Deck 16: Superposition and Standing Waves

Question 1

When a piano tuner strikes both the A above middle C on the piano and a 440 Hz tuning fork, he hears 4 beats each second. The frequency of the piano's A is

A) 440 Hz
B) 444 Hz
C) 880 Hz
D) 436 Hz
E) either 436 Hz or 444 Hz

Answer 1

either 436 Hz or 444 Hz

Question 2

Two wave trains of the same frequency are traveling in opposite directions down a string. When they meet, these wave trains will not

A) be described by the principle of superposition.
B) reflect from each other.
C) pass through one another.
D) continue to carry energy.
E) remain transverse.

Answer 2

reflect from each other.

Question 3

Two tones of equal amplitude but slightly different frequencies are emitted by a sound source. This gives rise to

A) standing waves.
B) destructive interference.
C) constructive interference.
D) beats.
E) amplification.

Answer 3

beats.

Question 4

A violinist is tuning the A string on her violin by listening for beats when this note is played simultaneously with a tuning fork of frequency 440 Hz. She hears a beat frequency of 4 Hz. She notices that, when she increases the tension in the string slightly, the beat frequency decreases. What was the frequency of the mistuned A string?

A) 448 Hz
B) 444 Hz
C) 436 Hz
D) 432 Hz
E) 438 Hz

Question 5

What is the phase difference at any given instant between two points on a wave which are 1.52 m apart if the wavelength of the wave is 2.13 m?

A) 0.430 rad
B) 2.70 rad
C) 4.48 rad
D) 44.0 rad
E) 119 rad

Question 6

In graph A, two waves are shown at a given instant. What is the number of the curve in graph B that represents the wave resulting from the superposition of the two waves in A at this instant?

A) 1
B) 2
C) 3
D) The resultant is zero for all values of x.
E) None of these represent the wave.

Question 7

The interference of waves refers to the

A) slowing down of one wave in the presence of another.
B) resultant disturbance of two or more waves at every point in the medium.
C) change in wavelength that occurs when two waves cross one another.
D) phase change of 180º that occurs on reflection of a wave at a fixed end.
E) ability of waves to go around corners.

Question 8

Two waves with the same frequency and wavelength but with different amplitudes are added. If A₁ = 2A₂ and the waves are 180º out of phase, then the amplitude of the resultant wave is

A) zero.
B) the same as A₁.
C) the same as A₂.
D) equal to A₁ + A₂.
E) coherent.

Question 9

Middle C on a piano has a frequency of 262 Hz. Sometimes it is said that middle C is actually 2⁸ = 256 Hz, and tuning forks are made with this frequency. How many beats per second would be heard if such a tuning fork were sounded simultaneously with the middle C of a (well-tuned) piano?

A) 3
B) 6
C) 12
D) 4
E) 8

Question 10

If two identical waves with the same phase are added, the result is

A) a wave with the same frequency but twice the amplitude.
B) a wave with the same amplitude but twice the frequency.
C) a wave with zero amplitude.
D) a wave with zero frequency.
E) This problem cannot be solved without knowing the wavelengths of the two waves.

Question 11

Two loudspeakers S₁ and S₂, 3.0 m apart, emit the same single-frequency tone in phase at the speakers. A listener L directly in front of speaker S₁ notices that the intensity is a minimum when she is 4.0 m from that speaker (see figure). What is the lowest frequency of the emitted tone? The speed of sound in air is 340 m/s.

A) 85 Hz
B) 0.17 kHz
C) 0.26 kHz
D) 0.34 kHz
E) 0.51 kHz

Question 12

Two loudspeakers S₁ and S₂, 3.0 m apart, emit the same single-frequency tone in phase at the speakers. A listener directly in front of speaker S₁ notices that the intensity is a minimum when she is 4.0 m from that speaker (see figure). The listener now walks around speaker S₁ in an arc of a circle, staying 4.0 m from that speaker but increasing her distance from the other speaker. How far is she from speaker S₂ when she notices the first maximum in the sound intensity? The speed of sound in air is 340 m/s.

A) 4.5 m
B) 5.0 m
C) 5.5 m
D) 6.0 m
E) 6.5 m

Question 13

If two identical waves with a phase difference of 3 $\pi$ are added, the result is

A) a wave with the same frequency but twice the amplitude.
B) a wave with the same amplitude but twice the frequency.
C) a wave with zero amplitude.
D) a wave with an intensity equal to the sum of the intensities of the two waves.
E) This problem cannot be solved without knowing the wavelengths of the two waves.

Question 14

If two identical waves with a phase difference of 6 $\pi$ are added, the result is

A) a wave with the same frequency but twice the amplitude.
B) a wave with the same amplitude but twice the frequency.
C) a wave with zero amplitude.
D) a wave with zero frequency.
E) This problem cannot be solved without knowing the wavelengths of the two waves.

Question 15

A wave on a string has a frequency of 100 Hz and travels at a speed of 24 m/s. The minimum distance between two points with a phase difference of 60º is

A) 0.040 m
B) 0.12 m
C) 0.14 m
D) 0.24 m
E) 25 m

Question 16

Tuning fork A has a frequency of 440 Hz. When A and a second tuning fork B are struck simultaneously, four beats per second are heard. When a small mass is added to one of the tines of B, the two forks struck simultaneously produce two beats per second. The original frequency of tuning fork B was

A) 448 Hz
B) 444 Hz
C) 438 Hz
D) 436 Hz
E) 432 Hz

Question 17

Sketch A shows two identical pulses traveling in opposite directions along a string, each with a velocity of 1.0 cm/s. After 4.0 s, the string will look like which of the other sketches?

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

Question 18

Two trumpet players are both playing a pitch with a frequency of 440 Hz, corresponding to the musical pitch A above middle C. However, one of the trumpet players is marching away from you so that you hear a beat frequency of 4 Hz from the two trumpets. With what speed is the departing trumpet player moving away from you? (The speed of sound in air is 340 m/s)

A) 3.12 m/s
B) 3.09 m/s
C) 3.06 m/s
D) 3.00 m/s
E) 2.95 m/s

Question 19

Two whistles produce sounds with wavelengths 3.40 m and 3.30 m. What is the beat frequency produced? (the speed of sound is 340 m/s)

A) 0.1 Hz
B) 1.0 Hz
C) 2.0 Hz
D) 3.0 Hz
E) 4.0 Hz

Question 20

The air columns in two identical pipes vibrate at frequencies of 150 Hz. The percentage of change needed in the length of one of the pipes to produce 3 beats per second is

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

Question 21

The figure shows two waves traveling in the positive-x direction. The amplitude of the resultant wave is closest to

A) 2.0 mm
B) 1.8 mm
C) 1.4 mm
D) 1.0 mm
E) zero

Question 22

Two wave trains travel on a string under a constant tension T. Which of the following statements is NOT correct?

A) The two waves can have different speed.
B) The two waves can have different frequency.
C) The two waves can have different wavelength.
D) The superposition principle applies for the two waves.
E) At any point on the string, the resultant amplitude is the algebraic sum of the amplitudes of the two waves.

Question 23

The sources S₁ and S₂ are coherent sources, and the circular arcs represent wave crests. The position that corresponds to a path difference of two wavelengths is

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

Question 24

The figure represents a string of length L, fixed at both ends, vibrating in several harmonics. The 4th harmonic is shown in

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

Question 25

The figure shows two waves traveling in the positive-x direction. The phase difference between these two waves is closest to

A) 1.0 radians.
B) 1.5 radians.
C) 2.0 radians.
D) 2.5 radians.
E) 3.0 radians.

Question 26

The figure represents a string of length L, fixed at both ends, vibrating in several harmonics. The 3rd harmonic is shown in

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

Question 27

At P_1, the waves from sources S₁ and S₂ shown in the figure

A) are out of phase.
B) have a path difference of one wavelength.
C) have a path difference of two wavelengths.
D) are interfering destructively.
E) None of these is correct.

Question 28

One source of sound is at A and another is at B. The two sources are in phase. The distance AB = 10.0 m. The frequency of the sound waves from both sources is 1000 Hz, and both have the same amplitude. The speed of sound in air is 330 m/s. A receiver is at point C, and AB is perpendicular to AC. The greatest distance AC for which the signal at C is a minimum is

A) 33.0 cm
B) 152 m
C) 330 m
D) 303 m
E) 100 m

Question 29

Two sound waves, one wave is given by y₁ = p_o sin (kx - $\omega$ t) and the other by y₂ = p_o sin (kx - $\omega$ t + $\pi$ /2). The amplitude resulting from the interference of the two waves is

A) 2p_o
B)
C) 1.25p_o
D)
E) 0

Question 30

At P₂ the waves from sources S₁ and S₂ shown in the figure

A) are in phase.
B) have a path difference of one wavelength.
C) have a path difference of one-half wavelength.
D) are interfering constructively.
E) None of these is correct.

Question 31

Two speakers face each other at a distance of 1 m and are driven by a common audio oscillator. A first minimum in sound intensity is found 16.1 cm from the midpoint. If the velocity of sound is 330 m/s, find the frequency of the oscillator.

A) 256 Hz
B) 1024 Hz
C) 512 Hz
D) 341 Hz
E) 683 Hz

Question 32

The figure shows two waves traveling in the positive-x direction. The amplitude of the resultant wave is

A) 2.0 mm
B) 1.8 mm
C) 1.4 mm
D) 1.0 mm
E) 0.83 mm

Question 33

The figure shows two waves traveling in the positive-x direction. The phase difference between these two waves is

A) $\pi$ radians.
B) $\pi$ /2 radians.
C) $\pi$ /4 radians.
D) $\pi$ /8 radians.
E) $\pi$ /16 radians.

Question 34

The figure shows two waves traveling in the positive-x direction. The amplitude of the resultant wave is

A) 2.0 mm
B) 1.8 mm
C) 1.4 mm
D) 1.0 mm
E) zero

Question 35

The figure shows two waves traveling in the positive-x direction. The phase difference between these two waves is closest to

A) 3.1 radians.
B) 1.6 radians.
C) 1.1 radians.
D) 2.4 radians.
E) 0.2 radians.

Question 36

Two sound waves, one wave is given by y₁ = p_o sin ( $\omega$ ₁t), and the other by y₂ = p_o sin ( $\omega$ ₂t), where $\omega$ ₁ differs from $\omega$ ₂ by a rad/s. The maximum sound intensity of the beat frequency is

A) p_o²
B) 2p_o²
C) 4p_o²
D) 8p_o²

Question 37

Two sound waves, one wave is given by y₁ = p_o sin (kx - $\omega$ t) and the other by y₂ = p_o sin (kx - $\omega$ t + $\pi$ /4). The phase constant resulting from the interference of the two waves is

A) $\pi$ /8
B) $\pi$ /4
C) $\pi$ /2
D) $\pi$
E) 0

Question 38

The figure shows two waves traveling in the positive-x direction. The phase difference between these two waves is

A) $\pi$ radians.
B) $\pi$ /2 radians.
C) $\pi$ /4 radians.
D) $\pi$ /8 radians.
E) $\pi$ /16 radians.

Question 39

Two sources are said to be coherent if

A) they are of the same frequency and has a phase of zero.
B) they are of the same frequency and maintain a constant non-zero phase.
C) they are of the same intensity but different frequency and has a phase of zero.
D) they are of the same intensity but different frequency and maintain a constant non-zero phase.
E) (A) and (B)

Question 40

The figure shows two waves traveling in the positive-x direction. The amplitude of the resultant wave is closest to

A) 2.0 mm
B) 1.8 mm
C) 1.4 mm
D) 1.0 mm
E) zero

Question 41

The fundamental frequency of a vibrating string is f₁. If the tension in the string is doubled, the fundamental frequency becomes

A) /2
B) 11eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
C) 11eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
D)
E) 211eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11

Question 42

A string whose length is 1 m is fixed at both ends and vibrates according to the equation
Y(x, t) = 0.04 sin $\pi$ x cos 2 $\pi$ t
Where the units are SI. The total number of nodes exhibited by the string is

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

Question 43

The fundamental frequency of a vibrating string is . If the tension in the string is decreased by 50% while the linear density is held constant, the fundamental frequency becomes

A) 0.511eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
B) 0.711eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
C) 0.911eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
D) 11eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
E) None of these is correct.

Question 44

If the amplitude of a standing wave is doubled, the energy in the wave increases by a factor of

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

Question 45

The fundamental frequency of a vibrating string is . If the tension in the string is quadrupled while the linear density is held constant, the fundamental frequency becomes

A) 11eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
B) 1.211eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
C) 1.511eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
D) 1.711eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
E) 211eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11

Question 46

The figure shows several modes of vibration of a string fixed at both ends. The mode of vibration that represents the fifth harmonic is

A) 1
B) 2
C) 3
D) 4
E) None of these is correct.

Question 47

The figure represents a wire of length L, fixed at both ends, vibrating in several harmonics. The 3rd harmonic is shown in

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

Question 48

A microphone is placed at the node of a standing sound wave. What does the microphone pick up?

A) A constant and very high intensity sound.
B) A constant and very low intensity sound.
C) A varying high intensity sound.
D) A varying low intensity sound.
E) Unable to tell.

Question 49

Which of the following equations represents a standing wave? (The symbols have their usual meaning.)

A)
B)
C)
D)
E) (B) and (D)

Question 50

If both the tension and the length of a vibrating string are doubled while the linear density remains constant, the fundamental frequency of the string is multiplied by

A) 1
B) 2
C)
D)
E)

Question 51

The two progressive waves are moving with equal velocities and wavelengths but in opposite directions in the string. Which of the following gives all of the points that will be nodes in the resultant standing wave?

A) 2, 4, 6, 8, and 10
B) 2, 6, and 10
C) 1, 5, and 9
D) 3 and 7
E) 1, 3, 5, 7, and 9

Question 52

The fundamental frequency of a vibrating string is f₁. If the tension in the string is increased by 50% while the linear density is held constant, the fundamental frequency becomes

A)
B) 1.211eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
C) 1.511eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
D) 1.711eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11
E) 211eaa4be_8550_d4cd_9e4a_9b9f5c46c44e_TB6080_11

Question 53

The figure shows a wave on a string approaching its fixed end at a wall. When the wave reaches the wall and is reflected, a standing wave will be set up in the string. One of the antinodes in the standing wave will be found at position

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

Question 54

A standing wave is shown in the figure on the right. If the period of the wave is T, the shortest time it takes for the wave to go from the solid curve to the dashed curve is

A) T/4
B) T/3
C) T/2
D) 3T/4
E) None of these is correct.

Question 55

A string of linear density $\mu$ and length L is under a constant tension T = mg. One end of the string is attached to a tunable harmonic oscillator. A resonant standing wave is observed

A) at any frequency.
B) when the frequency where n = 1, 2, 3, ...
C) when the frequency where n = 1, 2, 3, ...
D) when the frequency where n = 1, 2, 3, ...and v_s is the speed of sound.
E) unable to tell

Question 56

One wave moves to the right and a second wave (reflected) moves to the left to form a stationary wave. At which point(s) does the stationary wave have a node?

A) 1
B) 3 and 5
C) 2
D) 4 and 6
E) 2, 4, and 6

Question 57

Four pendulums are hung from a light rod that is free to rotate about its long axis. The pendulums have lengths L, 2L, L/2 and L, and masses m, m/2, 2m and 4m respectively. Pendulum 1 is set to swing at its natural frequency. Which of the other three will, over time, also oscillate at the same frequency?

A) (2)
B) (3)
C) (4)
D) (2) and (3)
E) all three

Question 58

The figure represents a wire of length L, fixed at both ends, vibrating in several harmonics. The 7th harmonic is shown in

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

Question 59

The figure shows a wave on a string approaching its fixed end at a wall. When the wave reaches the wall and is reflected, a standing wave is set up in the string. One will observe a node at position

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

Question 60

A standing wave is created by oscillating a taut string at a frequency that corresponds to one of the resonant frequencies. The amplitude of the antinodes is very much larger than the amplitude of the oscillator. Does this violate the conservation of energy principle? Explain why.

A) Yes, since E is proportional to amplitude squared.
B) Yes, since there is large kinetic energy of the string, and this is much bigger than the energy from the oscillator.
C) No, energy from waves does not obey the conservation of energy principle in the first place.
D) No, the energy at the antinodes builds up after the first few cycles, after which the dissipation due to friction equals the energy supplied by the oscillator.
E) Whether it obeys the conservation of energy principle depends on the tension in the string.

Question 61

The air column in an organ pipe, which is closed at one end, is vibrating in such a way as to produce the second harmonic. A pressure node and displacement node, respectively, occur at

A) 1 and 3
B) 1 and 5
C) 7 and 4
D) 7 and 5
E) 5 and 3

Question 62

A string fixed at both ends is driven by a tuning fork to produce standing waves. If the tension in the string is increased,

A) the frequency increases.
B) the frequency decreases and the wave velocity remains constant.
C) the wavelength decreases.
D) the wave velocity increases.
E) the wave velocity decreases.

Question 63

A string 2.0 m long has a mass of 2.4 $\times$ 10^-2 kg. When fixed at both ends, it vibrates with a fundamental frequency of 150 Hz. The speed of a transverse wave in the string is

A) 3.6 m/s
B) 75 m/s
C) 0.30 km/s
D) 0.60 km/s
E) 0.63 km/s

Question 64

A stretched string is fixed at points 1 and 5. When it is vibrating in its first harmonic frequency, the nodes are at points

A) 1 and 5 only.
B) 1, 3, and 5.
C) 2 and 4.
D) 2, 3, and 4.
E) 1, 2, 3, 4, and 5.

Question 65

A vibrating tuning fork of frequency 640 Hz is held above a tube filled with water. Assume the speed of sound to be 330 m/s. As the water level is lowered, consecutive maxima in intensity are observed at intervals of about

A) 12.9 cm
B) 19.4 cm
C) 25.8 cm
D) 51.7 cm
E) 194 cm

Question 66

A stretched string of length L, fixed at both ends, is vibrating in its third harmonic. How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration?

A) L/6
B) L/4
C) L/5
D) L/2
E) None of these is correct.

Question 67

Of the sound sources shown, that which is vibrating with its first harmonic is the

A) whistle.
B) organ pipe.
C) vibrating string.
D) vibrating rod.
E) vibrating spring.

Question 68

On a standing-wave pattern, the distance between two consecutive nodes is d. The wavelength is

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

Question 69

A string is connected to a tuning fork whose frequency is 80.0 Hz and is held under tension by 0.500 kg. The tuning fork causes the string to vibrate as shown. The mass per unit length for the string is

A) 9.45 $\times$ 10^-4 kg/m
B) 6.80 $\times$ 10^-3 kg/m
C) 4.34 kg/m
D) 6.00 $\times$ 10^-3 kg/m
E) 3.85 $\times$ 10^-2 kg/m

Question 70

The air in a closed organ pipe vibrates as shown. The length of the pipe is 3.0 m. The frequency of vibration is 80 Hz. The speed of sound in the pipe is approximately

A) 80 m/s
B) 0.16 km/s
C) 0.24 km/s
D) 0.32 km/s
E) 0.96 km/s

Question 71

The figure shows a standing wave in a pipe that is closed at one end. The frequency associated with this wave pattern is called the

A) first harmonic.
B) second harmonic.
C) third harmonic.
D) fourth harmonic.
E) fifth harmonic.

Question 72

Of the sound sources shown, that which is vibrating with its first harmonic is

A) the whistle.
B) the organ pipe.
C) the vibrating string.
D) the vibrating rod.
E) None of these is correct.

Question 73

In a pipe that is open at one end and closed at the other and that has a fundamental frequency of 256 Hz, which of the following frequencies cannot be produced?

A) 768 Hz
B) 1.28 kHz
C) 5.12 kHz
D) 19.7 kHz
E) all of these can be produced

Question 74

When an organ pipe, which is closed at one end only, vibrates with a frequency that is three times its fundamental (first harmonic) frequency,

A) the sound produced travels at three times its former speed.
B) the sound produced is its fifth harmonic.
C) beats are produced.
D) the sound produced has one-third its former wavelength.
E) the closed end is a displacement antinode.

Question 75

A vibrating tuning fork of frequency 1080 Hz is held above a tube filled with water. Assume the speed of sound to be 330 m/s. As the water level is lowered, consecutive maxima in intensity are observed at intervals of about

A) 7.65 cm
B) 15.3 cm
C) 23.0 cm
D) 30.6 cm
E) 53.6 cm

Question 76

In a vibrating-string experiment, three loops are observed between points A and B when the mass on one end of the string is 100 g. The number of loops between A and B can be changed to two by replacing the 100-g mass with a mass of

A) 150 g
B) 225 g
C) 44.4 g
D) 66.7 g
E) 300 g

Question 77

A string fixed at both ends is vibrating in a standing wave. There are three nodes between the ends of the string, not including those on the ends. The string is vibrating at a frequency that is its

A) fundamental.
B) second harmonic.
C) third harmonic.
D) fourth harmonic.
E) fifth harmonic.

Question 78

A stretched string is fixed at points 1 and 5. When it is vibrating at the second harmonic frequency, the nodes of the standing wave are at points

A) 1 and 5.
B) 1, 3, and 5.
C) 1 and 3.
D) 2 and 4.
E) 1, 2, 3, 4, and 5.

Question 79

A stationary wave of amplitude A and period T exists in a rope. At a particular instant, the configuration of the rope is as shown. At an instant T later, the configuration of the rope is

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

Question 80

A string 2.0 m long has a mass of 2.4 $\times$ 10^-2 kg. When fixed at both ends, it vibrates with a fundamental frequency of 150 Hz. The frequency of the third harmonic of this fundamental is

A) 50 Hz
B) 75 Hz
C) 0.15 kHz
D) 0.45 kHz
E) 1.1 kHz