1. 16 Superposition and Standing Waves
Slide 16-2

2. Slide 16-3

3. Slide 16-4

4. Principle of Superposition
Slide 16-11

5. Constructive and Destructive Interference
Displacements add
Destructive:
Displacements cancel
Slide 16-12

6. Standing Wave Modes
Slide 16-15

7. Standing Sound Waves
Slide 16-17

8. Interference of Spherical Waves
Slide 16-23

9. Beats
fbeat = | f1 – f2|
Slide 16-25

10. Physics of Speech and Hearing
Slide 16-29

11. Answer: B
Slide 16-31

12. Figure 16.4

13. Figure 16.5a

14. Figure 16.12

15. Figure 16.17

16. Figure 16.18

17. Figure 16.19

18. Figure 16.20

19. Figure 16.21

20. Figure 16.22

21. Figure 16.35

22. Additional Questions
The following tubes all support sound waves at their fundamental frequency. Which tube has the lowest fundamental frequency?
Answer: C
Slide 16-32

23. Answer
The following tubes all support sound waves at their fundamental frequency. Which tube has the lowest fundamental frequency?
(c)
Answer: C
Slide 16-33

24. Additional Questions
Which of the following changes will increase the frequency of the lowest-frequency standing sound wave on a stretched string? Choose all that apply.
Replacing the string with a thicker string.
Increasing the tension in the string.
Plucking the string harder.
Doubling the length of the string.
Answer: B
Slide 16-34

25. Answer
Which of the following changes will increase the frequency of the lowest-frequency standing sound wave on a stretched string? Choose all that apply.
Replacing the string with a thicker string.
Increasing the tension in the string.
Plucking the string harder.
Doubling the length of the string.
Answer: B
Slide 16-35

26. Additional Questions
Which of the following changes will not increase the frequency of the lowest-frequency standing sound wave in an open-open tube?
Closing one end of the tube.
Replacing the air in the tube with helium.
Reducing the length of the tube.
Increasing the temperature of the air in the tube.
Answer: B
Slide 16-36

27. Answer
Which of the following changes will not increase the frequency of the lowest-frequency standing sound wave in an open-open tube?
Closing one end of the tube.
Replacing the air in the tube with helium.
Reducing the length of the tube.
Increasing the temperature of the air in the tube.
Answer: B
Slide 16-37

28. Additional Examples
1. A tube, open at both ends, is filled with an unknown gas. The tube is 190 cm in length and 3 cm in diameter. By using different tuning forks, it is found that resonances can be excited at frequencies of 315 Hz, 420 Hz, and 525 Hz, and at no frequencies in between these. What is the speed of sound in this gas?
2. Two loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity. As you walk parallel to the plane of the speakers, staying 3.0 m away, the sound intensity decreases until reaching a minimum when you are directly in front of one of the speakers. The speed of sound in the room is 340 m/s. What is the frequency of the sound?
Slide 16-38

29. Reading Quiz
When two waves overlap, the displacement of the medium is the sum of the displacements of the two individual waves. This is the principle of __________.
constructive interference
destructive interference
standing waves
superposition
Answer: D
Slide 16-5

30. Answer
When two waves overlap, the displacement of the medium is the sum of the displacements of the two individual waves. This is the principle of __________.
constructive interference
destructive interference
standing waves
superposition
Answer: D
Slide 16-6

31. Reading Quiz
A point on a standing wave that is always stationary is a _________.
maximum
minimum
node
antinode
Answer: C
Slide 16-7

32. Answer
A point on a standing wave that is always stationary is a _________.
maximum
minumum
node
antinode
Answer: C
Slide 16-8

33. Reading Quiz
You can decrease the frequency of a standing wave on a string by:
making the string longer.
using a thicker string.
decreasing the tension.
all of the above.
Answer: D
Slide 16-9

34. Answer
You can decrease the frequency of a standing wave on a string by:
making the string longer.
using a thicker string.
decreasing the tension.
all of the above.
Answer: D
Slide 16-10

35. How does the string appear at t = 2 s?
Checking Understanding
Two waves on a string are moving toward each other. A picture at t = 0 s appears as follows:
How does the string appear at t = 2 s?
Answer: A
Slide 16-13

36. How does the string appear at t = 2 s?
Answer
Two waves on a string are moving toward each other. A picture at t = 0 s appears as follows:
How does the string appear at t = 2 s?
(a)
Answer: A
Slide 16-14

37. Example Problems
A particular species of spider spins a web with silk threads of density 1300 kg/m3 and diameter 3.0 µm. A passing insect brushes a 12-cm-long strand of the web, which has a tension of 1.0 mN, and excites the lowest frequency standing wave. With what frequency will the strand vibrate?
Two strings with linear densities of 5.0 g/m are stretched over pulleys, adjusted to have vibrating lengths of 50 cm, and attached to hanging blocks. The block attached to string 1 has a mass of 20 kg and the block attached to wire 2 has mass M. When driven at the same frequency, the two strings support the standing waves shown.
What is the driving frequency?
What is the mass of the block suspended from String 2?
Slide 16-16

38. Example Problem: Physics of Music
The Egyptian tomb of Tutankhamun contained many treasures, including two trumpets. These trumpets were simple instruments, consisting of straight tubes with a mouthpiece at one end and a bell at the other. The smaller of the two was 58 cm long, made of silver. This trumpet, like other similar musical instruments, couldn’t make a reasonable sound at the fundamental frequency, but it could produce all of the harmonics. What frequencies could the instrument have produced?
Slide 16-18

39. Example Problem
Why do most people find that they sound better when they sing in the shower? Part of the answer comes from the enhancement produced by resonance of the shower enclosure, which typically has hard sides that reflect sound quite well. Suppose a shower enclosure is created by adding glass doors and tile walls to a standard bathtub, so the enclosure has the dimensions of a standard tub, 30" wide and 60" long. Standing waves can be set up along either axis of the enclosure.
What type of standing waves will be observed?
What possible frequencies of standing waves will be observed?
Slide 16-19

40. Checking Understanding: Interference Along a Line
Two speakers are emitting identical sound waves with a wavelength of 4.0 m. The speakers are 8.0 m apart, directed toward each other, as in the diagram below.
At each of the noted points in the above diagram, is the interference
constructive? (a), (c), (e)
destructive? (b), (d)
something in between?
Answers: (a) A; (b) B; (c) A; (d) B; (e) A
Slide 16-20

41. Answer
Two speakers are emitting identical sound waves with a wavelength of 4.0 m. The speakers are 8.0 m apart, directed toward each other, as in the diagram below.
At each of the noted points in the above diagram, is the interference
constructive? (a), (c), (e)
destructive? (b), (d)
something in between?
Answers: (a) A; (b) B; (c) A; (d) B; (e) A
Slide 16-21

42. Example Problem
Two speakers on stands 10 m apart are aimed directly at each other. They are emitting a pure tone at 440 Hz. Alice stands at a point exactly between the two speakers; this is a point of constructive interference, so the sound is quite loud. How far must she walk toward one or the other speaker in order to stand at a quiet spot?
Slide 16-22

43. Example Problem: Interference of Sound Waves
Two speakers emit identical sinusoidal waves. The speakers are placed 4.0 m apart. A listener moving along a line in front of the two speakers finds loud and quiet spots as shown below. The grid lines are spaced at 1.0 m. What is the frequency of the sound from the two speakers?
Slide 16-24

44. Checking Understanding
Two speakers emit sounds of nearly-equal frequency, as shown. At a point between the two speakers, the sound varies from loud to soft. How much time elapses between two successive loud moments?
0.5 s
1.0 s
2.0 s
4.0 s
Answer: B
Slide 16-26

45. Answer
Two speakers emit sounds of nearly-equal frequency, as shown. At a point between the two speakers, the sound varies from loud to soft. How much time elapses between two successive loud moments?
0.5 s
1.0 s
2.0 s
4.0 s
Answer: B
Slide 16-27

46. Example Problem
A typical police radar sends out microwaves at 10.5 GHz. The unit combines the wave reflected from a car with the original signal and determines the beat frequency. This beat frequency is converted into a speed. If a car is moving at 20 m/s toward the detector, what will be the measured bear frequency?
Answer: B
Slide 16-28

47. Example Problem
Sopranos sing at much higher pitches than other signers. A high C note is over 1000 Hz. This makes sopranos easier to hear—as the pitch rises to a few thousand Hz, our hearing becomes more and more sensitive—but it makes them harder to understand. There is good anecdotal and empirical data that shows that the words sung at very high pitches are, in fact, harder to decipher. Explain.
Answer: B
Slide 16-30