Deck 17: The Principle of Linear Superposition and Interference Phenomena

Question 1

Consider the figures depicting rectangular pulses on a string.
Which pulses must be superimposed to give the situation shown in 5.

A)1 and 2
B)1 and 3
C)2 and 4
D)1, 2, and 4
E)2, 3, and 4.

Answer 1

1 and 3

Question 2

Two pulses of identical shape travel toward each other in opposite directions on a string, as shown in the drawing. Which one of the following statements concerning this situation is true?

A)The pulses will reflect from each other.
B)The pulses will diffract from each other.
C)The pulses will interfere to produce a standing wave.
D)The pulses will pass through each other and produce beats.
E)As the pulses pass through each other, they will interfere destructively.

Answer 2

As the pulses pass through each other, they will interfere destructively.

Question 3

Two identical tuning forks vibrate at 587 Hz. After a small piece of clay is placed on one of them, eight beats per second are heard. What is the period of the tuning fork that holds the clay?

A)1.68 × 10^-3 s
B)1.70 × 10^-3 s
C)1.73 × 10^-3 s
D)1.76 × 10^-3 s
E)1.80 × 10^-3 s

Answer 3

1.73 × 10^-3 s

Question 4

Water waves approach an aperture. The resulting patterns are shown for two different cases, A and B, in which the wavelength and aperture size are varied. Which one of the following statements concerning these cases is true?

A)Neither figure shows diffraction. In both cases, the wavelength is much smaller than the aperture.
B)Diffraction occurs in A, but not in B because the wavelength in A is much smaller than the aperture.
C)Diffraction occurs in B, but not in A because the wavelength in B is much smaller than the aperture.
D)Both figures show diffraction. In both cases, the wavelengths are approximately the same size as the aperture.
E)Diffraction occurs in B, but not in A because the wavelength in B is approximately the same size as the aperture.

Question 5

Sound waves are emitted from two speakers. Which one of the following statements about sound wave interference is false?

A)In a region where both destructive and constructive interference occur, energy is not conserved.
B)Destructive interference occurs when two waves are exactly out of phase when they meet.
C)Interference redistributes the energy carried by the individual waves.
D)Constructive interference occurs when two waves are exactly in phase when they meet.
E)Sound waves undergo diffraction as they exit each speaker.

Question 6

Two loudspeakers are located 3.0 m apart on the stage of an auditorium. A listener at point P is seated 19.0 m from one speaker and 15.0 m from the other. A signal generator drives the speakers in phase with the same amplitude and frequency. The wave amplitude at P due to each speaker alone is A. The frequency is then varied between 30 Hz and 400 Hz. The speed of sound is 343 m/s.
At what frequency or frequencies will the listener at P hear a maximum intensity?

A)170 Hz only
B)113 Hz and 226 Hz
C)86 Hz, 170 Hz, 257, and 343 Hz
D)57 Hz, 114 Hz, 171 Hz, 228 Hz, 285, 342, and 399 Hz
E)43 Hz, 85 Hz, 128 Hz, 170 Hz, 213 Hz, 257 Hz, 298, 344, and 387 Hz

Question 7

Two loudspeakers, A and B, are separated by a distance of 2.0 m. The speakers emit sound waves at a frequency of 680 Hz that are exactly out of phase. The speed of sound is 343 m/s. How far from speaker A along the +x axis will a point of constructive interference occur?

A)0.25 m
B)0.30 m
C)0.46 m
D)0.88 m
E)0.98 m

Question 8

A pebble is dropped in a lake; and it produces ripples with a frequency of 0.50 Hz. When should a second pebble be dropped at the same place to produce destructive interference?

A)0.50 s after the first
B)0.75 s after the first
C)1.0 s after the first
D)1.5 s after the first
E)2.0 s after the first

Question 9

For a diffraction horn loudspeaker, the sound emerges through a rectangular opening. The width of a diffraction horn is 0.14 m. If the speed of sound in air is 343 m/s, at what frequency is the diffraction angle $\theta$ equal to 35°?

A)3300 Hz
B)4300 Hz
C)6300 Hz
D)7700 Hz
E)9200 Hz

Question 10

For a diffraction horn loudspeaker, sound emerges through a rectangular opening. The opening of a diffraction horn has a width of 0.12 m. If the speaker emits a continuous tone with a wavelength of 0.09 m, at what angle does the first minimum occur?

A)47°
B)39°
C)23°
D)12°
E)66°

Question 11

Two timpani (tunable drums) are played at the same time. One is correctly tuned so that when it is struck, sound is produced that has a wavelength of 2.20 m. The second produces sound with a wavelength of 2.10 m. If the speed of sound is 343 m/s, what beat frequency is heard?

A)7 beats/s
B)9 beats/s
C)11 beats/s
D)13 beats/s
E)15 beats/s

Question 12

A guitar string has a linear density of 8.30 × 10^-4 kg/m and a length of 0.660 m. The tension in the string is 56.7 N. When the fundamental frequency of the string is sounded with a 196.0-Hz tuning fork, what beat frequency is heard?

A)6 Hz
B)4 Hz
C)12 Hz
D)8 Hz
E)2 Hz

Question 13

A speaker generates a continuous tone of 440 Hz. In the drawing, sound travels into a tube that splits into two segments, one longer than the other. The sound waves recombine before being detected by a microphone. The speed of sound in air is 339 m/s. What is the minimum difference in the lengths of the two paths for sound travel if the waves arrive exactly out of phase at the microphone?

A)0.10 m
B)0.39 m
C)0.77 m
D)1.11 m
E)1.54 m

Question 14

A guitar string produces 4 beats/s when sounded with a 250 Hz tuning fork and 9 beats per second when sounded with a 255 Hz tuning fork. What is the vibrational frequency of the string?

A)240 Hz
B)246 Hz
C)254 Hz
D)259 Hz
E)263 Hz

Question 15

A speaker is designed for wide dispersion for a high frequency sound. What should the diameter of the circular opening be for a speaker where the desired diffraction angle is 11° and a 9100 Hz sound is generated? The speed of sound is 343 m/s.

A)0.039 m
B)0.077 m
C)0.24 m
D)0.13 m
E)0.048 m

Question 16

Two loudspeakers are located 3.0 m apart on the stage of an auditorium. A listener at point P is seated 19.0 m from one speaker and 15.0 m from the other. A signal generator drives the speakers in phase with the same amplitude and frequency. The wave amplitude at P due to each speaker alone is A. The frequency is then varied between 30 Hz and 400 Hz. The speed of sound is 343 m/s.
Determine the value of the maximum amplitude in terms of A.

A)2.0A
B)2.5A
C)3.0A
D)4.0A
E)5.0A

Question 17

Which one of the following superpositions will result in beats?

A)the superposition of waves that travel with different speeds
B)the superposition of identical waves that travel in the same direction
C)the superposition of identical waves that travel in opposite directions
D)the superposition of waves that are identical except for slightly different amplitudes
E)the superposition of waves that are identical except for slightly different frequencies

Question 18

Consider the figures depicting rectangular pulses on a string.
Complete the following statement: If pulse 2 were superimposed on pulse 3,

A)constructive interference would occur.
B)the resulting pattern would be represented by 1.
C)the resulting pattern would be represented by 4.
D)the resulting pattern would be represented by 5.
E)the resulting pattern would be different than 1, 4, and 5.

Question 19

Two traveling pulses on a rope move toward each other at a speed of 1.0 m/s. The waves have the same amplitude. The drawing shows the position of the waves at time t = 0 s. Which one of the following drawings depicts the waves on the rope at t = 2.0 s?

A)a
B)b
C)c
D)d
E)e

Question 20

Consider the figures depicting rectangular pulses on a string.
If pulse 1 were reflected from a wall, which one of the patterns above would represent the reflected pulse?

A)1
B)2
C)3
D)4
E)5

Question 21

A 4.00-m long string, clamped at both ends, vibrates at 2.00 × 10² Hz. If the string resonates in six segments, what is the speed of transverse waves on the string?

A)100 m/s
B)133 m/s
C)267 m/s
D)328 m/s
E)400 m/s

Question 22

The beat period occurring when two tuning forks are vibrating is 0.333 s. One of the forks is known to vibrate at 588.0 Hz. What are the possible vibration frequencies of the second tuning fork?

A)587.7 or 588.3 Hz
B)586.0 or 592.0 Hz
C)580.3 or 596.7 Hz
D)585.0 or 591.0 Hz
E)584.5 Hz or 591.5 Hz

Question 23

A string with a length of 3.0 m has two adjacent resonances at frequencies 112 Hz and 140 Hz.
How many antinodes will be found on the 112 Hz resonance?

A)1
B)2
C)3
D)4
E)5

Question 24

A string with a length of 3.0 m has two adjacent resonances at frequencies 112 Hz and 140 Hz.
Determine the wavelength of the 140 Hz resonance.

A)0.30 m
B)0.60 m
C)1.2 m
D)1.5 m
E)1.8 m

Question 25

Which one of the following statements is true concerning the points on a string that sustain a standing wave pattern?

A)All points vibrate with the same energy.
B)All points undergo the same displacements.
C)All points vibrate with different frequencies.
D)All points vibrate with different amplitudes.
E)All points undergo motion that is purely longitudinal.

Question 26

A string with a length of 3.0 m has two adjacent resonances at frequencies 112 Hz and 140 Hz.
Determine the fundamental frequency of the string.

A)14 Hz
B)28 Hz
C)42 Hz
D)56 Hz
E)70 Hz

Question 27

A string with a length of 3.0 m has two adjacent resonances at frequencies 112 Hz and 140 Hz.
Determine the speed of the waves on the string vibrating at 140 Hz..

A)28 m/s
B)42 m/s
C)56 m/s
D)140 m/s
E)168 m/s

Question 28

What is the distance to the nearest antinode from the fixed end of a guitar string vibrating at its first harmonic?

A) $\lambda$
B)2 $\lambda$
C) $\lambda$ /2
D) $\lambda$ /4
E)3 $\lambda$ /4

Question 29

Which one of the following will result in standing waves?

A)the superposition of waves that travel with different speeds
B)the superposition of identical waves that travel in the same direction
C)the superposition of identical waves that travel in opposite directions
D)the superposition of nearly identical waves of slightly different amplitudes
E)the superposition of nearly identical waves of slightly different frequencies

Question 30

A string with a length of 3.0 m has two adjacent resonances at frequencies 112 Hz and 140 Hz.
What is the wavelength of the first harmonic?

A)0.75 m
B)1.50 m
C)3.00 m
D)6.00 m
E)12.0 m

Question 31

When plucked, a 0.62-m guitar string produces a sound wave with a fundamental frequency of 196 Hz. The speed of sound in air is 343 m/s. Determine the ratio of the wavelength of the sound wave to the wavelength of the waves that travel on the string.

A)0.071
B)0.28
C)0.49
D)1.4
E)2.0

Question 32

Four standing wave segments, or loops, are observed on a string fixed at both ends as it vibrates at a frequency of 240 Hz. What is the fundamental frequency of the string?

A)23 Hz
B)28 Hz
C)35 Hz
D)60 Hz
E)80 Hz

Question 33

A 6.00-m long string sustains a three-loop standing wave pattern as shown. The wave speed is 2.00 × 102 m/s.
What is the lowest possible frequency for standing waves on this string?

A)50.0 Hz
B)33.3 Hz
C)25.0 Hz
D)16.7 Hz
E)8.33 Hz

Question 34

Vibrations with frequency 6.00 × 102 Hz are established on a 1.33-m length of string that is clamped at both ends. The speed of waves on the string is 4.00 × 102 m/s.
How many antinodes are observed for the resulting standing wave pattern?

A)2
B)3
C)4
D)5
E)6

Question 35

A 6.00-m long string sustains a three-loop standing wave pattern as shown. The wave speed is 2.00 × 102 m/s.
What is the frequency of vibration?

A)25.0 Hz
B)33.0 Hz
C)50.0 Hz
D)75.0 Hz
E)1.00 × 10² Hz

Question 36

One string on a guitar is exactly in tune. The guitarist uses this string to produce a tone with a frequency of 196 Hz by pressing down at the proper fret. An adjacent string can also be used to produce this tone without being pressed against a fret. However, this adjacent string is out of tune and produces a tone that sounds lower in frequency than the other tone. When the tones are produced simultaneously, the beat frequency is 5.0 Hz. What frequency does the adjacent string produce?

A)196 Hz
B)191 Hz
C)171 Hz
D)201 Hz
E)186 Hz

Question 37

A string with a linear density of 0.035 kg/m and a mass of 0.014 kg is clamped at both ends. Under what tension in the string will it have a fundamental frequency of 110 Hz?

A)270 N
B)310 N
C)450 N
D)580 N
E)690 N

Question 38

A rope of length L is clamped at both ends. Which one of the following is not a possible wavelength for standing waves on this rope?

A)L/2
B)2L/3
C)L
D)2L
E)4L

Question 39

A certain string, clamped at both ends, vibrates in seven segments at a frequency of 2.40 × 10² Hz. What frequency will cause it to vibrate in four segments?

A)89 Hz
B)137 Hz
C)274 Hz
D)411 Hz
E)420 Hz

Question 40

Vibrations with frequency 6.00 × 102 Hz are established on a 1.33-m length of string that is clamped at both ends. The speed of waves on the string is 4.00 × 102 m/s.
How far from either end of the string does the first node occur?

A)0.17 m
B)0.33 m
C)0.49 m
D)0.66 m
E)0.75 m

Question 41

Pipe A is 0.50 m long and open at both ends. Pipe B is open at one end and closed at the other end. Determine the length of pipe B so that it has the same fundamental frequency as A.

A)0.25 m
B)0.50 m
C)0.75 m
D)1.0 m
E)2.0 m

Question 42

A cylindrical tube sustains standing waves at the following frequencies: 600 Hz, 800 Hz, and 1000 Hz. The tube does not sustain standing waves at 500 Hz, at 900 Hz, at any frequencies between 600 and 800 Hz, or at any frequencies between 800 and 1000 Hz. Determine the fundamental frequency of the tube and whether the tube is open at both ends or has only one end open.

A)50 Hz, both ends
B)100 Hz, one end
C)100 Hz, both ends
D)200 Hz, one end
E)200 Hz, both ends

Question 43

When a tuba is played, the player makes a buzzing sound and blows into one end of a tube that has an effective length of 3.50 m. The other end of the tube is open. If the speed of sound in air is 343 m/s, what is the lowest frequency the tuba can produce?

A)8.00 Hz
B)12.0 Hz
C)16.0 Hz
D)24.0 Hz
E)49.0 Hz

Question 44

A cylindrical tube sustains a fundamental frequency f₁ when both of its ends are open. A second cylindrical tube with one end closed has the same fundamental frequency. What is the ratio of the length of the second tube to that of the first, L₂/L₁?

A)0.5
B)4
C)2
D)0.25
E)1

Question 45

Some of the lowest pitches attainable on a musical instrument are achieved on the world's largest pipe organs. What is the length of an organ pipe that is open on both ends and has a fundamental frequency of 8.75 Hz when the speed of sound in air is 341 m/s?

A)9.83 m
B)19.5 m
C)21.2 m
D)29.3 m
E)32.4 m

Question 46

Determine the shortest length of pipe, open at both ends, which will resonate at 256 Hz. The speed of sound is 343 m/s.

A)0.330 m
B)0.670 m
C)0.990 m
D)1.32 m
E)1.67 m