Section: 16-1 Topic: Superposition of Waves Type: Factual Web viewSection: 7–1 Topic:...

Lecture 7: Superposition and Standing Waves

Section: 7–1 Topic: Superposition of Waves Type: FactualThe 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. Ans: B

Section: 7–1 Topic: Superposition of Waves Type: Conceptual

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

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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 Ans: B

Section: 7–1 Topic: Superposition of Waves Type: FactualTwo 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. Ans: B

Section: 7–1 Topic: Superposition of Waves Type: NumericalTuning 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 Ans: B

Section: 7–1 Topic: Superposition of Waves Type: NumericalThe 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% Ans: B

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Section: 7–1 Topic: Superposition of Waves Type: Numerical

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 Ans: B

Section: 7–1 Topic: Superposition of Waves Type: ConceptualIf 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. Ans: A

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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 Ans: B

Section: 7–1 Topic: Superposition of Waves Type: ConceptualIf two identical waves with a phase difference of 6 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. Ans: A

Section: 7–1 Topic: Superposition of Waves Type: ConceptualIf two identical waves with a phase difference of 3 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. Ans: C

Section: 7–1 Topic: Superposition of Waves Type: NumericalWhat 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 Ans: C

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Section: 7–1 Topic: Superposition of Waves Type: NumericalA 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 Ans: A

Section: 7–1 Topic: Superposition of Waves Type: ConceptualTwo 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. D) equal to A1 + A2. B) the same as A1. E) coherent. C) the same as A2. Ans: C

Section: 7–1 Topic: Superposition of Waves Type: NumericalTwo 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 Ans: D

Section: 7–1 Topic: Superposition of Waves Type: NumericalMiddle 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 Ans: B

Section: 7–1 Topic: Superposition of Waves Type: NumericalA 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 Ans: C

Section: 7–1 Topic: Superposition of Waves Type: NumericalTwo 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 Ans: A

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Section: 7–1 Topic: Superposition of Waves Type: ConceptualWhen 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 Ans: E

Section: 7–1 Topic: Superposition of Waves Type: ConceptualTwo tones of equal amplitude but slightly different frequencies are emitted by a sound source. This gives rise to A) standing waves. D) beats. B) destructive interference. E) amplification. C) constructive interference. Ans: D


At P1, 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. Ans: B

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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. Ans: C


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 Ans: D

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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 Ans: D


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 Ans: C

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The figure shows two waves traveling in the positive-x direction. The phase difference between these two waves is A) radians. D) /8 radians. B) /2 radians. E) /16 radians. C) /4 radians. Ans: B


The figure shows two waves traveling in the positive-x direction. The phase difference between these two waves is A) radians. D) /8 radians. B) /2 radians. E) /16 radians. C) /4 radians. Ans: A

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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 Ans: E


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 Ans: D

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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. D) 2.4 radians. B) 1.6 radians. E) 0.2 radians. C) 1.1 radians. Ans: D


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. D) 2.5 radians. B) 1.5 radians. E) 3.0 radians. C) 2.0 radians. Ans: B

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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 Ans: B

Section: 7–1 Topic: Superposition of Waves Type: NumericalTwo 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 Ans: C

Section: 7–1 Topic: Superposition of Waves Type: ConceptualTwo 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.Ans: A

Section: 7–1 Topic: Superposition of Waves Type: NumericalTwo sound waves, one wave is given by y1 = po sin (kx – t) and the other by y2 = po sin (kx – t + /2). The amplitude resulting from the interference of the two waves is

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

Ans: D

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Section: 7–1 Topic: Superposition of Waves Type: NumericalTwo sound waves, one wave is given by y1 = po sin (kx – t) and the other by y2 = po sin (kx – t + /4). The phase constant resulting from the interference of the two waves isA) /8 B) /4 C) /2 D) E) 0 Ans: A

Section: 7–1 Topic: Superposition of Waves Type: NumericalTwo sound waves, one wave is given by y1 = po sin (t), and the other by y2 = po sin (t), where 1 differs from 2 by a rad/s. The maximum sound intensity of the beat frequency is A) po

2 B) 2po2 C) 4po

2 D) 8po2 A) 0

Ans: C

Section: 7–1 Topic: Superposition of Waves Type: FactualTwo sources are said to be coherent ifA) 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) Ans: E

Section: 7–2 Topic: Standing Waves Type: Conceptual

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 Ans: D

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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 Ans: C


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 Ans: D

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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 Ans: B


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 Ans: D

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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 Ans: C


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 Ans: E


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 D) 3 and 7 B) 2, 6, and 10 E) 1, 3, 5, 7, and 9 C) 1, 5, and 9 Ans: E

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Section: 7–2 Topic: Standing Waves Type: NumericalA string whose length is 1 m is fixed at both ends and vibrates according to the equation

y(x, t) = 0.04 sin x cos 2t

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 Ans: B

Section: 7–2 Topic: Standing Waves Type: ConceptualIf 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

Ans: E

Section: 7–2 Topic: Standing Waves Type: NumericalIf 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) Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalThe fundamental frequency of a vibrating string is f1. If the tension in the string is doubled, the fundamental frequency becomes A) f1/2 B) C) f1 D) E) 2f1

Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalThe 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) f1 B) 1.2f1 C) 1.5f1 D) 1.7f1 E) 2f1

Ans: B

Section: 7–2 Topic: Standing Waves Type: NumericalThe fundamental frequency of a vibrating string is f1. If the tension in the string is decreased by 50% while the linear density is held constant, the fundamental frequency becomes A) 0.5f1 B) 0.7f1 C) 0.9f1 D) f1 E) None of these is correct. Ans: B

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Section: 7–2 Topic: Standing Waves Type: NumericalThe fundamental frequency of a vibrating string is f1. If the tension in the string is quadrupled while the linear density is held constant, the fundamental frequency becomes A) f1 B) 1.2f1 C) 1.5f1 D) 1.7f1 E) 2f1

Ans: E


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. Ans: E

Section: 7–2 Topic: Standing Waves Type: FactualWhich of the following equations represents a standing wave? (The symbols have their usual meaning.)A) C)B) )sin()(cos tkxAy D)E) (B) and (D)Ans: E

Section: 7–2 Topic: Standing Waves Type: ConceptualA 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. Ans: C

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Section: 7–2 Topic: Standing Waves Type: ConceptualA string of linear density 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 vs is the speed of sound.

E) unable to tellAns: B

Section: 7–2 Topic: Standing Waves Type: ConceptualA 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.Ans: D

Section: 7–2 Topic: Standing Waves Type: ConceptualA 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.Ans: B

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Section: 7–2 Topic: Standing Waves Type: ConceptualFour 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 threeAns: C

Section: 7–2 Topic: Standing Waves Type: NumericalIn 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 Ans: B

Section: 7–2 Topic: Standing Waves Type: NumericalA string 2.0 m long has a mass of 2.4 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 Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalA string 2.0 m long has a mass of 2.4 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 Ans: D

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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 Ans: E

Section: 7–2 Topic: Standing Waves Type: Numerical

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 10–4 kg/m D) 6.00 10–3 kg/m B) 6.80 10–3 kg/m E) 3.85 10–2 kg/m C) 4.34 kg/m Ans: A

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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. Ans: B


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

Section: 7–2 Topic: Standing Waves Type: ConceptualA 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. Ans: D

Section: 7–2 Topic: Standing Waves Type: Conceptual 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. D) fourth harmonic. B) second harmonic. E) fifth harmonic. C) third harmonic. Ans: E

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Of the sound sources shown, that which is vibrating with its first harmonic is A) the whistle. D) the vibrating rod. B) the organ pipe. E) None of these is correct. C) the vibrating string. Ans: E


Of the sound sources shown, that which is vibrating with its first harmonic is the A) whistle. D) vibrating rod. B) organ pipe. E) vibrating spring. C) vibrating string. Ans: A

Section: 7–2 Topic: Standing Waves Type: ConceptualWhen 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. Ans: D

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Section: 7–2 Topic: Standing Waves Type: Numerical

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 Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalA 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 Ans: C

Section: 7–2 Topic: Standing Waves Type: NumericalA 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 Ans: B


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 3Ans: D

Section: 7–2 Topic: Standing Waves Type: ConceptualA 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. D) fourth harmonic. B) second harmonic. E) fifth harmonic. C) third harmonic.

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Ans: D

Section: 7–2 Topic: Standing Waves Type: ConceptualOn 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 Ans: D

Section: 7–2 Topic: Standing Waves Type: ConceptualA 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. Ans: E

Section: 7–2 Topic: Standing Waves Type: ConceptualIn 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 D) 19.7 kHz B) 1.28 kHz E) all of these can be produced C) 5.12 kHz Ans: C

Section: 7–2 Topic: Standing Waves Type: ConceptualThe fundamental frequency of a pipe that has one end closed is 256 Hz. When both ends of the same pipe are opened, the fundamental frequency is A) 64.0 Hz B) 128 Hz C) 256 Hz D) 512 Hz E) 1.02 kHz Ans: B

Section: 7–2 Topic: Standing Waves Type: ConceptualThe standing waves on a string of length L that is fixed at both ends have a speed v. The three lowest frequencies of vibration are A) v/L, 2v/L, and 3v/L D) L/v, 2L/v, and 3L/v B) v/2L, v/L, and 3v/2L E) /3, 2/3, and 3/3 C) /2, , and 3/2 Ans: B

Section: 7–2 Topic: Standing Waves Type: ConceptualStanding waves exist in a string of length L that is fixed at one end and free at the other. The speed of the waves on the string is v. The three lowest frequencies of vibration are A) v/4L, v/2L, and 3v/4L D) v/4L, 3v/4L, and 5v/4L B) v/2L, v/L, and 3v/2L E) /3, 2/3, and 3/3 C) /4, /2, and 3/4 Ans: D

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Section: 7–2 Topic: Standing Waves Type: ConceptualThe standing waves in air in a pipe of length L that is open at both ends have a speed v. The frequencies of the three lowest harmonics are A) v/L, 2v/L, and 3v/L D) L/v, 2L/v, and 3L/v B) v/2L, v/L, and 3v/2L E) /3, 2/3, and 3/3 C) /2, , and 3/2 Ans: B

Section: 7–2 Topic: Standing Waves Type: ConceptualThe standing waves in air in a pipe of length L that is open at one end and closed at the other have a speed v. The frequencies of the three lowest harmonics are A) v/4L, v/2L, and 3v/4L D) v/4L, 3v/4L, and 5v/4L B) v/2L, v/L, and 3v/2L E) /3, 2/3, and 3/3 C) /4, /2, and 3/4 Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalA 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz. What is the speed of the waves on this string? A) 220 m/s B) 440 m/s C) 660 m/s D) 880 m/s E) 1.10 km/s Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalThe human vocal tract can be thought of as a tube that is open at one end. If the length of this tube is 17 cm (about average for an adult male), what are the lowest two harmonics? A) 500 Hz, 1500 Hz D) 1000 Hz, 3000 Hz B) 500 Hz, 1000 Hz E) 1500 Hz, 2500 Hz C) 1000 Hz, 2000 Hz Ans: A

Section: 7–2 Topic: Standing Waves Type: NumericalFor a tube of length 57.0 cm that is open at both ends, what is the frequency of the fundamental mode? (the speed of sound in air is 340 m/s) A) 149 Hz B) 447 Hz C) 596 Hz D) 298 Hz E) 746 Hz Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalA string fixed at both ends is 50.0 cm long and has a tension that causes the frequency of its fundamental to be 262 Hz. If the tension is increased by 4%, what does the fundamental frequency become? A) 252 Hz B) 257 Hz C) 264 Hz D) 267 Hz E) 272 Hz Ans: D

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Section: 7–2 Topic: Standing Waves Type: NumericalA clarinet, which is essentially a tube that is open at one end, is properly tuned to concert A (440 Hz) indoors, where the temperature is 20ºC and the speed of sound is 340 m/s. The musician then takes the instrument to play an outdoor concert, where the temperature is 0ºC and the speed of sound is 331 m/s. What is the frequency of the A played on the cold clarinet? (Ignore any thermal changes in the body of the clarinet itself.) A) 417 Hz B) 428 Hz C) 434 Hz D) 445 Hz E) 451 Hz Ans: B

Section: 7–2 Topic: Standing Waves Type: NumericalSound has a velocity of 335 m/s in air. For an air column that is closed at both ends to resonate to a frequency of 528 Hz, the length of the air column could be A) 79.2 cm B) 55.5 cm C) 47.5 cm D) 31.7 cm E) 15.8 cm Ans: D


The sound wave in an organ tube has a wavelength that is equal to the distance between A) A and B. D) the antinodes farthest apart. B) A and C. E) None of these is correct. C) the nodes farthest apart. Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalThe third harmonic of a tube closed at one end is 735 Hz. If the speed of sound in air is 335 m/s, the length of the tube must be A) 11.6 cm B) 22.9 cm C) 34.1 cm D) 45.7 cm E) 57.3 cm Ans: C

Section: 7–2 Topic: Standing Waves Type: NumericalThe ratio of the fundamental frequency (first harmonic) of an open pipe to that of a closed pipe of the same length is A) 2:1 B) 7:8 C) 4:5 D) 3:2 E) 1:2 Ans: A

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Section: 7–2 Topic: Standing Waves Type: NumericalThe wave function y(x,t) for a standing wave on a string fixed at both ends is given by y(x,t) = 0.080 sin 6.0x cos 600t where the units are SI. The amplitudes of the traveling wave that result in this standing wave are A) 0.04 m B) 0.08 m C) 0.02 m D) 0.16 m E) impossible to tell given this information about the standing wave. Ans: A

Section: 7–2 Topic: Standing Waves Type: NumericalThe wave function y(x,t) for a standing wave on a string fixed at both ends is given by y(x,t) = 0.080 sin 6.0x cos 600t where the units are SI. The wavelength of this wave is A) 6.00 m B) 1.05 m C) 600 m D) 0.010 m E) impossible to tell given this information about the standing wave. Ans: B

Section: 7–2 Topic: Standing Waves Type: NumericalThe wave function y(x,t) for a standing wave on a string fixed at both ends is given by y(x,t) = 0.080 sin 6.0x cos 600t where the units are SI. The speed of the traveling waves that result in this standing wave is A) 6.00 m B) 1.05 m C) 600 m D) 0.010 m E) impossible to tell given this information about the standing wave. Ans: C

Section: 7–2 Topic: Standing Waves Type: NumericalThe wave function y(x,t) for a standing wave on a string fixed at both ends is given by y(x,t) = 0.080 sin 6.0x cos 600t where the units are SI. The distance between successive nodes on the string is A) 0.24 m B) 0.08 m C) 0.02 m D) 0.52 m E) impossible to tell given this information about the standing wave. Ans: C

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Section: 7–2 Topic: Standing Waves Type: NumericalA string with mass density equal to 0.0025 kg/m is fixed at both ends and at a tension of 290 N. Resonant frequencies are found at 558 Hz and the next one at 744 Hz. What is the fundamental frequency of the string? A) 558 Hz B) 372 Hz C) 93 Hz D) 186 Hz E) none of the above Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalA string with mass density equal to 0.0025 kg/m is fixed at both ends and at a tension of 290 N. Resonant frequencies are found at 558 Hz and the next one at 744 Hz. To what harmonic does the 558 Hz resonance correspond? A) 1 B) 2 C) 3 D) 4 E) 5 Ans: C

Section: 7–2 Topic: Standing Waves Type: NumericalA string with mass density equal to 0.0025 kg/m is fixed at both ends and at a tension of 290 N. Resonant frequencies are found at 558 Hz and the next one at 744 Hz. What is the length of the wire? A) 0.8 m B) 1.6 m C) 3.2 m D) 1.2 m E) 0.4 m Ans: A

Section: 7–2 Topic: Standing Waves Type: NumericalA wire of mass 1.1 g is under a tension of 100 N. If its third overtone is at a frequency of 750 Hz, calculate the length of the wire. A) 72 cm B) 101 cm C) 36 cm D) 65 cm E) None of the above Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalA vibrating tuning fork is held above a tube filled with water. The first two resonances occur when the water level is lowered by 14.2 cm and 44.2 cm from the top of the tube. If there is a small end correction that adds a small extra length L? to the effective length of the air column, calculate the frequency of the tuning fork. Assume the speed of sound to be 330 m/s. A) 560 Hz B) 581 Hz C) 550 Hz D) 1100 Hz E) 1120 Hz Ans: C

Section: 7–2 Topic: Standing Waves Type: NumericalA vibrating tuning fork of 850 Hz is held above a tube filled with water. The first and third resonances occur when the water level is lowered by 8.8 cm and 47.6 cm from the top of the tube. If there is a small end correction that adds a small extra length L to the effective length of the air column, calculate L. Assume the speed of sound to be 330 m/s. A) 0.2 cm B) 0.9 cm C) 0.4 cm D) 0.6 cm E) 1.1 cm Ans: B

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Section: 7–2 Topic: Standing Waves Type: NumericalA vibrating tuning fork of 725 Hz is held above a tube filled with water. Successive resonances are heard when the water level is lowered by 11.5 cm and 34.5 cm from the top of the tube. Calculate a value for the speed of sound. (Hint: remember the small end correction L at the top of the tube.) A) 333 m/s B) 343 m/s C) 325 m/s D) 315 m/s E) 338 m/s Ans: A

Section: 7–2 Topic: Standing Waves Type: ConceptualTwo pipes closed at one end of length L1 and L2 are excited at their resonant frequencies. If the beat period is Bf Hz, then the velocity of sound is given by: A) Bf L1 L2 / (4L1 4L2) D) 4 Bf L1 L2 / (L1 L2) B) 4 Bf L1 L2 / (4L1 4L2) E) 4 Bf L1 L2 / (L1 + L2) C) 16 Bf L1 L2 / (4L1 + 4L2) Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalA vibrating tuning fork of 300 Hz is held above a tube filled with water. The first resonance is heard when the water level is lowered by 26.1 cm. A second tuning fork of 400 Hz is held above the tube, and its first resonance occurs when the water level is lowered by 19.3 cm from the top. Calculate a value for the speed of sound. (Hint: remember the small end correction L at the top of the tube.) A) 333 m/s B) 343 m/s C) 325 m/s D) 315 m/s E) 338 m/s Ans: C

Section: 7–2 Topic: Standing Waves Type: NumericalA guitar string of length 105 cm is in resonance with a tuning fork of frequency f. Using the fret board the length of the string is shortened by 1.5 cm while keeping the tension in the string constant. Now a beat frequency of 10 Hz is heard between the string and the tuning fork. What is the frequency of the tuning fork? A) 230 Hz B) 1380 Hz C) 345 Hz D) 690 Hz E) none of the above Ans: D

Section: 7–2 Topic: Standing Waves Type: NumericalWire A is the same mass per unit length as wire B. However wire A is twice as long as wire B and has three times as much tension on it. Calculate the fundamental frequency of wire A divided by wire B. A) 0.87 B) 0.66 C) 0.43 D) 0.75 E) 1.50 Ans: A

Section: 7–2 Topic: Standing Waves Type: NumericalWhat is the third harmonic of an open-both-ends organ pipe of length 1.5 m? Assume the speed of sound to be 340 m/s. A) 229 Hz B) 340 Hz C) 457 Hz D) 686 Hz E) none of the above Ans: B

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Section: 7–2 Topic: Standing Waves Type: NumericalA piano tuner hears a beat every 0.33 seconds when he hits a note and compares it to his reference tone at 163 Hz. What is the lowest possible frequency of the piano note? A) 44.9 Hz B) 166.0 Hz C) 162.7 Hz D) 163.3 Hz E) 160.0 Hz Ans: E

Section: 7–2 Topic: Standing Waves Type: NumericalTwo identical loudspeakers are driven in phase by the same amplifier. The speakers are positioned a distance of 3.2 m apart. A person stands 4.1 m away from one speaker and 4.8 m away from the other. Calculate the second lowest frequency that results in destructive interference at the point where the person is standing. Assume the speed of sound to be 340 m/s. A) 245 Hz B) 735 Hz C) 1225 Hz D) 490 Hz E) 1470 Hz Ans: B

Section: 7–2 Topic: Standing Waves Type: NumericalA pipe produces successive harmonics at 300 Hz and 350 Hz. Calculate the length of the pipe and state whether it is closed at one end or not. Assume the speed of sound to be 340 m/s. A) 1.7 m closed one end B) 3.4 m open both ends C) 4.0 m closed one end D) 8.0 m closed one end E) 4.0 m open both ends Ans: B

Section: 7–3 Topic: Additional Topics Type: ConceptualThe reason we can tell the difference between a trumpet and a clarinet when they both play the same pitch is that they have A) the same overtones. D) different waveforms. B) the same harmonics. E) harmonic syntheses. C) different fundamental frequencies. Ans: D

Section: 7–3 Topic: Additional Topics Type: ConceptualThe electronic music synthesizer is based on the results of A) harmonic synthesis. D) Fourier analysis. B) overtones. E) all of these factors. C) tone quality. Ans: E

Section: 7–3 Topic: Additional Topics Type: FactualA string with length L is fixed on both ends. If o = 2L and fo = v/, the wave function for the harmonic shown is

A) D)

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B) E)C)Ans: E

Section: 7–3 Topic: Additional Topics Type: Conceptual

The complex wave whose frequency spectrum is shown in the figure is made up of waves whose frequencies are A) 1, 2, and 4. D) 1 and 4. B) 100, 200, and 400. E) 100 and 400. C) 100, 100, and 400. Ans: B


The complex wave whose frequency spectrum is shown in the figure is made up of waves whose relative amplitudes are A) 1, 2, and 4. D) 200 and 400. B) 100, 200, and 400. E) 1 and 2. C) 1 and 4. Ans: A

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The frequency spectrum of the composite wave 1 + 3 + 5 shown is best represented by

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


An examination of this frequency spectrum allows you to conclude that A) the odd harmonics 1 through 19 are present in the composite wave. B) the even harmonics 2 through 20 are present in the composite wave. C) the amplitudes of the component waves are equal. D) the wave form is a simple sinusoid. E) None of these is correct. Ans: A

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Section: 7–3 Topic: Additional Topics Type: ConceptualThe three curves show the harmonics of a pipe that is closed one end and open the other end. The fundamental frequency is fo. The three harmonics are

A) fo, 2fo, 3fo D) fo, 3fo, 4fo

B) fo, 5fo, 7fo E) fo, 6fo, 8fo

C) fo, 4fo, 6fo

Ans: B

Use the figure to the right to answer the next two problems.

The graph shows three harmonics.

-2

-1

0

1

2

0 1

Section: 7–3 Topic: Additional Topics Type: ConceptualThe frequency spectrum of the composite wave is best represented by

-0.5

0.5

1.5

2.5

3.5

0 1

A)

-0.5

0.5

1.5

2.5

0 1

B)

-0.5

0.5

1.5

2.5

3.5

0 1

C)

-0.5

0.5

1.5

2.5

3.5

0 1

D)E) None of the aboveAns: B

Section: 7–3 Topic: Additional Topics Type: ConceptualIf fo is the fundamental frequency, the three harmonics and their relative intensities can best be written asA) fo, 2fo, 3fo, and 2:1:0.5 D) fo, 3fo, 4fo, and 2:1:0.5B) fo, 3fo, 5fo, and 2:1:1 E) fo, 6fo, 8fo, and 2:1:1C) fo, 3fo, 5fo, and 2:1:0.5Ans: C

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Use the figure to the right to answer the next two problems.

The graph shows a wave pulse of width w = 5 cm and speed v = 100 m/s.

Section: 7–3 Topic: Additional Topics Type: NumericalThe duration of the wave pulse is A) 0.005 s B) 0.0005 s C) 0.001 s D) 0.02 s E) 0.5 sAns: B

Section: 7–3 Topic: Additional Topics Type: NumericalThe range of frequencies isA) 2000 s 1 B) 200 s 1 C) 1000 s 1 D) 50 s 1 E) 5 s 1 Ans: A

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