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superposition of two or more waves moving in opposite directions. The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. Standing waves created by the superposition of two identical waves moving in opposite directions are illustrated in Figure 13.14. Figure 13.14 A standing wave is created by the superposition of two identical waves moving in opposite directions. The oscillations are at fixed locations in space and result from alternating constructive and destructive interferences. As an example, standing waves can be seen on the surface of a glass of milk in a refrigerator. The vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. The two waves that Access for free at openstax.org. 13.3 • Wave Interaction: Superposition and Interference 403 produce standing waves may be due to the reflections from the side of the glass. Earthquakes can create standing waves and cause constructive and destructive interferences. As the earthquake waves travel along the surface of Earth and reflect off denser rocks, constructive interference occurs at certain points. As a result, areas closer to the epicenter are not damaged while areas farther from the epicenter are damaged. Standing waves are also found on the strings of musical instruments and are due to reflections of waves from the ends of the string. Figure 13.15 and Figure 13.16 show three standing waves that can be created on a string that is fixed at both ends. When the wave reaches the fixed end, it has nowhere else to go but back where it came from, causing the reflection. The nodes are the points where the string does not move; more generally, the nodes are the points where the wave disturbance is zero in a standing wave. The fixed ends of strings must be nodes, too, because the string cannot move there. The antinode is the location of maximum amplitude in standing waves. The standing waves on a string have a frequency that is of the disturbance on the string. The wavelength is determined by the distance between related to the propagation speed the points where the string is fixed in place. Figure 13.15 The figure shows a string oscillating with its maximum disturbance as the antinode. Figure 13.16 The figure shows a string oscillating with multiple nodes. Reflection and Refraction of Waves As we saw in the case of standing waves on the strings of a musical instrument, reflection is the change in
direction of a wave when it bounces off a barrier, such as a fixed end. When the wave hits the fixed end, it changes direction, returning to its source. As it is reflected, the wave experiences an inversion, which means that it flips vertically. If a wave hits the fixed end with a crest, it will return as a trough, and vice versa (Henderson 2015). Refer to Figure 13.17. Figure 13.17 A wave is inverted after reflection from a fixed end. TIPS FOR SUCCESS If the end is not fixed, it is said to be a free end, and no inversion occurs. When the end is loosely attached, it reflects without 404 Chapter 13 • Waves and Their Properties inversion, and when the end is not attached to anything, it does not reflect at all. You may have noticed this while changing the settings from Fixed End to Loose End to No End in the Waves on a String PhET simulation. Rather than encountering a fixed end or barrier, waves sometimes pass from one medium into another, for instance, from air into water. Different types of media have different properties, such as density or depth, that affect how a wave travels through them. At the boundary between media, waves experience refraction—they change their path of propagation. As the wave bends, it also changes its speed and wavelength upon entering the new medium. Refer to Figure 13.18. Figure 13.18 A wave refracts as it enters a different medium. For example, water waves traveling from the deep end to the shallow end of a swimming pool experience refraction. They bend in a path closer to perpendicular to the surface of the water, propagate slower, and decrease in wavelength as they enter shallower water. Check Your Understanding 13. What is the superposition of waves? a. When a single wave splits into two different waves at a point b. When two waves combine at the same place at the same time 14. How do waves superimpose on one another? a. By adding their frequencies b. By adding their wavelengths c. By adding their disturbances d. By adding their speeds 15. What is interference of waves? a. b. c. d. Interference is a superposition of two waves to form a resultant wave with higher or lower frequency. Interference is a superposition of two waves to form a wave of larger or smaller amplitude. Interference is a superposition of two waves to form a resultant wave with higher or lower velocity. Interference is
a superposition of two waves to form a resultant wave with longer or shorter wavelength. 16. Is the following statement true or false? The two types of interference are constructive and destructive interferences. a. True b. False 17. What are standing waves? a. Waves that appear to remain in one place and do not seem to move b. Waves that seem to move along a trajectory 18. How are standing waves formed? a. Standing waves are formed by the superposition of two or more waves moving in opposite directions. b. Standing waves are formed by the superposition of two or more waves moving in the same direction. c. Standing waves are formed by the superposition of two or more waves moving in perpendicular directions. d. Standing waves are formed by the superposition of two or more waves moving in any arbitrary directions. 19. What is the reflection of a wave? a. The reflection of a wave is the change in amplitude of a wave when it bounces off a barrier. b. The reflection of a wave is the change in frequency of a wave when it bounces off a barrier. c. The reflection of a wave is the change in velocity of a wave when it bounces off a barrier. d. The reflection of a wave is the change in direction of a wave when it bounces off a barrier. Access for free at openstax.org. 13.3 • Wave Interaction: Superposition and Interference 405 20. What is inversion of a wave? a. b. c. d. Inversion occurs when a wave reflects off a fixed end, and the wave amplitude changes sign. Inversion occurs when a wave reflects off a loose end, and the wave amplitude changes sign. Inversion occurs when a wave reflects off a fixed end without the wave amplitude changing sign. Inversion occurs when a wave reflects off a loose end without the wave amplitude changing sign. 406 Chapter 13 • Key Terms KEY TERMS antinode location of maximum amplitude in standing motion waves pulse wave sudden disturbance with only one wave or a few constructive interference when two waves arrive at the waves generated same point exactly in phase; that is, the crests of the two waves are precisely aligned, as are the troughs reflection change in direction of a wave at a boundary or fixed end destructive interference when two identical waves arrive at refraction bending of a wave as it passes from one medium the same point exactly out of phase that is precisely aligned crest to trough inversion vertical flipping of a wave after reflection from a fixed end longitudinal wave wave
in which the disturbance is parallel to another medium with a different density standing wave wave made by the superposition of two waves of the same amplitude and wavelength moving in opposite directions and which appears to vibrate in place superposition phenomenon that occurs when two or more to the direction of propagation waves arrive at the same point mechanical wave wave that requires a medium through which it can travel transverse wave wave in which the disturbance is perpendicular to the direction of propagation medium solid, liquid, or gas material through which a wave disturbance that moves from its source and carries wave propagates energy nodes points where the string does not move; more generally, points where the wave disturbance is zero in a standing wave periodic wave wave that repeats the same oscillation for several cycles and is associated with simple harmonic SECTION SUMMARY 13.1 Types of Waves • A wave is a disturbance that moves from the point of creation and carries energy but not mass. • Mechanical waves must travel through a medium. • Sound waves, water waves, and earthquake waves are all examples of mechanical waves. • Light is not a mechanical wave since it can travel through a vacuum. • A periodic wave is a wave that repeats for several cycles, whereas a pulse wave has only one crest or a few crests and is associated with a sudden disturbance. • Periodic waves are associated with simple harmonic motion. • A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation. 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period • A wave is a disturbance that moves from the point of creation at a wave velocity vw. • A wave has a wavelength, which is the distance between adjacent identical parts of the wave. • The wave velocity and the wavelength are related to the wave’s frequency and period by or Access for free at openstax.org. wave velocity speed at which the disturbance moves; also called the propagation velocity or propagation speed wavelength distance between adjacent identical parts of a wave • The time for one complete wave cycle is the period T. • The number of waves per unit time is the frequency ƒ. • The wave frequency and the period are inversely related to one another. 13.3 Wave Interaction: Superposition and Interference • Superposition is the combination of two waves at the same location. • Constructive interference occurs when two identical waves are superimposed exactly in phase. • Destructive interference occurs when two identical waves are superimposed exactly out of phase
. • A standing wave is a wave produced by the superposition of two waves. It varies in amplitude but does not propagate. • The nodes are the points where there is no motion in standing waves. • An antinode is the location of maximum amplitude of a standing wave. Inversion occurs when a wave reflects from a fixed end. • Reflection causes a wave to change direction. • • Refraction causes a wave’s path to bend and occurs when a wave passes from one medium into another medium with a different density. KEY EQUATIONS 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period wave velocity or CHAPTER REVIEW Concept Items 13.1 Types of Waves 1. Do water waves push water from one place to another? Explain. a. No, water waves transfer only energy from one place to another. b. Yes, water waves transfer water from one place to another. 2. With reference to waves, what is a trough? a. b. c. d. the lowermost position of a wave the uppermost position of a wave the final position of a wave the initial position of the wave 3. Give an example of longitudinal waves. light waves a. b. water waves in a lake sound waves in air c. seismic waves in Earth’s surface d. 4. What does the speed of a mechanical wave depend on? a. b. c. d. the properties of the material through which it travels the shape of the material through which it travels the size of the material through which it travels the color of the material through which it travels 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period 5. Which characteristic of a transverse wave is measured along the direction of propagation? a. The amplitude of a transverse wave is measured along the direction of propagation. b. The amplitude and the wavelength of a transverse wave are measured along the direction of propagation. c. The wavelength of a transverse wave is measured along the direction of propagation. d. The displacement of the particles of the medium in a transverse wave is measured along the direction of propagation. 6. Which kind of seismic waves cannot travel through Chapter 13 • Key Equations 407 period frequency liquid? a. b. P-waves c. d. S-waves compressional waves longitudinal waves 7. What is the period of a wave? a. b. c. d. the time that a wave takes to complete a half cycle the time that a wave takes to complete
one cycle the time that a wave takes to complete two cycles the time that a wave takes to complete four cycles 8. When the period of a wave increases, what happens to its frequency? a. b. c. Its frequency decreases. Its frequency increases. Its frequency remains the same. 13.3 Wave Interaction: Superposition and Interference 9. Is this statement true or false? The amplitudes of waves add up only if they are propagating in the same line. a. True b. False 10. Why is sound from a stereo louder in one part of the room and softer in another? a. Sound is louder in parts of the room where the density is greatest. Sound is softer in parts of the room where density is smallest. b. Sound is louder in parts of the room where the density is smallest. Sound is softer in parts of the room where density is greatest. c. Sound is louder in parts of the room where constructive interference occurs and softer in parts where destructive interference occurs. d. Sound is louder in parts of the room where destructive interference occurs and softer in parts where constructive interference occurs. 11. In standing waves on a string, what does the frequency depend on? a. The frequency depends on the propagation speed and the density of the string. 408 Chapter 13 • Chapter Review b. The frequency depends on the propagation speed a. Refraction is the phenomenon in which waves and the length of the string. c. The frequency depends on the density and the length of the string. d. The frequency depends on the propagation speed, the density, and the length of the string. 12. Is the following statement true or false? Refraction is useful in fiber optic cables for transmitting signals. a. False b. True 13. What is refraction? Critical Thinking Items 13.1 Types of Waves 14. Give an example of a wave that propagates only through a solid. a. Light wave b. Sound wave c. Seismic wave d. Surface wave 15. Can mechanical waves be periodic waves? a. No, mechanical waves cannot be periodic waves. b. Yes, mechanical waves can be periodic. 16. In a sound wave, which parameter of the medium varies with every cycle? a. The density of the medium varies with every cycle. b. The mass of the medium varies with every cycle. c. The resistivity of the medium varies with every cycle. d. The volume of the medium varies with every cycle. 17. What is a transverse wave in an earthquake
called? a. L-wave b. P-wave c. S-wave d. R-wave 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period 18. If the horizontal distance, that is, the distance in the direction of propagation, between a crest and the adjacent trough of a sine wave is 1 m, what is the wavelength of the wave? a. 0.5 m b. 1 m c. 2 m d. 4 m 19. How is the distance to the epicenter of an earthquake determined? Access for free at openstax.org. change their path of propagation at the interface of two media with different densities. b. Refraction is the phenomenon in which waves change their path of propagation at the interface of two media with the same density. c. Refraction is the phenomenon in which waves become non-periodic at the boundary of two media with different densities. d. Refraction is the phenomenon in which waves become non-periodic at the boundary of two media with the same density. a. The wavelength difference between P-waves and S- waves is used to measure the distance to the epicenter. b. The time difference between P-waves and S-waves is used to measure the distance to the epicenter. c. The frequency difference between P-waves and Swaves is used to measure the distance to the epicenter. d. The phase difference between P-waves and S-waves is used to measure the distance to the epicenter. 20. Two identical waves superimpose in pure constructive interference. What is the height of the resultant wave if the amplitude of each of the waves is 1 m? a. 1 m b. 2 m 3 m c. d. 4 m 13.3 Wave Interaction: Superposition and Interference 21. Two identical waves with an amplitude superimpose in a way that pure constructive interference occurs. What is the amplitude of the resultant wave? a. b. c. d. 22. In which kind of wave is the amplitude at each point constant? a. Seismic waves b. Pulse wave c. Standing waves d. Electromagnetic waves 23. Which property of a medium causes refraction? a. Conductivity b. Opacity c. Ductility d. Density 24. What is added together when two waves superimpose? a. Amplitudes Problems 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period 25. If a seag
ull sitting in water bobs up and down once every 2 seconds and the distance between two crests of the water wave is 3 m, what is the velocity of the wave? 1.5 m/s a. 3 m/s b. c. 6 m/s Performance Task 13.3 Wave Interaction: Superposition and Interference 27. Ocean waves repeatedly crash against beaches and coasts. Their energy can lead to erosion and collapse of land. Scientists and engineers need to study how waves interact with beaches in order to assess threats to coastal communities and construct breakwater systems. In this task, you will construct a wave tank and fill it with water. Simulate a beach by placing sand at one end. Create waves by moving a piece of wood or plastic up and down in the water. Measure or estimate the TEST PREP Multiple Choice 13.1 Types of Waves 28. What kind of waves are sound waves? a. Mechanical waves b. Electromagnetic waves Chapter 13 • Test Prep 409 b. Wavelengths c. Velocities d. 12 m/s 26. A boat in the trough of a wave takes 3 seconds to reach the highest point of the wave. The velocity of the wave is 5 m/s. What is its wavelength? a. 0.83 m 15 m b. 30 m c. 180 m d. wavelength, period, frequency, and amplitude of the wave, and observe the effect of the wave on the sand. Produce waves of different amplitudes and frequencies, and record your observations each time. Use mathematical representations to demonstrate the relationships between different wave properties. Change the position of the sand to create a steeper beach and record your observations. Give a qualitative analysis of the effects of the waves on the beach. What kind of wave causes the most damage? At what height, wavelength, and frequency do waves break? How does the steepness of the beach affect the waves? that creates a wave. It refers to the wavelength of the wave. It refers to the speed of the wave. c. d. 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period 29. What kind of a wave does a tuning fork create? 32. Which of these is not a characteristic of a wave? a. Pulse wave b. Periodic wave c. Electromagnetic wave 30. What kind of waves are electromagnetic waves? a. amplitude b. period c. mass d. velocity a. Longitudinal waves b. Transverse waves c. Mechanical waves d.
P-waves a. 31. With reference to waves, what is a disturbance? It refers to the resistance produced by some particles of a material. It refers to an oscillation produced by some energy b. 33. If you are in a boat at a resting position, how much will your height change when you are hit by the peak of a wave with a height of 2 m? a. 0 m 1 m b. c. 2 m d. 4 m 34. What is the period of a wave with a frequency of 0.5 Hz? 410 Chapter 13 • Test Prep a. 0.5 s 1 s b. c. 2 s 3 s d. 35. What is the relation between the amplitude of a wave and its speed? a. The amplitude of a wave is independent of its speed. b. The amplitude of a wave is directly proportional to its speed. c. The amplitude of a wave is directly proportional to the square of the inverse of its speed. d. The amplitude of a wave is directly proportional to the inverse of its speed. 36. What does the speed of seismic waves depend on? a. The speed of seismic waves depends on the size of the medium. b. The speed of seismic waves depends on the shape of the medium. c. The speed of seismic waves depends on the rigidity of the medium. 13.3 Wave Interaction: Superposition and Interference 37. What is added together when two waves superimpose? a. amplitudes b. wavelengths c. velocities 38. Pure constructive interference occurs between two waves when they have the same _____. Short Answer 13.1 Types of Waves frequency and are in phase frequency and are out of phase a. b. c. amplitude and are in phase d. amplitude and are out of phase 39. What kind(s) of interference can occur between two identical waves moving in opposite directions? a. Constructive interference only b. Destructive interference only c. Both constructive and destructive interference d. Neither constructive nor destructive interference 40. What term refers to the bending of light at the junction interference of two media? a. b. diffraction scattering c. refraction d. 41. Which parameter of a wave gets affected after superposition? a. wavelength b. direction c. amplitude frequency d. 42. When do the amplitudes of two waves get added? a. When their amplitudes are the same b. When their amplitudes are different c. When they propagate in perpendicular directions d. When they
are propagating along the same line in opposite directions d. The cone of a speaker vibrates to create small changes in the resistance of the air. 43. Give an example of a non-mechanical wave. a. A radio wave is an example of a nonmechanical wave. b. A sound wave is an example of a nonmechanical wave. 45. What kind of wave is thunder? a. Transverse wave b. Pulse wave c. R-wave d. P-wave c. A surface wave is an example of a nonmechanical 46. Are all ocean waves perfectly sinusoidal? wave. d. A seismic wave is an example of a nonmechanical wave. 44. How is sound produced by an electronic speaker? a. The cone of a speaker vibrates to create small changes in the temperature of the air. b. The cone of a speaker vibrates to create small changes in the pressure of the air. c. The cone of a speaker vibrates to create small changes in the volume of the air. a. No, all ocean waves are not perfectly sinusoidal. b. Yes, all ocean waves are perfectly sinusoidal. 47. What are orbital progressive waves? a. Waves that force the particles of the medium to follow a linear path from the crest to the trough b. Waves that force the particles of the medium to follow a circular path from the crest to the trough c. Waves that force the particles of the medium to follow a zigzag path from the crest to the trough d. Waves that force the particles of the medium to Access for free at openstax.org. follow a random path from the crest to the trough 48. Give an example of orbital progressive waves. a. light waves b. ocean waves sound waves c. seismic waves d. 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period 49. What is the relation between the amplitude and height of a transverse wave? a. The height of a wave is half of its amplitude. b. The height of a wave is equal to its amplitude. c. The height of a wave is twice its amplitude. d. The height of a wave is four times its amplitude. 50. If the amplitude of a water wave is 0.2 m and its frequency is 2 Hz, how much distance would a bird sitting on the water’s surface move with every wave? How many times will it do this every second? a. The
bird will go up and down a distance of 0.4 m. It will do this twice per second. b. The bird will go up and down a distance of 0.2 m. It will do this twice per second. c. The bird will go up and down a distance of 0.4 m. It will do this once per second. d. The bird will go up and down a distance of 0.2 m. It will do this once per second. 51. What is the relation between the amplitude and the frequency of a wave? a. The amplitude and the frequency of a wave are independent of each other. b. The amplitude and the frequency of a wave are equal. c. The amplitude decreases with an increase in the frequency of a wave. d. The amplitude increases with an increase in the frequency of a wave. 52. What is the relation between a wave’s energy and its amplitude? a. There is no relation between the energy and the amplitude of a wave. b. The magnitude of the energy is equal to the magnitude of the amplitude of a wave. c. The energy of a wave increases with an increase in the amplitude of the wave. d. The energy of a wave decreases with an increase in the amplitude of a wave. 53. A wave travels every 2 cycles. What is its wavelength? a. Chapter 13 • Test Prep 411 b. c. d. 54. A water wave propagates in a river at 6 m/s. If the river moves in the opposite direction at 3 m/s, what is the effective velocity of the wave? a. 3 m/s b. 6 m/s c. 9 m/s 18 m/s d. 13.3 Wave Interaction: Superposition and Interference 55. Is this statement true or false? Spherical waves can superimpose. a. True b. False 56. Is this statement true or false? Waves can superimpose if their frequencies are different. a. True b. False 57. When does pure destructive interference occur? a. When two waves with equal frequencies that are perfectly in phase and propagating along the same line superimpose. b. When two waves with unequal frequencies that are perfectly in phase and propagating along the same line superimpose. c. When two waves with unequal frequencies that are perfectly out of phase and propagating along the same line superimpose. d. When two waves with equal frequencies that are perfectly out of phase and propagating
along the same line superimpose. 58. Is this statement true or false? The amplitude of one wave is affected by the amplitude of another wave only when they are precisely aligned. a. True b. False 59. Why does a standing wave form on a guitar string? a. due to superposition with the reflected waves from the ends of the string b. due to superposition with the reflected waves from the walls of the room c. due to superposition with waves generated from the body of the guitar 60. Is the following statement true or false? A standing wave is a superposition of two identical waves that are in phase and propagating in the same direction. 412 Chapter 13 • Test Prep a. True b. False 61. Why do water waves traveling from the deep end to the shallow end of a swimming pool experience refraction? a. Because the pressure of water at the two ends of the c. Because the density of water at the two ends of the pool is same d. Because the density of water at the two ends of the pool is different 62. Is the statement true or false? Waves propagate faster in pool is same b. Because the pressures of water at the two ends of the pool are different a less dense medium if the stiffness is the same. a. True b. False Extended Response 13.1 Types of Waves 63. Why can light travel through outer space while sound cannot? a. Sound waves are mechanical waves and require a medium to propagate. Light waves can travel through a vacuum. b. Sound waves are electromagnetic waves and require a medium to propagate. Light waves can travel through a vacuum. c. Light waves are mechanical waves and do not require a medium to propagate; sound waves require a medium to propagate. d. Light waves are longitudinal waves and do not require a medium to propagate; sound waves require a medium to propagate. 64. Do periodic waves require a medium to travel through? a. No, the requirement of a medium for propagation does not depend on whether the waves are pulse waves or periodic waves. b. Yes, the requirement of a medium for propagation depends on whether the waves are pulse waves or periodic waves. 65. How is the propagation of sound in solids different from that in air? a. Sound waves in solids are transverse, whereas in and moves with the wave in its direction. b. The gull experiences mostly side-to-side motion but does not move with the wave in its direction. c. The gull experiences mostly up-and-down
motion and moves with the wave in its direction. d. The gull experiences mostly up-and-down motion but does not move in the direction of the wave. 67. Why does a good-quality speaker have a woofer and a tweeter? a. In a good-quality speaker, sounds with high frequencies or short wavelengths are reproduced accurately by woofers, while sounds with low frequencies or long wavelengths are reproduced accurately by tweeters. b. Sounds with high frequencies or short wavelengths are reproduced more accurately by tweeters, while sounds with low frequencies or long wavelengths are reproduced more accurately by woofers. 68. The time difference between a 2 km/s S-wave and a 6 km/s P-wave recorded at a certain point is 10 seconds. How far is the epicenter of the earthquake from that point? a. b. c. d. 15 m 30 m 15 km 30 km air, they are longitudinal. b. Sound waves in solids are longitudinal, whereas in 13.3 Wave Interaction: Superposition and Interference air, they are transverse. c. Sound waves in solids can be both longitudinal and transverse, whereas in air, they are longitudinal. d. Sound waves in solids are longitudinal, whereas in air, they can be both longitudinal and transverse. 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period 66. A seagull is sitting in the water surface and a simple water wave passes under it. What sort of motion does the gull experience? Why? a. The gull experiences mostly side-to-side motion 69. Why do water waves sometimes appear like a complex criss-cross pattern? a. The crests and the troughs of waves traveling in the same direction combine to form a criss-cross pattern. b. The crests and the troughs of waves traveling in different directions combine to form a criss-cross pattern. 70. What happens when two dissimilar waves interfere? a. pure constructive interference b. pure destructive interference c. a combination of constructive and destructive Access for free at openstax.org. interference 71. Occasionally, during earthquakes, areas near the epicenter are not damaged while those farther away are damaged. Why could this occur? a. Destructive interference results in waves with greater amplitudes being formed in places farther away from the epicenter. b. Constructive interference results in waves with greater amplitudes being formed in places farther away from the epicenter. c
. The standing waves of great amplitudes are formed in places farther away from the epicenter. d. The pulse waves of great amplitude are formed in places farther away from the epicenter. 72. Why does an object appear to be distorted when you Chapter 13 • Test Prep 413 view it through a glass of water? a. The glass and the water reflect the light in different directions. Hence, the object appears to be distorted. b. The glass and the water absorb the light by different amounts. Hence, the object appears to be distorted. c. Water, air, and glass are media with different densities. Light rays refract and bend when they pass from one medium to another. Hence, the object appears to be distorted. d. The glass and the water disperse the light into its components. Hence, the object appears to be distorted. 414 Chapter 13 • Test Prep Access for free at openstax.org. CHAPTER 14 Sound Figure 14.1 This tree fell some time ago. When it fell, particles in the air were disturbed by the energy of the tree hitting the ground. This disturbance of matter, which our ears have evolved to detect, is called sound. (B.A. Bowen Photography) Chapter Outline 14.1 Speed of Sound, Frequency, and Wavelength 14.2 Sound Intensity and Sound Level 14.3 Doppler Effect and Sonic Booms 14.4 Sound Interference and Resonance INTRODUCTION If a tree falls in a forest (see Figure 14.1) and no one is there to hear it, does it make a sound? The answer to this old philosophical question depends on how you define sound. If sound only exists when someone is around to perceive it, then the falling tree produced no sound. However, in physics, we know that colliding objects can disturb the air, water or other matter surrounding them. As a result of the collision, the surrounding particles of matter began vibrating in a wave-like fashion. This is a sound wave. Consequently, if a tree collided with another object in space, no one would hear it, because no sound would be produced. This is because, in space, there is no air, water or other matter to be disturbed and produce sound waves. In this chapter, we’ll learn more about the wave properties of sound, and explore hearing, as well as some special uses for sound. 416 Chapter 14 • Sound 14.1 Speed of Sound, Frequency, and Wavelength Section Learning Objectives By
the end of this section, you will be able to do the following: • Relate the characteristics of waves to properties of sound waves • Describe the speed of sound and how it changes in various media • Relate the speed of sound to frequency and wavelength of a sound wave Section Key Terms rarefaction sound Properties of Sound Waves Sound is a wave. More specifically, sound is defined to be a disturbance of matter that is transmitted from its source outward. A disturbance is anything that is moved from its state of equilibrium. Some sound waves can be characterized as periodic waves, which means that the atoms that make up the matter experience simple harmonic motion. A vibrating string produces a sound wave as illustrated in Figure 14.2, Figure 14.3, and Figure 14.4. As the string oscillates back and forth, part of the string’s energy goes into compressing and expanding the surrounding air. This creates slightly higher and lower pressures. The higher pressure... regions are compressions, and the low pressure regions are rarefactions. The pressure disturbance moves through the air as longitudinal waves with the same frequency as the string. Some of the energy is lost in the form of thermal energy transferred to the air. You may recall from the chapter on waves that areas of compression and rarefaction in longitudinal waves (such as sound) are analogous to crests and troughs in transverse waves. Figure 14.2 A vibrating string moving to the right compresses the air in front of it and expands the air behind it. Figure 14.3 As the string moves to the left, it creates another compression and rarefaction as the particles on the right move away from the string. Access for free at openstax.org. 14.1 • Speed of Sound, Frequency, and Wavelength 417 Figure 14.4 After many vibrations, there is a series of compressions and rarefactions that have been transmitted from the string as a sound wave. The graph shows gauge pressure (Pgauge) versus distance xfrom the source. Gauge pressure is the pressure relative to atmospheric pressure; it is positive for pressures above atmospheric pressure, and negative for pressures below it. For ordinary, everyday sounds, pressures vary only slightly from average atmospheric pressure. The amplitude of a sound wave decreases with distance from its source, because the energy of the wave is spread over a larger and larger area. But some of the energy is also absorbed by objects, such as the eardrum in Figure 14.5, and some of the
energy is converted to thermal energy in the air. Figure 14.4 shows a graph of gauge pressure versus distance from the vibrating string. From this figure, you can see that the compression of a longitudinal wave is analogous to the peak of a transverse wave, and the rarefaction of a longitudinal wave is analogous to the trough of a transverse wave. Just as a transverse wave alternates between peaks and troughs, a longitudinal wave alternates between compression and rarefaction. Figure 14.5 Sound wave compressions and rarefactions travel up the ear canal and force the eardrum to vibrate. There is a net force on the eardrum, since the sound wave pressures differ from the atmospheric pressure found behind the eardrum. A complicated mechanism converts the vibrations to nerve impulses, which are then interpreted by the brain. The Speed of Sound The speed of sound varies greatly depending upon the medium it is traveling through. The speed of sound in a medium is determined by a combination of the medium’s rigidity (or compressibility in gases) and its density. The more rigid (or less compressible) the medium, the faster the speed of sound. The greater the density of a medium, the slower the speed of sound. The speed of sound in air is low, because air is compressible. Because liquids and solids are relatively rigid and very difficult to compress, the speed of sound in such media is generally greater than in gases. Table 14.1 shows the speed of sound in various media. Since temperature affects density, the speed of sound varies with the temperature of the medium through which it’s traveling to some extent, especially for gases. Medium vw (m/s) Gases at 0 °C Air Carbon dioxide Oxygen Helium 331 259 316 965 Hydrogen 1290 Liquids at 20 °C Ethanol Mercury Water, fresh Sea water 1160 1450 1480 1540 Human tissue 1540 Solids (longitudinal or bulk) Vulcanized rubber 54 Polyethylene 920 Marble Glass, Pyrex Lead Aluminum Steel 3810 5640 1960 5120 5960 Table 14.1 Speed of Sound in Various Media 418 Chapter 14 • Sound Access for free at openstax.org. The Relationship Between the Speed of Sound and the Frequency and Wavelength of a Sound Wave 14.1 • Speed of Sound, Frequency, and Wavelength 419 Figure 14.6 When fireworks explode in the sky, the light energy is perceived before the sound energy. Sound travels
more slowly than light does. (Dominic Alves, Flickr) Sound, like all waves, travels at certain speeds through different media and has the properties of frequency and wavelength. Sound travels much slower than light—you can observe this while watching a fireworks display (see Figure 14.6), since the flash of an explosion is seen before its sound is heard. The relationship between the speed of sound, its frequency, and wavelength is the same as for all waves: where vis the speed of sound (in units of m/s), fis its frequency (in units of hertz), and is its wavelength (in units of meters). Recall that wavelength is defined as the distance between adjacent identical parts of a wave. The wavelength of a sound, therefore, is the distance between adjacent identical parts of a sound wave. Just as the distance between adjacent crests in a transverse wave is one wavelength, the distance between adjacent compressions in a sound wave is also one wavelength, as shown in Figure 14.7. The frequency of a sound wave is the same as that of the source. For example, a tuning fork vibrating at a given frequency would produce sound waves that oscillate at that same frequency. The frequency of a sound is the number of waves that pass a point per unit time. 14.1 Figure 14.7 A sound wave emanates from a source vibrating at a frequency f, propagates at v, and has a wavelength. One of the more important properties of sound is that its speed is nearly independent of frequency. If this were not the case, and high-frequency sounds traveled faster, for example, then the farther you were from a band in a football stadium, the more the sound from the low-pitch instruments would lag behind the high-pitch ones. But the music from all instruments arrives in cadence independent of distance, and so all frequencies must travel at nearly the same speed. Recall that between fand is inverse: The higher the frequency, the shorter the wavelength of a sound wave., and in a given medium under fixed temperature and humidity, vis constant. Therefore, the relationship The speed of sound can change when sound travels from one medium to another. However, the frequency usually remains the same because it is like a driven oscillation and maintains the frequency of the original source. If vchanges and fremains the same, then the wavelength must change. Since frequency., the higher the speed of a sound, the greater its wavelength for a given Virtual Physics Sound Click
to view content (https://www.openstax.org/l/28sound) 420 Chapter 14 • Sound This simulation lets you see sound waves. Adjust the frequency or amplitude (volume) and you can see and hear how the wave changes. Move the listener around and hear what she hears. Switch to the Two Source Interference tab or the Interference by Reflection tab to experiment with interference and reflection. TIPS FOR SUCCESS Make sure to have audio enabled and set to Listener rather than Speaker, or else the sound will not vary as you move the listener around. GRASP CHECK In the first tab, Listen to a Single Source, move the listener as far away from the speaker as possible, and then change the frequency of the sound wave. You may have noticed that there is a delay between the time when you change the setting and the time when you hear the sound get lower or higher in pitch. Why is this? a. Because, intensity of the sound wave changes with the frequency. b. Because, the speed of the sound wave changes when the frequency is changed. c. Because, loudness of the sound wave takes time to adjust after a change in frequency. d. Because it takes time for sound to reach the listener, so the listener perceives the new frequency of sound wave after a delay. Is there a difference in the amount of delay depending on whether you make the frequency higher or lower? Why? a. Yes, the speed of propagation depends only on the frequency of the wave. b. Yes, the speed of propagation depends upon the wavelength of the wave, and wavelength changes as the frequency changes. c. No, the speed of propagation depends only on the wavelength of the wave. d. No, the speed of propagation is constant in a given medium; only the wavelength changes as the frequency changes. Snap Lab Voice as a Sound Wave In this lab you will observe the effects of blowing and speaking into a piece of paper in order to compare and contrast different sound waves. • • • sheet of paper tape table Instructions Procedure 1. Suspend a sheet of paper so that the top edge of the paper is fixed and the bottom edge is free to move. You could tape the top edge of the paper to the edge of a table, for example. 2. Gently blow air near the edge of the bottom of the sheet and note how the sheet moves. 3. Speak softly and then louder such that the sounds hit the edge of the bottom of the paper, and note
how the sheet moves. Interpret the results. 4. GRASP CHECK Which sound wave property increases when you are speaking more loudly than softly? a. amplitude of the wave frequency of the wave b. speed of the wave c. Access for free at openstax.org. 14.1 • Speed of Sound, Frequency, and Wavelength 421 d. wavelength of the wave WORKED EXAMPLE What Are the Wavelengths of Audible Sounds? Calculate the wavelengths of sounds at the extremes of the audible range, 20 and 20,000 Hz, in conditions where sound travels at 348.7 m/s. STRATEGY To find wavelength from frequency, we can use. Solution (1) Identify the knowns. The values for vand fare given. (2) Solve the relationship between speed, frequency and wavelength for. (3) Enter the speed and the minimum frequency to give the maximum wavelength. (4) Enter the speed and the maximum frequency to give the minimum wavelength. 14.2 14.3 14.4 Discussion Because the product of fmultiplied by be, and vice versa. Note that you can also easily rearrange the same formula to find frequency or velocity. equals a constant velocity in unchanging conditions, the smaller fis, the larger must Practice Problems 1. What is the speed of a sound wave with frequency and wavelength? a. b. c. d. 2. Dogs can hear frequencies of up to. What is the wavelength of a sound wave with this frequency traveling in air at? a. b. c. d. 422 Chapter 14 • Sound LINKS TO PHYSICS Echolocation Figure 14.8 A bat uses sound echoes to find its way about and to catch prey. The time for the echo to return is directly proportional to the distance. Echolocation is the use of reflected sound waves to locate and identify objects. It is used by animals such as bats, dolphins and whales, and is also imitated by humans in SONAR—Sound Navigation and Ranging—and echolocation technology. Bats, dolphins and whales use echolocation to navigate and find food in their environment. They locate an object (or obstacle) by emitting a sound and then sensing the reflected sound waves. Since the speed of sound in air is constant, the time it takes for the sound to travel to the object and back gives the animal a sense of the distance between itself and the object. This is called ranging. Figure 14
.8 shows a bat using echolocation to sense distances. Echolocating animals identify an object by comparing the relative intensity of the sound waves returning to each ear to figure out the angle at which the sound waves were reflected. This gives information about the direction, size and shape of the object. Since there is a slight distance in position between the two ears of an animal, the sound may return to one of the ears with a bit of a delay, which also provides information about the position of the object. For example, if a bear is directly to the right of a bat, the echo will return to the bat’s left ear later than to its right ear. If, however, the bear is directly ahead of the bat, the echo would return to both ears at the same time. For an animal without a sense of sight such as a bat, it is important to know whereother animals are as well as whatthey are; their survival depends on it. Principles of echolocation have been used to develop a variety of useful sensing technologies. SONAR, is used by submarines to detect objects underwater and measure water depth. Unlike animal echolocation, which relies on only one transmitter (a mouth) and two receivers (ears), manmade SONAR uses many transmitters and beams to get a more accurate reading of the environment. Radar technologies use the echo of radio waves to locate clouds and storm systems in weather forecasting, and to locate aircraft for air traffic control. Some new cars use echolocation technology to sense obstacles around the car, and warn the driver who may be about to hit something (or even to automatically parallel park). Echolocation technologies and training systems are being developed to help visually impaired people navigate their everyday environments. GRASP CHECK If a predator is directly to the left of a bat, how will the bat know? a. The echo would return to the left ear first. b. The echo would return to the right ear first. Check Your Understanding 3. What is a rarefaction? a. Rarefaction is the high-pressure region created in a medium when a longitudinal wave passes through it. b. Rarefaction is the low-pressure region created in a medium when a longitudinal wave passes through it. c. Rarefaction is the highest point of amplitude of a sound wave. d. Rarefaction is the lowest point of amplitude of a sound wave. 4. What sort of motion do the particles of a medium experience
when a sound wave passes through it? a. Simple harmonic motion Access for free at openstax.org. 14.2 • Sound Intensity and Sound Level 423 b. Circular motion c. Random motion d. Translational motion 5. What does the speed of sound depend on? a. The wavelength of the wave b. The size of the medium c. The frequency of the wave d. The properties of the medium 6. What property of a gas would affect the speed of sound traveling through it? a. The volume of the gas b. The flammability of the gas c. The mass of the gas d. The compressibility of the gas 14.2 Sound Intensity and Sound Level Section Learning Objectives By the end of this section, you will be able to do the following: • Relate amplitude of a wave to loudness and energy of a sound wave • Describe the decibel scale for measuring sound intensity • Solve problems involving the intensity of a sound wave • Describe how humans produce and hear sounds Section Key Terms amplitude decibel hearing loudness pitch sound intensity sound intensity level Amplitude, Loudness and Energy of a Sound Wave Figure 14.9 Noise on crowded roadways like this one in Delhi makes it hard to hear others unless they shout. (Lingaraj G J, Flickr) In a quiet forest, you can sometimes hear a single leaf fall to the ground. But in a traffic jam filled with honking cars, you may have to shout just so the person next to you can hear Figure 14.9.The loudness of a sound is related to how energetically its source is vibrating. In cartoons showing a screaming person, the cartoonist often shows an open mouth with a vibrating uvula (the hanging tissue at the back of the mouth) to represent a loud sound coming from the throat. Figure 14.10 shows such a cartoon depiction of a bird loudly expressing its opinion. A useful quantity for describing the loudness of sounds is called sound intensity. In general, the intensity of a wave is the power per unit area carried by the wave. Power is the rate at which energy is transferred by the wave. In equation form, intensity Iis where Pis the power through an area A. The SI unit for Iis W/m2. The intensity of a sound depends upon its pressure amplitude. 14.5 424 Chapter 14 • Sound The relationship between the intensity of a sound wave and its pressure amplitude (or pressure variation Δp)
is 14.6 where ρis the density of the material in which the sound wave travels, in units of kg/m3, and vis the speed of sound in the medium, in units of m/s. Pressure amplitude has units of pascals (Pa) or N/m2. Note that Δpis half the difference between the maximum and minimum pressure in the sound wave. We can see from the equation that the intensity of a sound is proportional to its amplitude squared. The pressure variation is proportional to the amplitude of the oscillation, and so Ivaries as (Δp)2. This relationship is consistent with the fact that the sound wave is produced by some vibration; the greater its pressure amplitude, the more the air is compressed during the vibration. Because the power of a sound wave is the rate at which energy is transferred, the energy of a sound wave is also proportional to its amplitude squared. TIPS FOR SUCCESS Pressure is usually denoted by capital P, but we are using a lowercase pfor pressure in this case to distinguish it from power Pabove. Figure 14.10 Graphs of the pressures in two sound waves of different intensities. The more intense sound is produced by a source that has larger-amplitude oscillations and has greater pressure maxima and minima. Because pressures are higher in the greater-intensity sound, it can exert larger forces on the objects it encounters. The Decibel Scale You may have noticed that when people talk about the loudness of a sound, they describe it in units of decibels rather than watts per meter squared. While sound intensity (in W/m2) is the SI unit, the sound intensity level in decibels (dB) is more relevant for how humans perceive sounds. The way our ears perceive sound can be more accurately described by the logarithm of the intensity of a sound rather than the intensity of a sound directly. The sound intensity level βis defined to be 14.7 where Iis sound intensity in watts per meter squared, and I0 = 10–12 W/m2 is a reference intensity. I0 is chosen as the reference point because it is the lowest intensity of sound a person with normal hearing can perceive. The decibel level of a sound having an intensity of 10–12 W/m2 is β= 0 dB, because log10 1 = 0. That is, the threshold of human hearing is 0 decibels. Each factor of 10 in intensity corresponds to
10 dB. For example, a 90 dB sound compared with a 60 dB sound is 30 dB greater, or three factors of 10 (that is, 103 times) as intense. Another example is that if one sound is 107 as intense as another, it is 70 dB higher. Since βis defined in terms of a ratio, it is unit-less. The unit called decibel (dB) is used to indicate that this ratio is multiplied by 10. The sound intensity level is not the same as sound intensity—it tells you the levelof the sound relative to a reference intensity rather than the actual intensity. Access for free at openstax.org. 14.2 • Sound Intensity and Sound Level 425 Snap Lab Feeling Sound In this lab, you will play music with a heavy beat to literally feel the vibrations and explore what happens when the volume changes. • CD player or portable electronic device connected to speakers • • a lightweight table rock or rap music CD or mp3 Procedure 1. Place the speakers on a light table, and start playing the CD or mp3. 2. Place your hand gently on the table next to the speakers. 3. 4. Increase the volume and note the level when the table just begins to vibrate as the music plays. Increase the reading on the volume control until it doubles. What has happened to the vibrations? GRASP CHECK Do you think that when you double the volume of a sound wave you are doubling the sound intensity level (in dB) or the sound intensity (in a. The sound intensity in b. The sound intensity level in c. The sound intensity in d. The sound intensity level in, because it is a closer measure of how humans perceive sound. because it is a closer measure of how humans perceive sound. because it is the only unit to express the intensity of sound. because it is the only unit to express the intensity of sound. )? Why? Solving Sound Wave Intensity Problems WORKED EXAMPLE Calculating Sound Intensity Levels: Sound Waves Calculate the sound intensity level in decibels for a sound wave traveling in air at 0 ºC and having a pressure amplitude of 0.656 Pa. STRATEGY We are given Δp, so we can calculate Iusing the equation. Using I, we can calculate βstraight from its definition in. Solution (1) Identify knowns: Sound travels at 331 m/s in air at 0 °C. Air has a density of 1.29 kg/m3 at atmospheric
pressure and 0ºC. (2) Enter these values and the pressure amplitude into. (3) Enter the value for Iand the known value for I0 into decibels.. Calculate to find the sound intensity level in 426 Chapter 14 • Sound Discussion This 87.0 dB sound has an intensity five times as great as an 80 dB sound. So a factor of five in intensity corresponds to a difference of 7 dB in sound intensity level. This value is true for any intensities differing by a factor of five. WORKED EXAMPLE Change Intensity Levels of a Sound: What Happens to the Decibel Level? Show that if one sound is twice as intense as another, it has a sound level about 3 dB higher. STRATEGY You are given that the ratio of two intensities is 2 to 1, and are then asked to find the difference in their sound levels in decibels. You can solve this problem using of the properties of logarithms. Solution (1) Identify knowns: The ratio of the two intensities is 2 to 1, or: We want to show that the difference in sound levels is about 3 dB. That is, we want to show Note that (2) Use the definition of βto get Therefore, 14.8 14.9 14.10 Discussion This means that the two sound intensity levels differ by 3.01 dB, or about 3 dB, as advertised. Note that because only the ratio I2/I1 is given (and not the actual intensities), this result is true for any intensities that differ by a factor of two. For example, a 56.0 dB sound is twice as intense as a 53.0 dB sound, a 97.0 dB sound is half as intense as a 100 dB sound, and so on. Practice Problems 7. Calculate the intensity of a wave if the power transferred is 10 W and the area through which the wave is transferred is 5 square meters. a. 200 W / m2 50 W / m2 b. c. 0.5 W / m2 d. 2 W / m2 8. Calculate the sound intensity for a sound wave traveling in air at and having a pressure amplitude of. a. b. c. d. Hearing and Voice People create sounds by pushing air up through their lungs and through elastic folds in the throat called vocal cords. These folds Access for free at openstax.org. 14.2 • Sound Intensity and Sound Level 427 open
and close rhythmically, creating a pressure buildup. As air travels up and past the vocal cords, it causes them to vibrate. This vibration escapes the mouth along with puffs of air as sound. A voice changes in pitch when the muscles of the larynx relax or tighten, changing the tension on the vocal chords. A voice becomes louder when air flow from the lungs increases, making the amplitude of the sound pressure wave greater. Hearing is the perception of sound. It can give us plenty of information—such as pitch, loudness, and direction. Humans can normally hear frequencies ranging from approximately 20 to 20,000 Hz. Other animals have hearing ranges different from that of humans. Dogs can hear sounds as high as 45,000 Hz, whereas bats and dolphins can hear up to 110,000 Hz sounds. You may have noticed that dogs respond to the sound of a dog whistle which produces sound out of the range of human hearing. Sounds below 20 Hz are called infrasound, whereas those above 20,000 Hz are ultrasound. The perception of frequency is called pitch, and the perception of intensity is called loudness. The way we hear involves some interesting physics. The sound wave that hits our ear is a pressure wave. The ear converts sound waves into electrical nerve impulses, similar to a microphone. Figure 14.11 shows the anatomy of the ear with its division into three parts: the outer ear or ear canal; the middle ear, which runs from the eardrum to the cochlea; and the inner ear, which is the cochlea itself. The body part normally referred to as the ear is technically called the pinna. Figure 14.11 The illustration shows the anatomy of the human ear. The outer ear, or ear canal, carries sound to the eardrum protected inside of the ear. The middle ear converts sound into mechanical vibrations and applies these vibrations to the cochlea. The lever system of the middle ear takes the force exerted on the eardrum by sound pressure variations, amplifies it and transmits it to the inner ear via the oval window. Two muscles in the middle ear protect the inner ear from very intense sounds. They react to intense sound in a few milliseconds and reduce the force transmitted to the cochlea. This protective reaction can also be triggered by your own voice, so that humming during a fireworks display, for example, can reduce noise damage. Figure 14.12 shows the middle and inner ear in greater detail. As the middle ear bones vibr
ate, they vibrate the cochlea, which contains fluid. This creates pressure waves in the fluid that cause the tectorial membrane to vibrate. The motion of the tectorial membrane stimulates tiny cilia on specialized cells called hair cells. These hair cells, and their attached neurons, transform the motion of the tectorial membrane into electrical signals that are sent to the brain. The tectorial membrane vibrates at different positions based on the frequency of the incoming sound. This allows us to detect the pitch of sound. Additional processing in the brain also allows us to determine which direction the sound is coming from (based on comparison of the sound’s arrival time and intensity between our two ears). 428 Chapter 14 • Sound Figure 14.12 The inner ear, or cochlea, is a coiled tube about 3 mm in diameter and 3 cm in length when uncoiled. As the stapes vibrates against the oval window, it creates pressure waves that travel through fluid in the cochlea. These waves vibrate the tectorial membrane, which bends the cilia and stimulates nerves in the organ of Corti. These nerves then send information about the sound to the brain. FUN IN PHYSICS Musical Instruments Figure 14.13 Playing music, also known as “rocking out”, involves creating vibrations using musical instruments. (John Norton) Yet another way that people make sounds is through playing musical instruments (see the previous figure). Recall that the perception of frequency is called pitch. You may have noticed that the pitch range produced by an instrument tends to depend upon its size. Small instruments, such as a piccolo, typically make high-pitch sounds, while larger instruments, such as a tuba, typically make low-pitch sounds. High-pitch means small wavelength, and the size of a musical instrument is directly related to the wavelengths of sound it produces. So a small instrument creates short-wavelength sounds, just as a large instrument creates long-wavelength sounds. Most of us have excellent relative pitch, which means that we can tell whether one sound has a different frequency from another. We can usually distinguish one sound from another if the frequencies of the two sounds differ by as little as 1 Hz. For example, 500.0 and 501.5 Hz are noticeably different. Musical notes are particular sounds that can be produced by most instruments, and are the building blocks of a song. In Western music, musical notes have particular names, such as
A-sharp, C, or E-flat. Some people can identify musical notes just by listening to them. This rare ability is called perfect, or absolute, pitch. When a violin plays middle C, there is no mistaking it for a piano playing the same note. The reason is that each instrument produces a distinctive set of frequencies and intensities. We call our perception of these combinations of frequencies and intensities the timbreof the sound. It is more difficult to quantify timbre than loudness or pitch. Timbre is more subjective. Evocative adjectives such as dull, brilliant, warm, cold, pure, and rich are used to describe the timbre of a sound rather than Access for free at openstax.org. 14.2 • Sound Intensity and Sound Level 429 quantities with units, which makes for a difficult topic to dissect with physics. So the consideration of timbre takes us into the realm of perceptual psychology, where higher-level processes in the brain are dominant. This is also true for other perceptions of sound, such as music and noise. But as a teenager, you are likely already aware that one person’s music may be another person’s noise. GRASP CHECK If you turn up the volume of your stereo, will the pitch change? Why or why not? a. No, because pitch does not depend on intensity. b. Yes, because pitch is directly related to intensity. Check Your Understanding 9. What is sound intensity? a. b. c. d. Intensity is the energy per unit area carried by a wave. Intensity is the energy per unit volume carried by a wave. Intensity is the power per unit area carried by a wave. Intensity is the power per unit volume carried by a wave. 10. How is power defined with reference to a sound wave? a. Power is the rate at which energy is transferred by a sound wave. b. Power is the rate at which mass is transferred by a sound wave. c. Power is the rate at which amplitude of a sound wave changes. d. Power is the rate at which wavelength of a sound wave changes. 11. What word or phrase is used to describe the loudness of sound? frequency or oscillation intensity level or decibel timbre a. b. c. d. pitch 12. What is the mathematical expression for sound intensity level? a. b. c. d. 13. What is the range frequencies that humans are capable of hearing? a.
b. c. d. to to to 14. How do humans change the pitch of their voice? a. Relaxing or tightening their glottis b. Relaxing or tightening their uvula c. Relaxing or tightening their tongue d. Relaxing or tightening their larynx References Nave, R. Vocal sound production—HyperPhysics. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/music/voice.html 430 Chapter 14 • Sound 14.3 Doppler Effect and Sonic Booms Section Learning Objectives By the end of this section, you will be able to do the following: • Describe the Doppler effect of sound waves • Explain a sonic boom • Calculate the frequency shift of sound from a moving object by the Doppler shift formula, and calculate the speed of an object by the Doppler shift formula Section Key Terms Doppler effect sonic boom The Doppler Effect of Sound Waves The Doppler effect is a change in the observed pitch of a sound, due to relative motion between the source and the observer. An example of the Doppler effect due to the motion of a source occurs when you are standing still, and the sound of a siren coming from an ambulance shifts from high-pitch to low-pitch as it passes by. The closer the ambulance is to you, the more sudden the shift. The faster the ambulance moves, the greater the shift. We also hear this shift in frequency for passing race cars, airplanes, and trains. An example of the Doppler effect with a stationary source and moving observer is if you ride a train past a stationary warning bell, you will hear the bell’s frequency shift from high to low as you pass by. What causes the Doppler effect? Let’s compare three different scenarios: Sound waves emitted by a stationary source (Figure 14.14), sound waves emitted by a moving source (Figure 14.15), and sound waves emitted by a stationary source but heard by moving observers (Figure 14.16). In each case, the sound spreads out from the point where it was emitted. If the source and observers are stationary, then observers on either side see the same wavelength and frequency as emitted by the source. But if the source is moving and continues to emit sound as it travels, then the air compressions (crests) become closer together in the direction in which it’s traveling and farther apart
in the direction it’s traveling away from. Therefore, the wavelength is shorter in the direction the source is moving (on the right in Figure 14.15), and longer in the opposite direction (on the left in Figure 14.15). Finally, if the observers move, as in Figure 14.16, the frequency at which they receive the compressions changes. The observer moving toward the source receives them at a higher frequency (and therefore shorter wavelength), and the person moving away from the source receives them at a lower frequency (and therefore longer wavelength). Figure 14.14 Sounds emitted by a source spread out in spherical waves. Because the source, observers, and air are stationary, the wavelength and frequency are the same in all directions and to all observers. Figure 14.15 Sounds emitted by a source moving to the right spread out from the points at which they were emitted. The wavelength is Access for free at openstax.org. reduced and, consequently, the frequency is increased in the direction of motion, so that the observer on the right hears a higher-pitch sound. The opposite is true for the observer on the left, where the wavelength is increased and the frequency is reduced. 14.3 • Doppler Effect and Sonic Booms 431 Figure 14.16 The same effect is produced when the observers move relative to the source. Motion toward the source increases frequency as the observer on the right passes through more wave crests than she would if stationary. Motion away from the source decreases frequency as the observer on the left passes through fewer wave crests than he would if stationary. We know that wavelength and frequency are related by medium and has the same speed vin that medium whether the source is moving or not. Therefore, fmultiplied by constant. Because the observer on the right in Figure 14.15 receives a shorter wavelength, the frequency she perceives must be higher. Similarly, the observer on the left receives a longer wavelength and therefore perceives a lower frequency. where vis the fixed speed of sound. The sound moves in a is a The same thing happens in Figure 14.16. A higher frequency is perceived by the observer moving toward the source, and a lower frequency is perceived by an observer moving away from the source. In general, then, relative motion of source and observer toward one another increases the perceived frequency. Relative motion apart decreases the perceived frequency. The greater the relative speed is, the greater the effect. WATCH PHYSICS Introduction to the Doppler Effect This video explains the Do
ppler effect visually. Click to view content (https://www.openstax.org/l/28doppler) GRASP CHECK If you are standing on the sidewalk facing the street and an ambulance drives by with its siren blaring, at what point will the frequency that you observe most closely match the actual frequency of the siren? a. when it is coming toward you b. when it is going away from you c. when it is in front of you For a stationary observer and a moving source of sound, the frequency (fobs) of sound perceived by the observer is 14.11 where fs is the frequency of sound from a source, vs is the speed of the source along a line joining the source and observer, and vw is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away from the observer. TIPS FOR SUCCESS Rather than just memorizing rules, which are easy to forget, it is better to think about the rules of an equation intuitively. Using a minus sign in will decrease the denominator and increase the observed frequency, which is consistent with the expected outcome of the Doppler effect when the source is moving toward the observer. Using a plus sign will increase the denominator and decrease the observed frequency, consistent with what you would expect for the source 432 Chapter 14 • Sound moving away from the observer. This may be more helpful to keep in mind rather than memorizing the fact that “the minus sign is used for motion toward the observer and the plus sign for motion away from the observer.” Note that the greater the speed of the source, the greater the Doppler effect. Similarly, for a stationary source and moving observer, the frequency perceived by the observer fobs is given by 14.12 where vobs is the speed of the observer along a line joining the source and observer. Here the plus sign is for motion toward the source, and the minus sign is for motion away from the source. Sonic Booms What happens to the sound produced by a moving source, such as a jet airplane, that approaches or even exceeds the speed of sound? Suppose a jet airplane is coming nearly straight at you, emitting a sound of frequency fs. The greater the plane’s speed, vs, the greater the Doppler shift and the greater the value of fobs. Now, as vs approaches the speed of sound, vw, fobs approaches infinity, because the denominator
in approaches zero. This result means that at the speed of sound, in front of the source, each wave is superimposed on the previous one because the source moves forward at the speed of sound. The observer gets them all at the same instant, and so the frequency is theoretically infinite. If the source exceeds the speed of sound, no sound is received by the observer until the source has passed, so that the sounds from the source when it was approaching are stacked up with those from it when receding, creating a sonic boom. A sonic boom is a constructive interference of sound created by an object moving faster than sound. An aircraft creates two sonic booms, one from its nose and one from its tail (see Figure 14.17). During television coverage of space shuttle landings, two distinct booms could often be heard. These were separated by exactly the time it would take the shuttle to pass by a point. Observers on the ground often do not observe the aircraft creating the sonic boom, because it has passed by before the shock wave reaches them. If the aircraft flies close by at low altitude, pressures in the sonic boom can be destructive enough to break windows. Because of this, supersonic flights are banned over populated areas of the United States. Figure 14.17 Two sonic booms, created by the nose and tail of an aircraft, are observed on the ground after the plane has passed by. Solving Problems Using the Doppler Shift Formula WATCH PHYSICS Doppler Effect Formula for Observed Frequency This video explains the Doppler effect formula for cases when the source is moving toward the observer. Click to view content (https://www.openstax.org/l/28dopplerform) GRASP CHECK Let’s say that you have a rare phobia where you are afraid of the Doppler effect. If you see an ambulance coming your way, what would be the best strategy to minimize the Doppler effect and soothe your Doppleraphobia? Access for free at openstax.org. 14.3 • Doppler Effect and Sonic Booms 433 a. Stop moving and become stationary till it passes by. b. Run toward the ambulance. c. Run alongside the ambulance. WATCH PHYSICS Doppler Effect Formula When Source is Moving Away This video explains the Doppler effect formula for cases when the source is moving away from the observer. Click to view content (https://www.openstax.org/
l/28doppleraway) GRASP CHECK Sal uses two different formulas for the Doppler effect-one for when the source is moving toward the observer and another for when the source is moving away. However, in this textbook we use only one formula. Explain. a. The combined formula that can be used is, Use ( source is moving away from the observer. b. The combined formula that can be used is, ) when the source is moving toward the observer and ( ) when the source is moving away from the ) when the. Use ( observer and ( ) when the source is moving toward the observer. c. The combined formula that can be used is,. Use ( ) when the source is moving toward the observer and ( ) when the source is moving away from the observer. d. The combined formula that can be used is,. Use ( ) when the source is moving away from the observer and ( ) when the source is moving toward the observer. WORKED EXAMPLE Calculate Doppler Shift: A Train Horn Suppose a train that has a 150 Hz horn is moving at 35 m/s in still air on a day when the speed of sound is 340 m/s. What frequencies are observed by a stationary person at the side of the tracks as the train approaches and after it passes? Strategy To find the observed frequency, must be used because the source is moving. The minus sign is used for the approaching train, and the plus sign for the receding train. Solution (1) Enter known values into to calculate the frequency observed by a stationary person as the train approaches: (2) Use the same equation but with the plus sign to find the frequency heard by a stationary person as the train recedes. Discussion The numbers calculated are valid when the train is far enough away that the motion is nearly along the line joining the train and the observer. In both cases, the shift is significant and easily noticed. Note that the shift is approximately 20 Hz for motion toward and approximately 10 Hz for motion away. The shifts are not symmetric. 434 Chapter 14 • Sound Practice Problems 15. What is the observed frequency when the source having frequency is moving towards the observer at a speed of and the speed of sound is? a. b. c. d. 16. A train is moving away from you at a speed of. If you are standing still and hear the whistle at a frequency of, what is the actual frequency of the produced whistle? (Assume speed of sound
to be.) a. b. c. d. Check Your Understanding 17. What is the Doppler effect? a. The Doppler effect is a change in the observed speed of a sound due to the relative motion between the source and the observer. b. The Doppler effect is a change in the observed frequency of a sound due to the relative motion between the source and the observer. c. The Doppler effect is a change in the observed intensity of a sound due to the relative motion between the source and the observer. d. The Doppler effect is a change in the observed timbre of a sound, due to the relative motion between the source and the observer. 18. Give an example of the Doppler effect caused by motion of the source. a. The sound of a vehicle horn shifts from low-pitch to high-pitch as we move towards it. b. The sound of a vehicle horn shifts from low-pitch to high-pitch as we move away from it. c. The sound of a vehicle horn shifts from low-pitch to high-pitch as it passes by. d. The sound of a vehicle horn shifts from high-pitch to low-pitch as it passes by. 19. What is a sonic boom? a. b. c. d. It is a destructive interference of sound created by an object moving faster than sound. It is a constructive interference of sound created by an object moving faster than sound. It is a destructive interference of sound created by an object moving slower than sound. It is a constructive interference of sound created by an object moving slower than sound. 20. What is the relation between speed of source and value of observed frequency when the source is moving towards the observer? a. They are independent of each other. b. The greater the speed, the greater the value of observed frequency. c. The greater the speed, the smaller the value of observed frequency. d. The speed of the sound is directly proportional to the square of the frequency observed. 14.4 Sound Interference and Resonance Section Learning Objectives By the end of this section, you will be able to do the following: • Describe resonance and beats • Define fundamental frequency and harmonic series • Contrast an open-pipe and closed-pipe resonator • Solve problems involving harmonic series and beat frequency Access for free at openstax.org. 14.4 • Sound Interference and Resonance 435 Section Key Terms beat beat
frequency damping fundamental harmonics natural frequency overtones resonance resonate Resonance and Beats Sit in front of a piano sometime and sing a loud brief note at it while pushing down on the sustain pedal. It will sing the same note back at you—the strings that have the same frequencies as your voice, are resonating in response to the forces from the sound waves that you sent to them. This is a good example of the fact that objects—in this case, piano strings—can be forced to oscillate but oscillate best at their natural frequency. A driving force (such as your voice in the example) puts energy into a system at a certain frequency, which is not necessarily the same as the natural frequency of the system. Over time the energy dissipates, and the amplitude gradually reduces to zero- this is called damping. The natural frequency is the frequency at which a system would oscillate if there were no driving and no damping force. The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance, and a system being driven at its natural frequency is said to resonate. Most of us have played with toys where an object bobs up and down on an elastic band, something like the paddle ball suspended from a finger in Figure 14.18. At first you hold your finger steady, and the ball bounces up and down with a small amount of damping. If you move your finger up and down slowly, the ball will follow along without bouncing much on its own. As you increase the frequency at which you move your finger up and down, the ball will respond by oscillating with increasing amplitude. When you drive the ball at its natural frequency, the ball’s oscillations increase in amplitude with each oscillation for as long as you drive it. As the driving frequency gets progressively higher than the resonant or natural frequency, the amplitude of the oscillations becomes smaller, until the oscillations nearly disappear and your finger simply moves up and down with little effect on the ball. Figure 14.18 The paddle ball on its rubber band moves in response to the finger supporting it. If the finger moves with the natural frequency of the ball on the rubber band, then a resonance is achieved, and the amplitude of the ball’s oscillations increases dramatically. At higher and lower driving frequencies, energy is transferred to the ball less efficiently, and it responds with lower-amplitude oscillations. Another example is that when you tune a radio, you adjust its resonant frequency so that
it oscillates only at the desired station’s broadcast (driving) frequency. Also, a child on a swing is driven (pushed) by a parent at the swing’s natural frequency to reach the maximum amplitude (height). In all of these cases, the efficiency of energy transfer from the driving force into the oscillator is best at resonance. 436 Chapter 14 • Sound Figure 14.19 Some types of headphones use the phenomena of constructive and destructive interference to cancel out outside noises. All sound resonances are due to constructive and destructive interference. Only the resonant frequencies interfere constructively to form standing waves, while others interfere destructively and are absent. From the toot made by blowing over a bottle to the recognizability of a great singer’s voice, resonance and standing waves play a vital role in sound. Interference happens to all types of waves, including sound waves. In fact, one way to support that something is a waveis to observe interference effects. Figure 14.19 shows a set of headphones that employs a clever use of sound interference to cancel noise. To get destructive interference, a fast electronic analysis is performed, and a second sound is introduced with its maxima and minima exactly reversed from the incoming noise. In addition to resonance, superposition of waves can also create beats. Beats are produced by the superposition of two waves with slightly different frequencies but the same amplitude. The waves alternate in time between constructive interference and destructive interference, giving the resultant wave an amplitude that varies over time. (See the resultant wave in Figure 14.20). This wave fluctuates in amplitude, or beats, with a frequency called the beat frequency. The equation for beat frequency is where f1 and f2 are the frequencies of the two original waves. If the two frequencies of sound waves are similar, then what we hear is an average frequency that gets louder and softer at the beat frequency. TIPS FOR SUCCESS Don’t confuse the beat frequency with the regular frequency of a wave resulting from superposition. While the beat frequency is given by the formula above, and describes the frequency of the beats, the actual frequency of the wave resulting from superposition is the average of the frequencies of the two original waves. 14.13 Figure 14.20 Beats are produced by the superposition of two waves of slightly different frequencies but identical amplitudes. The waves alternate in time between constructive interference and destructive interference, giving the resulting wave a time-varying amplitude. Virtual Physics Wave Interference Click to
view content (https://www.openstax.org/l/28interference) Access for free at openstax.org. 14.4 • Sound Interference and Resonance 437 For this activity, switch to the Sound tab. Turn on the Sound option, and experiment with changing the frequency and amplitude, and adding in a second speaker and a barrier. GRASP CHECK According to the graph, what happens to the amplitude of pressure over time. What is this phenomenon called, and what causes it? a. The amplitude decreases over time. This phenomenon is called damping. It is caused by the dissipation of energy. b. The amplitude increases over time. This phenomenon is called feedback. It is caused by the gathering of energy. c. The amplitude oscillates over time. This phenomenon is called echoing. It is caused by fluctuations in energy. Fundamental Frequency and Harmonics Suppose we hold a tuning fork near the end of a tube that is closed at the other end, as shown in Figure 14.21, Figure 14.22, and Figure 14.23. If the tuning fork has just the right frequency, the air column in the tube resonates loudly, but at most frequencies it vibrates very little. This means that the air column has only certain natural frequencies. The figures show how a resonance at the lowest of these natural frequencies is formed. A disturbance travels down the tube at the speed of sound and bounces off the closed end. If the tube is just the right length, the reflected sound arrives back at the tuning fork exactly half a cycle later, and it interferes constructively with the continuing sound produced by the tuning fork. The incoming and reflected sounds form a standing wave in the tube as shown. Figure 14.21 Resonance of air in a tube closed at one end, caused by a tuning fork. A disturbance moves down the tube. Figure 14.22 Resonance of air in a tube closed at one end, caused by a tuning fork. The disturbance reflects from the closed end of the tube. Figure 14.23 Resonance of air in a tube closed at one end, caused by a tuning fork. If the length of the tube Lis just right, the disturbance gets back to the tuning fork half a cycle later and interferes constructively with the continuing sound from the tuning fork. This interference forms a standing wave, and the air column resonates. The standing wave formed in the tube has its maximum air displacement (an antinode) at the open end
, and no displacement (a 438 Chapter 14 • Sound node) at the closed end. Recall from the last chapter on waves that motion is unconstrained at the antinode, and halted at the node. The distance from a node to an antinode is one-fourth of a wavelength, and this equals the length of the tube; therefore,. This same resonance can be produced by a vibration introduced at or near the closed end of the tube, as shown in Figure 14.24. Figure 14.24 The same standing wave is created in the tube by a vibration introduced near its closed end. Since maximum air displacements are possible at the open end and none at the closed end, there are other, shorter wavelengths that can resonate in the tube see Figure 14.25). Here the standing wave has three-fourths of its wavelength in the tube, or, so that. There is a whole series of shorter-wavelength and higher-frequency sounds that resonate in the tube. We use specific terms for the resonances in any system. The lowest resonant frequency is called the fundamental, while all higher resonant frequencies are called overtones. All resonant frequencies are multiples of the fundamental, and are called harmonics. The fundamental is the first harmonic, the first overtone is the second harmonic, and so on. Figure 14.26 shows the fundamental and the first three overtones (the first four harmonics) in a tube closed at one end. Figure 14.25 Another resonance for a tube closed at one end. This has maximum air displacements at the open end, and none at the closed end. The wavelength is shorter, with three-fourths equaling the length of the tube, so that. This higher-frequency vibration is the first overtone. Figure 14.26 The fundamental and three lowest overtones for a tube closed at one end. All have maximum air displacements at the open end and none at the closed end. The fundamental and overtones can be present at the same time in a variety of combinations. For example, the note middle C on a trumpet sounds very different from middle C on a clarinet, even though both instruments are basically modified versions of a Access for free at openstax.org. 14.4 • Sound Interference and Resonance 439 tube closed at one end. The fundamental frequency is the same (and usually the most intense), but the overtones and their mix of intensities are different. This mix is what gives musical instruments (
and human voices) their distinctive characteristics, whether they have air columns, strings, or drumheads. In fact, much of our speech is determined by shaping the cavity formed by the throat and mouth and positioning the tongue to adjust the fundamental and combination of overtones. Open-Pipe and Closed-Pipe Resonators The resonant frequencies of a tube closed at one end (known as a closed-pipe resonator) are where f1 is the fundamental, f3 is the first overtone, and so on. Note that the resonant frequencies depend on the speed of sound vand on the length of the tube L. Another type of tube is one that is openat both ends (known as an open-pipe resonator). Examples are some organ pipes, flutes, and oboes. The air columns in tubes open at both ends have maximum air displacements at both ends. (See Figure 14.27). Standing waves form as shown. Figure 14.27 The resonant frequencies of a tube open at both ends are shown, including the fundamental and the first three overtones. In all cases the maximum air displacements occur at both ends of the tube, giving it different natural frequencies than a tube closed at one end. The resonant frequencies of an open-pipe resonator are where f1 is the fundamental, f2 is the first overtone, f3 is the second overtone, and so on. Note that a tube open at both ends has a fundamental frequency twice what it would have if closed at one end. It also has a different spectrum of overtones than a tube closed at one end. So if you had two tubes with the same fundamental frequency but one was open at both ends and the other was closed at one end, they would sound different when played because they have different overtones. Middle C, for example, would sound richer played on an open tube since it has more overtones. An open-pipe resonator has more overtones than a closed-pipe resonator because it has even multiples of the fundamental as well as odd, whereas a closed tube has only odd multiples. In this section we have covered resonance and standing waves for wind instruments, but vibrating strings on stringed instruments also resonate and have fundamentals and overtones similar to those for wind instruments. Solving Problems Involving Harmonic Series and Beat Frequency WORKED EXAMPLE Finding the Length of a Tube for a Closed-Pipe Resonator If sound travels through the air at a speed of 344 m/
s, what should be the length of a tube closed at one end to have a fundamental frequency of 128 Hz? 440 Chapter 14 • Sound Strategy The length Lcan be found by rearranging the equation. Solution (1) Identify knowns. • The fundamental frequency is 128 Hz. • The speed of sound is 344 m/s. (2) Use to find the fundamental frequency (n= 1). (3) Solve this equation for length. (4) Enter the values of the speed of sound and frequency into the expression for L. 14.14 14.15 14.16 Discussion Many wind instruments are modified tubes that have finger holes, valves, and other devices for changing the length of the resonating air column and therefore, the frequency of the note played. Horns producing very low frequencies, such as tubas, require tubes so long that they are coiled into loops. WORKED EXAMPLE Finding the Third Overtone in an Open-Pipe Resonator If a tube that’s open at both ends has a fundamental frequency of 120 Hz, what is the frequency of its third overtone? Strategy Since we already know the value of the fundamental frequency (n = 1), we can solve for the third overtone (n = 4) using the equation. Solution Since fundamental frequency (n = 1) is and 14.17 14.18 Discussion To solve this problem, it wasn’t necessary to know the length of the tube or the speed of the air because of the relationship between the fundamental and the third overtone. This example was of an open-pipe resonator; note that for a closed-pipe resonator, the third overtone has a value of n = 7 (not n = 4). WORKED EXAMPLE Using Beat Frequency to Tune a Piano Piano tuners use beats routinely in their work. When comparing a note with a tuning fork, they listen for beats and adjust the string until the beats go away (to zero frequency). If a piano tuner hears two beats per second, and the tuning fork has a Access for free at openstax.org. 14.4 • Sound Interference and Resonance 441 frequency of 256 Hz, what are the possible frequencies of the piano? Strategy Since we already know that the beat frequency fBis 2, and one of the frequencies (let’s say f2) is 256 Hz, we can use the equation to solve for the frequency of the piano f1. Solution Since, we know that
either or. Solving for f1, Substituting in values, So, 14.19 14.20 14.21 Discussion The piano tuner might not initially be able to tell simply by listening whether the frequency of the piano is too high or too low and must tune it by trial and error, making an adjustment and then testing it again. If there are even more beats after the adjustment, then the tuner knows that he went in the wrong direction. Practice Problems 21. Two sound waves have frequencies and. What is the beat frequency produced by their superposition? a. b. c. d. 22. What is the length of a pipe closed at one end with fundamental frequency? (Assume the speed of sound in air is.) a. b. c. d. Check Your Understanding 23. What is damping? a. Over time the energy increases and the amplitude gradually reduces to zero. This is called damping. b. Over time the energy dissipates and the amplitude gradually increases. This is called damping. c. Over time the energy increases and the amplitude gradually increases. This is called damping. d. Over time the energy dissipates and the amplitude gradually reduces to zero. This is called damping. 24. What is resonance? When can you say that the system is resonating? a. The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance, and a system being driven at its natural frequency is said to resonate. b. The phenomenon of driving a system with a frequency higher than its natural frequency is called resonance, and a system being driven at its natural frequency does not resonate. c. The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance, and a system being driven at its natural frequency does not resonate. d. The phenomenon of driving a system with a frequency higher than its natural frequency is called resonance, and a system being driven at its natural frequency is said to resonate. 442 Chapter 14 • Sound 25. In the tuning fork and tube experiment, in case a standing wave is formed, at what point on the tube is the maximum disturbance from the tuning fork observed? Recall that the tube has one open end and one closed end. a. At the midpoint of the tube b. Both ends of the tube c. At the closed end of the tube d. At the open end of the tube 26. In the tuning fork and tube experiment, when will the air column produce the loudest sound?
a. b. c. d. If the tuning fork vibrates at a frequency twice that of the natural frequency of the air column. If the tuning fork vibrates at a frequency lower than the natural frequency of the air column. If the tuning fork vibrates at a frequency higher than the natural frequency of the air column. If the tuning fork vibrates at a frequency equal to the natural frequency of the air column. 27. What is a closed-pipe resonator? a. A pipe or cylindrical air column closed at both ends b. A pipe with an antinode at the closed end c. A pipe with a node at the open end d. A pipe or cylindrical air column closed at one end 28. Give two examples of open-pipe resonators. a. piano, violin b. drum, tabla c. d. rlectric guitar, acoustic guitar flute, oboe Access for free at openstax.org. KEY TERMS amplitude the amount that matter is disrupted during a sound wave, as measured by the difference in height between the crests and troughs of the sound wave. beat a phenomenon produced by the superposition of two waves with slightly different frequencies but the same amplitude beat frequency the frequency of the amplitude fluctuations of a wave damping the reduction in amplitude over time as the energy of an oscillation dissipates decibel a unit used to describe sound intensity levels Doppler effect an alteration in the observed frequency of a sound due to relative motion between the source and the observer fundamental harmonics the term used to refer to the fundamental and the lowest-frequency resonance its overtones hearing the perception of sound SECTION SUMMARY 14.1 Speed of Sound, Frequency, and Wavelength • Sound is one type of wave. • Sound is a disturbance of matter that is transmitted from its source outward in the form of longitudinal waves. • The relationship of the speed of sound v, its frequency f, and its wavelength is given by same relationship given for all waves., which is the • The speed of sound depends upon the medium through • which the sound wave is travelling. In a given medium at a specific temperature (or density), the speed of sound vis the same for all frequencies and wavelengths. 14.2 Sound Intensity and Sound Level • The intensity of a sound is proportional to its amplitude squared. • The energy of a sound wave is also proportional to its amplitude squared. • Sound intensity level in decibels (dB) is more relevant for how humans perceive sounds
than sound intensity (in W/m2), even though sound intensity is the SI unit. • Sound intensity level is not the same as sound intensity—it tells you the levelof the sound relative to a reference intensity rather than the actual intensity. • Hearing is the perception of sound and involves that transformation of sound waves into vibrations of parts within the ear. These vibrations are then transformed Chapter 14 • Key Terms 443 loudness the perception of sound intensity natural frequency the frequency at which a system would oscillate if there were no driving and no damping forces overtones all resonant frequencies higher than the fundamental pitch the perception of the frequency of a sound rarefaction a low-pressure region in a sound wave resonance the phenomenon of driving a system with a frequency equal to the system's natural frequency resonate to drive a system at its natural frequency sonic boom a constructive interference of sound created by an object moving faster than sound sound a disturbance of matter that is transmitted from its source outward by longitudinal waves sound intensity the power per unit area carried by a sound wave sound intensity level the level of sound relative to a fixed standard related to human hearing into neural signals that are interpreted by the brain. • People create sounds by pushing air up through their lungs and through elastic folds in the throat called vocal cords. 14.3 Doppler Effect and Sonic Booms • The Doppler effect is a shift in the observed frequency of a sound due to motion of either the source or the observer. • The observed frequency is greater than the actual source’s frequency when the source and the observer are moving closer together, either by the source moving toward the observer or the observer moving toward the source. • A sonic boom is constructive interference of sound created by an object moving faster than sound. 14.4 Sound Interference and Resonance • A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces. • A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate. • Beats occur when waves of slightly different frequencies • are superimposed. In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are 444 Chapter 14 • Key Equations called harmonics. • The resonant frequencies of a tube closed at one end are, where f1is the fundamental and L is the length of the tube. • The resonant frequencies of a tube open at both ends
are KEY EQUATIONS 14.1 Speed of Sound, Frequency, and Wavelength speed of sound 14.2 Sound Intensity and Sound Level intensity sound intensity sound intensity level CHAPTER REVIEW Concept Items 14.1 Speed of Sound, Frequency, and Wavelength 1. What is the amplitude of a sound wave perceived by the loudness human ear? a. b. pitch c. d. intensity timbre 2. The compressibility of air and hydrogen is almost the same. Which factor is the reason that sound travels faster in hydrogen than in air? a. Hydrogen is more dense than air. b. Hydrogen is less dense than air. c. Hydrogen atoms are heavier than air molecules. d. Hydrogen atoms are lighter than air molecules. 14.3 Doppler Effect and Sonic Booms Doppler effect observed frequency (moving source) Doppler effect observed frequency (moving observer) 14.4 Sound Interference and Resonance beat frequency resonant frequencies of a closed-pipe resonator resonant frequencies of an open-pipe resonator a. b. c. d. 4. How does the "decibel" get its name? a. The meaning of deci is “hundred” and the number of decibels is one-hundredth of the logarithm to base 10 of the ratio of two sound intensities. b. The meaning of deci is "ten" and the number of decibels is one-tenth of the logarithm to base 10 of the ratio of two sound intensities. c. The meaning of deci is “one-hundredth” and the number of decibels is hundred times the logarithm to base 10 of the ratio of two sound intensities. d. The meaning of deci is “one-tenth” and the number of decibels is ten times the logarithm to base 10 of the ratio of two sound intensities. 14.2 Sound Intensity and Sound Level 5. What is “timbre” of sound? a. Timbre is the quality of the sound that distinguishes 3. What is the mathematical relationship between intensity, it from other sound power, and area? Access for free at openstax.org. Chapter 14 • Chapter Review 445 b. Timbre is the loudness of the sound that distinguishes it from other sound. c. Timbre is the pitch of the sound that distinguishes it
from other sound. down is greater than the amplitude of the yo-yo b. when the amplitude of the finger moving up and down is less than the amplitude of the yo-yo c. when the frequency of the finger moving up and d. Timbre is the wavelength of the sound that down is equal to the resonant frequency of the yo-yo distinguishes it from other sound. d. when the frequency of the finger moving up and 14.3 Doppler Effect and Sonic Booms 6. Two sources of sound producing the same frequency are moving towards you at different speeds. Which one would sound more high-pitched? the one moving slower a. the one moving faster b. 7. When the speed of the source matches the speed of sound, what happens to the amplitude of the sound wave? Why? a. It approaches zero. This is because all wave crests are superimposed on one another through constructive interference. It approaches infinity. This is because all wave crests are superimposed on one another through constructive interference. It approaches zero, because all wave crests are superimposed on one another through destructive interference. It approaches infinity, because all wave crests are superimposed on one another through destructive interference. b. c. d. 8. What is the mathematical expression for the frequency perceived by the observer in the case of a stationary observer and a moving source? a. b. c. d. 14.4 Sound Interference and Resonance 9. When does a yo-yo travel the farthest from the finger? a. when the amplitude of the finger moving up and down is different from the resonant frequency of the yo-yo 10. What is the difference between harmonics and overtones? a. Harmonics are all multiples of the fundamental frequency. The first overtone is actually the first harmonic. b. Harmonics are all multiples of the fundamental frequency. The first overtone is actually the second harmonic. c. Harmonics are all multiples of the fundamental frequency. The second overtone is actually the first harmonic. d. Harmonics are all multiples of the fundamental frequency. The third overtone is actually the second harmonic. 11. What kind of waves form in pipe resonators? a. damped waves b. propagating waves c. high-frequency waves d. standing waves 12. What is the natural frequency of a system? a. The natural frequency is the frequency at which a system oscillates when it undergoes forced vibration. b. The natural
frequency is the frequency at which a system oscillates when it undergoes damped oscillation. c. The natural frequency is the frequency at which a system oscillates when it undergoes free vibration without a driving force or damping. d. The natural frequency is the frequency at which a system oscillates when it undergoes forced vibration with damping. Critical Thinking Items d. It remains constant. 14.1 Speed of Sound, Frequency, and Wavelength 13. What can be said about the frequency of a monotonous sound? a. b. c. It decreases with time. It decreases with distance. It increases with distance. 14. A scientist notices that a sound travels faster through a solid material than through the air. Which of the following can explain this? a. Solid materials are denser than air. b. Solid materials are less dense than air. c. A solid is more rigid than air. d. A solid is easier to compress than air. 446 Chapter 14 • Chapter Review 14.2 Sound Intensity and Sound Level 15. Which property of the wave is related to its intensity? How? a. The frequency of the wave is related to the intensity of the sound. The larger-frequency oscillations indicate greater pressure maxima and minima, and the pressure is higher in greater-intensity sound. b. The wavelength of the wave is related to the intensity of the sound. The longer-wavelength oscillations indicate greater pressure maxima and minima, and the pressure is higher in greaterintensity sound. c. The amplitude of the wave is related to the intensity of the sound. The larger-amplitude oscillations indicate greater pressure maxima and minima, and the pressure is higher in greater-intensity sound. d. The speed of the wave is related to the intensity of the sound. The higher-speed oscillations indicate greater pressure maxima and minima, and the pressure is higher in greater-intensity sound. 16. Why is decibel (dB) used to describe loudness of sound? a. Because, human ears have an inverse response to the amplitude of sound. b. Because, human ears have an inverse response to the intensity of sound. c. Because, the way our ears perceive sound can be more accurately described by the amplitude of a sound rather than the intensity of a sound directly. d. Because, the way our ears perceive sound can be more accurately described by the logarithm of the intensity of a sound rather than the intensity of a sound
directly. 17. How can humming while shooting a gun reduce ear damage? a. Humming can trigger those two muscles in the outer ear that react to intense sound produced while shooting and reduce the force transmitted to the cochlea. b. Humming can trigger those three muscles in the outer ear that react to intense sound produced while shooting and reduce the force transmitted to the cochlea. c. Humming can trigger those two muscles in the middle ear that react to intense sound produced while shooting and reduce the force transmitted to the cochlea. d. Humming can trigger those three muscles in the middle ear that react to intense sound produced while shooting and reduce the force transmitted to the cochlea. 18. A particular sound, S1, has an intensity times that of Access for free at openstax.org. another sound, S2. What is the difference in sound intensity levels measured in decibels? a. b. c. d. 14.3 Doppler Effect and Sonic Booms 19. When the source of sound is moving through the air, does the speed of sound change with respect to a stationary person standing nearby? a. Yes b. No 20. Why is no sound heard by the observer when an object approaches him at a speed faster than that of sound? a. If the source exceeds the speed of sound, then destructive interference occurs and no sound is heard by the observer when an object approaches him. If the source exceeds the speed of sound, the frequency of sound produced is beyond the audible range of sound. If the source exceeds the speed of sound, all the sound waves produced approach minimum intensity and no sound is heard by the observer when an object approaches him. If the source exceeds the speed of sound, all the sound waves produced are behind the source. Hence, the observer hears the sound only after the source has passed. b. c. d. 21. Does the Doppler effect occur when the source and observer are both moving towards each other? If so, how would this affect the perceived frequency? a. Yes, the perceived frequency will be even lower in this case than if only one of the two were moving. b. No, the Doppler effect occurs only when an observer is moving towards a source. c. No, the Doppler effect occurs only when a source is moving towards an observer. d. Yes, the perceived frequency will be even higher in this case than if only one of the two were moving. 14
.4 Sound Interference and Resonance 22. When does the amplitude of an oscillating system become maximum? a. When two sound waves interfere destructively. b. When the driving force produces a transverse wave in the system. c. When the driving force of the oscillator to the oscillating system is at a maximum amplitude. d. When the frequency of the oscillator equals the natural frequency of the oscillating system. 23. How can a standing wave be formed with the help of a tuning fork and a closed-end tube of appropriate length? a. If the tube is just the right length, the reflected sound arrives back at the tuning fork exactly half a cycle later, and it interferes constructively with the continuing sound produced by the tuning fork. If the tube is just the right length, the reflected sound arrives back at the tuning fork exactly half a cycle later, and it interferes destructively with the continuing sound produced by the tuning fork. If the tube is just the right length, the reflected sound arrives back at the tuning fork exactly one b. c. Problems 14.1 Speed of Sound, Frequency, and Wavelength 25. A bat produces a sound at and wavelength. What is the speed of the sound? a. b. c. d. 26. A sound wave with frequency of is traveling. By how much will its wavelength through air at change when it enters aluminum? a. b. c. d. 14.2 Sound Intensity and Sound Level 27. Calculate the sound intensity for a sound wave traveling through air at 15° C and having a pressure amplitude of 0.80 Pa. (Hint—Speed of sound in air at 15° C is 340 m/s.) a. 9.6×10−3 W / m2 7.7×10−3 W / m2 b. c. 9.6×10−4 W / m2 7.7×10−4 W / m2 d. Chapter 14 • Chapter Review 447 d. full cycle later, and it interferes constructively with the continuing sound produced by the tuning fork. If the tube is just the right length, the reflected sound arrives back at the tuning fork exactly one full cycle later, and it interferes destructively with the continuing sound produced by the tuning fork. 24. A tube open at both ends has a fundamental frequency. What will the frequency be if one end is of closed? a. b. c. d. 14.
3 Doppler Effect and Sonic Booms 29. An ambulance is moving away from you. You are standing still and you hear its siren at a frequency of. You know that the actual frequency of the siren. What is the speed of the ambulance?.) is (Assume the speed of sound to be a. b. c. d. 30. An ambulance passes you at a speed of. If its siren has a frequency of, what is difference in the frequencies you perceive before and after it passes you? (Assume the speed of sound in air is a. b. c. d..) 14.4 Sound Interference and Resonance 31. What is the length of an open-pipe resonator with a? (Assume the.) fundamental frequency of speed of sound is a. b. c. d. 28. The sound level in dB of a sound traveling through air at is. Calculate its pressure amplitude. 32. An open-pipe resonator has a fundamental frequency of. By how much would its length have to be a. b. c. d. changed to get a fundamental frequency of (Assume the speed of sound is a. b. c. d..)? 448 Chapter 14 • Test Prep Performance Task 14.4 Sound Interference and Resonance 33. Design and make an open air resonator capable of playing at least three different pitches (frequencies) of sound using a selection of bamboo of varying widths and lengths, which can be obtained at a local hardware store. Choose a piece of bamboo for creating a musical TEST PREP Multiple Choice 14.1 Speed of Sound, Frequency, and Wavelength 34. What properties does a loud, shrill whistle have? a. high amplitude, high frequency b. high amplitude, low frequency low amplitude, high frequency c. low amplitude, low frequency d. 35. What is the speed of sound in fresh water at degrees Celsius? a. b. c. d. 36. A tuning fork oscillates at a frequency of, creating sound waves. How many waves will reach the eardrum of a person near that fork in seconds? a. b. c. d. 37. Why does the amplitude of a sound wave decrease with distance from its source? a. The amplitude of a sound wave decreases with distance from its source, because the frequency of the sound wave decreases. b. The amplitude of a sound wave decreases with distance from its source, because the speed of the sound wave decreases. c
. The amplitude of a sound wave decreases with distance from its source, because the wavelength of the sound wave increases. d. The amplitude of a sound wave decreases with distance from its source, because the energy of the wave is spread over a larger and larger area. 38. Does the elasticity of the medium affect the speed of sound? How? a. No, there is no relationship that exists between the speed of sound and elasticity of the medium. Access for free at openstax.org. pipe. Calculate the length required for a certain frequency to resonate and then mark the locations where holes should be placed in the pipe to achieve their desired pitches. Use a simple hand drill or ask your wood shop department for help drilling holes. Use tuning forks to test and calibrate your instrument. Demonstrate your pipe for the class. b. Yes. When particles are more easily compressed in a medium, sound does not travel as quickly through the medium. c. Yes. When the particles in a medium do not compress much, sound does not travel as quickly through the medium. d. No, the elasticity of a medium affects frequency and wavelength, not wave speed. 14.2 Sound Intensity and Sound Level 39. Which of the following terms is a useful quantity to intensity frequency describe the loudness of a sound? a. b. c. pitch d. wavelength 40. What is the unit of sound intensity level? a. decibels b. hertz c. watts 41. If a particular sound S1 is times more intense than another sound S2, then what is the difference in sound intensity levels in dB for these two sounds? a. b. c. 42. By what minimum amount should frequencies vary for humans to be able to distinguish two separate sounds? a. b. c. d. 43. Why is I0chosen as the reference for sound intensity? a. Because, it is the highest intensity of sound a person with normal hearing can perceive at a frequency of 100 Hz. b. Because, it is the lowest intensity of sound a person with normal hearing can perceive at a frequency of 100 Hz. c. Because, it is the highest intensity of sound a person with normal hearing can perceive at a frequency of 1000 Hz. d. Because, it is the lowest intensity of sound a person with normal hearing can perceive at a frequency of 1000 Hz. 14.3 Doppler Effect and Sonic Booms 44. In which of the following situations is the Doppler effect absent?
a. The source and the observer are moving towards each other. b. The observer is moving toward the source. c. The source is moving away from the observer. d. Neither the source nor the observer is moving relative to one another. 45. What does the occurrence of the sonic boom depend on? speed of the source a. b. frequency of source c. amplitude of source d. distance of observer from the source 46. What is the observed frequency when the observer is? The source moving away from the source at frequency is. and the speed of sound is a. b. c. d. 47. How will your perceived frequency change if the source is moving towards you? Short Answer 14.1 Speed of Sound, Frequency, and Wavelength 52. What component of a longitudinal sound wave is analogous to a trough of a transverse wave? a. b. c. node d. antinode compression rarefaction 53. What is the frequency of a sound wave as perceived by the human ear? timbre a. loudness b. intensity c. d. pitch 54. What properties of a solid determine the speed of sound traveling through it? Chapter 14 • Test Prep 449 a. The frequency will become lower. b. The frequency will become higher. 14.4 Sound Interference and Resonance 48. Observation of which phenomenon can be considered interference proof that something is a wave? a. b. noise c. d. reflection conduction 49. Which of the resonant frequencies has the greatest amplitude? a. The first harmonic b. The second harmonic c. The first overtone d. The second overtone 50. What is the fundamental frequency of an open-pipe resonator? a. b. c. d.? 51. What is the beat frequency produced by the superposition of two waves with frequencies and a. b. c. d. a. mass and density b. rigidity and density c. volume and density d. shape and rigidity 55. Does the density of a medium affect the speed of sound? a. No b. Yes 56. Does a bat make use of the properties of sound waves to locate its prey? a. No b. Yes 57. Do the properties of a sound wave change when it travels from one medium to another? a. No b. Yes 450 Chapter 14 • Test Prep 14.2 Sound Intensity and Sound Level meter squared. 58. When a passing driver has his stereo turned up, you 61. Why is the reference
intensity cannot even hear what the person next to you is saying. Why is this so? a. The sound from the passing car’s stereo has a higher amplitude and hence higher intensity compared to the intensity of the sound coming from the person next to you. The higher intensity corresponds to greater loudness, so the first sound dominates the second. b. The sound from the passing car’s stereo has a higher amplitude and hence lower intensity compared to the intensity of the sound coming from the person next to you. The lower intensity corresponds to greater loudness, so the first sound dominates the second. c. The sound from the passing car’s stereo has a higher frequency and hence higher intensity compared to the intensity of the sound coming from the person next to you. The higher frequency corresponds to greater loudness so the first sound dominates the second. d. The sound from the passing car’s stereo has a lower frequency and hence higher intensity compared to the intensity of the sound coming from the person next to you. The lower frequency corresponds to greater loudness, so the first sound dominates the second. 59. For a constant area, what is the relationship between intensity of a sound wave and power? a. The intensity is inversely proportional to the power transmitted by the wave, for a constant area. b. The intensity is inversely proportional to the square of the power transmitted by the wave, for a constant area. c. The intensity is directly proportional to the square of the power transmitted by the wave, for a constant area. d. The intensity is directly proportional to the power transmitted by the wave, for a constant area.? decibels, a. The upper limit of human hearing is, i.e.. For. b. The lower threshold of human hearing is decibels, i.e.. For, c. The upper limit of human hearing is decibels, i.e.. For, d. The lower threshold of human hearing is decibels, i.e.,. For, 62. Given that the sound intensity level of a particular wave, what will be the sound intensity for that wave? is a. b. c. d. 63. For a sound wave with intensity, calculate the pressure amplitude given that the sound. travels through air at a. b. c. d. 64. Which nerve carries auditory information to the brain? a. buccal nerve b. peroneal nerve c. cochlear nerve d. mandibular
nerve 65. Why do some smaller instruments, such as piccolos, produce higher-pitched sounds than larger instruments, such as tubas? a. Smaller instruments produce sounds with shorter wavelengths, and thus higher frequencies. b. Smaller instruments produce longer wavelength, and thus higher amplitude, sounds. c. Smaller instruments produce lower amplitude, and thus longer wavelength sounds. 60. What does stand for in the equation d. Smaller instruments produce higher amplitude,? What is its unit? and thus lower frequency, sounds. a. Yes, is the sound intensity in watts per meter squared in the equation,. 14.3 Doppler Effect and Sonic Booms is the sound illuminance and its unit is lumen 66. How will your perceived frequency change if you move b. c. per meter squared. is the sound intensity and its unit is watts per meter cubed. d. is the sound intensity and its unit is watts per Access for free at openstax.org. away from a stationary source of sound? a. The frequency will become lower. b. The frequency will be doubled. c. The frequency will be tripled. d. The frequency will become higher. 67. True or false—The Doppler effect also occurs with waves other than sound waves. a. False b. True 68. A source of sound is moving towards you. How will what you hear change if the speed of the source increases? a. The sound will become more high-pitched. b. The sound will become more low-pitched. c. The pitch of the sound will not change. 69. Do sonic booms continue to be created when an object is traveling at supersonic speeds? a. No, a sonic boom is created only when the source exceeds the speed of sound. b. Yes, sonic booms continue to be created when an Chapter 14 • Test Prep 451 a. Human speech is produced by shaping the cavity formed by the throat and mouth, the vibration of vocal cords, and using the tongue to adjust the fundamental frequency and combination of overtones. b. Human speech is produced by shaping the cavity formed by the throat and mouth into a closed pipe and using tongue to adjust the fundamental frequency and combination of overtones. c. Human speech is produced only by the vibrations of the tongue. d. Human speech is produced by elongating the vocal cords. 75. What is the possible number of nodes and antinodes along one full wavelength of a standing wave?
a. nodes and antinodes or antinodes and object is traveling at supersonic speeds. nodes. 70. Suppose you are driving at a speed of and you.? hear the sound of a bell at a frequency of What is the actual frequency of the bell if the speed of sound is a. b. c. d. 71. What is the frequency of a stationary sound source if you hear it at 1200.0 Hz while moving towards it at a speed of 50.0 m/s? (Assume speed of sound to be 331 m/s.) a. b. c. d. 1410 Hz 1380 Hz 1020 Hz 1042 Hz 14.4 Sound Interference and Resonance 72. What is the actual frequency of the wave produced as a result of superposition of two waves? a. It is the average of the frequencies of the two original waves that were superimposed. It is the difference between the frequencies of the two original waves that were superimposed. It is the product of the frequencies of the two original waves that were superimposed. It is the sum of the frequencies of the two original waves that were superimposed. b. c. d. 73. Can beats be produced through a phenomenon different from resonance? How? a. No, beats can be produced only by resonance. b. Yes, beats can be produced by superimposition of any two waves having slightly different frequencies. 74. How is human speech produced? b. c. nodes and antinodes or antinodes and nodes. nodes and antinodes or antinodes and nodes. d. nodes and antinodes or antinodes and nodes. 76. In a pipe resonator, which frequency will be the least second overtone frequency intense of those given below? a. b. first overtone frequency fundamental frequency c. third overtone frequency d. 77. A flute is an open-pipe resonator. If a flute is long, what is the longest wavelength it can produce? a. b. c. d. 78. What is the frequency of the second overtone of a? closed-pipe resonator with a length of (Assume the speed of sound is a. b. c. d..) when the speed of sound is 79. An open-pipe resonator has a fundamental frequency of. What will its fundamental frequency be when the speed of sound is a. b. c. d.? 452 Chapter 14 • Test Prep Extended Response e
ardrum. 14.1 Speed of Sound, Frequency, and Wavelength 80. How is a human able to hear sounds? a. Sound waves cause the eardrum to vibrate. A complicated mechanism converts the vibrations to nerve impulses, which are perceived by the person as sound. b. Sound waves cause the ear canal to vibrate. A complicated mechanism converts the vibrations to nerve impulses, which are perceived by the person as sound. c. Sound waves transfer electrical impulses to the eardrum. A complicated mechanism converts the electrical impulses to sound. d. Sound waves transfer mechanical vibrations to the ear canal, and the eardrum converts them to electrical impulses. 81. Why does sound travel faster in iron than in air even though iron is denser than air? a. The density of iron is greater than that of air. However, the rigidity of iron is much greater than that of air. Hence, sound travels faster in it. b. The density of iron is greater than that of air. However, the rigidity of iron is much less than that of air. Hence, sound travels faster in it. c. The density of iron is greater than that of air. However, the rigidity of iron is equal to that of air. Hence, sound travels faster in it. d. The mass of iron is much less than that of air and the rigidity of iron is much greater than that of air. Hence, sound travels faster in it. 82. Is the speed of sound dependent on its frequency? a. No b. Yes 14.2 Sound Intensity and Sound Level 83. Why is the sound from a tire burst louder than that from a finger snap? a. The sound from the tire burst has higher pressure amplitudes, hence it can exert smaller force on the eardrum. b. The sound from the tire burst has lower pressure amplitudes, hence it can exert smaller force on the eardrum. c. The sound from the tire burst has lower pressure amplitudes, hence it can exert larger force on the ear drum. d. The sound from the tire burst has higher pressure amplitudes, hence it can exert larger force on the Access for free at openstax.org. 84. Sound A is times more intense than Sound B. What will be the difference in decibels in their sound intensity levels? a. b. c. d. 85. The ratio of the pressure amplitudes of two sound waves. What will be the is traveling through water at difference
in their sound intensity levels in dB? a. b. c. d. 86. Which of the following most closely models how sound is produced by the vocal cords? a. A person plucks a string. b. A person blows over the mouth of a half-filled glass bottle. c. A person strikes a hammer against a hard surface. d. A person blows through a small slit in a wide, stretched rubber band. 14.3 Doppler Effect and Sonic Booms 87. True or false—The Doppler effect occurs only when the sound source is moving. a. False b. True 88. True or false—The observed frequency becomes infinite when the source is moving at the speed of sound. a. False b. True 89. You are driving alongside a train. You hear its horn at a pitch that is lower than the actual frequency. What should you do to match the speed of the train? Why? a. In order to match the speed of the train, one would need to increase or decrease the speed of his/her car because a lower pitch means that either the train (the source) is moving away or that you (the observer) are moving away. In order to match the speed of the train, one would need to drive at a constant speed because a lower pitch means that the train and the car are at the same speed. b. 14.4 Sound Interference and Resonance 90. How are the beat frequency and the regular frequency of a wave resulting from superposition of two waves different? a. Beat frequency is the sum of two frequencies and regular frequency is the difference between frequencies of two original waves. overtone so resonance will occur. c. The frequency formed is a harmonic and third overtone so resonance will occur. b. Beat frequency is the difference between the d. The frequency formed is a harmonic and fourth Chapter 14 • Test Prep 453 constituent frequencies, but the regular frequency is the average of the frequencies of the two original waves. c. Beat frequency is the sum of two frequencies and regular frequency is the average of frequencies of two original waves. d. Beat frequency is the average of frequencies of two original waves and regular frequency is the sum of two original frequencies. 91. In the tuning fork and tube experiment, if resonance is is the length of the tube formed for, where and is the wavelength of the sound wave, can resonance also be formed for a wavelength Why? a. The frequency formed is a harmonic and first? overtone so resonance will
occur. b. The frequency formed is a harmonic and second overtone so resonance will occur. 92. True or false—An open-pipe resonator has more overtones than a closed-pipe resonator. a. False b. True 93. A flute has finger holes for changing the length of the resonating air column, and therefore, the frequency of the note played. How far apart are two holes that, when closed, play two frequencies that are apart, if the first hole is the flute? a. b. c. d. away from the mouthpiece of 454 Chapter 14 • Test Prep Access for free at openstax.org. CHAPTER 15 Light Figure 15.1 Human eyes detect these orange sea goldiefish swimming over a coral reef in the blue waters of the Gulf of Eilat, in the Red Sea, using visible light. (credit: David Darom, Wikimedia Commons) Chapter Outline 15.1 The Electromagnetic Spectrum 15.2 The Behavior of Electromagnetic Radiation INTRODUCTION The beauty of a coral reef, the warm radiance of sunshine, the sting of sunburn, the X-ray revealing a broken bone, even microwave popcorn—all are brought to us by electromagnetic waves. The list of the various types of electromagnetic waves, ranging from radio transmission waves to nuclear gamma-ray (γ-ray) emissions, is interesting in itself. Even more intriguing is that all of these different phenomena are manifestations of the same thing—electromagnetic waves (see Figure 15.1). What are electromagnetic waves? How are they created, and how do they travel? How can we understand their widely varying properties? What is the relationship between electric and magnetic effects? These and other questions will be explored. 15.1 The Electromagnetic Spectrum Section Learning Objectives By the end of this section, you will be able to do the following: • Define the electromagnetic spectrum, and describe it in terms of frequencies and wavelengths • Describe and explain the differences and similarities of each section of the electromagnetic spectrum and the applications of radiation from those sections Section Key Terms electric field electromagnetic radiation (EMR) magnetic field Maxwell’s equations 456 Chapter 15 • Light The Electromagnetic Spectrum We generally take light for granted, but it is a truly amazing and mysterious form of energy. Think about it: Light travels to Earth across millions of kilometers of empty space. When it reaches us, it interacts with matter in various ways to generate almost all the energy needed to
support life, provide heat, and cause weather patterns. Light is a form of electromagnetic radiation (EMR). The term lightusually refers to visible light, but this is not the only form of EMR. As we will see, visible light occupies a narrow band in a broad range of types of electromagnetic radiation. Electromagnetic radiation is generated by a moving electric charge, that is, by an electric current. As you will see when you study electricity, an electric current generates both an electric field, E, and a magnetic field, B. These fields are perpendicular to each other. When the moving charge oscillates, as in an alternating current, an EM wave is propagated. Figure 15.2 shows how an electromagnetic wave moves away from the source—indicated by the ~ symbol. WATCH PHYSICS Electromagnetic Waves and the Electromagnetic Spectrum This video, link below, is closely related to the following figure. If you have questions about EM wave properties, the EM spectrum, how waves propagate, or definitions of any of the related terms, the answers can be found in this video (http://www.openstax.org/l/28EMWaves). Click to view content (https://www.openstax.org/l/28EMWaves) GRASP CHECK In an electromagnetic wave, how are the magnetic field, the electric field, and the direction of propagation oriented to each other? a. All three are parallel to each other and are along the x-axis. b. All three are mutually perpendicular to each other. c. The electric field and magnetic fields are parallel to each other and perpendicular to the direction of propagation. d. The magnetic field and direction of propagation are parallel to each other along the y-axis and perpendicular to the electric field. Virtual Physics Radio Waves and Electromagnetic Fields Click to view content (https://www.openstax.org/l/28Radiowaves) This simulation demonstrates wave propagation. The EM wave is propagated from the broadcast tower on the left, just as in Figure 15.2. You can make the wave yourself or allow the animation to send it. When the wave reaches the antenna on the right, it causes an oscillating current. This is how radio and television signals are transmitted and received. GRASP CHECK Where do radio waves fall on the electromagnetic spectrum? a. Radio waves have the same wavelengths as visible light. b. Radio waves fall on the high-frequency side of visible
light. c. Radio waves fall on the short-wavelength side of visible light. d. Radio waves fall on the low-frequency side of visible light. Access for free at openstax.org. 15.1 • The Electromagnetic Spectrum 457 Figure 15.2 A part of the electromagnetic wave sent out from an oscillating charge at one instant in time. The electric and magnetic fields (E and B) are in phase, and they are perpendicular to each other and to the direction of propagation. For clarity, the waves are shown only along one direction, but they propagate out in other directions too. From your study of sound waves, recall these features that apply to all types of waves: • Wavelength—The distance between two wave crests or two wave troughs, expressed in various metric measures of distance • Frequency—The number of wave crests that pass a point per second, expressed in hertz (Hz or s–1) • Amplitude: The height of the crest above the null point As mentioned, electromagnetic radiation takes several forms. These forms are characterized by a range of frequencies. Because frequency is inversely proportional to wavelength, any form of EMR can also be represented by its range of wavelengths. Figure 15.3 shows the frequency and wavelength ranges of various types of EMR. With how many of these types are you familiar? Figure 15.3 The electromagnetic spectrum, showing the major categories of electromagnetic waves. The range of frequencies and wavelengths is remarkable. The dividing line between some categories is distinct, whereas other categories overlap. Take a few minutes to study the positions of the various types of radiation on the EM spectrum, above. Sometimes all radiation with frequencies lower than those of visible light are referred to as infrared (IR) radiation. This includes radio waves, which overlap with the frequencies used for media broadcasts of TV and radio signals. The microwave radiation that you see on the diagram is the same radiation that is used in a microwave oven. What we feel as radiant heat is also a form of low-frequency EMR. All the high-frequency radiation to the right of visible light is sometimes referred to as ultraviolet (UV) radiation. This includes X-rays and gamma (γ) rays. The narrow band that is visible light extends from lower-frequency red light to higher-frequency violet light, thus the terms are infrared(below red) and ultraviolet(beyond violet). BOUNDLESS PHYSICS Maxwell’s Equations The Scottish physicist James Clerk Maxwell (1831–18
79) is regarded widely to have been the greatest theoretical physicist of the 458 Chapter 15 • Light nineteenth century. Although he died young, Maxwell not only formulated a complete electromagnetic theory, represented by Maxwell’s equations, he also developed the kinetic theory of gases, and made significant contributions to the understanding of color vision and the nature of Saturn’s rings. Maxwell brought together all the work that had been done by brilliant physicists, such as Ørsted, Coulomb, Ampere, Gauss, and Faraday, and added his own insights to develop the overarching theory of electromagnetism. Maxwell’s equations are paraphrased here in words because their mathematical content is beyond the level of this text. However, the equations illustrate how apparently simple mathematical statements can elegantly unite and express a multitude of concepts—why mathematics is the language of science. Maxwell’s Equations 1. Electric field lines originate on positive charges and terminate on negative charges. The electric field is defined as the force per unit charge on a test charge, and the strength of the force is related to the electric constant, ε0. 2. Magnetic field lines are continuous, having no beginning or end. No magnetic monopoles are known to exist. The strength of the magnetic force is related to the magnetic constant, μ0. 3. A changing magnetic field induces an electromotive force (emf) and, hence, an electric field. The direction of the emf opposes the change, changing direction of the magnetic field. 4. Magnetic fields are generated by moving charges or by changing electric fields. Maxwell’s complete theory shows that electric and magnetic forces are not separate, but different manifestations of the same thing—the electromagnetic force. This classical unification of forces is one motivation for current attempts to unify the four basic forces in nature—the gravitational, electromagnetic, strong nuclear, and weak nuclear forces. The weak nuclear and electromagnetic forces have been unified, and further unification with the strong nuclear force is expected; but, the unification of the gravitational force with the other three has proven to be a real head-scratcher. One final accomplishment of Maxwell was his development in 1855 of a process that could produce color photographic images. In 1861, he and photographer Thomas Sutton worked together on this process. The color image was achieved by projecting red, blue, and green light through black-and-white photographs of a tartan ribbon, each photo itself exposed in different-colored light. The final image was projected onto a
screen (see Figure 15.4). Figure 15.4 Maxwell and Sutton’s photograph of a colored ribbon. This was the first durable color photograph. The plaid tartan of the Scots made a colorfulphotographic subject. GRASP CHECK Describe electromagnetic force as explained by Maxwell’s equations. a. According to Maxwell’s equations, electromagnetic force gives rise to electric force and magnetic force. b. According to Maxwell’s equations, electric force and magnetic force are different manifestations of electromagnetic force. c. According to Maxwell’s equations, electric force is the cause of electromagnetic force. d. According to Maxwell’s equations, magnetic force is the cause of electromagnetic force. Characteristics of Electromagnetic Radiation All the EM waves mentioned above are basically the same form of radiation. They can all travel across empty space, and they all Access for free at openstax.org. 15.1 • The Electromagnetic Spectrum 459 travel at the speed of light in a vacuum. The basic difference between types of radiation is their differing frequencies. Each frequency has an associated wavelength. As frequency increases across the spectrum, wavelength decreases. Energy also increases with frequency. Because of this, higher frequencies penetrate matter more readily. Some of the properties and uses of the various EM spectrum bands are listed in Table 15.1. Types of EM Waves Radio and TV Production Applications Life Sciences Aspect Issues Accelerating charges Communications, remote controls MRI Requires controls for band use Microwaves Accelerating charges & thermal agitation Communications, microwave ovens, radar Deep heating Cell phone use Infrared Thermal agitation & electronic transitions Thermal imaging, heating Visible Light Thermal agitation & electronic transitions All pervasive Greenhouse effect Absorption by atmosphere Photosynthesis, human vision Ultraviolet Thermal agitation & electronic transitions Sterilization, slowing abnormal growth of cells Vitamin D production Ozone depletion, causes cell damage X-rays Gamma Rays Inner electronic transitions & fast collisions Medical, security Medical diagnosis, cancer therapy Causes cell damage Nuclear decay Nuclear medicine, security Medical diagnosis, cancer therapy Causes cell damage, radiation damage Table 15.1 Electromagnetic Waves This table shows how each type of EM radiation is produced, how it is applied, as well as environmental and health issues associated with it. The narrow band of visible light is a combination of the colors of the rainbow. Figure 15.5 shows the section of the EM spectrum that includes visible light. The frequencies corresponding to these wavelengths are at the red end to at the violet end. This is
a very narrow range, considering that the EM spectrum spans about 20 orders of magnitude. Figure 15.5 A small part of the electromagnetic spectrum that includes its visible components. The divisions between infrared, visible, and ultraviolet are not perfectly distinct, nor are the divisions between the seven rainbow colors TIPS FOR SUCCESS Wavelengths of visible light are often given in nanometers, nm. One nm equals wavelength of about 600 nm, or m. m. For example, yellow light has a 460 Chapter 15 • Light As a child, you probably learned the color wheel, shown on the left in Figure 15.6. It helps if you know what color results when you mix different colors of paint together. Mixing two of the primary pigmentcolors—magenta, yellow, or cyan—together results in a secondary color. For example, mixing cyan and yellow makes green. This is called subtractivecolor mixing. Mixing different colors of lighttogether is quite different. The diagram on the right shows additivecolor mixing. In this case, the primary colors are red, green, and blue, and the secondary colors are cyan, magenta, and yellow. Mixing pigments and mixing light are different because materials absorb light by a different set of rules than does the perception of light by the eye. Notice that, when all colors are subtracted, the result is no color, or black. When all colors are added, the result is white light. We see the reverse of this when white sunlight is separated into the visible spectrum by a prism or by raindrops when a rainbow appears in the sky. Figure 15.6 Mixing colored pigments follows the subtractive color wheel, and mixing colored light follows the additive color wheel. Virtual Physics Color Vision Click to view content (https://www.openstax.org/l/28Colorvision) This video demonstrates additive color and color filters. Try all the settings except Photons. GRASP CHECK Explain why only light from a blue bulb passes through the blue filter. a. A blue filter absorbs blue light. b. A blue filter reflects blue light. c. A blue filter absorbs all visible light other than blue light. d. A blue filter reflects all of the other colors of light and absorbs blue light. LINKS TO PHYSICS Animal Color Perception The physics of color perception has interesting links to zoology. Other animals have very different views of the world than humans, especially with respect to which colors can be seen. Color is detected by cells in the
eye called cones. Humans have three cones that are sensitive to three different ranges of electromagnetic wavelengths. They are called red, blue, and green cones, although these colors do not correspond exactly to the centers of the three ranges. The ranges of wavelengths that each cone detects are red, 500 to 700 nm; green, 450 to 630 nm; and blue, 400 to 500 nm. Most primates also have three kinds of cones and see the world much as we do. Most mammals other than primates only have two cones and have a less colorful view of things. Dogs, for example see blue and yellow, but are color blind to red and green. You might think that simplerspecies, such as fish and insects, would have less sophisticated vision, but this is not the case. Many birds, reptiles, amphibians, and insects have four or five different cones in their eyes. These species don’t have a wider range of perceived colors, but they see more hues, or combinations of colors. Also, some animals, such as bees or rattlesnakes, see a Access for free at openstax.org. 15.1 • The Electromagnetic Spectrum 461 range of colors that is as broad as ours, but shifted into the ultraviolet or infrared. These differences in color perception are generally adaptations that help the animals survive. Colorful tropical birds and fish display some colors that are too subtle for us to see. These colors are believed to play a role in the species mating rituals. Figure 15.7 shows the colors visible and the color range of vision in humans, bees, and dogs. Figure 15.7 Humans, bees, and dogs see colors differently. Dogs see fewer colors than humans, and bees see a different range of colors. GRASP CHECK The belief that bulls are enraged by seeing the color red is a misconception. What did you read in this Links to Physics that shows why this belief is incorrect? a. Bulls are color-blind to every color in the spectrum of colors. b. Bulls are color-blind to the blue colors in the spectrum of colors. c. Bulls are color-blind to the red colors in the spectrum of colors. d. Bulls are color-blind to the green colors in the spectrum of colors. Humans have found uses for every part of the electromagnetic spectrum. We will take a look at the uses of each range of frequencies, beginning with visible light. Most of our uses of visible light are obvious; without it our interaction with our surroundings would be much
different. We might forget that nearly all of our food depends on the photosynthesis process in plants, and that the energy for this process comes from the visible part of the spectrum. Without photosynthesis, we would also have almost no oxygen in the atmosphere. The low-frequency, infrared region of the spectrum has many applications in media broadcasting. Television, radio, cell phone, and remote-control devices all broadcast and/or receive signals with these wavelengths. AM and FM radio signals are both lowfrequency radiation. They are in different regions of the spectrum, but that is not their basic difference. AM and FM are abbreviations for amplitude modulationand frequency modulation. Information in AM signals has the form of changes in amplitudeof the radio waves; information in FM signals has the form of changes in wave frequency. Another application of long-wavelength radiation is found in microwave ovens. These appliances cook or warm food by irradiating it with EM radiation in the microwave frequency range. Most kitchen microwaves use a frequency of Hz. These waves have the right amount of energy to cause polar molecules, such as water, to rotate faster. Polar molecules are those that have a partial charge separation. The rotational energy of these molecules is given up to surrounding matter as heat. The first microwave ovens were called Radarangesbecause they were based on radar technology developed during World War II. Radar uses radiation with wavelengths similar to those of microwaves to detect the location and speed of distant objects, such as airplanes, weather formations, and motor vehicles. Radar information is obtained by receiving and analyzing the echoes of microwaves reflected by an object. The speed of the object can be measured using the Doppler shift of the returning waves. This is the same effect you learned about when you studied sound waves. Like sound waves, EM waves are shifted to higher frequencies by an object moving toward an observer, and to lower frequencies by an object moving away from the observer. Astronomers use this same Doppler effect to measure the speed at which distant galaxies are moving away from us. In this case, the shift in frequency is called the red shift, because visible frequencies are shifted toward the lower-frequency, red end of the spectrum. 462 Chapter 15 • Light Exposure to any radiation with frequencies greater than those of visible light carries some health hazards. All types of radiation in this range are known to cause cell damage. The danger is related to the high energy and penetrating ability of these EM waves. The likelihood of being harmed by any of this radiation depends largely on the
amount of exposure. Most people try to reduce exposure to UV radiation from sunlight by using sunscreen and protective clothing. Physicians still use X-rays to diagnose medical problems, but the intensity of the radiation used is extremely low. Figure 15.8 shows an X-ray image of a patient’s chest cavity. One medical-imaging technique that involves no danger of exposure is magnetic resonance imaging (MRI). MRI is an important imaging and research tool in medicine, producing highly detailed two- and three-dimensional images. Radio waves are broadcast, absorbed, and reemitted in a resonance process that is sensitive to the density of nuclei, usually hydrogen nuclei—protons. Figure 15.8 This shadow X-ray image shows many interesting features, such as artificial heart valves, a pacemaker, and wires used to close the sternum. (credit: P.P. Urone) Check Your Understanding 1. Identify the fields produced by a moving charged particle. a. Both an electric field and a magnetic field will be produced. b. Neither a magnetic field nor an electric field will be produced. c. A magnetic field, but no electric field will be produced. d. Only the electric field, but no magnetic field will be produced. 2. X-rays carry more energy than visible light. Compare the frequencies and wavelengths of these two types of EM radiation. a. Visible light has higher frequencies and shorter wavelengths than X-rays. b. Visible light has lower frequencies and shorter wavelengths than X-rays. c. Visible light has higher frequencies and longer wavelengths than X-rays. d. Visible light has lower frequencies and longer wavelengths than X-rays. 3. How does wavelength change as frequency increases across the EM spectrum? a. The wavelength increases. b. The wavelength first increases and then decreases. c. The wavelength first decreases and then increases. d. The wavelength decreases. 4. Why are X-rays used in imaging of broken bones, rather than radio waves? a. X-rays have higher penetrating energy than radio waves. b. X-rays have lower penetrating energy than radio waves. c. X-rays have a lower frequency range than radio waves. d. X-rays have longer wavelengths than radio waves. 5. Identify the fields that make up an electromagnetic wave. a. both an electric field and a magnetic field b. neither a magnetic field nor an electric field Access for free at openstax.org. 15.2 • The Behavior
of Electromagnetic Radiation 463 c. only a magnetic field, but no electric field d. only an electric field, but no magnetic field 15.2 The Behavior of Electromagnetic Radiation Section Learning Objectives By the end of this section, you will be able to do the following: • Describe the behavior of electromagnetic radiation • Solve quantitative problems involving the behavior of electromagnetic radiation Section Key Terms illuminance interference lumens luminous flux lux polarized light Types of Electromagnetic Wave Behavior In a vacuum, all electromagnetic radiation travels at the same incredible speed of 3.00 × 108 m/s, which is equal to 671 million miles per hour. This is one of the fundamental physical constants. It is referred to as the speed of light and is given the symbol c. The space between celestial bodies is a near vacuum, so the light we see from the Sun, stars, and other planets has traveled here at the speed of light. Keep in mind that all EM radiation travels at this speed. All the different wavelengths of radiation that leave the Sun make the trip to Earth in the same amount of time. That trip takes 8.3 minutes. Light from the nearest star, besides the Sun, takes 4.2 years to reach Earth, and light from the nearest galaxy—a dwarf galaxy that orbits the Milky Way—travels 25,000 years on its way to Earth. You can see why we call very long distances astronomical. When light travels through a physical medium, its speed is always less than the speed of light. For example, light travels in water at three-fourths the value of c. In air, light has a speed that is just slightly slower than in empty space: 99.97 percent of c. Diamond slows light down to just 41 percent of c. When light changes speeds at a boundary between media, it also changes direction. The greater the difference in speeds, the more the path of light bends. In other chapters, we look at this bending, called refraction, in greater detail. We introduce refraction here to help explain a phenomenon called thin-film interference. Have you ever wondered about the rainbow colors you often see on soap bubbles, oil slicks, and compact discs? This occurs when light is both refracted by and reflected from a very thin film. The diagram shows the path of light through such a thin film. The symbols n1, n2, and n3 indicate that light travels at different speeds in each of the three materials. Learn more about this
topic in the chapter on diffraction and interference. Figure 15.9 shows the result of thin film interference on the surface of soap bubbles. Because ray 2 travels a greater distance, the two rays become out of phase. That is, the crests of the two emerging waves are no longer moving together. This causes interference, which reinforces the intensity of the wavelengths of light that create the bands of color. The color bands are separated because each color has a different wavelength. Also, the thickness of the film is not uniform, and different thicknesses cause colors of different wavelengths to interfere in different places. Note that the film must be very, very thin—somewhere in the vicinity of the wavelengths of visible light. 464 Chapter 15 • Light Figure 15.9 Light striking a thin film is partially reflected (ray 1) and partially refracted at the top surface. The refracted ray is partially reflected at the bottom surface and emerges as ray 2. These rays will interfere in a way that depends on the thickness of the film and the indices of refraction of the various media. You have probably experienced how polarized sunglasses reduce glare from the surface of water or snow. The effect is caused by the wave nature of light. Looking back at, we see that the electric field moves in only one direction perpendicular to the direction of propagation. Light from most sources vibrates in all directions perpendicular to propagation. Light with an electric field that vibrates in only one direction is called polarized. A diagram of polarized light would look like. Polarized glasses are an example of a polarizing filter. These glasses absorb most of the horizontal light waves and transmit the vertical waves. This cuts down glare, which is caused by horizontal waves. Figure 15.10 shows how waves traveling along a rope can be used as a model of how a polarizing filter works. The oscillations in one rope are in a vertical plane and are said to be vertically polarized. Those in the other rope are in a horizontal plane and are horizontally polarized. If a vertical slit is placed on the first rope, the waves pass through. However, a vertical slit blocks the horizontally polarized waves. For EM waves, the direction of the electric field oscillation is analogous to the disturbances on the ropes. Figure 15.10 The transverse oscillations in one rope are in a vertical plane, and those in the other rope are in a horizontal plane. The first is said to be vertically polarized, and the other is said to be horizontally polarized. Vertical slits pass vertically polarized waves and block horizontally
polarized waves. Light can also be polarized by reflection. Most of the light reflected from water, glass, or any highly reflective surface is polarized horizontally. Figure 15.11 shows the effect of a polarizing lens on light reflected from the surface of water. Access for free at openstax.org. 15.2 • The Behavior of Electromagnetic Radiation 465 Figure 15.11 These two photographs of a river show the effect of a polarizing filter in reducing glare in light reflected from the surface of water. Part (b) of this figure was taken with a polarizing filter and part (a) was taken without. As a result, the reflection of clouds and sky observed in part (a) is not observed in part (b). Polarizing sunglasses are particularly useful on snow and water. WATCH PHYSICS Polarization of Light, Linear and Circular This video explains the polarization of light in great detail. Before viewing the video, look back at the drawing of an electromagnetic wave from the previous section. Try to visualize the two-dimensional drawing in three dimensions. Click to view content (https://www.openstax.org/l/28Polarization) GRASP CHECK How do polarized glasses reduce glare reflected from the ocean? a. They block horizontally polarized and vertically polarized light. b. They are transparent to horizontally polarized and vertically polarized light. c. They block horizontally polarized rays and are transparent to vertically polarized rays. d. They are transparent to horizontally polarized light and block vertically polarized light. Snap Lab Polarized Glasses • EYE SAFETY—Looking at the Sun directly can cause permanent eye damage. Avoid looking directly at the Sun. • • two pairs of polarized sunglasses OR two lenses from one pair of polarized sunglasses Procedure 1. Look through both or either polarized lens and record your observations. 2. Hold the lenses, one in front of the other. Hold one lens stationary while you slowly rotate the other lens. Record your observations, including the relative angles of the lenses when you make each observation. 3. Find a reflective surface on which the Sun is shining. It could be water, glass, a mirror, or any other similar smooth surface. The results will be more dramatic if the sunlight strikes the surface at a sharp angle. 4. Observe the appearance of the surface with your naked eye and through one of the polarized lenses. 5. Observe any changes as you slowly rotate the lens, and note the angles at which you see changes. 466 Chapter 15 • Light GRAS
P CHECK If you buy sunglasses in a store, how can you be sure that they are polarized? a. When one pair of sunglasses is placed in front of another and rotated in the plane of the body, the light passing through the sunglasses will be blocked at two positions due to refraction of light. b. When one pair of sunglasses is placed in front of another and rotated in the plane of the body, the light passing through the sunglasses will be blocked at two positions due to reflection of light. c. When one pair of sunglasses is placed in front of another and rotated in the plane of the body, the light passing through the sunglasses will be blocked at two positions due to the polarization of light. d. When one pair of sunglasses is placed in front of another and rotated in the plane of the body, the light passing through the sunglasses will be blocked at two positions due to the bending of light waves. Quantitative Treatment of Electromagnetic Waves We can use the speed of light, c, to carry out several simple but interesting calculations. If we know the distance to a celestial object, we can calculate how long it takes its light to reach us. Of course, we can also make the reverse calculation if we know the time it takes for the light to travel to us. For an object at a very great distance from Earth, it takes many years for its light to reach us. This means that we are looking at the object as it existed in the distant past. The object may, in fact, no longer exist. Very large distances in the universe are measured in light years. One light year is the distance that light travels in one year, which is kilometers or miles (…and 1012 is a trillion!). A useful equation involving cis where fis frequency in Hz, and is wavelength in meters. WORKED EXAMPLE Frequency and Wavelength Calculation For example, you can calculate the frequency of yellow light with a wavelength of STRATEGY Rearrange the equation to solve for frequency. Solution Substitute the values for the speed of light and wavelength into the equation. m. 15.2 15.1 15.3 Discussion Manipulating exponents of 10 in a fraction can be tricky. Be sure you keep track of the + and – exponents correctly. Checking back to the diagram of the electromagnetic spectrum in the previous section shows that 1014 is a reasonable order of magnitude for the frequency of yellow light. The frequency of a wave is proportional to the energy the wave carries. The actual proportionality constant will
be discussed in a later chapter. Since frequency is inversely proportional to wavelength, we also know that wavelength is inversely proportional to energy. Keep these relationships in mind as general rules. The rate at which light is radiated from a source is called luminous flux, P, and it is measured in lumens (lm). Energy-saving light bulbs, which provide more luminous flux for a given use of electricity, are now available. One of these bulbs is called a compact fluorescent lamp; another is an LED(light-emitting diode) bulb. If you wanted to replace an old incandescent bulb with an energy saving bulb, you would want the new bulb to have the same brightness as the old one. To compare bulbs accurately, you would need to compare the lumens each one puts out. Comparing wattage—that is, the electric power used—would be Access for free at openstax.org. 15.2 • The Behavior of Electromagnetic Radiation 467 misleading. Both wattage and lumens are stated on the packaging. The luminous flux of a bulb might be 2,000 lm. That accounts for all the light radiated in all directions. However, what we really need to know is how much light falls on an object, such as a book, at a specific distance. The number of lumens per square meter is called illuminance, and is given in units of lux (lx). Picture a light bulb in the middle of a sphere with a 1-m radius. The total surface of the sphere equals 4πr2 m2. The illuminance then is given by What happens if the radius of the sphere is increased 2 m? The illuminance is now only one-fourth as great, because the r2 term in the denominator is 4 instead of 1. Figure 15.12 shows how illuminance decreases with the inverse square of the distance. 15.4 Figure 15.12 The diagram shows why the illuminance varies inversely with the square of the distance from a source of light. WORKED EXAMPLE Calculating Illuminance A woman puts a new bulb in a floor lamp beside an easy chair. If the luminous flux of the bulb is rated at 2,000 lm, what is the illuminance on a book held 2.00 m from the bulb? STRATEGY Choose the equation and list the knowns. Equation: P= 2,000 lm π = 3
.14 r= 2.00 m Solution Substitute the known values into the equation. Discussion Try some other distances to illustrate how greatly light fades with distance from its source. For example, at 3 m the illuminance is only 17.7 lux. Parents often scold children for reading in light that is too dim. Instead of shouting, “You’ll ruin your eyes!” it might be better to explain the inverse square law of illuminance to the child. 468 Chapter 15 • Light Practice Problems 6. Red light has a wavelength of 7.0 × 10−7 m and a frequency of 4.3 × 1014 Hz. Use these values to calculate the speed of light in a vacuum. a. b. c. d. 3 × 1020 m/s 3 × 1015 m/s 3 × 1014 m/s 3 × 108 m/s 7. A light bulb has a luminous flux of 942 lumens. What is the illuminance on a surface from the bulb when it is lit? a. b. c. d. Check Your Understanding 8. Give an example of a place where light travels at the speed of 3.00 × 108 m/s. a. outer space b. water c. Earth’s atmosphere d. quartz glass 9. Explain in terms of distances and the speed of light why it is currently very unlikely that humans will visit planets that circle stars other than our Sun. a. The spacecrafts used for travel are very heavy and thus very slow. b. Spacecrafts do not have a constant source of energy to run them. c. If a spacecraft could attain a maximum speed equal to that of light, it would still be too slow to cover astronomical distances. d. Spacecrafts can attain a maximum speed equal to that of light, but it is difficult to locate planets around stars. Access for free at openstax.org. Chapter 15 • Key Terms 469 KEY TERMS electric field a field that tells us the force per unit charge at all locations in space around a charge distribution electromagnetic radiation (EMR) radiant energy that consists of oscillating electric and magnetic fields lux unit of measure for illuminance magnetic field the directional lines around a magnetic material that indicates the direction and magnitude of the magnetic force illuminance number of lumens per square meter, given in Maxwell’s equations equations that describe the units of lux (lx) interference increased or decreased light intensity caused by the phase
differences between waves lumens unit of measure for luminous flux luminous flux rate at which light is radiated from a source interrelationship between electric and magnetic fields, and how these fields combine to form electromagnetic radiation polarized light light whose electric field component vibrates in a specific plane SECTION SUMMARY 15.1 The Electromagnetic Spectrum • The electromagnetic spectrum is made up of a broad range of frequencies of electromagnetic radiation. • All frequencies of EM radiation travel at the same speed in a vacuum and consist of an electric field and a magnetic field. The types of EM radiation have different frequencies and wavelengths, and different energies and penetrating ability. 15.2 The Behavior of Electromagnetic Radiation • EM radiation travels at different speeds in different media, produces colors on thin films, and can be polarized to oscillate in only one direction. • Calculations can be based on the relationship among the speed, frequency, and wavelength of light, and on the relationship among luminous flux, illuminance, and distance. KEY EQUATIONS 15.2 The Behavior of Electromagnetic Radiation frequency and wavelength illuminance CHAPTER REVIEW Concept Items 15.1 The Electromagnetic Spectrum 1. Use the concepts on which Maxwell’s equations are based to explain why a compass needle is deflected when the compass is brought near a wire that is carrying an electric current. a. The charges in the compass needle and the charges in the electric current have interacting electric fields, causing the needle to deflect. b. The electric field from the moving charges in the current interacts with the magnetic field of the compass needle, causing the needle to deflect. c. The magnetic field from the moving charges in the current interacts with the electric field of the compass needle, causing the needle to deflect. d. The moving charges in the current produce a magnetic field that interacts with the compass needle’s magnetic field, causing the needle to deflect. 2. Consider these colors of light: yellow, blue, and red. Part A. Put these light waves in order according to wavelength, from shortest wavelength to longest wavelength. Part B. Put these light waves in order according to frequency, from lowest frequency to highest frequency. a. wavelength: blue, yellow, red frequency: blue, yellow, red b. wavelength: red, yellow, blue frequency: red, yellow, blue c. wavelength: red, yellow, blue frequency: blue, yellow, red d. wavelength: blue, yellow, red frequency: red, yellow, blue 3. Describe the location
of gamma rays on the electromagnetic spectrum. 470 Chapter 15 • Chapter Review a. At the high-frequency and long-wavelength end of the spectrum b. At the high-frequency and short-wavelength end of thickness of the wall of a soap bubble? Explain your answer. a. The thickness of the bubble wall is ten times that of the spectrum the wavelength of light. c. At the low-frequency and long-wavelength end of b. The thickness of the bubble wall is similar to that of the spectrum the wavelength of light. d. At the low-frequency and short-wavelength end of c. The thickness of the bubble wall is half the the spectrum wavelength of light. 4. In which region of the electromagnetic spectrum would you find radiation that is invisible to the human eye and has low energy? a. Long-wavelength and high-frequency region b. Long-wavelength and low-frequency region c. Short-wavelength and high-frequency region d. Short-wavelength and low-frequency region 15.2 The Behavior of Electromagnetic Radiation 5. Light travels at different speeds in different media. Put these media in order, from the slowest light speed to the fastest light speed: air, diamond, vacuum, water. a. diamond, water, air, vacuum b. vacuum, diamond, air, water c. diamond, air, water, vacuum d. air, diamond, water, vacuum 6. Visible light has wavelengths in the range of about 400 to 800 nm. What does this indicate about the approximate Critical Thinking Items 15.1 The Electromagnetic Spectrum 8. Standing in front of a fire, we can sense both its heat and its light. How are the light and heat radiated by the fire the same, and how are they different? a. Both travel as waves, but only light waves are a form of electromagnetic radiation. b. Heat and light are both forms of electromagnetic radiation, but light waves have higher frequencies. c. Heat and light are both forms of electromagnetic radiation, but heat waves have higher frequencies. d. Heat and light are both forms of electromagnetic radiation, but light waves have higher wavelengths. 9. Light shines on a picture of the subtractive color wheel. The light is a mixture of red, blue, and green light. Part A—Which part of the color wheel will look blue? Explain in terms of absorbed and reflected light. Part B—Which part of the color wheel will look yellow? Explain in terms of
absorbed and reflected light. a. A. The yellow section of the wheel will look blue because it will reflect blue light and absorb red Access for free at openstax.org. d. The thickness of the bubble wall equals the cube of the wavelength of light. 7. Bright sunlight is reflected from an icy pond. You look at the glare of the reflected light through polarized glasses. When you take the glasses off, rotate them 90°, and look through one of the lenses again, the light you see becomes brighter. Explain why the light you see changes. a. The glass blocks horizontally polarized light, and the light reflected from the icy pond is, in part, polarized horizontally. b. The glass blocks vertically polarized light, and the light reflected from the icy pond is, in part, polarized vertically. c. The glass allows horizontally polarized light to pass, and the light reflected from the icy pond is, in part, polarized vertically. d. The glass allows horizontally polarized light to pass, and the light reflected from the icy pond is, in part, polarized horizontally. and green. B. The blue section of the wheel will look yellow because it will reflect red and green light and absorb blue. b. A. The blue section of the wheel will look blue because it will absorb blue light and reflect red and green. B. The yellow section of the wheel will look yellow because it will absorb red and green light and reflect blue. c. A. The yellow section of the wheel will look blue because it will absorb blue light and reflect red and green. B. The blue section of the wheel will look yellow because it will absorb red and green light and reflect blue. d. A. The blue section of the wheel will look blue because it will reflect blue light and absorb red and green. B. The yellow section of the wheel will look yellow because it will reflect red and green light and absorb blue. 10. Part A. When you stand in front of an open fire, you can sense light waves with your eyes. You sense another type of electromagnetic radiation as heat. What is this other type of radiation? Part B. How is this other type of radiation different front light waves? a. A. X-rays B. The X-rays have higher frequencies and shorter wavelengths than the light waves. b. A. X-rays B. The X-rays have lower frequencies and longer wavelengths than the light waves. c. A. infrared rays B. The infrared rays have higher frequencies and shorter wavelengths than
the light waves. d. A. infrared rays B. The infrared rays have lower frequencies and longer wavelengths than the light waves. 11. Overexposure to this range of EM radiation is dangerous, and yet it is used by doctors to diagnose medical problems. Part A—Identify the type of radiation. Part B—Locate the position of this radiation on the EM spectrum by comparing its frequency and wavelength to visible light. Part C—Explain why this radiation is both dangerous and therapeutic in terms of its energy, based on your answer to Part B. a. A. X-rays B. X-rays have shorter wavelengths (1 × 10–8 – 5 × 10–12 m) and higher frequencies (3 × 1016 – 6 × 1019 Hz) than visible light (7.5 × 10–7 – 4.0 × 10–7 m; 4.0 × 1014 – 7.5 × 1014 Hz). C. X-rays have low energies because of their high frequencies, and so can penetrate matter to greater depths. b. A. X-rays B. X-rays have shorter wavelengths (1 × 10–8 – 5 × 10–12 m) and higher frequencies (3 × 1010 – 6 × 1013 Hz) than visible light (7.5 × 10–7 – 4.0 × 10–7 m; 4.0 × 1014 – 7.5 × 1014 Hz). C. X-rays have low energies because of their low frequencies, and so can penetrate matter to greater depths. c. A. X-rays B. X-rays have longer wavelengths (1 × 10–6 – 5 × 10–7 m) and higher frequencies (3 × 1015 – 6 × 1015 Hz) than visible light (7.5 × 10–7 – 4.0 × 10–7 m; 4.0 × 1014 – 7.5 × 1014 Hz). C. X-rays have high energies because of their high Chapter 15 • Chapter Review 471 frequencies, and therefore can penetrate matter to greater depths. d. A. X-rays B. X-rays have shorter wavelengths (1 × 10–8 – 5 × 10–12 m) and higher frequencies (3 × 1016 – 6 × 1019 Hz) than visible light (7.5 × 10–7 – 4.0 × 10–7 m; 4.0 × 1014 – 7.5 × 1014 Hz). C
. X-rays have high energies because of their high frequencies, and so can penetrate matter to greater depths. 15.2 The Behavior of Electromagnetic Radiation 12. Explain how thin-film interference occurs. Discuss in terms of the meaning of interference and the pathways of light waves. a. For a particular thickness of film, light of a given wavelength that reflects from the outer and inner film surfaces is completely in phase, and so undergoes constructive interference. b. For a particular thickness of film, light of a given wavelength that reflects from the outer and inner surfaces is completely in phase, and so undergoes destructive interference. c. For a particular thickness of film, light of a given wavelength that reflects from the outer and inner film surfaces is completely out of phase, and so undergoes constructive interference. d. For a particular thickness of film, light of a given wavelength that reflects from the outer and inner film surfaces is completely out of phase, and so undergoes no interference. 13. When you move a rope up and down, waves are created. If the waves pass through a slot, they will be affected differently, depending on the orientation of the slot. Using the rope waves and the slot as a model, explain how polarizing glasses affect light waves. a. If the wave—electric field—is vertical and slit—polarizing molecules in the glass—is horizontal, the wave will pass. If the wave—electric field— is vertical and slit—polarizing molecules in the glass—is vertical, the wave will not pass. If the wave—electric field—is horizontal and slit—polarizing molecules in the glass—is horizontal, the wave will pass. If the wave—electric field—is horizontal and slit—polarizing molecules in the glass—is horizontal, the wave will not pass. b. c. d. 472 Chapter 15 • Test Prep Problems 15.2 The Behavior of Electromagnetic Radiation 14. Visible light has a range of wavelengths from about 400 nm to 800 nm. What is the range of frequencies for visible light? a. b. c. d. 3.75 × 106 Hz to 7.50 × 106 Hz 3.75 Hz to 7.50 Hz 3.75 × 10−7 Hz to 7.50 × 10−7 Hz 3.75 × 1014 Hz to 7.50 × 1014 Hz Performance Task 15.2 The Behavior of Electromagnetic Radiation 16. Design an experiment to observe the phenomenon of thin-film interference
. Observe colors of visible light, and relate each color to its corresponding wavelength. Comparison with the magnitudes of visible light wavelength will give an appreciation of just how very thin a thin film is. Thin-film interference has a number of practical applications, such as anti-reflection coatings and optical filters. Thin films used in filters can be designed to reflect or transmit specific wavelengths of light. This is done by depositing a film one molecular layer at a time from a vapor, thus allowing the thickness of the film to be exactly controlled. • EYE SAFETY—Chemicals in this lab are poisonous if ingested. If chemicals are ingested, inform your teacher immediately. • FUMES—Certain chemicals or chemical reactions in this lab create a vapor that is harmful if inhaled. Follow your teacher's instructions for the use of fume hoods and other safety apparatus designed to prevent fume inhalation. Never smell or otherwise breath in any chemicals or vapors in the lab. • FLAMMABLE—Chemicals in this lab are highly flammable and can ignite, especially if exposed to a spark or open flame. Follow your teacher's TEST PREP Multiple Choice 15.1 The Electromagnetic Spectrum 17. Which type of EM radiation has the shortest wavelengths? a. gamma rays b. c. blue light d. microwaves infrared waves Access for free at openstax.org. 15. Light travels through the wall of a soap bubble that is 600 nm thick and is reflected from the inner surface back into the air. Assume the bubble wall is mostly water and that light travels in water at 75 percent of the speed of light in vacuum. How many seconds behind will the light reflected from the inner surface arrive compared to the light that was reflected from the outer surface? a. 4.0 × 10–8 s 5.3 × 10–6 s b. c. 2.65 × 10–15 s 5.3 × 10–15 s d. instructions carefully on how to handle flammable chemicals. Do not expose any chemical to a flame or other heat source unless specifically instructed by your teacher. • HAND WASHING—Some materials may be hazardous if in extended contact with the skin. Be sure to wash your hands with soap after handling and disposing of these materials during the lab. • WASTE—Some things in this lab are hazardous and need to be disposed of properly. Follow your teacher's instructions for disposal of all items. • A large flat tray with raised sides
, such as a baking tray • Small volumes of motor oil, lighter fluid or a penetrating oil of the type used to loosen rusty bolts, and cooking oil • Water • A camera a. Thin-film interference causes colors to appear on the surface of a thin transparent layer. Do you expect to see a pattern to the colors? b. How could you make a permanent record of your observations? c. What data would you need to look up to help explain any patterns that you see? d. What could explain colors failing to appear under some conditions? 18. Which form of EM radiation has the most penetrating ability? a. red light b. microwaves c. gamma rays d. infrared radiation 19. Why are high-frequency gamma rays more dangerous to humans than visible light? a. Gamma rays have a lower frequency range than visible light. 22. What is the wavelength of red light with a frequency of Chapter 15 • Test Prep 473 b. Gamma rays have a longer wavelength range than visible light. c. Gamma rays have greater energy than visible light for penetrating matter. d. Gamma rays have less energy than visible light for penetrating matter. 20. A dog would have a hard time stalking and catching a red bird hiding in a field of green grass. Explain this in terms of cone cells and color perception. a. Dogs are red-green color-blind because they can see only blue and yellow through two kinds of cone cells present in their eyes. b. Dogs are only red color-blind because they can see only blue and yellow through two kinds of cones cells present in their eyes. c. Dogs are only green color-blind because they can see only blue and yellow through two kinds of cones cells present in their eyes. d. Dogs are color-blind because they have only rods and no cone cells present in their eyes. 15.2 The Behavior of Electromagnetic Radiation 21. To compare the brightness of light bulbs for sale in a frequency store, you should look on the labels to see how they are rated in terms of ____. a. b. watts c. amps d. lumens 4.00 × 1014 Hz? a. 2.50 × 1014 m b. 4.00 × 1015 m c. 2.50 × 106 m d. 4.00 × 10-7 m 23. What is the distance of one light year in kilometers? a. 2.59 × 1010 km 1.58 × 1011 km b. c. 2.63 × 109 km
d. 9.46 × 1012 km 24. How does the illuminance of light change when the distance from the light source is tripled? Cite the relevant equation and explain how it supports your answer. a. if distance is tripled, then the illuminance increases by 19 times. b. c. d. if distance is tripled, then the illuminance decreases by 13 times. then the illuminance decreases by 9 times. if distance is tripled, if distance tripled, then the illuminance increases by 3 times. 25. A light bulb has an illuminance of 19.9 lx at a distance of 2 m. What is the luminous flux of the bulb? 500 lm a. b. 320 lm c. 250 lm d. 1,000 lm Short Answer 15.1 The Electromagnetic Spectrum 26. Describe one way in which heat waves—infrared radiation—are different from sound waves. a. Sound waves are transverse waves, whereas heat waves—infrared radiation—are longitudinal waves. b. Sound waves have shorter wavelengths than heat waves. c. Sound waves require a medium, whereas heat waves—infrared radiation—do not. d. Sound waves have higher frequencies than heat waves. 27. Describe the electric and magnetic fields that make up an electromagnetic wave in terms of their orientation relative to each other and their phases. a. They are perpendicular to and out of phase with each other. b. They are perpendicular to and in phase with each other. c. They are parallel to and out of phase with each other. d. They are parallel to and in phase with each other. 28. Explain how X-radiation can be harmful and how it can be a useful diagnostic tool. a. Overexposure to X-rays can cause HIV, though normal levels of X-rays can be used for sterilizing needles. b. Overexposure to X-rays can cause cancer, though in limited doses X-rays can be used for imaging internal body parts. c. Overexposure to X-rays causes diabetes, though normal levels of X-rays can be used for imaging internal body parts. d. Overexposure to X-rays causes cancer, though normal levels of X-rays can be used for reducing 474 Chapter 15 • Test Prep cholesterol in the blood. 31. What is it about the nature of light reflected from snow 29. Explain how sunlight is the original source of the energy
in the food we eat. a. Sunlight is converted into chemical energy by plants; this energy is released when we digest food. b. Sunlight is converted into chemical energy by animals; this energy is released when we digest food. c. Sunlight is converted into chemical energy by fish; this energy is released when we digest food. that causes skiers to wear polarized sunglasses? a. The reflected light is polarized in the vertical direction. b. The reflected light is polarized in the horizontal direction. c. The reflected light has less intensity than the incident light. d. The reflected light has triple the intensity of the incident light. d. Sunlight is converted into chemical energy by 32. How many lumens are radiated from a candle which has humans; this energy is released when we digest food. 15.2 The Behavior of Electromagnetic Radiation 30. Describe what happens to the path of light when the light slows down as it passes from one medium to another? a. The path of the light remains the same. b. The path of the light becomes circular. c. The path of the light becomes curved. d. The path of the light changes. an illuminance of 3.98 lx at a distance of 2.00 m? a. 400 lm 100 lm b. c. 50 lm d. 200 lm 33. Saturn is 1.43×1012 m from the Sun. How many minutes does it take the Sun’s light to reach Saturn? a. b. c. d. 7.94 × 109 minutes 3.4 × 104 minutes 3.4 × 10–6 minutes 79.4 minutes Extended Response C. ultraviolet radiation 15.1 The Electromagnetic Spectrum 35. A mixture of red and green light is shone on each of the 34. A frequency of red light has a wavelength of 700 nm. Part A—Compare the wavelength and frequency of violet light to red light. Part B—Identify a type of radiation that has lower frequencies than red light. Part C—Identify a type of radiation that has shorter wavelengths than violet light. a. A. Violet light has a lower frequency and longer wavelength than red light. B. ultraviolet radiation infrared radiation C. b. A. Violet light has a lower frequency and longer wavelength than red light. B. infrared radiation C. ultraviolet radiation c. A. Violet light has a higher frequency and shorter wavelength than red light. B. ultraviolet radiation infrared radiation C. d. A
. Violet light has a higher frequency and shorter wavelength than red light. infrared radiation B. Access for free at openstax.org. subtractive colors. Part A—Which of these colors of light are reflected from magenta? Part B—Which of these colors of light are reflected from yellow? Part C—Which these colors of light are reflected from cyan? a. Part A. red and green Part B. green Part C. red b. Part A. red and green Part B. red Part C. green c. Part A. green Part B. red and green Part C. red d. Part A. red Part B. red and green Part C. green 15.2 The Behavior of Electromagnetic Radiation 36. Explain why we see the colorful effects of thin-film interference on the surface of soap bubbles and oil slicks, but not on the surface of a window pane or clear plastic bag. a. The thickness of a window pane or plastic bag is more than the wavelength of light, and interference occurs for thicknesses smaller than the wavelength of light. b. The thickness of a window pane or plastic bag is less than the wavelength of light, and interference occurs for thicknesses similar to the wavelength of light. c. The thickness of a window pane or plastic bag is more than the wavelength of light, and interference occurs for thicknesses similar to the wavelength of light. d. The thickness of a window pane or plastic bag is Chapter 15 • Test Prep 475 less than the wavelength of light, and interference occurs for thicknesses larger than the wavelength of light. 37. The Occupational Safety and Health Administration (OSHA) recommends an illuminance of for desktop lighting. An office space has lighting hung. above desktop level that provides only To what height would the lighting fixtures have to be lowered to provide a. b. c. d. on desktops? 476 Chapter 15 • Test Prep Access for free at openstax.org. CHAPTER 16 Mirrors and Lenses Figure 16.1 Flat, smooth surfaces reflect light to form mirror images. (credit: NASA Goddard Photo and Video, via Flickr) Chapter Outline 16.1 Reflection 16.2 Refraction 16.3 Lenses INTRODUCTION “In another moment Alice was through the glass, and had jumped lightly down into the Looking-glass room.” —Through the Looking Glass by Lewis Carol Through the Looking Glasstells of the adventures of Alice after she steps from the real world, through a mirror
, and into the virtual world. In this chapter we examine the optical meanings of real and virtual, as well as other concepts that make up the field of optics. The light from this page or screen is formed into an image by the lens of your eyes, much as the lens of the camera that made the photograph at the beginning of this chapter. Mirrors, like lenses, can also form images, which in turn are captured by your eyes. Optics is the branch of physics that deals with the behavior of visible light and other electromagnetic waves. For now, we concentrate on the propagation of light and its interaction with matter. It is convenient to divide optics into two major parts based on the size of objects that light encounters. When light interacts with an object that is several times as large as the light’s wavelength, its observable behavior is similar to a ray; it does not display its 478 Chapter 16 • Mirrors and Lenses wave characteristics prominently. We call this part of optics geometric optics. This chapter focuses on situations for which geometric optics is suited. 16.1 Reflection Section Learning Objectives By the end of this section, you will be able to do the following: • Explain reflection from mirrors, describe image formation as a consequence of reflection from mirrors, apply ray diagrams to predict and interpret image and object locations, and describe applications of mirrors • Perform calculations based on the law of reflection and the equations for curved mirrors Section Key Terms angle of incidence angle of reflection central axis concave mirror convex mirror diffused focal length focal point geometric optics law of reflection law of refraction ray real image specular virtual image Characteristics of Mirrors There are three ways, as shown in Figure 16.2, in which light can travel from a source to another location. It can come directly from the source through empty space, such as from the Sun to Earth. Light can travel to an object through various media, such as air and glass. Light can also arrive at an object after being reflected, such as by a mirror. In all these cases, light is modeled as traveling in a straight line, called a ray. Light may change direction when it encounters the surface of a different material (such as a mirror) or when it passes from one material to another (such as when passing from air into glass). It then continues in a straight line—that is, as a ray. The word raycomes from mathematics. Here it means a straight line that originates from some point. It is acceptable to visualize light rays
as laser rays (or even science fiction depictions of ray guns). Figure 16.2 Three methods for light to travel from a source to another location are shown. (a) Light reaches the upper atmosphere of Earth by traveling through empty space directly from the source (the Sun). (b) This light can reach a person in one of two ways. It can travel through a medium, such as air or glass, and typically travels from one medium to another. It can also reflect from an object, such as a mirror. Because light moves in straight lines, that is, as rays, and changes directions when it interacts with matter, it can be described through geometry and trigonometry. This part of optics, described by straight lines and angles, is therefore called geometric optics. There are two laws that govern how light changes direction when it interacts with matter: the law of reflection, for situations in which light bounces off matter; and the law of refraction, for situations in which light passes through matter. In this section, we consider the geometric optics of reflection. Whenever we look into a mirror or squint at sunlight glinting from a lake, we are seeing a reflection. How does the reflected light travel from the object to your eyes? The law of reflection states: The angle of reflection,, equals the angle of incidence, Access for free at openstax.org. 16.1 • Reflection 479.This law governs the behavior of all waves when they interact with a smooth surface, and therefore describe the behavior of light waves as well. The reflection of light is simplified when light is treated as a ray. This concept is illustrated in Figure 16.3, which also shows how the angles are measured relative to the line perpendicular to the surface at the point where the light ray strikes it. This perpendicular line is also called the normal line, or just the normal. Light reflected in this way is referred to as specular (from the Latin word for mirror: speculum). We expect to see reflections from smooth surfaces, but Figure 16.4, illustrates how a rough surface reflects light. Because the light is reflected from different parts of the surface at different angles, the rays go in many different directions, so the reflected light is diffused. Diffused light allows you to read a printed page from almost any angle because some of the rays go in different directions. Many objects, such as people, clothing, leaves, and walls, have rough surfaces and can be seen from many angles. A mirror, on the other
hand, has a smooth surface and reflects light at specific angles. Figure 16.3 The law of reflection states that the angle of reflection, θr, equals the angle of incidence, θi. The angles are measured relative to the line perpendicular to the surface at the point where the ray strikes the surface. The incident and reflected rays, along with the normal, lie in the same plane. Figure 16.4 Light is diffused when it reflects from a rough surface. Here, many parallel rays are incident, but they are reflected at many different angles because the surface is rough. When we see ourselves in a mirror, it appears that our image is actually behind the mirror. We see the light coming from a direction determined by the law of reflection. The angles are such that our image is exactly the same distance behind the mirror, di, as the distance we stand away from the mirror, do. Although these mirror images make objects appear to be where they cannot be (such as behind a solid wall), the images are not figments of our imagination. Mirror images can be photographed and videotaped by instruments and look just as they do to our eyes, which are themselves optical instruments. An image in a mirror is said to be a virtual image, as opposed to a real image. A virtual image is formed when light rays appear to diverge from a point without actually doing so. Figure 16.5 helps illustrate how a flat mirror forms an image. Two rays are shown emerging from the same point, striking the mirror, and reflecting into the observer’s eye. The rays can diverge slightly, and both still enter the eye. If the rays are extrapolated backward, they seem to originate from a common point behind the mirror, allowing us to locate the image. The paths of the reflected rays into the eye are the same as if they had come directly from that point behind the mirror. Using the law of reflection—the angle of reflection equals the angle of incidence—we can see that the image and object are the same distance from the mirror. This is a virtual image, as defined earlier. 480 Chapter 16 • Mirrors and Lenses Figure 16.5 When two sets of rays from common points on an object are reflected by a flat mirror into the eye of an observer, the reflected rays seem to originate from behind the mirror, which determines the position of the virtual image. FUN IN PHYSICS Mirror Mazes Figure 16.6 is a chase scene from an old silent film called The Circus, starring Charlie Cha
plin. The chase scene takes place in a mirror maze. You may have seen such a maze at an amusement park or carnival. Finding your way through the maze can be very difficult. Keep in mind that only one image in the picture is real—the others are virtual. Figure 16.6 Charlie Chaplin is in a mirror maze. Which image is real? One of the earliest uses of mirrors for creating the illusion of space is seen in the Palace of Versailles, the former home of French royalty. Construction of the Hall of Mirrors (Figure 16.7) began in 1678. It is still one of the most popular tourist attractions at Versailles. Figure 16.7 Tourists love to wander in the Hall of Mirrors at the Palace of Versailles. (credit: Michal Osmenda, Flickr) GRASP CHECK Only one Charlie in this image (Figure 16.8) is real. The others are all virtual images of him. Can you tell which is real? Hint—His hat is tilted to one side. Access for free at openstax.org. 16.1 • Reflection 481 Figure 16.8 a. The virtual images have their hats tilted to the right. b. The virtual images have their hats tilted to the left. c. The real images have their hats tilted to the right. d. The real images have their hats tilted to the left. WATCH PHYSICS Virtual Image This video explains the creation of virtual images in a mirror. It shows the location and orientation of the images using ray diagrams, and relates the perception to the human eye. Click to view content (https://openstax.org/l/28Virtualimage) Compare the distance of an object from a mirror to the apparent distance of its virtual image behind the mirror. a. The distances of the image and the object from the mirror are the same. b. The distances of the image and the object from the mirror are always different. c. The image is formed at infinity if the object is placed near the mirror. d. The image is formed near the mirror if the object is placed at infinity. Some mirrors are curved instead of flat. A mirror that curves inward is called a concave mirror, whereas one that curves outward is called a convex mirror. Pick up a well-polished metal spoon and you can see an example of each type of curvature. The side of the spoon that holds the food is a concave mirror; the back
of the spoon is a convex mirror. Observe your image on both sides of the spoon. TIPS FOR SUCCESS You can remember the difference between concave and convex by thinking, Concave means caved in. Ray diagrams can be used to find the point where reflected rays converge or appear to converge, or the point from which rays appear to diverge. This is called the focal point, F. The distance from F to the mirror along the central axis (the line perpendicular to the center of the mirror’s surface) is called the focal length, f. Figure 16.9 shows the focal points of concave and convex mirrors. 482 Chapter 16 • Mirrors and Lenses Figure 16.9 (a, b) The focal length for the concave mirror in (a), formed by converging rays, is in front of the mirror, and has a positive value. The focal length for the convex mirror in (b), formed by diverging rays, appears to be behind the mirror, and has a negative value. Images formed by a concave mirror vary, depending on which side of the focal point the object is placed. For any object placed on the far side of the focal point with respect to the mirror, the rays converge in front of the mirror to form a real image, which can be projected onto a surface, such as a screen or sheet of paper However, for an object located inside the focal point with respect to the concave mirror, the image is virtual. For a convex mirror the image is always virtual—that is, it appears to be behind the mirror. The ray diagrams in Figure 16.10 show how to determine the nature of the image formed by concave and convex mirrors. Figure 16.10 (a) The image of an object placed outside the focal point of a concave mirror is inverted and real. (b) The image of an object Access for free at openstax.org. placed inside the focal point of a concave mirror is erect and virtual. (c) The image of an object formed by a convex mirror is erect and virtual. The information in Figure 16.10 is summarized in Table 16.1. Type of Mirror Object to Mirror Distance, do Image Characteristics 16.1 • Reflection 483 Concave Concave Convex Real and inverted Virtual and erect Virtual and erect Table 16.1 Curved Mirror Images This table details the type and orientation of images formed by concave and convex
mirrors. Snap Lab Concave and Convex Mirrors • Silver spoon and silver polish, or a new spoon made of any shiny metal Instructions Procedure 1. Choose any small object with a top and a bottom, such as a short nail or tack, or a coin, such as a quarter. Observe the object’s reflection on the back of the spoon. 2. Observe the reflection of the object on the front (bowl side) of the spoon when held away from the spoon at a distance of several inches. 3. Observe the image while slowly moving the small object toward the bowl of the spoon. Continue until the object is all the way inside the bowl of the spoon. 4. You should see one point where the object disappears and then reappears. This is the focal point. WATCH PHYSICS Parabolic Mirrors and Real Images This video uses ray diagrams to show the special feature of parabolic mirrors that makes them ideal for either projecting light energy in parallel rays, with the source being at the focal point of the parabola, or for collecting at the focal point light energy from a distant source. Click to view content (https://www.openstax.org/l/28Parabolic) Explain why using a parabolic mirror for a car headlight throws much more light on the highway than a flat mirror. a. The rays do not polarize after reflection. b. The rays are dispersed after reflection. c. The rays are polarized after reflection. d. The rays become parallel after reflection. You should be able to notice everyday applications of curved mirrors. One common example is the use of security mirrors in stores, as shown in Figure 16.11. 484 Chapter 16 • Mirrors and Lenses Figure 16.11 Security mirrors are convex, producing a smaller, upright image. Because the image is smaller, a larger area is imaged compared with what would be observed for a flat mirror; hence, security is improved. (credit: Laura D’Alessandro, Flickr) Some telescopes also use curved mirrors and no lenses (except in the eyepieces) both to magnify images and to change the path of light. Figure 16.12 shows a Schmidt-Cassegrain telescope. This design uses a spherical primary concave mirror and a convex secondary mirror. The image is projected onto the focal plane by light passing through the perforated primary mirror. The effective focal length of such a telescope is the focal length of the primary mirror multiplied by
the magnification of the secondary mirror. The result is a telescope with a focal length much greater than the length of the telescope itself. Figure 16.12 This diagram shows the design of a Schmidt–Cassegrain telescope. A parabolic concave mirror has the very useful property that all light from a distant source, on reflection by the mirror surface, is directed to the focal point. Likewise, a light source placed at the focal point directs all the light it emits in parallel lines away from the mirror. This case is illustrated by the ray diagram in Figure 16.13. The light source in a car headlight, for example, is located at the focal point of a parabolic mirror. Figure 16.13 The bulb in this ray diagram of a car headlight is located at the focal point of a parabolic mirror. Parabolic mirrors are also used to collect sunlight and direct it to a focal point, where it is transformed into heat, which in turn can be used to generate electricity. This application is shown in Figure 16.14. Figure 16.14 Parabolic trough collectors are used to generate electricity in southern California. (credit: kjkolb, Wikimedia Commons) Access for free at openstax.org. 16.1 • Reflection 485 Using a concave mirror, you look at the reflection of a faraway object. The image size changes if you move the object closer to the mirror. Why does the image disappear entirely when the object is at the mirror's focal point? a. The height of the image became infinite. b. The height of the object became zero. c. The intensity of intersecting light rays became zero. d. The intensity of intersecting light rays increased. The Application of the Curved Mirror Equations Curved mirrors and the images they create involve a fairly small number of variables: the mirror’s radius of curvature, R; the focal length, f; the distances of the object and image from the mirror, doand di, respectively; and the heights of the object and image, hoand hi, respectively. The signs of these values indicate whether the image is inverted, erect (upright), real, or virtual. We now look at the equations that relate these variables and apply them to everyday problems. Figure 16.15 shows the meanings of most of the variables we will use for calculations involving curved mirrors. The basic equation that describes both lenses and mirrors is the lens/mirror equation Figure 16.15 Look for the variables, do, di
, ho, hi,and fin this figure. This equation can be rearranged several ways. For example, it may be written to solve for focal length. Magnification, m, is the ratio of the size of the image, hi, to the size of the object, ho. The value of mcan be calculated in two ways. This relationship can be written to solve for any of the variables involved. For example, the height of the image is given by We saved the simplest equation for last. The radius of curvature of a curved mirror, R, is simply twice the focal length. We can learn important information from the algebraic sign of the result of a calculation using the previous equations: • A negative diindicates a virtual image; a positive value indicates a real image • A negative hiindicates an inverted image; a positive value indicates an erect image • For concave mirrors, fis positive; for convex mirrors, fis negative Now let’s apply these equations to solve some problems. 486 Chapter 16 • Mirrors and Lenses WORKED EXAMPLE Calculating Focal Length A person standing 6.0 m from a convex security mirror forms a virtual image that appears to be 1.0 m behind the mirror. What is the focal length of the mirror? STRATEGY The person is the object, so do= 6.0 m. We know that, for this situation, dois positive. The image is virtual, so the value for the image distance is negative, so di= –1.0 m. Now, use the appropriate version of the lens/mirror equation to solve for focal length by substituting the known values. Solution Discussion The negative result is expected for a convex mirror. This indicates the focal point is behind the mirror. WORKED EXAMPLE Calculating Object Distance Electric room heaters use a concave mirror to reflect infrared (IR) radiation from hot coils. Note that IR radiation follows the same law of reflection as visible light. Given that the mirror has a radius of curvature of 50.0 cm and produces an image of the coils 3.00 m in front of the mirror, where are the coils with respect to the mirror? STRATEGY We are told that the concave mirror projects a real image of the coils at an image distance di= 3.00 m. The coils are the object, and we are asked to find their location—that is, to find the object distance do. We are also given the radius
of curvature of the mirror, so that its focal length is f= R/2 = 25.0 cm (a positive value, because the mirror is concave, or converging). We can use the lens/mirror equation to solve this problem. Solution Because diand fare known, the lens/mirror equation can be used to find do. Rearranging to solve for do, we have Entering the known quantities gives us 16.1 16.2 16.3 Discussion Note that the object (the coil filament) is farther from the mirror than the mirror’s focal length. This is a case 1image (do > fand f positive), consistent with the fact that a real image is formed. You get the most concentrated thermal energy directly in front of the mirror and 3.00 m away from it. In general, this is not desirable because it could cause burns. Usually, you want the rays to emerge parallel, and this is accomplished by having the filament at the focal point of the mirror. Note that the filament here is not much farther from the mirror than the focal length, and that the image produced is considerably farther away. Access for free at openstax.org. 16.2 • Refraction 487 Practice Problems 1. A concave mirror has a radius of curvature of. What is the focal length of the mirror? a. b. c. d. 2. What is the focal length of a makeup mirror that produces a magnification of 1.50 when a person’s face is 12.0 cm away? Construct a ray diagram using paper, a pencil and a ruler to confirm your calculation. a. –36.0 cm b. –7.20 cm c. d. 7.20 cm 36.0 cm Check Your Understanding 3. How does the object distance, do, compare with the focal length, f, for a concave mirror that produces an image that is real and inverted? a. do > f, where do and f are object distance and focal length, respectively. b. do < f, where do and f are object distance and focal length, respectively. c. do = f, where do and f are object distance and focal length, respectively. d. do = 0, where do is the object distance. 4. Use the law of reflection to explain why it is not a good idea to polish a mirror with sandpaper. a. The surface becomes smooth, and a smooth surface produces a sharp image. b. The
surface becomes irregular, and an irregular surface produces a sharp image. c. The surface becomes smooth, and a smooth surface transmits light, but does not reflect it. d. The surface becomes irregular, and an irregular surface produces a blurred image. 5. An object is placed in front of a concave mirror at a distance that is greater than the focal length of the mirror. Will the image produced by the mirror be real or virtual? Will it be erect or inverted? a. b. c. d. It is real and erect. It is real and inverted. It is virtual and inverted. It is virtual and erect. 16.2 Refraction Section Learning Objectives By the end of this section, you will be able to do the following: • Explain refraction at media boundaries, predict the path of light after passing through a boundary (Snell’s law), describe the index of refraction of materials, explain total internal reflection, and describe applications of refraction and total internal reflection • Perform calculations based on the law of refraction, Snell’s law, and the conditions for total internal reflection Section Key Terms angle of refraction corner reflector critical angle dispersion incident ray index of refraction refracted ray Snell’s law total internal reflection The Law of Refraction You may have noticed some odd optical phenomena when looking into a fish tank. For example, you may see the same fish appear to be in two different places (Figure 16.16). This is because light coming to you from the fish changes direction when it 488 Chapter 16 • Mirrors and Lenses leaves the tank and, in this case, light rays traveling along two different paths both reach our eyes. The changing of a light ray’s direction (loosely called bending) when it passes a boundary between materials of different composition, or between layers in single material where there are changes in temperature and density, is called refraction. Refraction is responsible for a tremendous range of optical phenomena, from the action of lenses to voice transmission through optical fibers. Figure 16.16 Looking at the fish tank as shown, we can see the same fish in two different locations, because light changes directions when it passes from water to air. In this case, light rays traveling on two different paths change direction as they travel from water to air, and so reach the observer. Consequently, the fish appears to be in two different places. This bending of light is called refractionand is responsible for many optical phenomena. Why
does light change direction when passing from one material (medium) to another? It is because light changes speed when going from one material to another. This behavior is typical of all waves and is especially easy to apply to light because light waves have very small wavelengths, and so they can be treated as rays. Before we study the law of refraction, it is useful to discuss the speed of light and how it varies between different media. The speed of light is now known to great precision. In fact, the speed of light in a vacuum, c, is so important, and is so precisely known, that it is accepted as one of the basic physical quantities, and has the fixed value where the approximate value of 3.00 matter is less than it is in a vacuum, because light interacts with atoms in a material. The speed of light depends strongly on the type of material, given that its interaction with different atoms, crystal lattices, and other substructures varies. We define the index of refraction, n, of a material to be 108 m/s is used whenever three-digit precision is sufficient. The speed of light through 16.4 where vis the observed speed of light in the material. Because the speed of light is always less than cin matter and equals conly in a vacuum, the index of refraction (plural: indices of refraction) is always greater than or equal to one. Table 16.2 lists the indices of refraction in various common materials. Medium n Gases at 0 °C and 1 atm Table 16.2 Indices of Refraction The table lists the indices of refraction for various materials that are transparent to light. Note, that light travels the slowest in the materials with the greatest indices of refraction. Access for free at openstax.org. 16.2 • Refraction 489 Medium n Air 1.000293 Carbon dioxide 1.00045 Hydrogen Oxygen Liquids at 20 °C Benzene Carbon disulfide 1.000139 1.000271 1.501 1.628 Carbon tetrachloride 1.461 Ethanol Glycerin Water, fresh Solids at 20 °C Diamond Fluorite Glass, crown Glass, flint Ice at 0 °C Plexiglas Polystyrene 1.361 1.473 1.333 2.419 1.434 1.52 1.66 1.309 1.51 1.49 Quartz, crystalline 1.544 Quartz, fused Sodium chloride 1.458 1
.544 Table 16.2 Indices of Refraction The table lists the indices of refraction for various materials that are transparent to light. Note, that light travels the slowest in the materials with the greatest indices of refraction. 490 Chapter 16 • Mirrors and Lenses Medium n Zircon 1.923 Table 16.2 Indices of Refraction The table lists the indices of refraction for various materials that are transparent to light. Note, that light travels the slowest in the materials with the greatest indices of refraction. Figure 16.17 provides an analogy for and a description of how a ray of light changes direction when it passes from one medium to another. As in the previous section, the angles are measured relative to a perpendicular to the surface at the point where the light ray crosses it. The change in direction of the light ray depends on how the speed of light changes. The change in the speed of light is related to the indices of refraction of the media involved. In the situations shown in Figure 16.17, medium 2 has a greater index of refraction than medium 1. This difference in index of refraction means that the speed of light is less in medium 2 than in medium 1. Note that, in Figure 16.17(a), the path of the ray moves closer to the perpendicular when the ray slows down. Conversely, in Figure 16.17(b), the path of the ray moves away from the perpendicular when the ray speeds up. The path is exactly reversible. In both cases, you can imagine what happens by thinking about pushing a lawn mower from a footpath onto grass, and vice versa. Going from the footpath to grass, the right front wheel is slowed and pulled to the side as shown. This is the same change in direction for light when it goes from a fast medium to a slow one. When going from the grass to the footpath, the left front wheel moves faster than the others, and the mower changes direction as shown. This, too, is the same change in direction as light going from slow to fast. Figure 16.17 The change in direction of a light ray depends on how the speed of light changes when it crosses from one medium to another. For the situations shown here, the speed of light is greater in medium 1 than in medium 2. (a) A ray of light moves closer to the perpendicular when it slows down. This is analogous to what happens when a lawnmower goes from a footpath (medium 1)
to grass (medium 2). (b) A ray of light moves away from the perpendicular when it speeds up. This is analogous to what happens when a lawnmower goes from grass (medium 2) to the footpath (medium 1). The paths are exactly reversible. Snap Lab Bent Pencil A classic observation of refraction occurs when a pencil is placed in a glass filled halfway with water. Do this and observe the shape of the pencil when you look at it sideways through air, glass, and water. • A full-length pencil • A glass half full of water Instructions Procedure 1. Place the pencil in the glass of water. 2. Observe the pencil from the side. Access for free at openstax.org. 16.2 • Refraction 491 3. Explain your observations. Virtual Physics Bending Light Click to view content (https://www.openstax.org/l/28Bendinglight) The Bending Light simulation in allows you to show light refracting as it crosses the boundaries between various media (download animation first to view). It also shows the reflected ray. You can move the protractor to the point where the light meets the boundary and measure the angle of incidence, the angle of refraction, and the angle of reflection. You can also insert a prism into the beam to view the spreading, or dispersion, of white light into colors, as discussed later in this section. Use the ray option at the upper left. A light ray moving upward strikes a horizontal boundary at an acute angle relative to the perpendicular and enters the medium above the boundary. What must be true for the light to bend away from the perpendicular? a. The medium below the boundary must have a greater index of refraction than the medium above. b. The medium below the boundary must have a lower index of refraction than the medium above. c. The medium below the boundary must have an index of refraction of zero. d. The medium above the boundary must have an infinite index of refraction. The amount that a light ray changes direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in the angle of refraction. The exact mathematical relationship is the law of refraction, or Snell’s law, which is stated in equation form as In terms of speeds, Snell’s law becomes Here, n1 and n
2 are the indices of refraction for media 1 and 2, respectively, and θ1 and θ2 are the angles between the rays and the perpendicular in the respective media 1 and 2, as shown in Figure 16.17. The incoming ray is called the incident rayand the outgoing ray is called the refracted ray. The associated angles are called the angle of incidenceand the angle of refraction. Later, we apply Snell’s law to some practical situations. Dispersion is defined as the spreading of white light into the wavelengths of which it is composed. This happens because the index of refraction varies slightly with wavelength. Figure 16.18 shows how a prism disperses white light into the colors of the rainbow. 16.5 492 Chapter 16 • Mirrors and Lenses Figure 16.18 (a) A pure wavelength of light ( ) falls onto a prism and is refracted at both surfaces. (b) White light is dispersed by the prism (spread of light exaggerated). Because the index of refraction varies with wavelength, the angles of refraction vary with wavelength. A sequence of red to violet is produced, because the index of refraction increases steadily with decreasing wavelength. Rainbows are produced by a combination of refraction and reflection. You may have noticed that you see a rainbow only when you turn your back to the Sun. Light enters a drop of water and is reflected from the back of the drop, as shown in Figure 16.19. The light is refracted both as it enters and as it leaves the drop. Because the index of refraction of water varies with wavelength, the light is dispersed and a rainbow is observed. Figure 16.19 Part of the light falling on this water drop enters and is reflected from the back of the drop. This light is refracted and dispersed both as it enters and as it leaves the drop. WATCH PHYSICS Dispersion This video explains how refraction disperses white light into its composite colors. Click to view content (https://www.openstax.org/l/28Raindrop) Which colors of the rainbow bend most when refracted? a. Colors with a longer wavelength and higher frequency bend most when refracted. b. Colors with a shorter wavelength and higher frequency bend most when refracted. c. Colors with a shorter wavelength and lower frequency bend most when refracted. d. Colors with a longer wavelength and a lower frequency bend most when refracted. Access for free at openstax.org. 16.
2 • Refraction 493 A good-quality mirror reflects more than 90 percent of the light that falls on it; the mirror absorbs the rest. But, it would be useful to have a mirror that reflects all the light that falls on it. Interestingly, we can produce total reflection using an aspect of refraction. Consider what happens when a ray of light strikes the surface between two materials, such as is shown in Figure 16.20(a). Part of the light crosses the boundary and is refracted; the rest is reflected. If, as shown in the figure, the index of refraction for the second medium is less than the first, the ray bends away from the perpendicular. Because n1 > n2, the angle of refraction is greater than the angle of incidence—that is, increased. This causes The critical angle, refraction of 90°. That is, as shown in Figure 16.20(c), then all the light is reflected back into medium 1, a condition called total internal reflection., which produces an angle of, is greater than the critical angle,, for a combination of two materials is defined to be the incident angle, is the incident angle for which =90°. If the incident angle, to increase as well. The largest the angle of refraction,. Now, imagine what happens as the incident angle is, can be is 90°, as shown in Figure 16.20(b). > Figure 16.20 (a) A ray of light crosses a boundary where the speed of light increases and the index of refraction decreases—that is, n2 < n1. The refracted ray bends away from the perpendicular. (b) The critical angle,, is the one for which the angle of refraction is 90°. (c) Total internal reflection occurs when the incident angle is greater than the critical angle. Recall that Snell’s law states the relationship between angles and indices of refraction. It is given by 16.6 When the incident angle equals the critical angle ( Snell’s law in this case becomes = ), the angle of refraction is 90° ( = 90°). Noting that sin 90° = 1, 494 Chapter 16 • Mirrors and Lenses The critical angle,, for a given combination of materials is thus for n1 > n2. Total internal reflection occurs for any incident angle greater than the critical angle, medium has an index of refraction less than the first. Note that the previous equation is
written for a light ray that travels in medium 1 and reflects from medium 2, as shown in Figure 16.20., and it can only occur when the second There are several important applications of total internal reflection. Total internal reflection, coupled with a large index of refraction, explains why diamonds sparkle more than other materials. The critical angle for a diamond-to-air surface is only 24.4°; so, when light enters a diamond, it has trouble getting back out (Figure 16.21). Although light freely enters the diamond at different angles, it can exit only if it makes an angle less than 24.4° with the normal to a given surface. Facets on diamonds are specifically intended to make this unlikely, so that the light can exit only in certain places. Diamonds with very few impurities are very clear, so the light makes many internal reflections and is concentrated at the few places it can exit—hence the sparkle. Figure 16.21 Light cannot escape a diamond easily because its critical angle with air is so small. Most reflections are total and the facets are placed so that light can exit only in particular ways, thus concentrating the light and making the diamond sparkle. A light ray that strikes an object that consists of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came. This parallel reflection is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. Such an object is called a corner reflectorbecause the light bounces from its inside corner. Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originates. Corner reflectors are perfectly efficient when the conditions for total internal reflection are satisfied. With common materials, it is easy to obtain a critical angle that is less than 45°. One use of these perfect mirrorsis in binoculars, as shown in Figure 16.22. Another application is for periscopes used in submarines. Figure 16.22 These binoculars use corner reflectors with total internal reflection to get light to the observer’s eyes. Fiber optics are one common application of total internal reflection. In communications, fiber optics are used to transmit telephone, internet, and cable TV signals, and they use the transmission of light down fibers of plastic or glass. Because the Access for free at openstax.org. fibers are thin, light entering one is likely to strike the inside surface at an angle greater than the
critical angle and, thus, be totally reflected (Figure 16.23). The index of refraction outside the fiber must be smaller than inside, a condition that is satisfied easily by coating the outside of the fiber with a material that has an appropriate refractive index. In fact, most fibers have a varying refractive index to allow more light to be guided along the fiber through total internal reflection. Rays are reflected around corners as shown in the figure, making the fibers into tiny light pipes. 16.2 • Refraction 495 Figure 16.23 (a) Fibers in bundles are clad by a material that has a lower index of refraction than the core to ensure total internal reflection, even when fibers are in contact with one another. A single fiber with its cladding is shown. (b) Light entering a thin fiber may strike the inside surface at large, or grazing, angles, and is completely reflected if these angles exceed the critical angle. Such rays continue down the fiber, even following it around corners, because the angles of reflection and incidence remain large. LINKS TO PHYSICS Medicine: Endoscopes A medical device called an endoscopeis shown in Figure 16.24. Figure 16.24 Endoscopes, such as the one drawn here, send light down a flexible fiber optic tube, which sends images back to a doctor in charge of performing a medical procedure. The word endoscope means looking inside. Doctors use endoscopes to look inside hollow organs in the human body and inside body cavities. These devices are used to diagnose internal physical problems. Images may be transmitted to an eyepiece or sent to a video screen. Another channel is sometimes included to allow the use of small surgical instruments. Such surgical procedures include collecting biopsies for later testing, and removing polyps and other growths. Identify the process that allows light and images to travel through a tube that is not straight. a. The process is refraction of light. b. The process is dispersion of light. c. The process is total internal reflection of light. d. The process is polarization of light. Calculations with the Law of Refraction The calculation problems that follow require application of the following equations: 16.8 496 Chapter 16 • Mirrors and Lenses and These are the equations for refractive index, the mathematical statement of the law of refraction (Snell’s law), and the equation for the critical angle. WATCH PHYSICS Snell’s Law Example 1 This video leads
you through calculations based on the application of the equation that represents Snell’s law. Click to view content (https://www.openstax.org/l/28Snellslaw) Which two types of variables are included in Snell’s law? a. The two types of variables are density of a material and the angle made by the light ray with the normal. b. The two types of variables are density of a material and the thickness of a material. c. The two types of variables are refractive index and thickness of each material. d. The two types of variables are refractive index of a material and the angle made by a light ray with the normal. WORKED EXAMPLE Calculating Index of Refraction from Speed Calculate the index of refraction for a solid medium in which the speed of light is 2.012 substance, based on the previous table of indicies of refraction. STRATEGY We know the speed of light, c, is 3.00 of refraction, n. 108 m/s, and we are given v. We can simply plug these values into the equation for index 108 m/s, and identify the most likely Solution This value matches that of polystyrene exactly, according to the table of indices of refraction (Table 16.2). Discussion The three-digit approximation for cis used, which in this case is all that is needed. Many values in the table are only given to three significant figures. Note that the units for speed cancel to yield a dimensionless answer, which is correct. 16.9 WORKED EXAMPLE Calculating Index of Refraction from Angles Suppose you have an unknown, clear solid substance immersed in water and you wish to identify it by finding its index of refraction. You arrange to have a beam of light enter it at an angle of 45.00°, and you observe the angle of refraction to be 40.30°. What are the index of refraction of the substance and its likely identity? STRATEGY We must use the mathematical expression for the law of refraction to solve this problem because we are given angle data, not speed data. The subscripts 1 and 2 refer to values for water and the unknown, respectively, where 1 represents the medium from which the 16.10 Access for free at openstax.org. light is coming and 2 is the new medium it is entering. We are given the angle values, and the table of indicies of refraction
gives us nfor water as 1.333. All we have to do before solving the problem is rearrange the equation 16.2 • Refraction 497 Solution 16.11 16.12 The best match from Table 16.2 is fused quartz, with n= 1.458. Discussion Note the relative sizes of the variables involved. For example, a larger angle has a larger sine value. This checks out for the two indicates the ray has bent towardnormal. This result is to angles involved. Note that the smaller value of compared with be expected if the unknown substance has a greater nvalue than that of water. The result shows that this is the case. WORKED EXAMPLE Calculating Critical Angle Verify that the critical angle for light going from water to air is 48.6°. (See Table 16.2, the table of indices of refraction.) STRATEGY First, choose the equation for critical angle Then, look up the nvalues for water, n1, and air, n2. Find the value of and it compare with the given angle of 48.6°.. Last, find the angle that has a sine equal to this value Solution For water, n1 = 1.333; for air, n2 = 1.0003. So, 16.13 16.14 Discussion Remember, when we try to find a critical angle, we look for the angle at which light can no longer escape past a medium boundary by refraction. It is logical, then, to think of subscript 1 as referring to the medium the light is trying to leave, and subscript 2 as where it is trying (unsuccessfully) to go. So water is 1 and air is 2. Practice Problems 6. The refractive index of ethanol is 1.36. What is the speed of light in ethanol? a. 2.25×108 m/s b. 2.21×107 m/s c. 2.25×109 m/s d. 2.21×108 m/s 7. The refractive index of air is and the refractive index of crystalline quartz is. What is the critical angle for a ray of light going from crystalline quartz into air? a. b. c. d. 498 Chapter 16 • Mirrors and Lenses Check Your Understanding 8. Which law is expressed by the equation? a. This is Ohm’s law. b. This is Wien’s displacement law. c. This is Snell�
�s law. d. This is Newton’s law. 9. Explain why the index of refraction is always greater than or equal to one. a. The formula for index of refraction,, of a material is where, so is always greater than one. b. The formula for index of refraction,, of a material is where, so is always greater than one. c. The formula for index of refraction,, of a material is than one. d. The formula for refractive index,, of a material is, so is always greater than one. 10. Write an equation that expresses the law of refraction. a. b. c. d. where,, so is always greater where 16.3 Lenses Section Learning Objectives By the end of this section, you will be able to do the following: • Describe and predict image formation and magnification as a consequence of refraction through convex and concave lenses, use ray diagrams to confirm image formation, and discuss how these properties of lenses determine their applications • Explain how the human eye works in terms of geometric optics • Perform calculations, based on the thin-lens equation, to determine image and object distances, focal length, and image magnification, and use these calculations to confirm values determined from ray diagrams Section Key Terms aberration chromatic aberration concave lens converging lens convex lens diverging lens eyepiece objective ocular parfocal Characteristics of Lenses Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom lens. In this section, we use the law of refraction to explore the properties of lenses and how they form images. Some of what we learned in the earlier discussion of curved mirrors also applies to the study of lenses. Concave, convex, focal point F, and focal length fhave the same meanings as before, except each measurement is made from the center of the lens instead of the surface of the mirror. The convex lens shown in Figure 16.25 has been shaped so that all light rays that enter it parallel to its central axis cross one another at a single point on the opposite side of the lens. The central axis, or axis, is defined to be a line normal to the lens at its center. Such a lens is called a converging lens because of the converging effect it has on light rays. An expanded view of the path of one ray through the lens is shown

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