chapter (18)

Chapter 18

Superposition and Standing Waves

Multiple Choice 1. Two harmonic waves are described by

⎟⎠⎞

⎜⎝⎛ −= txy

s700

m4

sin)m 3(1

⎟⎠⎞

⎜⎝⎛ −−= 2

s700

m4

sin)m 3(2 txy

What is the amplitude of the resultant wave?

a. 8.0 m b. 4.3 m c. 6.0 m d. 3.2 m e. 3.0 m

2. Two harmonic waves are described by

⎟⎠⎞

⎜⎝⎛ −= txy

s300

m8

sin)m 4(1

⎟⎠⎞

⎜⎝⎛ −−= 2

s300

m8

sin)m 4(2 txy

What is the frequency of the resultant wave?

a. 300 b. 48 c. 8 d. 0.8 e. 150


⎟⎠⎞

⎜⎝⎛ −= txy

s900

m6

sin)m 5(1

⎟⎠⎞

⎜⎝⎛ −−= 2

s900

m6

sin)m 5(2 txy

What is the wavelength of the resultant wave?

a. 3 m b. 2 m c. 1 m d. 4 m e. 6 m

321

322 CHAPTER 18


⎟⎠⎞

⎜⎝⎛ −= txy

s100

m5

sin)m 7(1

⎟⎠⎞

⎜⎝⎛ −−= 2

s100

m5

sin)m 7(2 txy

What is the phase of the resultant wave when x = t = 0?

a. 3 b. 0 c. 2 d. 1 e. 4

5. The path difference between two waves is 5m. If the wavelength of the waves emitted by the two sources is 4m, what is the phase difference (in degrees)?

a. 90 b. 400 c. 1.57 d. 7.85 e. 15


⎟⎠⎞

⎜⎝⎛ += txy

s2

m8

sin)cm 3(1

⎟⎠⎞

⎜⎝⎛ −= txy

s2

m8

sin)cm 3(2

What is the magnitude of the speed (in m/s) of the two traveling waves?

a. 16 b. 4 c. 8 d. 0.25 e. 2

Superposition and Standing Waves 323


⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ += txy

s2

m8

sin)cm 6(1 π

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −= txy

s2

m8

sin)cm 6(2 π

From the choices given, determine the smallest positive value of x (in cm) corresponding to a node of the resultant standing wave.

a. 3 b. 0.25 c. 0 d. 6 e. 1.5


⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ += txy

s00.3

m00.2

sin)cm 00.6(1 π

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −= txy

s00.3

m00.2

sin)cm 00.6(2 π

What is the magnitude of the displacement (in cm) of this wave at x = 3 cm and t = 5 sec?

a. 12.0 b. 3.00 c. 6.00 d. 2.25 e. 0

9. Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 3 sin (2x) cos 5t where x is in m and t is in s. What is the wavelength of the interfering waves?

a. 3.14 m b. 1.00 m c. 6.28 m d. 12.0 m e. 2.00 m

10. Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 4 sin (5x) cos (6t) where x is in m and t is in s. What is the approximate frequency of the interfering waves?

a. 3 Hz b. 1 Hz c. 6 Hz d. 12 Hz e. 5 Hz

324 CHAPTER 18

11. Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (4x) cos (3t) where x is in m and t is in s. What is the speed (in m/s) of the interfering waves?

a. 0.75 b. 0.25 c. 1.3 d. 12 e. 3.0

12. Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (πx) cos (3πt) where x is in m and t is in s. What is the distance (in m) between the first two antinodes?

a. 8 b. 2 c. 4 d. 1 e. 0.5

13. A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the wavelength (in cm) of the first harmonic?

a. 600 b. 400 c. 800 d. 1000 e. 200

14. A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the fundamental frequency?


15. A stretched string is observed to vibrate in three equal segments when driven by a 480 Hz oscillator. What is the fundamental frequency of vibration for this string?



16. A clarinet behaves like a tube closed at one end. If its length is 1.0 m, and the velocity of sound is 344 m/s, what is its fundamental frequency (in Hz)?

a. 264 b. 140 c. 86 d. 440 e. 172

17. An organ pipe open at both ends has a radius of 4.0 cm and a length of 6.0 m. What is the frequency (in Hz) of the third harmonic? (Assume the velocity of sound is 344 m/s.)

a. 76 b. 86 c. 54 d. 28 e. 129

18. A vertical tube one meter long is open at the top. It is filled with 75 cm of water. If the velocity of sound is 344 m/s, what will the fundamental resonant frequency be (in Hz)?

a. 3.4 b. 172 c. 344 d. 1.7 e. 688

19. A length of organ pipe is closed at one end. If the speed of sound is 344 m/s, what length of pipe (in cm) is needed to obtain a fundamental frequency of 50 Hz?

a. 28 b. 86 c. 344 d. 172 e. 688

20. Two tuning forks with frequencies 264 and 262 Hz produce “beats”. What is the beat frequency (in Hz)?

a. 4 b. 2 c. 1 d. 3 e. 0 (no beats are produced)

21. Two instruments produce a beat frequency of 5 Hz. If one has a frequency of 264 Hz, what could be the frequency of the other instrument?


326 CHAPTER 18

22. Two waves are described by

y1 = 6 cos 180t and y2 = 6 cos 186t, (both in meters).

With what angular frequency does the maximum amplitude of the resultant wave vary with time?

a. 366 rad/s b. 6 rad/s c. 3 rad/s d. 92 rad/s e. 180 rad/s

23. Two waves are described by

y1 = 6 cos 180t and y2 = 6 cos 186t, (both in meters).

What effective frequency does the resultant vibration have at a point?


24. An organ pipe open at both ends is 1.5 m long. A second organ pipe that is closed at one end and open at the other is 0.75 m long. The speed of sound in the room is 330 m/s. Which of the following sets of frequencies consists of frequencies which can be produced by both pipes?

a. 110 Hz, 220 Hz, 330 Hz b. 220 Hz, 440 Hz, 660 Hz c. 110 Hz, 330 Hz, 550 Hz d. 330 Hz, 440 Hz, 550 Hz e. 220 Hz, 660 Hz, 1100 Hz

25. Two strings are respectively 1.00 m and 2.00 m long. Which of the following wavelengths, in meters, could represent harmonics present on both strings?

a. 0.800, 0.670, 0.500 b. 1.33, 1.00, 0.500 c. 2.00, 1.00, 0.500 d. 2.00, 1.33, 1.00 e. 4.00, 2.00, 1.00


26. Two point sources emit sound waves of 1.0-m wavelength. The sources, 2.0 m apart, as shown below, emit waves which are in phase with each other at the instant of emission. Where, along the line between the sources, are the waves out of phase with each other by π radians?

0.5 1.0 1.5 2.00

S1 S2x

a. x = 0, 1.0 m, 2.0 m b. x = 0.50 m, 1.5 m c. x = 0.50 m, 1.0 m, 1.5 m d. x = 0.75 m, 1.25 m e. x = 0.25 m, 0.75 m, 1.25 m, 1.75 m

27. Two identical strings have the same length and same mass per unit length. String B is stretched with four times as great a tension as that applied to string A. Which statement is correct for all n harmonics on the two strings, n = 1, 2, 3…?

a. fn,B = 41

fn,A.

b. fn,B = 21

fn,A.

c. fn,B = 2 fn,A.d. fn,B = 2 fn,A. e. fn,B = 4 fn,A.

28. The superposition of two waves

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛= ty

s156

2cos)cm 006.0(1 π and ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛= ty

s150

2cos)cm 004.0(2 π

at the location x = 0 in space results in

a. beats at a beat frequency of 3 Hz. b. a pure tone at a frequency of 153 Hz. c. a pure tone at a frequency of 156 Hz. d. beats at a beat frequency of 6 Hz in a 153 Hz tone. e. a tone at a frequency of 156 Hz, as well as beats at a beat frequency of 6 Hz

in a 153 Hz tone.

328 CHAPTER 18

29. The superposition of two waves,

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −×= − t

xy

s170

m 2sin)m 102( 8

1 π

and

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−×= −

21

s170

m 2sin)m 102( 8

2 tx

y π ,

results in a wave with a phase angle of

a. 0 rad. b. 0.5 rad.

c. 4π

rad.

d. 2π

rad.

e. π rad.


⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −×= − t

xy

s170

m 2sin)m 102( 8

1 π

and

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−×= −

21

s170

m 2sin)m 102( 8

2 tx

y π ,

results in a wave with a wavelength of

a. 2π

m.

b. 2 m. c. π m. d. 4 m. e. 4π m.


⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −×= − t

xy

s170

m 2sin)m 102( 8

1 π

and

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−×= −

21

s170

m 2sin)m 102( 8

2 tx

y π ,

results in a wave with a frequency of

a. 85 Hz. b. 170 Hz. c. 85π Hz. d. 340 Hz. e. 170π Hz.


32. In a standing wave, not necessarily at the fundamental frequency, on a string of length L, the distance between nodes is

a. λ/4. b. λ/2. c. λ. d. L/4. e. L/2.

33. Which of the following wavelengths could NOT be present as a harmonic on a 2 m long string?

a. 4 m b. 2 m c. 1 m d. 0.89 m e. 0.5 m

34. Which of the following wavelengths could NOT be present as a standing wave in a 2 m long organ pipe open at both ends?

a. 4 m b. 2 m c. 1 m d. 0.89 m e. 0.5 m

35. Which of the following frequencies could NOT be present as a standing wave in a 2m long organ pipe open at both ends. The fundamental frequency is 85 Hz.

a. 85 Hz. b. 170 Hz. c. 255 Hz. d. 340 Hz. e. 382 Hz.

330 CHAPTER 18

36. An observer stands 3 m from speaker A and 4 m from speaker B. Both speakers, oscillating in phase, produce 170 Hz waves. The speed of sound in air is 340 m/s. What is the phase difference (in radians) between the waves from A and B at the observer’s location, point P?

4 m

B

A

P

3 m

a. 0

b. 2π

c. π d. 2π e. 4π

37. As shown below, a garden room has three walls, a floor and a roof, but is open to the garden on one side. The wall widths are L and w. The roof height is h. When traveling sound waves enter the room, standing sound waves can be present in the room if the wavelength of the standing waves is

w

L

a. Ln

, where n is a positive integer.

b. wn

, where n is an odd integer.

c. hn

, where n is an even integer.

d. in all cases listed above. e. given by (a) or (b) above, but not by (c).


38. Transverse waves y1 = A1 sin(k1x −ω1t) and y2 = A2 sin(k2x + ω2t), with start at opposite ends of a long rope when t

A2 > A1= 0. The magnitude of the maximum

displacement, y, of the rope at any point is

a. . A1 − A2b. . A2 − A1c. . A1 + A2d. A1 − A2( ) k1

k2.

e. A2 − A1( )k2k1

.

39. Two speakers in an automobile emit sound waves that are in phase at the speakers. One speaker is 40 cm ahead of and 30 cm to the left of the driver’s left ear. The other speaker is 50 cm ahead of and 120 cm to the right of the driver’s right ear. Which of the following wavelengths is(are) in phase at the left ear for the speaker on the left and the right ear for the speaker on the right?

a. 10 cm b. 20 cm c. 650 cm d. All of the wavelengths listed above. e. Only the wavelengths listed in (a) and (c).

40. A very long string is tied to a rigid wall at one end while the other end is attached to a simple harmonic oscillator. Which of the following can be changed by changing the frequency of the oscillator?

a. The speed of the waves traveling along the string. b. The tension in the string. c. The wavelength of the waves on the string. d. All of the above. e. None of the above.

41. When two organ pipes open at both ends sound a perfect fifth, such as two notes with fundamental frequencies at 440 Hz and 660 Hz, both pipes produce overtones. Which choice below correctly describes overtones present in both pipes?

a. 440, 880 and 1320 Hz. b. 660, 1320 and 1980 Hz. c. 880, 1320 and 1760 Hz. d. 1320, 2640 and 3960 Hz. e. They have no overtones in common.

332 CHAPTER 18

42. When two organ pipes closed at both ends sound a perfect fifth, such as two notes with fundamental frequencies at 440 Hz and 660 Hz, both pipes produce overtones. Which choice below correctly describes overtones present in both pipes?

a. 440, 880 and 1320 Hz. b. 660, 1320 and 1980 Hz. c. 880, 1320 and 1760 Hz. d. 1320, 2640 and 3960 Hz. e. They have no overtones in common.

43. Two organ pipes, a pipe of fundamental frequency 440 Hz, closed at one end, and a pipe of fundamental frequency 660 Hz, open at both ends, produce overtones. Which choice below correctly describes overtones present in both pipes?

a. After the first overtone of each pipe, every second overtone of the first pipe matches every second overtone of the second pipe.

b. After the first overtone of each pipe, every second overtone of the first pipe matches every third overtone of the second pipe.

c. After the first overtone of each pipe, every third overtone of the first pipe matches every second overtone of the second pipe.

d. After the first overtone of each pipe, every second overtone of the first pipe matches every fourth overtone of the second pipe.

e. After the first overtone of each pipe, every third overtone of the first pipe matches every fourth overtone of the second pipe.

44. The figure below shows wave crests after a stone is thrown into a pond.

A

B CD

The phase difference in radians between points A and B is

a. 0.

b. π4

.

c. π2

.

d. π .

e. 3π2

.



A

B CD

The phase difference in radians between points A and C is

a. 0.

b. π2

.

c. π .

d. 3π2

.

e. 2π


A

B CD

The phase difference in radians between points A and D is

a. π . b. 2π . c. 3π . d. 4π . e. 5π.

Open-Ended Problems 47. A student wants to establish a standing wave on a wire 1.8 m long clamped at

both ends. The wave speed is 540 m/s. What is the minimum frequency she should apply to set up standing waves?

48. Find the frequencies of the first three harmonics of a 1.0-m long string which has a mass per unit length of 2.0 × 10–3 kg/m and a tension of 80 N when both ends are fixed in place.

49. A steel wire in a piano has a length of 0.70 m and a mass of 4.3 grams. To what tension must this wire be stretched to make the fundamental frequency correspond to middle C, (fc = 261.6 Hz)?

334 CHAPTER 18


Chapter 18

Superposition and Standing Waves

1. d

2. b

3. c

4. d

5. a

6. d

7. c

8. d

9. a

10. b

11. a

12. d

13. b

14. c

15. c

16. c

17. b

18. c

19. d

20. b

21. a

22. c

23. b

24. c

25. c

26. d

27. d

28. e

29. c

30. d

31. a

32. b

33. d

34. d

35. e

36. c

37. e

38. c

39. e

40. c

41. d

42. e

43. e

44. a

45. e

46. c

47. 150 Hz

48. 100 Hz, 200 Hz, 300 Hz

49. 824 N

336 CHAPTER 18

chapter (18)

Documents

opposite directions interfere

shows wave crests

harmonic waves traveling

organ pipe open

pipes produce overtones

a1 a2

a2 a1

resultant wave