College Physics: A Strategic Approach

Randall D. Knight, Brian Jones, Stuart Field

Chapter 16

Superposition and Standing Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

Figure $\mathrm{P} 16.1$ is a snapshot graph at $t=0$ s of two waves on a taut string approaching each other at $1 \mathrm{m} / \mathrm{s} .$ Draw six snapshot graphs, stacked vertically, showing the string at 1 s intervals from $t=1$ s to $t=6 \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

Figure $\mathrm{P} 16.2$ is a snapshot graph at $t=0$ s of two waves approaching each other at $1 \mathrm{m} / \mathrm{s}$. Draw two snapshot graphs, stacked vertically, showing the string at $t=2 \mathrm{s}$ and $3 \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 3

Figure $\mathrm{P} 16.2$ is a snapshot graph at $t=0 \mathrm{s}$ of two waves approaching each other at $1 \mathrm{m} / \mathrm{s}$. Draw a history graph of the point of the string at $x=4 \mathrm{cm} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

Figure $\mathrm{P} 16.4 \mathrm{a}$ is a snapshot graph at $t=0 \mathrm{s}$ of two waves on a string approaching each other at $1 \mathrm{m} / \mathrm{s}$. At what time was the snapshot graph in Figure $\mathrm{P} 16.4 \mathrm{b}$ taken?
(a) $y(\mathrm{cm})$ at $t=0 \mathrm{s}$
(b) $y(\mathrm{cm})$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

Figure $\mathrm{P} 16.5$ is a snapshot graph at $t=0$ s of two waves on a string approaching each other at $1 \mathrm{m} / \mathrm{s} .$ List the values of the displacement of the string at $x=5.0 \mathrm{cm}$ at $1 \mathrm{s}$ intervals from $t=0 \mathrm{s}$ to $t=6 \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 6

Figure $\mathrm{P} 16.6$ is a snapshot graph at $t=0 \mathrm{s}$ of a pulse on a string moving to the right at $1 \mathrm{m} / \mathrm{s}$. The string is fixed at $x=3 \mathrm{m} .$ Draw a snapshot graph of the string at time $t=1.25 \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

At $t=0$ s, a small "upward" (positive $y$ ) pulse centered at $x=6.0 \mathrm{m}$ is moving to the right on a string with fixed ends at $x=0.0 \mathrm{m}$ and $x=10.0 \mathrm{m} .$ The wave speed on the string is $4.0 \mathrm{m} / \mathrm{s}$. At what time will the string next have the same appearance that it did at $t=0$ s?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

You are holding one end of an elastic cord that is fastened to a wall $3.0 \mathrm{m}$ away. You begin shaking the end of the cord at $3.5 \mathrm{Hz},$ creating a continuous sinusoidal wave of wavelength $1.0 \mathrm{m} .$ How much time will pass until a standing wave fills the entire length of the string?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

A 2.0-m-long string is fixed at both ends and tightened until the wave speed is $40 \mathrm{m} / \mathrm{s}$. What is the frequency of the standing wave shown in Figure $\mathrm{P} 16.9 ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

Figure P16.10 shows a standing wave oscillating at $100 \mathrm{Hz}$ on a string. What is the wave speed?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

A bass guitar string is 89 cm long with a fundamental frequency of $30 \mathrm{Hz}$. What is the wave speed on this string?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

The four strings of a bass guitar are $0.865 \mathrm{m}$ long and are tuned to the notes $\mathrm{G}(98 \mathrm{Hz}), \mathrm{D}(73.4 \mathrm{Hz}), \mathrm{A}(55 \mathrm{Hz}),$ and $\mathrm{E}(41.2 \mathrm{Hz}) .$ In one bass guitar, the $\mathrm{G}$ and $\mathrm{D}$ strings have a linear mass density of $5.8 \mathrm{g} / \mathrm{m},$ and the $\mathrm{A}$ and $\mathrm{E}$ strings have a linear mass density of $26.8 \mathrm{g} / \mathrm{m} .$ What is the total force exerted by the strings on the neck?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 13

What are the three longest wavelengths for standing waves on a 240 -cm-long string that is fixed at both ends?
b. If the frequency of the second-longest wavelength is $50.0 \mathrm{Hz},$ what is the frequency of the third-longest wavelength?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

The G string on a guitar is $59 \mathrm{cm}$ long and has a fundamental frequency of 196 Hz. A guitarist can play different notes by pushing the string against various frets, which changes the string's length. The first fret from the neck gives $A b(207.65 \mathrm{Hz})$ the second fret gives $\mathrm{A}(220 \mathrm{Hz}) .$ How far apart are the first and second frets?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

A guitar string with a linear density of $2.0 \mathrm{g} / \mathrm{m}$ is stretched between supports that are $60 \mathrm{cm}$ apart. The string is observed to form a standing wave with three antinodes when driven at a frequency of $420 \mathrm{Hz}$. What are (a) the frequency of the fifth harmonic of this string and (b) the tension in the string?

Donald Albin

Numerade Educator

Problem 16

Some guitarists like the feel of a set of strings that all have the same tension. For such a guitar, the G string $(196 \mathrm{Hz})$ has a mass density of $0.31 \mathrm{g} / \mathrm{m} .$ What is the mass density of the $\mathrm{A}$ string $(110 \mathrm{Hz}) ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

The lowest note on a grand piano has a frequency of $27.5 \mathrm{Hz}$. The entire string is $2.00 \mathrm{m}$ long and has a mass of $400 \mathrm{g}$. The vibrating section of the string is $1.90 \mathrm{m}$ long. What tension is needed to tune this string properly?

Donald Albin

Numerade Educator

Problem 18

An experimenter finds that standing waves on a string fixed at both ends occur at $24 \mathrm{Hz}$ and $32 \mathrm{Hz},$ but at no frequencies in between.
a. What is the fundamental frequency?
b. Draw the standing-wave pattern for the string at 32 Hz.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 19

Ocean waves of wavelength $26 \mathrm{m}$ are moving directly toward a concrete barrier wall at $4.4 \mathrm{m} / \mathrm{s}$. The waves reflect from the wall, and the incoming and reflected waves overlap to make a lovely standing wave with an antinode at the wall. (Such waves are a common occurrence in certain places.) A kayaker is bobbing up and down with the water at the first antinode out from the wall. How far from the wall is she? What is the period of her up-and-down motion?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

The lowest frequency in the audible range is $20 \mathrm{Hz}$. What are the lengths of (a) the shortest open-open tube and (b) the shortest open-closed tube needed to produce this frequency?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

To study the physical basis of underwater hearing in frogs, scientists used a vertical tube filled with water to a depth of $1.4 \mathrm{m} .$ A microphone at the bottom of the tube was used to create standing sound waves in the water column. Frogs were lowered to different depths where the standing waves created large or small pressure variations. Because the microphone creates the sound, the bottom of the tube is a pressure antinode; the water's surface, fixed at atmospheric pressure, is a node.
a. What is the fundamental frequency of this water-filled tube?
b. A frog sits on a platform located $0.28 \mathrm{m}$ from the bottom. What is the lowest frequency that would result in a sound node at this point?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

The vuvuzela is a simple horn, typically 0.65 m long, that fans use to make noise at sporting events. What is the frequency of the fundamental note produced by a vuvuzela?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

The world's longest organ pipe, in the Boardwalk Hall Auditorium in Atlantic City, is 64 feet long. What is the fundamental frequency of this open-open pipe?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

An organ pipe is made to play a low note at $27.5 \mathrm{Hz}$, the same as the lowest note on a piano. Assuming a sound speed of $343 \mathrm{m} / \mathrm{s},$ what length open-open pipe is needed? What length open-closed pipe would suffice?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 25

The speed of sound in room temperature $\left(20^{\circ} \mathrm{C}\right)$ air is $343 \mathrm{m} / \mathrm{s} ;$ in room temperature helium, it is $1010 \mathrm{m} / \mathrm{s} .$ The fundamental frequency of an open-closed tube is 315 Hz when the tube is filled with air. What is the fundamental frequency if the air is replaced with helium?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

Parasaurolophus was a dinosaur whose distinguishing feature was a hollow crest on the head. The 1.5 -m-long hollow tube in the crest had connections to the nose and throat, leading some investigators to hypothesize that the tube was a resonant chamber for vocalization.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

A drainage pipe running under a freeway is $30.0 \mathrm{m}$ long. Both ends of the pipe are open, and wind blowing across one end causes the air inside to vibrate.
a. If the speed of sound on a particular day is $340 \mathrm{m} / \mathrm{s},$ what will be the fundamental frequency of air vibration in this pipe?
b. What is the frequency of the lowest harmonic that would be audible to the human ear?
c. What will happen to the frequency in the later afternoon as the air begins to cool?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

The pan flute is a musical instrument consisting of a number of closed-end tubes of different lengths. When the musician blows over the open ends, each tube plays a different note. The longest pipe is $0.33 \mathrm{m}$ long. What is the frequency of the note it plays?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

Although the vocal tract is quite complicated, we can make a simple model of it as an open-closed tube extending from the opening of the mouth to the diaphragm, the large muscle separating the abdomen and the chest cavity. What is the length of this tube if its fundamental frequency equals a typical speech frequency of $200 \mathrm{Hz}$ ? Assume a sound speed of $350 \mathrm{m} / \mathrm{s}$. Does this result for the tube length seem reasonable, based on observations on your own body?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

You know that you sound better when you sing in the shower. This has to do with the amplification of frequencies that correspond to the standing-wave resonances of the shower enclosure. A shower enclosure is created by adding glass doors and tile walls to a standard bathtub, so the enclosure has the dimensions of a standard tub, $0.75 \mathrm{m}$ wide and $1.5 \mathrm{m}$ long. Standing sound waves can be set up along either axis of the enclosure. What are the lowest two frequencies that correspond to resonances on each axis of the shower? These frequencies will be especially amplified. Assume a sound speed of $343 \mathrm{m} / \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

$\mathrm{A}$ child has an ear canal that is $1.3 \mathrm{cm}$ long. At what sound frequencies in the audible range will the child have increased hearing sensitivity?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

When a sound wave travels directly toward a hard wall, the incoming and reflected waves can combine to produce a standing wave. There is an antinode right at the wall, just as at the end of a closed tube, so the sound near the wall is loud. You are standing beside a brick wall listening to a $50 \mathrm{Hz}$ tone from a distant loudspeaker. How far from the wall must you move to find the first quiet spot? Assume a sound speed of $340 \mathrm{m} / \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

The first formant of your vocal system can be modeled as the resonance of an open-closed tube, the closed end being your vocal cords and the open end your lips. Estimate the frequency of the first formant from the graph of Figure $16.23,$ and then estimate the length of the tube of which this is a resonance. Does your result seem reasonable?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

When you voice the vowel sound in "hat," you narrow the opening where your throat opens into the cavity of your mouth so that your vocal tract appears as two connected tubes. The first is in your throat, closed at the vocal cords and open at the back of the mouth. The second is the mouth itself, open at the lips and closed at the back of the mouth-a different condition than for the throat because of the relatively larger size of the cavity. The corresponding formant frequencies are $800 \mathrm{Hz}$ (for the throat) and $1500 \mathrm{Hz}$ (for the mouth). What are the lengths of these two cavities? Assume a sound speed of $350 \mathrm{m} / \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 35

The first and second formants when you make an "ee" vowel sound are approximately $270 \mathrm{Hz}$ and $2300 \mathrm{Hz} .$ The speed of sound in your vocal tract is approximately $350 \mathrm{m} / \mathrm{s}$. If you breathe a mix of oxygen and helium (as deep-sea divers often do), the speed increases to $750 \mathrm{m} / \mathrm{s}$. With this mix of gases, what are the frequencies of the two formants?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

Figure $\mathrm{P} 16.36$ shows the two lowest resonances recorded in the vocal tract of the eastern towhee, a small songbird.
a. Is this bird's vocal tract better modeled as an open-open tube or an open-closed tube?
b. Estimate the length of the towhee's vocal tract.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

Two loudspeakers emit identical sound waves along the $x$ -axis. The sound at a point on the axis has maximum intensity when the speakers are $20 \mathrm{cm}$ apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of $30 \mathrm{cm} .$
a. What is the wavelength of the sound?
b. If the distance between the speakers continues to increase, at what separation will the sound intensity again be a maximum?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

Two loudspeakers in a $20^{\circ} \mathrm{C}$ room emit $686 \mathrm{Hz}$ sound waves along the $x$ -axis. What is the smallest distance between the speakers for which the interference of the sound waves for a point on the axis is destructive?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 39

Two loudspeakers, $1.0 \mathrm{m}$ apart, emit sound waves with the same frequency along the positive $x$ -axis. Victor, standing on the axis to the right of the speakers, hears no sound. As the frequency is slowly tripled, Victor hears the sound go through the sequence loud-soft-loud-soft-loud before becoming quiet again. What was the original sound frequency?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

In noisy factory environments, it's possible to use a loudspeaker to cancel persistent low-frequency machine noise at the position of one worker. The details of practical systems are complex, but we can present a simple example that gives you the idea. Suppose a machine $5.0 \mathrm{m}$ away from a worker emits a persistent $80 \mathrm{Hz}$ hum. To cancel the sound at the worker's location with a speaker that exactly duplicates the machine's hum, how far from the worker should the speaker be placed? Assume a sound speed of $340 \mathrm{m} / \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

Two identical loudspeakers separated by distance $d$ emit $170 \mathrm{Hz}$ sound waves along the $x$ -axis. As you walk along the axis, away from the speakers, you don't hear anything even though both speakers are on. What are three possible values for d? Assume a sound speed of $340 \mathrm{m} / \mathrm{s}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 42

Figure P16.42 shows the circular wave fronts emitted by two sources. Make a table with rows
labeled P, Q and R and columns labeled $r_{1}, r_{2}, \Delta r,$ and $C / D .$ Fill in the table for points $P, Q,$ and $\mathbf{R},$ giving the distances as multiples of $\lambda$ and indicating, with a $C$ or a $D,$ whether the interference at that point is constructive or destructive.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 43

Two identical loudspeakers $2.0 \mathrm{m}$ apart are emitting $1800 \mathrm{Hz}$ sound waves into a room where the speed of sound is $340 \mathrm{m} / \mathrm{s} .$ Is the point $4.0 \mathrm{m}$ directly in front of one of the speakers, perpendicular to the line joining the speakers, a point of maximum constructive interference, perfect destructive interference, or something in between?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

Two identical loudspeakers $2.0 \mathrm{m}$ apart are emitting sound waves into a room where the speed of sound is $340 \mathrm{m} / \mathrm{s}$. Abby is standing $5.0 \mathrm{m}$ in front of one of the speakers, perpendicular to the line joining the speakers, and hears a maximum in the intensity of the sound. What is the lowest possible frequency of sound for which this is possible?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

Musicians can use beats to tune their instruments. One flute is properly tuned and plays the musical note A at exactly $440 \mathrm{Hz}$. A second player sounds the same note and hears that her instrument is slightly "flat" (that is, at too low a frequency). Playing at the same time as the first flute, she hears two loud-soft-loud beats per second. What is the frequency of her instrument?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

A student waiting at a stoplight notices that her turn signal, which has a period of $0.85 \mathrm{s},$ makes one blink exactly in sync with the turn signal of the car in front of her. The blinker of the car ahead then starts to get ahead, but 17 s later the two are exactly in sync again. What is the period of the blinker of the other car?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

Two strings are adjusted to vibrate at exactly $200 \mathrm{Hz}$. Then the tension in one string is increased slightly. Afterward, three beats per second are heard when the strings vibrate at the same time. What is the new frequency of the string that was tightened?

Donald Albin

Numerade Educator

Problem 48

Figure P16.48 shows the superposition of two sound waves. What are their frequencies?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 49

A flute player hears four beats per second when she compares her note to a $523 \mathrm{Hz}$ tuning fork (the note C). She can match the frequency of the tuning fork by pulling out the "tuning joint" to lengthen her flute slightly. What was her initial frequency?

Donald Albin

Numerade Educator

Problem 50

The fundamental frequency of a standing wave on a $1.0-\mathrm{m}-$ long string is $440 \mathrm{Hz} .$ What would be the wave speed of a pulse moving along this string?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

In addition to producing images, ultrasound can be used to heat tissues of the body for therapeutic purposes. An emitter is placed against the surface of the skin; the amplitude of the ultrasound wave at this point is quite large. When a sound wave hits the boundary between soft tissue and bone, most of the energy is reflected. The boundary acts like the closed end of a tube, which can lead to standing waves. Suppose $0.70 \mathrm{MHz}$ ultrasound is directed through a layer of tissue with a bone $0.55 \mathrm{cm}$ below the surface. Will standing waves be created? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

An $80-\mathrm{cm}$ -long steel string with a linear density of $1.0 \mathrm{g} / \mathrm{m}$ is under $200 \mathrm{N}$ tension. It is plucked and vibrates at its fundamental frequency. What is the wavelength of the sound wave that reaches your ear in a $20^{\circ} \mathrm{C}$ room?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

Tendons are, essentially, elastic cords stretched between two fixed ends; as such, they can support standing waves. These resonances can be undesirable. The Achilles tendon connects the heel with a muscle in the calf. A woman has a $20-\mathrm{cm}$ -long tendon with a cross-section area of $110 \mathrm{mm}^{2} .$ The density of tendon tissue is $1100 \mathrm{kg} / \mathrm{m}^{3} .$ For a reasonable tension of $500 \mathrm{N}$, what will be the resonant frequencies of her Achilles tendon?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

Two loudspeakers directly face each other $30 \mathrm{m}$ apart, with the left speaker positioned at $x=0 \mathrm{m}$. The pressure of the sound wave emitted by the left speaker is described by the equation $\Delta p_{\mathrm{L}}=p_{0} \cos (1.90 x-630 t),$ while that from the right speaker is given by $\Delta p_{\mathrm{R}}=p_{0} \cos (1.90 x+630 t),$ where $x$ is measured in $\mathrm{m}$ and $t$ in $\mathrm{s}$. What is the point nearest to the left speaker at which there is a node in the sound wave?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 55

Spiders may "tune" strands of their webs to give enhanced response at frequencies corresponding to the frequencies at which desirable prey might struggle. Orb web silk has a typical diameter of $0.0020 \mathrm{mm},$ and spider silk has a density of $1300 \mathrm{kg} / \mathrm{m}^{3} .$ To give a resonance at $100 \mathrm{Hz},$ to what tension must a spider adjust a $12-\mathrm{cm}$ -long strand of silk?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

A particularly beautiful note reaching your ear from a rare Stradivarius violin has a wavelength of $39.1 \mathrm{cm} .$ The room is slightly warm, so the speed of sound is $344 \mathrm{m} / \mathrm{s}$. If the string's linear density is $0.600 \mathrm{g} / \mathrm{m}$ and the tension is $150 \mathrm{N},$ how long is the vibrating section of the violin string?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

A 12 kg hanging sculpture is suspended by a $90-\mathrm{cm}$ -long, $5.0 \mathrm{g}$ steel wire. When the wind blows hard, the wire hums at its fundamental frequency. What is the frequency of the hum?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

Lake Erie is prone to remarkable seiches - standing waves that slosh water back and forth in the lake basin from the west end at Toledo to the east end at Buffalo. Figure $\mathrm{P} 16.58$ shows smoothed data for the displacement from normal water levels along the lake at the high point of one particular seiche. 3 hours later the water was at normal levels throughout the basin; 6 hours later the water was high in Toledo and low in Buffalo.
a. What is the wavelength of this standing wave?
b. What is the frequency?
c. What is the wave speed?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

A guitar player can change the frequency of a string by "bending" it-pushing it along a fret that is perpendicular to its length. This stretches the string. increasing its tension and its frequency. The G string on a guitar is $64 \mathrm{cm}$ long and has a tension of $74 \mathrm{N}$. The guitarist pushes this string down against a fret located at the center of the string, which gives it a frequency of 392 Hz. He then bends the string, pushing with a force of $4.0 \mathrm{N}$ so that it moves $8.0 \mathrm{mm}$ along the fret. What is the new frequency?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 60

A carbon-dioxide laser emits infrared light with a wavelength of $10.6 \mu \mathrm{m}$
a. What is the length of a tube that will oscillate in the $m=100,000$ mode?
b. What is the frequency?
c. Imagine a pulse of light bouncing back and forth between the ends of the tube. How many round trips will the pulse make in each second?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 61

A $40-\mathrm{cm}$ -long tube has a $40-\mathrm{cm}-$ long insert that can be pulled in and out, as shown in Figure $\mathrm{P} 16.61 .$ A vibrating tuning fork is held next to the tube. As the insert is slowly pulled out, the sound from the tuning fork creates standing waves in the tube when the total length $L$ is $42.5 \mathrm{cm}, 56.7 \mathrm{cm},$ and $70.9 \mathrm{cm} .$ What is the frequency of the tuning fork? The air temperature is $20^{\circ} \mathrm{C}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 62

The width of a particular microwave oven is exactly right to support a standing-wave mode. Measurements of the temperature across the oven show that there are cold spots at each edge of the oven and at three spots in between. The wavelength of the microwaves is $12 \mathrm{cm} .$ How wide is the oven?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 63

Two loudspeakers located along the $x$ -axis as shown in Figure $P 16.63$ produce sounds of equal frequency. Speaker 1 is at the origin, while the location of speaker 2 can be varied by a remote control wielded by the listener. He notices maxima in the sound intensity when speaker 2 is located at $x=0.75 \mathrm{m}$ and $1.00 \mathrm{m},$ but at no points in between. What is the frequency of the sound? Assume the speed of sound is $340 \mathrm{m} / \mathrm{s}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

Two loudspeakers $42.0 \mathrm{m}$ apart and facing each other emit identical 115 Hz sinusoidal sound waves in a room where the sound speed is $345 \mathrm{m} / \mathrm{s} .$ Susan is walking along a line between the speakers. As she walks, she finds herself moving through loud and quiet spots. If Susan stands $19.5 \mathrm{m}$ from one speaker, is she standing at a quiet spot or a loud spot?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

You are standing $2.50 \mathrm{m}$ directly in front of one of the two loudspeakers shown in Figure P16.65. They are $3.00 \mathrm{m}$ apart and both are playing a $686 \mathrm{Hz}$ tone in phase. As you begin to walk directly away from the speaker, at what distances from the speaker do you hear a minimum sound intensity? The room temperature is $20^{\circ} \mathrm{C}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

Two loudspeakers, $4.0 \mathrm{m}$ apart and facing each other, play identical sounds of the same frequency. You stand halfway between them, where there is a maximum of sound intensity. Moving from this point toward one of the speakers, you encounter a minimum of sound intensity when you have moved $0.25 \mathrm{m}$.
a. What is the frequency of the sound?
b. If the frequency is then increased while you remain $0.25 \mathrm{m}$ from the center, what is the first frequency for which that location will be a maximum of sound intensity?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have the frequency $440 \mathrm{Hz}$ and the note E should be at $659 \mathrm{Hz}$. The tuner can determine this by listening to the beats between the third harmonic of the A and the second harmonic of the E. A tuner first tunes the A string very precisely by matching it to a $440 \mathrm{Hz}$ tuning fork. She then strikes the A and $\mathrm{E}$ strings simultaneously and listens for beats between the harmonics. What beat frequency indicates that the E string is properly tuned?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

A flutist assembles her flute in a room where the speed of sound is $342 \mathrm{m} / \mathrm{s} .$ When she plays the note $\mathrm{A},$ it is in perfect tune with a $440 \mathrm{Hz}$ tuning fork. After a few minutes, the air inside her flute has warmed to where the speed of sound is $346 \mathrm{m} / \mathrm{s}$.
a. How many beats per second will she hear if she now plays the note $A$ as the tuning fork is sounded?
b. How far does she need to extend the "tuning joint" of her flute to be in tune with the tuning fork?

Donald Albin

Numerade Educator

Problem 69

Police radars determine speed by measuring the shift of radio waves reflected by a moving vehicle. They do so by determining the beat frequency between the reflected wave and the 10.5 GHz emitted wave. Some units can be calibrated by using a tuning fork; holding a vibrating fork in front of the unit causes the display to register a speed corresponding to the vibration frequency. A tuning fork is labeled "55 mph." What is the frequency of the tuning fork?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

A Doppler blood flowmeter emits ultrasound at a frequency of $5.0 \mathrm{MHz} .$ What is the beat frequency between the emitted waves and the waves reflected from blood cells moving away from the emitter at $0.15 \mathrm{m} / \mathrm{s} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

An ultrasound unit is being used to measure a patient's heartbeat by combining the emitted $2.0 \mathrm{MHz}$ signal with the sound waves reflected from the moving tissue of one point on the heart. The beat frequency between the two signals has a maximum value of $520 \mathrm{Hz}$. What is the maximum speed of the heart tissue?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

What is the beat frequency between the second harmonic of G and the third harmonic of C?
A. $1 \mathrm{Hz}$
B. $2 \mathrm{Hz}$
$\mathrm{C} .4 \mathrm{Hz}$
D. $6 \mathrm{Hz}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

Would a G-flat (frequency $370 \mathrm{Hz}$ ) and a C played together be consonant or dissonant?
A. Consonant
B. Dissonant

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 74

An organ pipe open at both ends is tuned so that its fundamental frequency is a G. How long is the pipe?
A. $43 \mathrm{cm}$
B. $87 \mathrm{cm}$
C. 130 cm
D. $173 \mathrm{cm}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

If the $C$ were played on an organ pipe that was open at one end and closed at the other, which of the harmonic frequencies in Figure P16.72 would be present?
A. All of the harmonics in the figure would be present.
B. $262,786,$ and $1310 \mathrm{Hz}$
$\mathrm{C} .524,1048,$ and $1572 \mathrm{Hz}$
D. $262,524,$ and $1048 \mathrm{Hz}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator