Queens College, CUNY - PHYSICS 152

Chapter 14—Superposition and Standing Waves

MULTIPLE CHOICE

1)Two harmonic waves are described by

What is the amplitude of the resultant wave?

1. 1. 8.0 m
   2. 4.3 m
   3. 6.0 m
   4. 3.2 m
   5. 3.0 m

2. Two harmonic waves are described by

What is the frequency (in Hz) of the resultant wave? a.      300

2. 48
3. 8

d. 0.8

e.   150

3. Two harmonic waves are described by

What is the wavelength of the resultant wave?

1. 1. 3 m
   2. 2 m
   3. 1 m
   4. 4 m
   5. 6 m

4. Two harmonic waves are described by

What is the phase (in rad) of the resultant wave when x = t = 0?

1. 1. 3
   2. 0
   3. 2
   4. 1
   5. 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)?
   1. 90

b. 400

c.   1.57

d.   7.85

e.   15

6. Two harmonic waves are described by

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

1. 1. 16

b.   4.0

c.   8.0

d. 0.25

e.   2.0

7. 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?

1. 1. 366 rad/s
   2. 6 rad/s

8. 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?

1. 1. 92 Hz
   2. 183 Hz
   3. 6 Hz
   4. 3 Hz
   5. 366 Hz

9. 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 ? radians?

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

10. The superposition of two waves,

and

,

results in a wave with a phase angle of

1. 1. 0 rad.
   2. 0.5 rad. c.

rad.

d.

rad.

e.   ? rad.

11. The superposition of two waves,

and

,

results in a wave with a wavelength of a.

m.

2. 2 m.
3. ? m.
4. 4 m.
5. 4? m.

12. The superposition of two waves,

and

,

results in a wave with a frequency of

1. 1. 85 Hz.
   2. 170 Hz.
   3. 85? Hz.
   4. 340 Hz.

e.   170? Hz.

13. 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?

1. 1. 0 b.

3. ?
4. 2?
5. 4?

14. Transverse waves y₁ = A₁ sin(k₁x ? ?₁t) and y₂ = A₂ sin(k₂x ? ?₂t), with A₂ > A₁, start at opposite ends of a long rope when t = 0. The magnitude of the maximum displacement, y, of the rope at any point is
    1. A₁ ? A₂.
    2. A₂ ? A₁.
    3. A₁ + A₂. d.

.

e.

.

15. 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?
    1. 10 cm
    2. 20 cm
    3. 650 cm
    4. All of the wavelengths listed above.
    5. Only the wavelengths listed in (a) and (b).

NARRBEGIN: Exhibit 18-01

Exhibit 18-1

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

Use this exhibit to answer the following question(s). NARREND

16. Refer to Exhibit 18-1. The phase difference in radians between points A and B is
    1. 0.

b.

.

c.

.

d. ?. e.

.

17. Refer to Exhibit 18-1. The phase difference in radians between points A and C is
    1. 0. b.

.

c.   ?. d.

.

e.   2?

18. Refer to Exhibit 18-1. The phase difference in radians between points A and D is
    1. ?.

b. 2?.

c.   3?.

d. 4?.

e.   5?.

19. Superposition of waves can occur
    1. in transverse waves.
    2. in longitudinal waves.
    3. in sinusoidal waves.
    4. in all of the above.
    5. only in (a) and (c) above.

20. Two pulses are traveling towards each other at 10 cm/s on a long string at t = 0 s, as shown below.

Which diagram below correctly shows the shape of the string at 0.5 s?

a.

b.

c.

d.

e.

21. Two ropes are spliced together as shown.

A short time after the incident pulse shown in the diagram reaches the splice, the rope's appearance will be that in

a.  

b.

c.

d.

e.

22. Two ropes are spliced together as shown.

A short time after the incident pulse shown in the diagram reaches the splice, the rope's appearance will be that in

a.

b.

c.

d.

e.

23. Two harmonic waves are described by

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

24. 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

25. 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?
    1. 3 Hz
    2. 1 Hz
    3. 6 Hz
    4. 12 Hz
    5. 5 Hz

26. 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

27. Two harmonic waves are described by

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

1. 1. 3

b. 0.25

3. 0
4. 6

e.   1.5

28. Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin (?x) cos (3?t) where x is in m and t is in s. What is the distance (in m) between the first two antinodes?
    1. 8
    2. 2
    3. 4
    4. 1

e.   0.5

29. 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. 1 000

e.   200

30. 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?

31. 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?
    1. 480 Hz
    2. 320 Hz
    3. 160 Hz
    4. 640 Hz
    5. 240 Hz

32. 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

33. 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.

.

b.

.

c.

.

4. f_n,B = 2f_n,A.
5. f_n,B = 4f_n,A.

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

a.   ?/4.

b. ?/2.

3. ?.
4. L/4.
5. L/2.

35. Which of the following wavelengths could NOT be present as a harmonic on a 2 m long string?
    1. 4 m
    2. 2 m

36. 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

a.

, where n is a positive integer.

b.

, where n is an odd integer.

c.

, where n is an even integer.

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

37. 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?
    1. The speed of the waves traveling along the string.
    2. The tension in the string.
    3. The wavelength of the waves on the string.
    4. All of the above.
    5. None of the above.

38. A string with a fixed frequency vibrator at one end is undergoing resonance with 4 antinodes when under tension T₁. When the tension is slowly increased the resonance condition disappears until tension T₂ is reached, there being no resonances occurring between these two tensions. How many antinodes are there in this new resonance?
    1. 3
    2. 4, since all resonances in this situation have the same number of nodes
    3. 5
    4. 2, since resonances only involve whole wavelengths
    5. 6, since resonances only involve whole wavelength       

39. A string with a fixed frequency vibrator at one end is subjected to varying tensions. When the tension is 20 N, a resonance with 3 antinodes results. What tension would cause a resonance with 2 antinodes in this string?
    1. 30 N
    2. 45 N
    3. 80 N
    4. 8.9 N
    5. 13 N

40. 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

41. 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.)
    1. 76
    2. 86
    3. 54
    4. 28

e.   129

42. 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

43. 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?
    1. 28
    2. 86

c.   344

d.   172

e.   688

44. 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, 1 100 Hz

45. Which of the following wavelengths could NOT be present as a standing wave in a 2 m long organ pipe open at both ends?
    1. 4 m
    2. 2 m
    3. 1 m

d. 0.89 m

e.   0.5 m

46. 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.
    1. 85 Hz.
    2. 170 Hz.
    3. 255 Hz.
    4. 340 Hz.
    5. 382 Hz.

47. 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 1 320 Hz.

b.   660, 1 320 and 1 980 Hz.

c.   880, 1 320 and 1 760 Hz.

d. 1 320, 2 640 and 3 960 Hz.

e.   They have no overtones in common.

48. When two organ pipes each closed at one end 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 1 320 Hz.

b.   660, 1 320 and 1 980 Hz.

c.   880, 1 320 and 1 760 Hz.

d. 1 320, 2 640 and 3 960 Hz.

e.   They have no overtones in common.

49. 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?
    1. After the first overtone of each pipe, every second overtone of the first pipe matches every second overtone of the second pipe.
    2. After the first overtone of each pipe, every second overtone of the first pipe matches every

third overtone of the second pipe.

1. 3. After the first overtone of each pipe, every third overtone of the first pipe matches every second overtone of the second pipe.
   4. After the first overtone of each pipe, every second overtone of the first pipe matches every fourth overtone of the second pipe.
   5. After the first overtone of each pipe, every third overtone of the first pipe matches every fourth overtone of the second pipe.

50. A harmonic longitudinal wave propagating down a tube filled with a compressible gas has the form

s(x, t) = s_m cos (kx ? ?t). Its velocity can be obtained from

1. 1. ?/k
   2. k/?
   3. k
   4. ?
   5. ?k

51. The figure below shows the positions of particles in a longitudinal standing wave. One quarter period later the particle distribution is shown in

a.

b.

c.

d.

e.

52. Two tuning forks with frequencies 264 and 262 Hz produce "beats". What is the beat frequency (in Hz)?
    1. 4
    2. 2
    3. 1
    4. 3
    5. 0 (no beats are produced)

53. 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?
    1. 269 Hz
    2. 254 Hz
    3. 264 Hz
    4. 5 Hz
    5. 274 Hz

54. The superposition of two waves

at the location x = 0 in space results in

1. 1. beats at a beat frequency of 3 Hz.
   2. a pure tone at a frequency of 153 Hz.
   3. a pure tone at a frequency of 156 Hz.
   4. beats at a beat frequency of 6 Hz in a 153 Hz tone.
   5. a tone at a frequency of 156 Hz, as well as beats at a beat frequency of 6 Hz in a 153 Hz tone.

55. A fire engine approaches a wall at 5 m/s while the siren emits a tone of 500 Hz frequency. At the time, the speed of sound in air is 340 m/s. How many beats per second do the people on the fire engine hear?
    1. 0
    2. 15
    3. 29
    4. 63

e.   250

56. Tuning forks #1, #2, and #3 each have slightly different frequencies.When #1 and #2 are sounded together, a beat frequency of 3 Hz results. When #2 and #3 are sounded together, a beat frequency of 5 Hz results. If the frequency of #1 is 100 Hz, which of the following cannot be the frequency of #3?
    1. 92 Hz
    2. 95 Hz
    3. 98 Hz
    4. 102 Hz
    5. 108 Hz

PROBLEM

57. 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?

58. 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.