and v0 are positive numbers.)

Moving away from the source, the relative velocity of the detectorto the source decreases v ′ = v − v0.

f ′ =

(v − v0

v

)f

Page 14: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

The Doppler Effect

A similar thing happens if the source of the waves is moving.

17.4 The Doppler Effect 519

toward the source, and a negative value is substituted when the observer moves away from the source. Now suppose the source is in motion and the observer is at rest. If the source moves directly toward observer A in Figure 17.10a, each new wave is emitted from a position to the right of the origin of the previous wave. As a result, the wave fronts heard by the observer are closer together than they would be if the source were not moving. (Fig. 17.10b shows this effect for waves moving on the surface of water.) As a result, the wavelength l9 measured by observer A is shorter than the wave-length l of the source. During each vibration, which lasts for a time interval T (the period), the source moves a distance vST 5 vS /f and the wavelength is shortened by this amount. Therefore, the observed wavelength l9 is

l r 5 l 2 Dl 5 l 2vS

f

Because l 5 v/f, the frequency f 9 heard by observer A is

f r 5vl r

5v

l 2 1vS /f 2 5v1v/f 2 2 1vS /f 2

f r 5 a vv 2 vS

b f (source moving toward observer) (17.17)

That is, the observed frequency is increased whenever the source is moving toward the observer. When the source moves away from a stationary observer, as is the case for observer B in Figure 17.10a, the observer measures a wavelength l9 that is greater than l and hears a decreased frequency:

f r 5 a vv 1 vS

b f (source moving away from observer) (17.18)

We can express the general relationship for the observed frequency when a source is moving and an observer is at rest as Equation 17.17, with the same sign convention applied to vS as was applied to vO : a positive value is substituted for vS when the source moves toward the observer, and a negative value is substituted when the source moves away from the observer. Finally, combining Equations 17.15 and 17.17 gives the following general rela-tionship for the observed frequency that includes all four conditions described by Equations 17.15 through 17.18:

f r 5 av 1 vO

v 2 vSb f (17.19) �W General Doppler-shift

expression

Figure 17.10 (a) A source S mov-ing with a speed vS toward a sta-tionary observer A and away from a stationary observer B. Observer A hears an increased frequency, and observer B hears a decreased frequency. (b) The Doppler effect in water, observed in a ripple tank. Letters shown in the photo refer to Quick Quiz 17.4.S

S Observer AObserver B v

l!

a

S A C

B

b

A point source is moving to the right with speed vS.

Cour

tesy

of t

he E

duca

tiona

l De

velo

pmen

t Cen

ter,

New

ton,

MA

Pitfall Prevention 17.1Doppler Effect Does Not Depend on Distance Some people think that the Doppler effect depends on the distance between the source and the observer. Although the intensity of a sound varies as the distance changes, the apparent frequency depends only on the relative speed of source and observer. As you listen to an approaching source, you will detect increasing intensity but constant frequency. As the source passes, you will hear the frequency suddenly drop to a new constant value and the intensity begin to decrease.

In the diagram, the source is moving toward the wavefronts it hascreated on the right and away from the wavefronts it has createdon the left.

This changes the wavelength of the waves around the source. Theyare shorter on the right, and longer on the left.

Page 15: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

The Doppler Effect

17.4 The Doppler Effect 519

toward the source, and a negative value is substituted when the observer moves away from the source. Now suppose the source is in motion and the observer is at rest. If the source moves directly toward observer A in Figure 17.10a, each new wave is emitted from a position to the right of the origin of the previous wave. As a result, the wave fronts heard by the observer are closer together than they would be if the source were not moving. (Fig. 17.10b shows this effect for waves moving on the surface of water.) As a result, the wavelength l9 measured by observer A is shorter than the wave-length l of the source. During each vibration, which lasts for a time interval T (the period), the source moves a distance vST 5 vS /f and the wavelength is shortened by this amount. Therefore, the observed wavelength l9 is

l r 5 l 2 Dl 5 l 2vS

f

Because l 5 v/f, the frequency f 9 heard by observer A is

f r 5vl r

5v

l 2 1vS /f 2 5v1v/f 2 2 1vS /f 2

f r 5 a vv 2 vS

b f (source moving toward observer) (17.17)

That is, the observed frequency is increased whenever the source is moving toward the observer. When the source moves away from a stationary observer, as is the case for observer B in Figure 17.10a, the observer measures a wavelength l9 that is greater than l and hears a decreased frequency:

f r 5 a vv 1 vS

b f (source moving away from observer) (17.18)

We can express the general relationship for the observed frequency when a source is moving and an observer is at rest as Equation 17.17, with the same sign convention applied to vS as was applied to vO : a positive value is substituted for vS when the source moves toward the observer, and a negative value is substituted when the source moves away from the observer. Finally, combining Equations 17.15 and 17.17 gives the following general rela-tionship for the observed frequency that includes all four conditions described by Equations 17.15 through 17.18:

f r 5 av 1 vO

v 2 vSb f (17.19) �W General Doppler-shift

expression

Figure 17.10 (a) A source S mov-ing with a speed vS toward a sta-tionary observer A and away from a stationary observer B. Observer A hears an increased frequency, and observer B hears a decreased frequency. (b) The Doppler effect in water, observed in a ripple tank. Letters shown in the photo refer to Quick Quiz 17.4.S

S Observer AObserver B v

l!

a

S A C

B

b

A point source is moving to the right with speed vS.

Cour

tesy

of t

he E

duca

tiona

l De

velo

pmen

t Cen

ter,

New

ton,

MA

Pitfall Prevention 17.1Doppler Effect Does Not Depend on Distance Some people think that the Doppler effect depends on the distance between the source and the observer. Although the intensity of a sound varies as the distance changes, the apparent frequency depends only on the relative speed of source and observer. As you listen to an approaching source, you will detect increasing intensity but constant frequency. As the source passes, you will hear the frequency suddenly drop to a new constant value and the intensity begin to decrease.

Observer A detects the wavelength as λ ′ = λ− vsT = λ− vsf .

For A:

f ′ =v

λ ′=

(v

v/f − vs/f

)=

(v

v − vs

)f

For Observer B:

f ′ =

(v

v + vs

)f

Page 16: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

The Doppler Effect

17.4 The Doppler Effect 519

toward the source, and a negative value is substituted when the observer moves away from the source. Now suppose the source is in motion and the observer is at rest. If the source moves directly toward observer A in Figure 17.10a, each new wave is emitted from a position to the right of the origin of the previous wave. As a result, the wave fronts heard by the observer are closer together than they would be if the source were not moving. (Fig. 17.10b shows this effect for waves moving on the surface of water.) As a result, the wavelength l9 measured by observer A is shorter than the wave-length l of the source. During each vibration, which lasts for a time interval T (the period), the source moves a distance vST 5 vS /f and the wavelength is shortened by this amount. Therefore, the observed wavelength l9 is

l r 5 l 2 Dl 5 l 2vS

f

Because l 5 v/f, the frequency f 9 heard by observer A is

f r 5vl r

5v

l 2 1vS /f 2 5v1v/f 2 2 1vS /f 2

f r 5 a vv 2 vS

b f (source moving toward observer) (17.17)

That is, the observed frequency is increased whenever the source is moving toward the observer. When the source moves away from a stationary observer, as is the case for observer B in Figure 17.10a, the observer measures a wavelength l9 that is greater than l and hears a decreased frequency:

f r 5 a vv 1 vS

b f (source moving away from observer) (17.18)

We can express the general relationship for the observed frequency when a source is moving and an observer is at rest as Equation 17.17, with the same sign convention applied to vS as was applied to vO : a positive value is substituted for vS when the source moves toward the observer, and a negative value is substituted when the source moves away from the observer. Finally, combining Equations 17.15 and 17.17 gives the following general rela-tionship for the observed frequency that includes all four conditions described by Equations 17.15 through 17.18:

f r 5 av 1 vO

v 2 vSb f (17.19) �W General Doppler-shift

expression

Figure 17.10 (a) A source S mov-ing with a speed vS toward a sta-tionary observer A and away from a stationary observer B. Observer A hears an increased frequency, and observer B hears a decreased frequency. (b) The Doppler effect in water, observed in a ripple tank. Letters shown in the photo refer to Quick Quiz 17.4.S

S Observer AObserver B v

l!

a

S A C

B

b

A point source is moving to the right with speed vS.

Cour

tesy

of t

he E

duca

tiona

l De

velo

pmen

t Cen

ter,

New

ton,

MA

Pitfall Prevention 17.1Doppler Effect Does Not Depend on Distance Some people think that the Doppler effect depends on the distance between the source and the observer. Although the intensity of a sound varies as the distance changes, the apparent frequency depends only on the relative speed of source and observer. As you listen to an approaching source, you will detect increasing intensity but constant frequency. As the source passes, you will hear the frequency suddenly drop to a new constant value and the intensity begin to decrease.

Observer A detects the wavelength as λ ′ = λ− vsT = λ− vsf .

For A:

f ′ =v

λ ′=

(v

v/f − vs/f

)=

(v

v − vs

)f

For Observer B:

f ′ =

(v

v + vs

)f

Page 17: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

The Doppler Effect

In general:

f ′ =

(v ± v0v ∓ vs

)f

The top sign in the numerator and denominator corresponds to thedetector and/or source moving towards on another.

The bottom signs correspond to the detector and/or sourcemoving away from on another.

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The Doppler EffectQuick Quiz 17.42 Consider detectors of water waves at threelocations A, B, and C in the picture. Which of the followingstatements is true?

17.4 The Doppler Effect 519

toward the source, and a negative value is substituted when the observer moves away from the source. Now suppose the source is in motion and the observer is at rest. If the source moves directly toward observer A in Figure 17.10a, each new wave is emitted from a position to the right of the origin of the previous wave. As a result, the wave fronts heard by the observer are closer together than they would be if the source were not moving. (Fig. 17.10b shows this effect for waves moving on the surface of water.) As a result, the wavelength l9 measured by observer A is shorter than the wave-length l of the source. During each vibration, which lasts for a time interval T (the period), the source moves a distance vST 5 vS /f and the wavelength is shortened by this amount. Therefore, the observed wavelength l9 is

l r 5 l 2 Dl 5 l 2vS

f

Because l 5 v/f, the frequency f 9 heard by observer A is

f r 5vl r

5v

l 2 1vS /f 2 5v1v/f 2 2 1vS /f 2

f r 5 a vv 2 vS

b f (source moving toward observer) (17.17)

That is, the observed frequency is increased whenever the source is moving toward the observer. When the source moves away from a stationary observer, as is the case for observer B in Figure 17.10a, the observer measures a wavelength l9 that is greater than l and hears a decreased frequency:

f r 5 a vv 1 vS

b f (source moving away from observer) (17.18)

We can express the general relationship for the observed frequency when a source is moving and an observer is at rest as Equation 17.17, with the same sign convention applied to vS as was applied to vO : a positive value is substituted for vS when the source moves toward the observer, and a negative value is substituted when the source moves away from the observer. Finally, combining Equations 17.15 and 17.17 gives the following general rela-tionship for the observed frequency that includes all four conditions described by Equations 17.15 through 17.18:

f r 5 av 1 vO

v 2 vSb f (17.19) �W General Doppler-shift

expression

Figure 17.10 (a) A source S mov-ing with a speed vS toward a sta-tionary observer A and away from a stationary observer B. Observer A hears an increased frequency, and observer B hears a decreased frequency. (b) The Doppler effect in water, observed in a ripple tank. Letters shown in the photo refer to Quick Quiz 17.4.S

S Observer AObserver B v

l!

a

S A C

B

b

A point source is moving to the right with speed vS.

Cour

tesy

of t

he E

duca

tiona

l De

velo

pmen

t Cen

ter,

New

ton,

MA

Pitfall Prevention 17.1Doppler Effect Does Not Depend on Distance Some people think that the Doppler effect depends on the distance between the source and the observer. Although the intensity of a sound varies as the distance changes, the apparent frequency depends only on the relative speed of source and observer. As you listen to an approaching source, you will detect increasing intensity but constant frequency. As the source passes, you will hear the frequency suddenly drop to a new constant value and the intensity begin to decrease.

(A) The wave speed is highest at location C.

(B) The detected wavelength is largest at location C.

(C) The detected frequency is highest at location C.

(D) The detected frequency is highest at location A.

2Serway & Jewett, page 520.

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The Doppler Effect Question

A police car has a siren tone with a frequency at 2.0 kHz.

It is approaching you at 28 m/s. What frequency do you hear thesiren tone as?

Now it has passed by and is moving away from you. Whatfrequency do you hear the siren tone as now?

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The Doppler Effect for Light

Electromagnetic radiation also exhibits the Doppler effect,however, the equation needed to describe the effect is different.

For em radiation:

f ′ =

√1 + v/c√1 − v/c

f

Relativity is needed to derive this expression (hold on for Phys4D).

This can be used to determine how fast distant objects in spaceare moving toward or away from us.

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The Doppler Effect and Astronomy

1Image from Wikipedia by Georg Wiora.

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Bow Waves and Shock Waves

Bow waves and shock waves can be detected by nearby observerswhen the speed of the wave source exceeds the speed of the waves.

This effect happens when an aircraft transitions from subsonicflight to supersonic flight.

1Figure from Hewitt, 11ed.

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Bow waves

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Supersonic transition

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Shock Waves

522 Chapter 17 Sound Waves

▸ 17.5 c o n t i n u e d

Finalize This technique is used by police officers to measure the speed of a moving car. Microwaves are emitted from the police car and reflected by the moving car. By detecting the Doppler-shifted frequency of the reflected micro-waves, the police officer can determine the speed of the moving car.

Shock WavesNow consider what happens when the speed vS of a source exceeds the wave speed v. This situation is depicted graphically in Figure 17.11a. The circles represent spheri-cal wave fronts emitted by the source at various times during its motion. At t 5 0, the source is at S0 and moving toward the right. At later times, the source is at S1, and then S2, and so on. At the time t, the wave front centered at S0 reaches a radius of vt. In this same time interval, the source travels a distance vSt. Notice in Figure 17.11a that a straight line can be drawn tangent to all the wave fronts generated at various times. Therefore, the envelope of these wave fronts is a cone whose apex half-angle u (the “Mach angle”) is given by

sin u 5vtvS t

5vvS

The ratio vS/v is referred to as the Mach number, and the conical wave front pro-duced when vS . v (supersonic speeds) is known as a shock wave. An interesting anal-ogy to shock waves is the V-shaped wave fronts produced by a boat (the bow wave) when the boat’s speed exceeds the speed of the surface-water waves (Fig. 17.12). Jet airplanes traveling at supersonic speeds produce shock waves, which are responsible for the loud “sonic boom” one hears. The shock wave carries a great deal of energy concentrated on the surface of the cone, with correspondingly great pressure variations. Such shock waves are unpleasant to hear and can cause dam-age to buildings when aircraft fly supersonically at low altitudes. In fact, an air-plane flying at supersonic speeds produces a double boom because two shock waves are formed, one from the nose of the plane and one from the tail. People near the path of a space shuttle as it glides toward its landing point have reported hearing what sounds like two very closely spaced cracks of thunder.

Q uick Quiz 17.6 An airplane flying with a constant velocity moves from a cold air mass into a warm air mass. Does the Mach number (a) increase, (b) decrease, or (c) stay the same?

Figure 17.11 (a) A representa-tion of a shock wave produced when a source moves from S0 to the right with a speed vS that is greater than the wave speed v in the medium. (b) A stroboscopic photograph of a bullet moving at supersonic speed through the hot air above a candle.

vSt

vSS

S1 S2S0

The envelope of the wave fronts forms a cone whose apex half-angle is given bysin u ! v/vS.

vtu

a

01

2

b

Notice the shock wave in the vicinity of the bullet.

Omik

ron/

Phot

o Re

sear

cher

s/Ge

tty I

mag

es

Figure 17.12 The V-shaped bow wave of a boat is formed because the boat speed is greater than the speed of the water waves it gener-ates. A bow wave is analogous to a shock wave formed by an airplane traveling faster than sound.

Robe

rt Ho

lland

/Sto

ne/G

etty

Imag

es

The angle which the shockwave makes is called the Mach angle.

sin θ =v

vs

The ratio vs/v is called the Mach number.

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Shock Waves Question

Quick Quiz 17.63 An airplane flying with a constant velocitymoves from a cold air mass into a warm air mass. What happensto the Mach number?

(A) it increases

(B) it decreases

(C) it stays the same

3Serway & Jewett, page 522.

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Waves

Any questions about waves?

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What is Light?

Physicists have long been interested in the nature and uses oflight.

Egyptians and Mesopotamians developed lenses. Later Greeks andIndians began to develop a theory of geometric optics

Geometric optics was greatly advanced in 800-1000 by Arabphilosophers, especially Ibn al-Haytham (called Alhazen).

Newton developed a particle model of light, which explainedreflection and refraction.

Christian Huygens proposed a wave model of light (1678) andpointed out that it could also explain reflection and refraction, butit was less popular.

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What is Light?

Thomas Young experimentally demonstrated the interference oflight, which confirmed that it needed to be considered as beingwave-like.

This fit with the understanding of Maxwell’s equations.

Hertz then discovered the Photoelectric effect and was unable toexplain it with a wave model of light.

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Photoelectric Effect

Metal rod

Glass container

Zinc plate

Foil leaf

Incoming radiation

Even very intense light at a low frequency will not allow the plateto discharge. As soon as just a little light at a high frequency fallson the plate it begins discharging.

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What is Light?

Einstein resolved the issue by taking literally Max Planck’squantization model and showing that light behaves like a wave,but also like a particle.

The “particles” of light are called photons.

The energy of a photon depends on its frequency:

E = hf

where h = 6.63× 10−34 J s is Planck’s constant.

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Speed of Light

Light travels very fast.

We can figure out how fast it goes from Maxwell’s laws, byderiving a wave equation from them.

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Maxwell’s Equations

∮E · dA =

qencε0∮

B · dA = 0∮E · ds = −

dΦB

dt∮B · ds = µ0ε0

dΦE

dt+µ0Ienc

In free space (a vacuum) with no charges qenc = 0 and Ienc = 0.

1Strictly, these are Maxwell’s equations in a vacuum.

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Maxwell’s Equations Differential Form

∇ · E =ρ

ε0

∇ · B = 0

∇× E = −∂B

∂t

∇× B = µ0ε0∂E

∂t+ µ0J

In free space with no charges ρ = 0 and J = 0.

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Notation

The differential operators in 3 dimensions.

Gradient of a scalar field at a point, f :

∇f =∂f

∂xi +

∂f

∂yj +

∂f

∂zk

Divergence of a vector field at a point v = [vx , vy , vz ]:

∇ · v =∂vx∂x

+∂vy∂y

+∂vz∂z

Curl of a vector field at a point v:

∇× v =

(∂vz∂y

−∂vy∂z

)i +

(∂vx∂z

−∂vz∂x

)j +

(∂vy∂x

−∂vx∂y

)k

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Maxwell’s Equations and the Wave Equation

∇ · E = 0

∇ · B = 0

∇× E = −∂B

∂t

∇× B = µ0ε0∂E

∂t

Take the curl of both sides of equation 3:

∇× (∇× E) = −∂

∂t(∇× B)

Using the vector triple product rule, a× (b× c) = b(a · c)− c(a ·b)

∇(∇ · E) − (∇ ·∇)E = −∂

∂t(∇× B)

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Maxwell’s Equations and the Wave Equation

∇(∇ · E) −∇2E = −∂

∂t(∇× B)

Using the 1st equation:

∇2E =∂

∂t(∇× B)

Using the 4th equation,

∇2E = µ0ε0∂2E

∂t2

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Another Implication of Maxwell’s Equations

Using all 4 equations it is possible to reach a pair of waveequations for the electric and magnetic fields:

∇2E =1

c2∂2E

∂t2

∇2B =1

c2∂2B

∂t2

with wave solutions:

E = E0 sin(k · r −ωt)

B = B0 sin(k · r −ωt)

where c = ωk .

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Another Implication of Maxwell’s Equations

For E-field varying in x direction:

∂2Ex

∂x2=

1

c2∂2Ex

∂t2

The constant c appears as the wave speed and

c =1

√µ0ε0

c = 3.00× 108 m/s, is the speed of light.

The values of ε0 and µ0 together predict the speed of light!

ε0 = 8.85× 10−12 C2 N−1m−2 and µ0 = 4π× 10−7 kg m C−2

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Another Implication of Maxwell’s Equations

Wave solutions:

E = E0 sin(k · r −ωt)

B = B0 sin(k · r −ωt)

where c = ωk .

These two solutions are in phase. There is no offset in the anglesinside the sine functions.

The two fields peak at the same point in space and time.

At all times:E

B= c

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Measurements of the Speed of Light

Since light propagates so quickly it is difficult to measure its speedin practice.

Galileo and others tried to measure it with procedures that reliedon human reactions.

Human reactions are way too slow! This error dominates the data.

Many clever alternative methods were developed.

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Roemer’s Method

Ole Roemer observed the orbit of Io, a moon of Jupiter.

1060 Chapter 35 The Nature of Light and the Principles of Ray Optics

Roemer’s MethodIn 1675, Danish astronomer Ole Roemer (1644–1710) made the first successful esti-mate of the speed of light. Roemer’s technique involved astronomical observations of Io, one of the moons of Jupiter. Io has a period of revolution around Jupiter of approximately 42.5 h. The period of revolution of Jupiter around the Sun is about 12 yr; therefore, as the Earth moves through 90° around the Sun, Jupiter revolves through only ( 1

12)90° 5 7.5° (Fig. 35.1). An observer using the orbital motion of Io as a clock would expect the orbit to have a constant period. After collecting data for more than a year, however, Roemer observed a systematic variation in Io’s period. He found that the periods were lon-ger than average when the Earth was receding from Jupiter and shorter than aver-age when the Earth was approaching Jupiter. Roemer attributed this variation in period to the distance between the Earth and Jupiter changing from one observa-tion to the next. Using Roemer’s data, Huygens estimated the lower limit for the speed of light to be approximately 2.3 3 108 m/s. This experiment is important historically because it demonstrated that light does have a finite speed and gave an estimate of this speed.

Fizeau’s MethodThe first successful method for measuring the speed of light by means of purely ter-restrial techniques was developed in 1849 by French physicist Armand H. L. Fizeau (1819–1896). Figure 35.2 represents a simplified diagram of Fizeau’s apparatus. The basic procedure is to measure the total time interval during which light travels from some point to a distant mirror and back. If d is the distance between the light source (considered to be at the location of the wheel) and the mirror and if the time interval for one round trip is Dt, the speed of light is c 5 2d/Dt. To measure the transit time, Fizeau used a rotating toothed wheel, which con-verts a continuous beam of light into a series of light pulses. The rotation of such a wheel controls what an observer at the light source sees. For example, if the pulse traveling toward the mirror and passing the opening at point A in Figure 35.2 should return to the wheel at the instant tooth B had rotated into position to cover the return path, the pulse would not reach the observer. At a greater rate of rota-tion, the opening at point C could move into position to allow the reflected pulse to reach the observer. Knowing the distance d, the number of teeth in the wheel, and the angular speed of the wheel, Fizeau arrived at a value of 3.1 3 108 m/s. Similar measurements made by subsequent investigators yielded more precise values for c, which led to the currently accepted value of 2.997 924 58 3 108 m/s.

A

BC

Mirror

d

Toothedwheel

Figure 35.2 Fizeau’s method for measuring the speed of light using a rotating toothed wheel. The light source is considered to be at the location of the wheel; there-fore, the distance d is known.

Example 35.1 Measuring the Speed of Light with Fizeau’s Wheel

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/s when a pulse of light passing through opening A in Fig-ure 35.2 is blocked by tooth B on its return. If the distance to the mirror is 7 500 m, what is the speed of light?

Conceptualize Imagine a pulse of light passing through opening A in Figure 35.2 and reflecting from the mirror. By the time the pulse arrives back at the wheel, tooth B has rotated into the position previously occupied by opening A.

Categorize The wheel is a rigid object rotating at constant angular speed. We model the pulse of light as a particle under constant speed.

Analyze The wheel has 360 teeth, so it must have 360 openings. Therefore, because the light passes through opening A but is blocked by the tooth immediately adjacent to A, the wheel must rotate through an angular displacement of 1

720 rev in the time interval during which the light pulse makes its round trip.

AM

S O L U T I O N

Use Equation 10.2, with the angular speed constant, to find the time interval for the pulse’s round trip:

Dt 5Du

v5

1720 rev

27.5 rev/s5 5.05 3 1025 s

J1

E1Sun

E2

J2

Io

90!

7.5!

In the time interval during which the Earth travels 90! around the Sun (three months), Jupiter travels only about 7.5!.

Figure 35.1 Roemer’s method for measuring the speed of light (drawing not to scale).

If light travels infinitely fast, the orbit should always be observed tohave the same period. Instead it appears to have a slightly shorterperiod as Earth approaches Jupiter and longer when Earth movesaway.

Roemer’s 1675 lower limit for c : 2.3× 108 m/s.

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Wheatstone’s Rotating Mirror

Charles Wheatstone created a rotating mirror arrangement tostudy fast phenomena in electricity.

He told Francios Arago that he thought this could be used tomeasure the speed of light.

Arago passed on the suggestion to Armand Fizeau and LeonFoucault who were collaborating on various optical studies. Hesuggested it might be useful to measure the speed of light in wateras well as air to compare them.

Fizeau and Foucault then fell out and stopped working together.Both pursued their investigation separately.

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Fizeau’s Method

Fizeau sent a beam of light through a gear-tooth wheel toward amirror 5 miles (8 km) away.

1060 Chapter 35 The Nature of Light and the Principles of Ray Optics

Roemer’s MethodIn 1675, Danish astronomer Ole Roemer (1644–1710) made the first successful esti-mate of the speed of light. Roemer’s technique involved astronomical observations of Io, one of the moons of Jupiter. Io has a period of revolution around Jupiter of approximately 42.5 h. The period of revolution of Jupiter around the Sun is about 12 yr; therefore, as the Earth moves through 90° around the Sun, Jupiter revolves through only ( 1

12)90° 5 7.5° (Fig. 35.1). An observer using the orbital motion of Io as a clock would expect the orbit to have a constant period. After collecting data for more than a year, however, Roemer observed a systematic variation in Io’s period. He found that the periods were lon-ger than average when the Earth was receding from Jupiter and shorter than aver-age when the Earth was approaching Jupiter. Roemer attributed this variation in period to the distance between the Earth and Jupiter changing from one observa-tion to the next. Using Roemer’s data, Huygens estimated the lower limit for the speed of light to be approximately 2.3 3 108 m/s. This experiment is important historically because it demonstrated that light does have a finite speed and gave an estimate of this speed.

Fizeau’s MethodThe first successful method for measuring the speed of light by means of purely ter-restrial techniques was developed in 1849 by French physicist Armand H. L. Fizeau (1819–1896). Figure 35.2 represents a simplified diagram of Fizeau’s apparatus. The basic procedure is to measure the total time interval during which light travels from some point to a distant mirror and back. If d is the distance between the light source (considered to be at the location of the wheel) and the mirror and if the time interval for one round trip is Dt, the speed of light is c 5 2d/Dt. To measure the transit time, Fizeau used a rotating toothed wheel, which con-verts a continuous beam of light into a series of light pulses. The rotation of such a wheel controls what an observer at the light source sees. For example, if the pulse traveling toward the mirror and passing the opening at point A in Figure 35.2 should return to the wheel at the instant tooth B had rotated into position to cover the return path, the pulse would not reach the observer. At a greater rate of rota-tion, the opening at point C could move into position to allow the reflected pulse to reach the observer. Knowing the distance d, the number of teeth in the wheel, and the angular speed of the wheel, Fizeau arrived at a value of 3.1 3 108 m/s. Similar measurements made by subsequent investigators yielded more precise values for c, which led to the currently accepted value of 2.997 924 58 3 108 m/s.

A

BC

Mirror

d

Toothedwheel

Figure 35.2 Fizeau’s method for measuring the speed of light using a rotating toothed wheel. The light source is considered to be at the location of the wheel; there-fore, the distance d is known.

Example 35.1 Measuring the Speed of Light with Fizeau’s Wheel

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/s when a pulse of light passing through opening A in Fig-ure 35.2 is blocked by tooth B on its return. If the distance to the mirror is 7 500 m, what is the speed of light?

Conceptualize Imagine a pulse of light passing through opening A in Figure 35.2 and reflecting from the mirror. By the time the pulse arrives back at the wheel, tooth B has rotated into the position previously occupied by opening A.

Categorize The wheel is a rigid object rotating at constant angular speed. We model the pulse of light as a particle under constant speed.

Analyze The wheel has 360 teeth, so it must have 360 openings. Therefore, because the light passes through opening A but is blocked by the tooth immediately adjacent to A, the wheel must rotate through an angular displacement of 1

720 rev in the time interval during which the light pulse makes its round trip.

AM

S O L U T I O N

Use Equation 10.2, with the angular speed constant, to find the time interval for the pulse’s round trip:

Dt 5Du

v5

1720 rev

27.5 rev/s5 5.05 3 1025 s

J1

E1Sun

E2

J2

Io

90!

7.5!

In the time interval during which the Earth travels 90! around the Sun (three months), Jupiter travels only about 7.5!.

Figure 35.1 Roemer’s method for measuring the speed of light (drawing not to scale).

The teeth broke up the beam into pulses as it rotated.

When the wheel rotates fast enough, the light passing through gapA is blocked by tooth B on it’s return from the mirror.

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Fizeau’s Wheel Example

1060 Chapter 35 The Nature of Light and the Principles of Ray Optics

Roemer’s MethodIn 1675, Danish astronomer Ole Roemer (1644–1710) made the first successful esti-mate of the speed of light. Roemer’s technique involved astronomical observations of Io, one of the moons of Jupiter. Io has a period of revolution around Jupiter of approximately 42.5 h. The period of revolution of Jupiter around the Sun is about 12 yr; therefore, as the Earth moves through 90° around the Sun, Jupiter revolves through only ( 1

12)90° 5 7.5° (Fig. 35.1). An observer using the orbital motion of Io as a clock would expect the orbit to have a constant period. After collecting data for more than a year, however, Roemer observed a systematic variation in Io’s period. He found that the periods were lon-ger than average when the Earth was receding from Jupiter and shorter than aver-age when the Earth was approaching Jupiter. Roemer attributed this variation in period to the distance between the Earth and Jupiter changing from one observa-tion to the next. Using Roemer’s data, Huygens estimated the lower limit for the speed of light to be approximately 2.3 3 108 m/s. This experiment is important historically because it demonstrated that light does have a finite speed and gave an estimate of this speed.

Fizeau’s MethodThe first successful method for measuring the speed of light by means of purely ter-restrial techniques was developed in 1849 by French physicist Armand H. L. Fizeau (1819–1896). Figure 35.2 represents a simplified diagram of Fizeau’s apparatus. The basic procedure is to measure the total time interval during which light travels from some point to a distant mirror and back. If d is the distance between the light source (considered to be at the location of the wheel) and the mirror and if the time interval for one round trip is Dt, the speed of light is c 5 2d/Dt. To measure the transit time, Fizeau used a rotating toothed wheel, which con-verts a continuous beam of light into a series of light pulses. The rotation of such a wheel controls what an observer at the light source sees. For example, if the pulse traveling toward the mirror and passing the opening at point A in Figure 35.2 should return to the wheel at the instant tooth B had rotated into position to cover the return path, the pulse would not reach the observer. At a greater rate of rota-tion, the opening at point C could move into position to allow the reflected pulse to reach the observer. Knowing the distance d, the number of teeth in the wheel, and the angular speed of the wheel, Fizeau arrived at a value of 3.1 3 108 m/s. Similar measurements made by subsequent investigators yielded more precise values for c, which led to the currently accepted value of 2.997 924 58 3 108 m/s.

A

BC

Mirror

d

Toothedwheel

Figure 35.2 Fizeau’s method for measuring the speed of light using a rotating toothed wheel. The light source is considered to be at the location of the wheel; there-fore, the distance d is known.

Example 35.1 Measuring the Speed of Light with Fizeau’s Wheel

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/s when a pulse of light passing through opening A in Fig-ure 35.2 is blocked by tooth B on its return. If the distance to the mirror is 7 500 m, what is the speed of light?

Conceptualize Imagine a pulse of light passing through opening A in Figure 35.2 and reflecting from the mirror. By the time the pulse arrives back at the wheel, tooth B has rotated into the position previously occupied by opening A.

Categorize The wheel is a rigid object rotating at constant angular speed. We model the pulse of light as a particle under constant speed.

Analyze The wheel has 360 teeth, so it must have 360 openings. Therefore, because the light passes through opening A but is blocked by the tooth immediately adjacent to A, the wheel must rotate through an angular displacement of 1

720 rev in the time interval during which the light pulse makes its round trip.

AM

S O L U T I O N

Use Equation 10.2, with the angular speed constant, to find the time interval for the pulse’s round trip:

Dt 5Du

v5

1720 rev

27.5 rev/s5 5.05 3 1025 s

J1

E1Sun

E2

J2

Io

90!

7.5!

In the time interval during which the Earth travels 90! around the Sun (three months), Jupiter travels only about 7.5!.

Figure 35.1 Roemer’s method for measuring the speed of light (drawing not to scale).

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/swhen a pulse of light passing through opening A is blocked bytooth B on its return. If the distance to the mirror is 7500 m, whatis the speed of light?

∆t =∆θ

ωc =

2d

∆t

c = 2.97× 108 m/s

Page 46: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Fizeau’s Wheel Example

1060 Chapter 35 The Nature of Light and the Principles of Ray Optics

Roemer’s MethodIn 1675, Danish astronomer Ole Roemer (1644–1710) made the first successful esti-mate of the speed of light. Roemer’s technique involved astronomical observations of Io, one of the moons of Jupiter. Io has a period of revolution around Jupiter of approximately 42.5 h. The period of revolution of Jupiter around the Sun is about 12 yr; therefore, as the Earth moves through 90° around the Sun, Jupiter revolves through only ( 1

12)90° 5 7.5° (Fig. 35.1). An observer using the orbital motion of Io as a clock would expect the orbit to have a constant period. After collecting data for more than a year, however, Roemer observed a systematic variation in Io’s period. He found that the periods were lon-ger than average when the Earth was receding from Jupiter and shorter than aver-age when the Earth was approaching Jupiter. Roemer attributed this variation in period to the distance between the Earth and Jupiter changing from one observa-tion to the next. Using Roemer’s data, Huygens estimated the lower limit for the speed of light to be approximately 2.3 3 108 m/s. This experiment is important historically because it demonstrated that light does have a finite speed and gave an estimate of this speed.

Fizeau’s MethodThe first successful method for measuring the speed of light by means of purely ter-restrial techniques was developed in 1849 by French physicist Armand H. L. Fizeau (1819–1896). Figure 35.2 represents a simplified diagram of Fizeau’s apparatus. The basic procedure is to measure the total time interval during which light travels from some point to a distant mirror and back. If d is the distance between the light source (considered to be at the location of the wheel) and the mirror and if the time interval for one round trip is Dt, the speed of light is c 5 2d/Dt. To measure the transit time, Fizeau used a rotating toothed wheel, which con-verts a continuous beam of light into a series of light pulses. The rotation of such a wheel controls what an observer at the light source sees. For example, if the pulse traveling toward the mirror and passing the opening at point A in Figure 35.2 should return to the wheel at the instant tooth B had rotated into position to cover the return path, the pulse would not reach the observer. At a greater rate of rota-tion, the opening at point C could move into position to allow the reflected pulse to reach the observer. Knowing the distance d, the number of teeth in the wheel, and the angular speed of the wheel, Fizeau arrived at a value of 3.1 3 108 m/s. Similar measurements made by subsequent investigators yielded more precise values for c, which led to the currently accepted value of 2.997 924 58 3 108 m/s.

A

BC

Mirror

d

Toothedwheel

Figure 35.2 Fizeau’s method for measuring the speed of light using a rotating toothed wheel. The light source is considered to be at the location of the wheel; there-fore, the distance d is known.

Example 35.1 Measuring the Speed of Light with Fizeau’s Wheel

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/s when a pulse of light passing through opening A in Fig-ure 35.2 is blocked by tooth B on its return. If the distance to the mirror is 7 500 m, what is the speed of light?

Conceptualize Imagine a pulse of light passing through opening A in Figure 35.2 and reflecting from the mirror. By the time the pulse arrives back at the wheel, tooth B has rotated into the position previously occupied by opening A.

Categorize The wheel is a rigid object rotating at constant angular speed. We model the pulse of light as a particle under constant speed.

Analyze The wheel has 360 teeth, so it must have 360 openings. Therefore, because the light passes through opening A but is blocked by the tooth immediately adjacent to A, the wheel must rotate through an angular displacement of 1

720 rev in the time interval during which the light pulse makes its round trip.

AM

S O L U T I O N

Use Equation 10.2, with the angular speed constant, to find the time interval for the pulse’s round trip:

Dt 5Du

v5

1720 rev

27.5 rev/s5 5.05 3 1025 s

J1

E1Sun

E2

J2

Io

90!

7.5!

In the time interval during which the Earth travels 90! around the Sun (three months), Jupiter travels only about 7.5!.

Figure 35.1 Roemer’s method for measuring the speed of light (drawing not to scale).

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/swhen a pulse of light passing through opening A is blocked bytooth B on its return. If the distance to the mirror is 7500 m, whatis the speed of light?

∆t =∆θ

ωc =

2d

∆t

c = 2.97× 108 m/s

Page 47: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Fizeau’s Wheel Example

1060 Chapter 35 The Nature of Light and the Principles of Ray Optics

Roemer’s MethodIn 1675, Danish astronomer Ole Roemer (1644–1710) made the first successful esti-mate of the speed of light. Roemer’s technique involved astronomical observations of Io, one of the moons of Jupiter. Io has a period of revolution around Jupiter of approximately 42.5 h. The period of revolution of Jupiter around the Sun is about 12 yr; therefore, as the Earth moves through 90° around the Sun, Jupiter revolves through only ( 1

12)90° 5 7.5° (Fig. 35.1). An observer using the orbital motion of Io as a clock would expect the orbit to have a constant period. After collecting data for more than a year, however, Roemer observed a systematic variation in Io’s period. He found that the periods were lon-ger than average when the Earth was receding from Jupiter and shorter than aver-age when the Earth was approaching Jupiter. Roemer attributed this variation in period to the distance between the Earth and Jupiter changing from one observa-tion to the next. Using Roemer’s data, Huygens estimated the lower limit for the speed of light to be approximately 2.3 3 108 m/s. This experiment is important historically because it demonstrated that light does have a finite speed and gave an estimate of this speed.

Fizeau’s MethodThe first successful method for measuring the speed of light by means of purely ter-restrial techniques was developed in 1849 by French physicist Armand H. L. Fizeau (1819–1896). Figure 35.2 represents a simplified diagram of Fizeau’s apparatus. The basic procedure is to measure the total time interval during which light travels from some point to a distant mirror and back. If d is the distance between the light source (considered to be at the location of the wheel) and the mirror and if the time interval for one round trip is Dt, the speed of light is c 5 2d/Dt. To measure the transit time, Fizeau used a rotating toothed wheel, which con-verts a continuous beam of light into a series of light pulses. The rotation of such a wheel controls what an observer at the light source sees. For example, if the pulse traveling toward the mirror and passing the opening at point A in Figure 35.2 should return to the wheel at the instant tooth B had rotated into position to cover the return path, the pulse would not reach the observer. At a greater rate of rota-tion, the opening at point C could move into position to allow the reflected pulse to reach the observer. Knowing the distance d, the number of teeth in the wheel, and the angular speed of the wheel, Fizeau arrived at a value of 3.1 3 108 m/s. Similar measurements made by subsequent investigators yielded more precise values for c, which led to the currently accepted value of 2.997 924 58 3 108 m/s.

A

BC

Mirror

d

Toothedwheel

Figure 35.2 Fizeau’s method for measuring the speed of light using a rotating toothed wheel. The light source is considered to be at the location of the wheel; there-fore, the distance d is known.

Example 35.1 Measuring the Speed of Light with Fizeau’s Wheel

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/s when a pulse of light passing through opening A in Fig-ure 35.2 is blocked by tooth B on its return. If the distance to the mirror is 7 500 m, what is the speed of light?

Conceptualize Imagine a pulse of light passing through opening A in Figure 35.2 and reflecting from the mirror. By the time the pulse arrives back at the wheel, tooth B has rotated into the position previously occupied by opening A.

Categorize The wheel is a rigid object rotating at constant angular speed. We model the pulse of light as a particle under constant speed.

Analyze The wheel has 360 teeth, so it must have 360 openings. Therefore, because the light passes through opening A but is blocked by the tooth immediately adjacent to A, the wheel must rotate through an angular displacement of 1

720 rev in the time interval during which the light pulse makes its round trip.

AM

S O L U T I O N

Use Equation 10.2, with the angular speed constant, to find the time interval for the pulse’s round trip:

Dt 5Du

v5

1720 rev

27.5 rev/s5 5.05 3 1025 s

J1

E1Sun

E2

J2

Io

90!

7.5!

In the time interval during which the Earth travels 90! around the Sun (three months), Jupiter travels only about 7.5!.

Figure 35.1 Roemer’s method for measuring the speed of light (drawing not to scale).

Assume Fizeau’s wheel has 360 teeth and rotates at 27.5 rev/swhen a pulse of light passing through opening A is blocked bytooth B on its return. If the distance to the mirror is 7500 m, whatis the speed of light?

∆t =∆θ

ωc =

2d

∆t

c = 2.97× 108 m/s

Page 48: Waves Wrap Up and Review Doppler Effect Optics Nature of …nebula2.deanza.edu/~lanasheridan/4C/Phys4C-Lecture16-san.pdf3Named after Austrian physicist Christian Johann Doppler (1803Ð1853),

Foucault’s Method

Foucault used a rotating mirror, to send light from a source to astationary mirror and back again.

(Foucault did not use a laser, obviously.)

1Figure from Wikipedia, by user Rhodesl.

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Foucault’s Method

The angle formed between the source and the returning light beamallowed him to figure out how much the mirror had rotated(therefore how much time had passed) while the light traveledfrom R to M and back.

Foucault could only separate the mirrors by a distance of 20m, dueto limitations on his mirrors and lenses.

1Figure from Wikipedia, by user Stigmatella aurantiaca.

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Michelson’s Refinement

Albert Michelson adapted Foucault’s apparatus to increase pathlength of the light to 22 miles!

He used two observatories on adjacent mountains.

In spite of a forest fire and an earthquake, he got the value of

299, 796± 4 km/s

This is only 4 km/s faster that the current accepted value.

He later worked on the famous Michelson-Morley experiment whichshowed that light needs no medium.

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Summary

• sound level

• the Doppler effect

• bow and shock waves

• nature of light

• speed of light

Homework Serway & Jewett:

• Ch 17, onward from page 523. Probs: 19, 25, 37, 41, 47, 54,71

• Ch 18, onward from page 555. OQs: 3, 5; CQs: 3; Probs: 59,77

• Ch 35, onward from page 1077. CQs: 15; Probs: 1, 3