Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as Wiley PLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
Section 17.1 The Principle of Linear Superposition,
Section 17.2 Constructive and Destructive Interference of Sound Waves

| 1 | ssm The drawing graphs a string on which two rectangular pulses are traveling at a constant speed of 1 cm/s at time t = 0 s. Using the principle of linear superposition, draw the shape of the string’s pulses at t = 1 s, 2 s, 3 s, and 4 s.

| 2 | Two speakers, one directly behind the other, are each generating a 245-Hz sound wave. What is the smallest separation distance between the speakers that will produce destructive interference at a listener standing in front of them? The speed of sound is 343 m/s.

| 3 | In Concept Simulation 17.1 at you can explore how two pulses, traveling in opposite directions, combine to form a resultant pulse. Two pulses are traveling toward each other, each having a speed of 1 cm/s. At t = 0 s, their positions are shown in the drawing. When t = 1 s, what is the height of the resultant pulse at (a) x = 3 cm and at (b) x = 4 cm?

| 4 | In Figure 17-7, suppose that the separation between speakers A and B is 5.00 m and the speakers are vibrating in phase. They are playing identical 125-Hz tones, and the speed of sound is 343 m/s. What is the largest possible distance between speaker B and the observer at C, such that he observes destructive interference?

| 5 | ssm Review Example 1 in the text. Speaker A is moved further to the left, while ABC remains a right triangle. What is the separation between the speakers when constructive interference occurs again at point C?

| 6 | Suppose that the two speakers in Figure 17-7 are separated by 2.50 m and are vibrating exactly out of phase at a frequency of 429 Hz. The speed of sound is 343 m/s. Does the observer at C observe constructive or destructive interference when his distance from speaker B is (a) 1.15 m and (b) 2.00 m?

| 7 | ssm www The drawing shows a loudspeaker A and point C, where a listener is positioned. A second loud-speaker B is located somewhere to the right of A. Both speakers vibrate in phase and are playing a 68.6-Hz tone. The speed of sound is 343 m/s. What is the closest to speaker A that speaker B can be located, so that the listener hears no sound?

* | 8 | Refer to Interactive Solution 17.8 17.8 at to review a method by which this problem can be solved. Two loudspeakers on a concert stage are vibrating in phase. A listener is 50.5 m from the left speaker and 26.0 m from the right one. The listener can respond to all frequencies from 20 to 20 000 Hz, and the speed of sound is 343 m/s. What are the two lowest frequencies that can be heard loudly due to constructive interference?

** | 9 | Speakers A and B are vibrating in phase. They are directly facing each other, are 7.80 m apart, and are each playing a 73.0-Hz tone. The speed of sound is 343 m/s. On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker A?
Section 17.3 Diffraction

| 10 | Multiple-Concept Example 3 reviews the concepts that are important in this problem. The entrance to a large lecture room consists of two side-by-side doors, one hinged on the left and the other hinged on the right. Each door is 0.700 m wide. Sound of frequency 607 Hz is coming through the entrance from within the room. The speed of sound is 343 m/s. What is the diffraction angle of the sound after it passes through the doorway when (a) one door is open and (b) both doors are open?

| 11 | ssm Consult Multiple-Concept Example 3 for background pertinent to this problem. A speaker has a diameter of 0.30 m. (a) Assuming that the speed of sound is 343 m/s, find the diffraction angle for a 2.0-kHz tone. (b) What speaker diameter D should be used to generate a 6.0-kHz tone whose diffraction angle is as wide as that for the 2.0-kHz tone in part (a)?

| 12 | Sound exits a diffraction horn loudspeaker through a rectangular opening like a small doorway. Such a loudspeaker is mounted outside on a pole. In winter, when the temperature is 273 K, the diffraction angle has a value of 15.0°. What is the diffraction angle for the same sound on a summer day when the temperature is 311 K?

| 13 | For one approach to problems such as this, see Multiple-Concept Example 3. Sound emerges through a doorway, as in Figure 17-11. The width of the doorway is 77 cm, and the speed of sound is 343 m/s. Find the diffraction angle when the frequency of the sound is (a) 5.0 kHz and (b) 5.0 × 102 Hz.

* | 14 | A row of seats is parallel to a stage at a distance of 8.7 m from it. At the center and front of the stage is a diffraction horn loudspeaker. This speaker sends out its sound through an opening that is like a small doorway with a width D of 7.5 cm. The speaker is playing a tone that has a frequency of 1.0 × 104 Hz. The speed of sound is 343 m/s. What is the distance between two seats, located near the center of the row, at which the tone cannot be heard?

* | 15 | A 3.00-kHz tone is being produced by a speaker with a diameter of 0.175 m. The air temperature changes from 0 to 29 °C. Assuming air to be an ideal gas, find the change in the diffraction angle . Section 17.4 Beats

| 16 | Two out-of-tune flutes play the same note. One produces a tone that has a frequency of 262 Hz, while the other produces 266 Hz. When a tuning fork is sounded together with the 262-Hz tone, a beat frequency of 1 Hz is produced. When the same tuning fork is sounded together with the 266-Hz tone, a beat frequency of 3 Hz is produced. What is the frequency of the tuning fork?

| 17 | ssm Two pure tones are sounded together. The drawing shows the pressure variations of the two sound waves, measured with respect to atmospheric pressure. What is the beat frequency?

| 18 | Two pianos each sound the same note simultaneously, but they are both out of tune. On a day when the speed of sound is 343 m/s, piano A produces a wavelength of 0.769 m, while piano B produces a wavelength of 0.776 m. How much time separates successive beats?

| 19 | ssm When a guitar string is sounded along with a 440-Hz tuning fork, a beat frequency of 5 Hz is heard. When the same string is sounded along with a 436-Hz tuning fork, the beat frequency is 9 Hz. What is the frequency of the string?

| 20 | In Concept Simulation 17.2 at you can explore the concepts that are important in this problem. A 440.0-Hz tuning fork is sounded together with an out-of-tune guitar string, and a beat frequency of 3 Hz is heard. When the string is tightened, the frequency at which it vibrates increases, and the beat frequency is heard to decrease. What was the original frequency of the guitar string?

* | 21 | A sound wave is traveling in seawater, where the adiabatic bulk modulus and density are 2.31 X 109 Pa and 1025 kg/m3, respectively. The wavelength of the sound is 3.35 m. A tuning fork is struck underwater and vibrates at 440.0 Hz. What would be the beat frequency heard by an underwater swimmer?

** | 22 | Two tuning forks X and Y have different frequencies and produce an 8-Hz beat frequency when sounded together. When X is sounded along with a 392-Hz tone, a 3-Hz beat frequency is detected. When Y is sounded along with the 392-Hz tone, a 5-Hz beat frequency is heard. What are the frequencies fX and fY when (a)fX is greater than fY and (b)fX is less than fY?
Section 17.5 Transverse Standing Waves

| 23 | ssm The approach to solving this problem is similar to that taken in Multiple-Concept Example 4. On a cello, the string with the largest linear density (1.56 × 10−2 kg/m) is the C string. This string produces a fundamental frequency of 65.4 Hz and has a length of 0.800 m between the two fixed ends. Find the tension in the string.

| 24 | A string of length 0.28 m is fixed at both ends. The string is plucked and a standing wave is set up that is vibrating at its second harmonic. The traveling waves that make up the standing wave have a speed of 140 m/s. What is the frequency of vibration?

| 25 | The fundamental frequency of a string fixed at both ends is 256 Hz. How long does it take for a wave to travel the length of this string?

| 26 | Multiple-Concept Example 4 deals with the same concepts as this problem. A 41-cm length of wire has a mass of 6.0 g. It is stretched between two fixed supports and is under a tension of 160 N. What is the fundamental frequency of this wire?

| 27 | ssm Suppose the strings on a violin are stretched with the same tension and each has the same length between its two fixed ends. The musical notes and corresponding fundamental frequencies of two of these strings are G (196.0 Hz) and E (659.3 Hz). The linear density of the E string is 3.47 × 10−4 kg/m. What is the linear density of the G string?

| 28 | To review the concepts that play roles in this problem, consult Multiple-Concept Example 4. Sometimes, when the wind blows across a long wire, a low-frequency “moaning” sound is produced. This sound arises because a standing wave is set up on the wire, like a standing wave on a guitar string. Assume that a wire (linear density = 0.0140 kg/m) sustains a tension of 323 N because the wire is stretched between two poles that are 7.60 m apart. The lowest frequency that an average, healthy human ear can detect is 20.0 Hz. What is the lowest harmonic number n that could be responsible for the “moaning” sound?

| 29 | A string has a linear density of 8.5 × 10−3 kg/m and is under a tension of 280 N. The string is 1.8 m long, is fixed at both ends, and is vibrating in the standing wave pattern shown in the drawing. Determine the (a) speed, (b) wavelength, and (c) frequency of the traveling waves that make up the standing wave.

* | 30 | The drawing shows two strings that have the same length and linear density. The left end of each string is attached to a wall, while the right end passes over a pulley and is connected to objects of different weights (WA and WB). Different standing waves are set up on each string, but their frequencies are the same. If WA = 44 N, what is WB?

Problem 30

* | 31 | ssm The E string on an electric bass guitar has a length of 0.628 m and, when producing the note E, vibrates at a fundamental frequency of 41.2 Hz. Players sometimes add to their instruments a device called a “D-tuner.” This device allows the E string to be used to produce the note D, which has a fundamental frequency of 36.7 Hz. The D-tuner works by extending the length of the string, keeping all other factors the same. By how much does a D-tuner extend the length of the E string?

** | 32 | Review Conceptual Example 5 before attempting this problem. As the drawing shows, the length of a guitar string is 0.628 m. The frets are numbered for convenience. A performer can play a musical scale on a single string because the spacing between the frets is designed according to the following rule: When the string is pushed against any fret j, the fundamental frequency of the shortened string is larger by a factor of the twelfth root of two () than it is when the string is pushed against the fret j − 1. Assuming that the tension in the string is the same for any note, find the spacing (a) between fret 1 and fret 0 and (b) between fret 7 and fret 6.

** | 33 | ssm www The note that is three octaves above middle C is supposed to have a fundamental frequency of 2093 Hz. On a certain piano the steel wire that produces this note has a cross-sectional area of 7.85 × 10−7 m2. The wire is stretched between two pegs. When the piano is tuned properly to produce the correct frequency at 25.0 °C, the wire is under a tension of 818.0 N. Suppose the temperature drops to 20.0 °C. In addition, as an approximation, assume that the wire is kept from contracting as the temperature drops. Consequently, the tension in the wire changes. What beat frequency is produced when this piano and another instrument (properly tuned) sound the note simultaneously?
Section 17.6 Longitudinal Standing Waves, Section 17.7 Complex Sound Waves

| 34 | An organ pipe is open at both ends. It is producing sound at its third harmonic, the frequency of which is 262 Hz. The speed of sound is 343 m/s. What is the length of the pipe?

| 35 | ssm A tube of air is open at only one end and has a length of 1.5 m. This tube sustains a standing wave at its third harmonic. What is the distance between one node and the adjacent antinode?

| 36 | The range of human hearing is roughly from twenty hertz to twenty kilohertz. Based on these limits and a value of 343 m/s for the speed of sound, what are the lengths of the longest and shortest pipes (open at both ends and producing sound at their fundamental frequencies) that you expect to find in a pipe organ?

| 37 | Refer to Interactive Solution 17.37 17.37 at to review a method by which this problem can be solved. The fundamental frequencies of two air columns are the same. Column A is open at both ends, while column B is open at only one end. The length of column A is 0.70 m. What is the length of column B?

| 38 | Divers working in underwater chambers at great depths must deal with the danger of nitrogen narcosis (the “bends”), in which nitrogen dissolves into the blood at toxic levels. One way to avoid this danger is for divers to breathe a mixture containing only helium and oxygen. Helium, however, has the effect of giving the voice a high-pitched quality, like that of Donald Duck’s voice. To see why this occurs, assume for simplicity that the voice is generated by the vocal cords vibrating above a gas-filled cylindrical tube that is open only at one end. The quality of the voice depends on the harmonic frequencies generated by the tube; larger frequencies lead to higher-pitched voices. Consider two such tubes at 20 °C. One is filled with air, in which the speed of sound is 343 m/s. The other is filled with helium, in which the speed of sound is 1.00 × 103 m/s. To see the effect of helium on voice quality, calculate the ratio of the nth natural frequency of the helium-filled tube to the nth natural frequency of the air-filled tube.

| 39 | ssm A tube is open only at one end. A certain harmonic produced by the tube has a frequency of 450 Hz. The next higher harmonic has a frequency of 750 Hz. The speed of sound in air is 343 m/s. (a) What is the integer n that describes the harmonic whose frequency is 450 Hz? (b) What is the length of the tube?

* | 40 | Two loudspeakers face each other, vibrate in phase, and produce identical 440-Hz tones. A listener walks from one speaker toward the other at a constant speed and hears the loudness change (loud–soft–loud) at a frequency of 3.0 Hz. The speed of sound is 343 m/s. What is the walking speed?

* | 41 | A person hums into the top of a well and finds that standing waves are established at frequencies of 42, 70.0, and 98 Hz. The frequency of 42 Hz is not necessarily the fundamental frequency. The speed of sound is 343 m/s. How deep is the well?

* | 42 | Review Multiple-Concept Example 7 for background that is relevant to the kind of approach needed to solve this problem. Two ideal gases have the same temperature and the same value for (the ratio of the specific heat capacities at constant pressure and constant volume). A molecule of gas A has a mass of 7.31 × 10−26 kg, and a molecule of gas B has a mass of 1.06 × 10−25 kg. When gas A (speed of sound = 259 m/s) fills a tube that is open at both ends, the first overtone frequency of the tube is 386 Hz. Gas B fills another tube open at both ends, and this tube also has a first overtone frequency of 386 Hz. What is the length of the tube filled with gas B?

* | 43 | ssm www A vertical tube is closed at one end and open to air at the other end. The air pressure is 1.01 X 105 Pa. The tube has a length of 0.75 m. Mercury (mass density = 13 600 kg/m3) is poured into it to shorten the effective length for standing waves. What is the absolute pressure at the bottom of the mercury column, when the fundamental frequency of the shortened, air-filled tube is equal to the third harmonic of the original tube?

** | 44 | A tube, open at only one end, is cut into two shorter (nonequal) lengths. The piece that is open at both ends has a fundamental frequency of 425 Hz, while the piece open only at one end has a fundamental frequency of 675 Hz. What is the fundamental frequency of the original tube?

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