1. Wave Unit 10 Wave Motion Transverse and longitudinal Waves
Energy and Waves
Interference
Standing Waves, Resonance
Refraction
Diffraction
Sound

2. Simple Harmonic Motion
Hooke’s Law
One type of periodic motion is the motion of a mass attached to a spring.
The direction of the force acting on the mass (Felastic) is always opposite the direction of the mass’s displacement from equilibrium (x = 0).

3. Hooke’s Law, continued At equilibrium:
Simple Harmonic Motion
Hooke’s Law, continued
At equilibrium:
The spring force and the mass’s acceleration become zero.
The speed reaches a maximum.
At maximum displacement:
The spring force and the mass’s acceleration reach a maximum.
The speed becomes zero.

4. spring force = –(spring constant  displacement)
Simple Harmonic Motion
Hooke’s Law, continued
Measurements show that the spring force, or restoring force, is directly proportional to the displacement of the mass.
This relationship is known as Hooke’s Law:
Felastic = –kx
spring force = –(spring constant  displacement)
The quantity k is a positive constant called the spring constant.

5. Sample Problem Hooke’s Law
Simple Harmonic Motion
Sample Problem
Hooke’s Law
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

6. Sample Problem, continued
Simple Harmonic Motion
Sample Problem, continued
1. Define
Given:
m = 0.55 kg
x = –2.0 cm = –0.20 m
g = 9.81 m/s2
Diagram:
Unknown:
k = ?

7. Sample Problem, continued
Simple Harmonic Motion
Sample Problem, continued
2. Plan
Choose an equation or situation: When the mass is attached to the spring,the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass.
Fnet = 0 = Felastic + Fg
Felastic = –kx
Fg = –mg
–kx – mg = 0

8. Sample Problem, continued
Simple Harmonic Motion
Sample Problem, continued
2. Plan, continued
Rearrange the equation to isolate the unknown:

9. Sample Problem, continued
Simple Harmonic Motion
Sample Problem, continued
3. Calculate
Substitute the values into the equation and solve:
4. Evaluate
The value of k implies that 270 N of force is required to displace the spring 1 m.

10. Simple Harmonic Motion
The motion of a vibrating mass-spring system is an example of simple harmonic motion.
Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement.
Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-and-forth motion over the same path.

11. Simple Harmonic Motion
The Simple Pendulum
A simple pendulum consists of a mass called a bob, which is attached to a fixed string.
At any displacement from equilibrium, the weight of the bob (Fg) can be resolved into two components.
The x component (Fg,x = Fg sin q) is the only force acting on the bob in the direction of its motion and thus is the restoring force.
The forces acting on the bob at any point are the force exerted by the
string and the
gravitational force.

12. The Simple Pendulum, continued
Simple Harmonic Motion
The Simple Pendulum, continued
The magnitude of the restoring force (Fg,x = Fg sin q) is proportional to sin q.
When the maximum angle of displacement q is relatively small (<15°), sin q is approximately equal to q in radians.
As a result, the restoring force is very nearly proportional to the displacement.
Thus, the pendulum’s motion is an excellent approximation of simple harmonic motion.

13. Simple Harmonic Motion

14. Amplitude, Period, and Frequency in SHM
Measuring Simple Harmonic Motion
Amplitude, Period, and Frequency in SHM
In SHM, the maximum displacement from equilibrium is defined as the amplitude of the vibration.
A pendulum’s amplitude can be measured by the angle between the pendulum’s equilibrium position and its maximum displacement.
For a mass-spring system, the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position.
The SI units of amplitude are the radian (rad) and the meter (m).

15. Amplitude, Period, and Frequency in SHM
Measuring Simple Harmonic Motion
Amplitude, Period, and Frequency in SHM
The period (T) is the time that it takes a complete cycle to occur.
The SI unit of period is seconds (s).
The frequency (f) is the number of cycles or vibrations per unit of time.
The SI unit of frequency is hertz (Hz).
Hz = s–1

16. Amplitude, Period, and Frequency in SHM, continued
Measuring Simple Harmonic Motion
Amplitude, Period, and Frequency in SHM, continued
Period and frequency are inversely related:
Thus, any time you have a value for period or frequency, you can calculate the other value.

17. Measures of Simple Harmonic Motion
Measuring Simple Harmonic Motion
Measures of Simple Harmonic Motion

18. Period of a Simple Pendulum in SHM
Measuring Simple Harmonic Motion
Period of a Simple Pendulum in SHM
The period of a simple pendulum depends on the length and on the free-fall acceleration.
The period does not depend on the mass of the bob or on the amplitude (for small angles).

19. Period of a Mass-Spring System in SHM
Measuring Simple Harmonic Motion
Period of a Mass-Spring System in SHM
The period of an ideal mass-spring system depends on the mass and on the spring constant.
The period does not depend on the amplitude.
This equation applies only for systems in which the spring obeys Hooke’s law.

20. Definition of a Wave
A wave is a disturbance that propagates through space and time, usually with transference of energy.
Waves travel and transfer energy from one point to another, often with little or no permanent displacement of the particles of the medium (that is, with little or no associated mass transport); instead they are oscillations around almost fixed positions.

21. Waves Continued A wave is not made of matter.
The source of all waves are moving objects (objects that are in a state of motion) Also known as vibrations.
Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement.
Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-and-forth motion over the same path.

22. Wave Motion A wave is the motion of a disturbance.
Properties of Waves
Wave Motion
A wave is the motion of a disturbance.
A medium is a physical environment through which a disturbance can travel. For example, water is the medium for ripple waves in a pond.
Waves that require a medium through which to travel are called mechanical waves. Water waves and sound waves are mechanical waves.
Electromagnetic waves such as visible light do not require a medium.

23. Properties of Waves
Wave Types
A wave that consists of a single traveling pulse is called a pulse wave.
Whenever the source of a wave’s motion is a periodic motion, such as the motion of your hand moving up and down repeatedly, a periodic wave is produced.
A wave whose source vibrates with simple harmonic motion is called a sine wave. Thus, a sine wave is a special case of a periodic wave in which the periodic motion is simple harmonic.

24. Relationship Between SHM and Wave Motion
Properties of Waves
Relationship Between SHM and Wave Motion
As the sine wave created by this vibrating blade travels to the right, a single point on the string vibrates up and down with simple harmonic motion.

25. Wave Types, continued Properties of Waves
A transverse wave is a wave whose particles vibrate perpendicularly to the direction of the wave motion.
The crest is the highest point above the equilibrium position, and the trough is the lowest point below the equilibrium position.
The wavelength (l) is the distance between two adjacent similar points of a wave.

26. Wave Types, continued Properties of Waves
A longitudinal wave is a wave whose particles vibrate parallel to the direction the wave is traveling.
A longitudinal wave on a spring at some instant t can be represented by a graph. The crests correspond to compressed regions, and the troughs correspond to stretched regions.
The crests are regions of high density and pressure (relative to the equilibrium density or pressure of the medium), and the troughs are regions of low density and pressure.

27. Period, Frequency, and Wave Speed
Properties of Waves
Period, Frequency, and Wave Speed
The frequency of a wave describes the number of waves that pass a given point in a unit of time.
The period of a wave describes the time it takes for a complete wavelength to pass a given point.
The relationship between period and frequency in SHM holds true for waves as well; the period of a wave is inversely related to its frequency.

28. Period, Frequency, and Wave Speed, continued
Properties of Waves
Period, Frequency, and Wave Speed, continued
The speed of a mechanical wave is constant for any given medium.
The speed of a wave is given by the following equation:
v = fl
wave speed = frequency  wavelength
This equation applies to both mechanical and electromagnetic waves.

29. Waves and Energy Transfer
Properties of Waves
Waves and Energy Transfer
Waves transfer energy by the vibration of matter.
Waves are often able to transport energy efficiently.
The rate at which a wave transfers energy depends on the amplitude.
The greater the amplitude, the more energy a wave carries in a given time interval.
For a mechanical wave, the energy transferred is proportional to the square of the wave’s amplitude.
The amplitude of a wave gradually diminishes over time as its energy is dissipated.

30. Wave Interactions
Wave Interference
Two different material objects can never occupy the same space at the same time.
Because mechanical waves are not matter but rather are displacements of matter, two waves can occupy the same space at the same time.
The combination of two overlapping waves is called superposition.

31. Wave Interference, continued
Wave Interactions
Wave Interference, continued
In constructive interference, individual displacements on the same side of the equilibrium position are added together to form the resultant wave.

32. Wave Interference, continued
Wave Interactions
Wave Interference, continued
In destructive interference, individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave.

33. Wave Interactions
Reflection
What happens to the motion of a wave when it reaches a boundary?
At a free boundary, waves are reflected.
At a fixed boundary, waves are reflected and inverted.
Free boundary Fixed boundary

34. Wave Interactions
Standing Waves
A standing wave is a wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.
Standing waves have nodes and antinodes.
A node is a point in a standing wave that maintains zero displacement.
An antinode is a point in a standing wave, halfway between two nodes, at which the largest displacement occurs.

35. Standing Waves, continued
Wave Interactions
Standing Waves, continued
Only certain wavelengths produce standing wave patterns.
The ends of the string must be nodes because these points cannot vibrate.
A standing wave can be produced for any wavelength that allows both ends to be nodes.
In the diagram, possible wavelengths include 2L (b), L (c), and 2/3L (d).

36. Standing Waves Wave Interactions
This photograph shows four possible standing waves that can exist on a given string. The diagram shows the progression
of the second standing wave for one-half of a cycle.

37. Multiple Choice
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
1. In what direction does the restoring force act?
A. to the left
B. to the right
C. to the left or to the right depending on whether the spring is stretched or compressed
D. perpendicular to the motion of the mass

38. Multiple Choice
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
1. In what direction does the restoring force act?
A. to the left
B. to the right
C. to the left or to the right depending on whether the spring is stretched or compressed
D. perpendicular to the motion of the mass

39. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
2. If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant?
F. –5.0  10–2 N/m H. 5.0  10–2 N/m
G. –2.0  101 N/m J. 2.0  101 N/m

40. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
2. If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant?
F. –5.0  10–2 N/m H. 5.0  10–2 N/m
G. –2.0  101 N/m J. 2.0  101 N/m

41. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
3. In what form is the energy in the system when the mass passes through the equilibrium point?
A. elastic potential energy
B. gravitational potential energy
C. kinetic energy
D. a combination of two or more of the above

42. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
3. In what form is the energy in the system when the mass passes through the equilibrium point?
A. elastic potential energy
B. gravitational potential energy
C. kinetic energy
D. a combination of two or more of the above

43. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
4. In what form is the energy in the system when the mass is at maximum displacement?
F. elastic potential energy
G. gravitational potential energy
H. kinetic energy
J. a combination of two or more of the above

44. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
4. In what form is the energy in the system when the mass is at maximum displacement?
F. elastic potential energy
G. gravitational potential energy
H. kinetic energy
J. a combination of two or more of the above

45. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
5. Which of the following does not affect the period of the mass-spring system?
A. mass
B. spring constant
C. amplitude of vibration
D. All of the above affect the period.

46. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
5. Which of the following does not affect the period of the mass-spring system?
A. mass
B. spring constant
C. amplitude of vibration
D. All of the above affect the period.

47. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?
F. 8p s H. p s
G. 4p s J. p/2 s

48. Multiple Choice, continued
Base your answers to questions 1–6 on the information below.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?
F. 8p s H. p s
G. 4p s J. p/2 s

49. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
7. What is the restoring force in the pendulum?
A. the total weight of the bob
B. the component of the bob’s weight tangent to the motion of the bob
C. the component of the bob’s weight perpendicular to the motion of the bob
D. the elastic force of the stretched string

50. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
7. What is the restoring force in the pendulum?
A. the total weight of the bob
B. the component of the bob’s weight tangent to the motion of the bob
C. the component of the bob’s weight perpendicular to the motion of the bob
D. the elastic force of the stretched string

51. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
8. Which of the following does not affect the
period of the pendulum?
F. the length of the string
G. the mass of the pendulum bob
H. the free-fall acceleration at the pendulum’s location
J. All of the above affect the period.

52. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
8. Which of the following does not affect the
period of the pendulum?
F. the length of the string
G. the mass of the pendulum bob
H. the free-fall acceleration at the pendulum’s location
J. All of the above affect the period.

53. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
9. If the pendulum completes exactly 12
cycles in 2.0 min, what is the frequency of
the pendulum?
A Hz
B Hz
C. 6.0 Hz
D. 10 Hz

54. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
9. If the pendulum completes exactly 12
cycles in 2.0 min, what is the frequency of
the pendulum?
A Hz
B Hz
C. 6.0 Hz
D. 10 Hz

55. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
10. If the pendulum’s length is 2.00 m and
ag = 9.80 m/s2, how many complete
oscillations does the pendulum make in
5.00 min?
F H. 106
G J. 239

56. Multiple Choice, continued
Base your answers to questions 7–10 on the information below.
A pendulum bob hangs from a string and moves with simple harmonic motion.
10. If the pendulum’s length is 2.00 m and
ag = 9.80 m/s2, how many complete
oscillations does the pendulum make in
5.00 min?
F H. 106
G J. 239

57. Multiple Choice, continued
Base your answers to questions 11–13 on the graph.
11. What kind of wave does this graph represent?
A. transverse wave C. electromagnetic wave
B. longitudinal wave D. pulse wave

58. Multiple Choice, continued
Base your answers to questions 11–13 on the graph.
11. What kind of wave does this graph represent?
A. transverse wave C. electromagnetic wave
B. longitudinal wave D. pulse wave

59. Multiple Choice, continued
Base your answers to questions 11–13 on the graph.
12. Which letter on the graph represents wavelength?
F. A H. C
G. B J. D

60. Multiple Choice, continued
Base your answers to questions 11–13 on the graph.
12. Which letter on the graph represents wavelength?
F. A H. C
G. B J. D

61. Multiple Choice, continued
Base your answers to questions 11–13 on the graph.
13. Which letter on the graph is used for a trough?
A. A C. C
B. B D. D

62. Multiple Choice, continued
Base your answers to questions 11–13 on the graph.
13. Which letter on the graph is used for a trough?
A. A C. C
B. B D. D

63. Multiple Choice, continued
Base your answers to questions 14–15 on the passage.
A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of
0.53 m. The two waves interfere.
14. What is the amplitude of the resultant wave if the interference is constructive?
F m
G m
H m
J m

64. Multiple Choice, continued
Base your answers to questions 14–15 on the passage.
A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of
0.53 m. The two waves interfere.
14. What is the amplitude of the resultant wave if the interference is constructive?
F m
G m
H m
J m

65. Multiple Choice, continued
Base your answers to questions 14–15 on the passage.
A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of
0.53 m. The two waves interfere.
15. What is the amplitude of the resultant wave if the interference is destructive?
A m
B m
C m
D m

66. Multiple Choice, continued
Base your answers to questions 14–15 on the passage.
A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of
0.53 m. The two waves interfere.
15. What is the amplitude of the resultant wave if the interference is destructive?
A m
B m
C m
D m

67. Multiple Choice, continued
16. Two successive crests of a transverse wave 1.20 m apart. Eight crests pass a given point 12.0 s. What is the wave speed?
F m/s
G m/s
H m/s
J m/s

68. Multiple Choice, continued
16. Two successive crests of a transverse wave 1.20 m apart. Eight crests pass a given point 12.0 s. What is the wave speed?
F m/s
G m/s
H m/s
J m/s

69. Short Response
17. Green light has a wavelength of 5.20  10–7 m and a speed in air of 3.00  108 m/s. Calculate the frequency and the period of the light.

70. Short Response
17. Green light has a wavelength of 5.20  10–7 m and a speed in air of 3.00  108 m/s. Calculate the frequency and the period of the light.
Answer: 5.77  1014 Hz, 1.73  10–15 s

71. Short Response, continued
18. What kind of wave does not need a medium through which to travel?

72. Short Response, continued
18. What kind of wave does not need a medium through which to travel?
Answer: electromagnetic waves

73. Short Response, continued
19. List three wavelengths that could form standing waves on a 2.0 m string that is fixed at both ends.

74. Short Response, continued
19. List three wavelengths that could form standing waves on a 2.0 m string that is fixed at both ends.
Answer: Possible correct answers include 4.0 m, 2.0 m, 1.3 m, 1.0 m, or other wavelengths such that nl = 4.0 m (where n is a positive integer).

75. Extended Response
20. A visitor to a lighthouse wishes to find out the height of the tower. The visitor ties a spool of thread to a small rock to make a simple pendulum. Then, the visitor hangs the pendulum down a spiral staircase in the center of the tower. The period of oscillation is 9.49 s. What is the height of the tower? Show all of your work.

76. Extended Response
20. A visitor to a lighthouse wishes to find out the height of the tower. The visitor ties a spool of thread to a small rock to make a simple pendulum. Then, the visitor hangs the pendulum down a spiral staircase in the center of the tower. The period of oscillation is 9.49 s. What is the height of the tower? Show all of your work.
Answer: 22.4 m

77. Extended Response, continued
21. A harmonic wave is traveling along a rope. The oscillator that generates the wave completes 40.0 vibrations in 30.0 s. A given crest of the wave travels 425 cm along the rope in a period of 10.0 s. What is the wavelength? Show all of your work.

78. Extended Response, continued
21. A harmonic wave is traveling along a rope. The oscillator that generates the wave completes 40.0 vibrations in 30.0 s. A given crest of the wave travels 425 cm along the rope in a period of 10.0 s. What is the wavelength? Show all of your work.
Answer: m

79. Simple Harmonic Motion
Hooke’s Law

80. Properties of Waves
Transverse Waves

81. Properties of Waves
Longitudinal Waves

82. Constructive Interference
Wave Interactions
Constructive Interference

83. Destructive Interference
Wave Interactions
Destructive Interference

84. Reflection of a Pulse Wave
Wave Interactions
Reflection of a Pulse Wave

85. The Production of Sound Waves
Every sound wave begins with a vibrating object, such as the vibrating prong of a tuning fork.
A compression is the region of a longitudinal wave in which the density and pressure are at a maximum.
A rarefaction is the region of a longitudinal wave in which the density and pressure are at a minimum.

86. The Production of Sound Waves, continued
Sound waves are longitudinal.
The simplest longitudinal wave produced by a vibrating object can be represented by a sine curve.
In the diagram, the crests of the sine curve correspond to compressions, and the troughs correspond to rarefactions.

87. Frequency of Sound Waves
As discussed earlier, frequency is defined as the number of cycles per unit of time.
Sound waves that the average human ear can hear, called audible sound waves, have frequencies between 20 and Hz.
Sound waves with frequencies less than 20 Hz are called infrasonic waves.
Sound waves with frequencies above Hz are called ultrasonic waves.

88. Sound Waves
Frequency and Pitch
The frequency of an audible sound wave determines how high or low we perceive the sound to be, which is known as pitch.
As the frequency of a sound wave increases, the pitch rises.
The frequency of a wave is an objective quantity that can be measured, while pitch refers to how different frequencies are perceived by the human ear.

89. The Speed of Sound The speed of sound depends on the medium.
Sound Waves
The Speed of Sound
The speed of sound depends on the medium.
Because waves consist of particle vibrations, the speed of a wave depends on how quickly one particle can transfer its motion to another particle.
For example, sound waves generally travel faster through solids than through gases because the molecules of a solid are closer together than those of a gas are.
The speed of sound also depends on the temperature of the medium. This is most noticeable with gases.

90. The Speed of Sound in Various Media
Sound Waves
The Speed of Sound in Various Media

91. The Propagation of Sound Waves
Sound waves propagate in three dimensions.
Spherical waves can be represented graphically in two dimensions, as shown in the diagram.
The circles represent the centers of compressions, called wave fronts.
The radial lines perpendicular to the wave fronts are called rays.
The sine curve used in our previous representation corresponds to a single ray.

92. The Propagation of Sound Waves, continued
At distances from the source that are great relative to the wavelength, we can approximate spherical wave fronts with parallel planes.
Such waves are called plane waves.
Plane waves can be treated as one-dimensional waves all traveling in the same direction.

93. Sound Waves
The Doppler Effect
The Doppler effect is an observed change in frequency when there is relative motion between the source of waves and an observer.
Because frequency determines pitch, the Doppler effect affects the pitch heard by each listener.
Although the Doppler effect is most commonly experienced with sound waves, it is a phenomenon common to all waves, including electromagnetic waves, such as visible light.

94. Sound Intensity and Resonance
As sound waves travel, energy is transferred from one molecule to the next. The rate at which this energy is transferred through a unit area of the plane wave is called the intensity of the wave.
Because power (P) is defined as the rate of energy transfer, intensity can also be described in terms of power.

95. Sound Intensity, continued
Sound Intensity and Resonance
Sound Intensity, continued
Intensity has units of watt per square meter (W/m2).
The intensity equation shows that the intensity decreases as the distance (r) increases.
This occurs because the same amount of energy is spread over a larger area.

96. Sound Intensity, continued
Sound Intensity and Resonance
Sound Intensity, continued
Human hearing depends on both the frequency and the intensity of sound waves.
Sounds in the middle of the spectrum of frequencies can be heard more easily (at lower intensities) than those at lower and higher frequencies.

97. Sound Intensity, continued
Sound Intensity and Resonance
Sound Intensity, continued
The intensity of a wave approximately determines its perceived loudness.
However, loudness is not directly proportional to intensity. The reason is that the sensation of loudness is approximately logarithmic in the human ear.
Relative intensity is the ratio of the intensity of a given sound wave to the intensity at the threshold of hearing.

98. Sound Intensity, continued
Sound Intensity and Resonance
Sound Intensity, continued
Because of the logarithmic dependence of perceived loudness on intensity, using a number equal to 10 times the logarithm of the relative intensity provides a good indicator for human perceptions of loudness.
This is referred to as the decibel level.
A dimensionless unit called the decibel (dB) is used for values on this scale.

99. Conversion of Intensity to Decibel Level
Sound Intensity and Resonance
Conversion of Intensity to Decibel Level

100. Forced Vibrations and Resonance
Sound Intensity and Resonance
Forced Vibrations and Resonance
If one of the pendulums is set in motion, its vibrations are transferred by the rubber band to the other pendulums, which will also begin vibrating. This is called a forced vibration.
Each pendulum has a natural frequency based on its length.

101. Forced Vibrations and Resonance, continued
Sound Intensity and Resonance
Forced Vibrations and Resonance, continued
Resonance is a phenomenon that occurs when the frequency of a force applied to a system matches the natural frequency of vibration of the system, resulting in a large amplitude of vibration.
If one blue pendulum is set in motion, only the other blue pendulum, whose length is the same, will eventually resonate.

102. Sound Intensity and Resonance
The Human Ear
The human ear is divided into three sections—outer, middle, and inner.
Sound waves travel through the three regions of the ear and are then transmitted to the brain as impulses through nerve endings on the basilar membrane.

103. Standing Waves on a Vibrating String
Harmonics
Standing Waves on a Vibrating String
The vibrations on the string of a musical instrument usually consist of many standing waves, each of which has a different wavelength and frequency.
The greatest possible wavelength on a string of length L is l = 2L.
The fundamental frequency, which corresponds to this wavelength, is the lowest frequency of vibration.

104. Standing Waves on a Vibrating String, continued
Harmonics
Standing Waves on a Vibrating String, continued
Each harmonic is an integral multiple of the fundamental frequency.
The harmonic series is a series of frequencies that includes the fundamental frequency and integral multiples of the fundamental frequency.
Harmonic Series of Standing Waves on a Vibrating String

105. Harmonics
The Harmonic Series

106. Standing Waves in an Air Column
Harmonics
Standing Waves in an Air Column
If both ends of a pipe are open, there is an antinode at each end.
In this case, all harmonics are present, and the earlier equation for the harmonic series of a vibrating string can be used.
Harmonic Series of a Pipe Open at Both Ends

107. Standing Waves in an Air Column, continued
Harmonics
Standing Waves in an Air Column, continued
If one end of a pipe is closed, there is a node at that end.
With an antinode at one end and a node at the other end, a different set of standing waves occurs.
In this case, only odd harmonics are present.
Harmonic Series of a Pipe Closed at One End

108. Harmonics of Open and Closed Pipes

109. Sample Problem Harmonics
What are the first three harmonics in a 2.45 m long pipe that is open at both ends? What are the first three harmonics when one end of the pipe is closed? Assume that the speed of sound in air is 345 m/s.
1. Define
Given:
L = 2.45 m v = 345 m/s
Unknown:
Case 1: f1, f2, f3 Case 2: f1, f3, f5

110. Sample Problem Plan Choose an equation or situation: Case 1: Case 2:
Harmonics
Sample Problem
Plan
Choose an equation or situation:
Case 1:
Case 2:
In both cases, the second two harmonics can be found by multiplying the harmonic numbers by the
fundamental frequency.

111. Sample Problem 3. Calculate
Harmonics
Sample Problem
3. Calculate
Substitute the values into the equation and solve:
Case 1:
The next two harmonics are the second and third:

112. Sample Problem Calculate, continued Case 2:
Harmonics
Sample Problem
Calculate, continued
Case 2:
The next two harmonics are the third and the fifth:
Tip: Use the correct harmonic numbers for each situation. For a pipe open at both ends, n = 1, 2, 3, etc. For a pipe closed at one end, only odd harmonics are present, so n = 1, 3, 5, etc.

113. Sample Problem 4. Evaluate
Harmonics
Sample Problem
4. Evaluate
In a pipe open at both ends, the first possible wavelength is 2L; in a pipe closed at one end, the first possible wavelength is 4L. Because frequency and wavelength are inversely proportional, the fundamental frequency of the open pipe should be twice that of the closed pipe, that is, 70.4 = (2)(35.2).

114. Harmonics
Timbre
Timbre is the the musical quality of a tone resulting from the combination of harmonics present at different intensities.
A clarinet sounds different from a viola because of differences in timbre, even when both instruments are sounding the same note at the same volume.
The rich harmonics of most instruments provide a much fuller sound than that of a tuning fork.

115. Harmonics of Musical Instruments

116. Harmonics
Beats
When two waves of slightly different frequencies interfere, the interference pattern varies in such a way that a listener hears an alternation between loudness and softness.
The variation from soft to loud and back to soft is called a beat.
In other words, a beat is the periodic variation in the amplitude of a wave that is the superposition of two waves of slightly different frequencies.

117. Harmonics
Beats

118. Multiple Choice
1. When a part of a sound wave travels from air into water, which property of the wave remains unchanged?
A. speed
B. frequency
C. wavelength
D. amplitude

119. Multiple Choice
1. When a part of a sound wave travels from air into water, which property of the wave remains unchanged?
A. speed
B. frequency
C. wavelength
D. amplitude

120. Multiple Choice, continued
2. What is the wavelength of the sound wave shown in the figure?
F m
G m
H m
J m

121. Multiple Choice, continued
2. What is the wavelength of the sound wave shown in the figure?
F m
G m
H m
J m

122. Multiple Choice, continued
3. If a sound seems to be getting louder, which of the following is probably increasing?
A. speed of sound
B. frequency
C. wavelength
D. intensity

123. Multiple Choice, continued
3. If a sound seems to be getting louder, which of the following is probably increasing?
A. speed of sound
B. frequency
C. wavelength
D. intensity

124. Multiple Choice, continued
4. The intensity of a sound wave increases by 1000 W/m2. What is this increase equal to in decibels?
F. 10
G. 20
H. 30
J. 40

125. Multiple Choice, continued
4. The intensity of a sound wave increases by 1000 W/m2. What is this increase equal to in decibels?
F. 10
G. 20
H. 30
J. 40

126. Multiple Choice, continued
5. The Doppler effect occurs in all but which of the following situations?
A. A source of sound moves toward a listener.
B. A listener moves toward a source of sound.
C. A listener and a source of sound remain at rest with respect to each other.
D. A listener and a source of sound move toward or away from each other.

127. Multiple Choice, continued
5. The Doppler effect occurs in all but which of the following situations?
A. A source of sound moves toward a listener.
B. A listener moves toward a source of sound.
C. A listener and a source of sound remain at rest with respect to each other.
D. A listener and a source of sound move toward or away from each other.

128. Multiple Choice, continued
6. If the distance from a point source of sound is tripled, by what factor is the sound intensity changed?
F. 1/9
G. 1/3
H. 3
J. 9

129. Multiple Choice, continued
6. If the distance from a point source of sound is tripled, by what factor is the sound intensity changed?
F. 1/9
G. 1/3
H. 3
J. 9

130. Multiple Choice, continued
7. Why can a dog hear a sound produced by a dog whistle, but its owner cannot?
A. Dogs detect sounds of less intensity than do humans.
B. Dogs detect sounds of higher frequency than do humans.
C. Dogs detect sounds of lower frequency than do humans.
D. Dogs detect sounds of higher speed than do humans.

131. Multiple Choice, continued
7. Why can a dog hear a sound produced by a dog whistle, but its owner cannot?
A. Dogs detect sounds of less intensity than do humans.
B. Dogs detect sounds of higher frequency than do humans.
C. Dogs detect sounds of lower frequency than do humans.
D. Dogs detect sounds of higher speed than do humans.

132. Multiple Choice, continued
8. The greatest value ever achieved for the speed of sound in air is about 1.0  104 m/s, and the highest frequency ever produced is about 2.0  1010 Hz. If a single sound wave with this speed and frequency were produced, what would its wavelength be?
F. 5.0  10–6 m
G. 5.0  10–7 m
H. 2.0  106 m
J. 2.0  1014 m

133. Multiple Choice, continued
8. The greatest value ever achieved for the speed of sound in air is about 1.0  104 m/s, and the highest frequency ever produced is about 2.0  1010 Hz. If a single sound wave with this speed and frequency were produced, what would its wavelength be?
F. 5.0  10–6 m
G. 5.0  10–7 m
H. 2.0  106 m
J. 2.0  1014 m

134. Multiple Choice, continued
9. The horn of a parked automobile is stuck. If you are in a vehicle that passes the automobile, as shown in the diagram, what is the nature of the sound that you hear?
A. The original sound of the horn rises in pitch
B. The original sound of the horn drops in pitch.
C. A lower pitch is heard rising to a higher pitch.
D. A higher pitch is heard dropping to a lower pitch.

135. Multiple Choice, continued
9. The horn of a parked automobile is stuck. If you are in a vehicle that passes the automobile, as shown in the diagram, what is the nature of the sound that you hear?
A. The original sound of the horn rises in pitch
B. The original sound of the horn drops in pitch.
C. A lower pitch is heard rising to a higher pitch.
D. A higher pitch is heard dropping to a lower pitch.

136. Multiple Choice, continued
10.The second harmonic of a guitar string has a frequency of 165 Hz. If the speed of waves on the string is 120 m/s, what is the string’s length?
F m
G m
H. 1.1 m
J. 1.4 m

137. Multiple Choice, continued
10.The second harmonic of a guitar string has a frequency of 165 Hz. If the speed of waves on the string is 120 m/s, what is the string’s length?
F m
G m
H. 1.1 m
J. 1.4 m

138. Short Response
11. Two wind instruments produce sound waves with frequencies of 440 Hz and 447 Hz, respectively. How many beats per second are heard from the superposition of the two waves?

139. Short Response
11. Two wind instruments produce sound waves with frequencies of 440 Hz and 447 Hz, respectively. How many beats per second are heard from the superposition of the two waves?
Answer: 7 beats per second (7 Hz)

140. Short Response, continued
12. If you blow across the open end of a soda bottle and produce a tone of 250 Hz, what will be the frequency of the next harmonic heard if you blow much harder?

141. Short Response, continued
12. If you blow across the open end of a soda bottle and produce a tone of 250 Hz, what will be the frequency of the next harmonic heard if you blow much harder?
Answer: 750 Hz

142. Short Response, continued
13. The figure shows a string vibrating in the sixth harmonic. The length of the string is 1.0 m. What is the wavelength of the wave on the string?

143. Short Response, continued
13. The figure shows a string vibrating in the sixth harmonic. The length of the string is 1.0 m. What is the wavelength of the wave on the string?
Answer: 0.33 m

144. Short Response, continued
14. The power output of a certain loudspeaker is W. If a person listening to the sound produced by the speaker is sitting 6.5 m away, what is the intensity of the sound?

145. Short Response, continued
14. The power output of a certain loudspeaker is W. If a person listening to the sound produced by the speaker is sitting 6.5 m away, what is the intensity of the sound?
Answer: 0.47 W/m2

146. Extended Response Use the following information to solve problems
15–16. Be sure to show all of your work.
The area of a typical eardrum is approximately
equal to 5.0  10–5 m2.
15. What is the sound power (the energy per second) incident on the eardrum at the threshold of pain (1.0 W/m2)?

147. Extended Response Use the following information to solve problems
15–16. Be sure to show all of your work.
The area of a typical eardrum is approximately
equal to 5.0  10–5 m2.
15. What is the sound power (the energy per second) incident on the eardrum at the threshold of pain (1.0 W/m2)?
Answer: 5.0  10–5 W

148. Extended Response, continued
Use the following information to solve problems
15–16. Be sure to show all of your work.
The area of a typical eardrum is approximately
equal to 5.0  10–5 m2.
16. What is the sound power (the energy per second) incident on the eardrum at the threshold of hearing (1.0  10–12 W/m2)?

149. Extended Response, continued
Use the following information to solve problems
15–16. Be sure to show all of your work.
The area of a typical eardrum is approximately
equal to 5.0  10–5 m2.
16. What is the sound power (the energy per second) incident on the eardrum at the threshold of hearing (1.0  10–12 W/m2)?
Answer: 5.0  10–17 W

150. Extended Response, continued
Use the following information to solve problems
17–19. Be sure to show all of your work.
A pipe that is open at both ends has a fundamental frequency of 456 Hz when the speed of sound in air is 331 m/s.
17. How long is the pipe?

151. Extended Response, continued
Use the following information to solve problems
17–19. Be sure to show all of your work.
A pipe that is open at both ends has a fundamental frequency of 456 Hz when the speed of sound in air is 331 m/s.
17. How long is the pipe?
Answer: m

152. Extended Response, continued
Use the following information to solve problems
17–19. Be sure to show all of your work.
A pipe that is open at both ends has a fundamental frequency of 456 Hz when the speed of sound in air is 331 m/s.
18. What is the frequency of the pipe’s second harmonic?

153. Extended Response, continued
Use the following information to solve problems
17–19. Be sure to show all of your work.
A pipe that is open at both ends has a fundamental frequency of 456 Hz when the speed of sound in air is 331 m/s.
18. What is the frequency of the pipe’s second harmonic?
Answer: 912 Hz

154. Extended Response, continued
Use the following information to solve problems
17–19. Be sure to show all of your work.
A pipe that is open at both ends has a fundamental frequency of 456 Hz when the speed of sound in air is 331 m/s.
19. What is the fundamental frequency of this pipe when the speed of sound in air is increased to 367 m/s as a result of a rise in the temperature of the air?

155. Extended Response, continued
Use the following information to solve problems
17–19. Be sure to show all of your work.
A pipe that is open at both ends has a fundamental frequency of 456 Hz when the speed of sound in air is 331 m/s.
19. What is the fundamental frequency of this pipe when the speed of sound in air is increased to 367 m/s as a result of a rise in the temperature of the air?
Answer: 506 Hz

156. The Production of Sound Waves

157. The Propagation of Sound Waves

158. Sound Intensity and Resonance

159. Sound Intensity and Resonance
The Human Ear