Physics for Scientist and Engineers: A Strategic Approach

Randall Knight

Chapter 21

Superposition - all with Video Answers

Educators

Chapter Questions

Problem 1

FIGURE EX21.1 is a snapshot graph at $t=0$ s of two waves approaching each other at $1.0 \mathrm{m} / \mathrm{s} .$ Draw six snapshot graphs, stacked vertically, showing the string at 1 s intervals from $t=1$ s to $t=6 \mathrm{s}$.
(FIGURE CAN'T COPY)

Donald Albin

Donald Albin

Numerade Educator

Problem 2

FIGURE EX21.2 is a snapshot graph at $t=0$ s of two waves approaching each other at $1.0 \mathrm{m} / \mathrm{s}$. Draw six snapshot graphs, stacked vertically, showing the string at 1 s intervals from $t=1 \mathrm{s}$ to $t=6 \mathrm{s}$
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 3

FIGURE EX21.3 is a snapshot graph at $t=0$ s of two waves approaching each other at $1.0 \mathrm{m} / \mathrm{s} .$ Draw four snapshot graphs. stacked vertically, showing the string at $t=2,4,6,$ and $8 \mathrm{s}$.
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 4

FIGURE ex21.4a is a snapshot graph at $t=0$ s of two waves approaching each other at $1.0 \mathrm{m} / \mathrm{s}$
a. At what time was the snapshot graph in FIGURE 21.4B taken?
b. Draw a history graph of the string at $x=5.0 \mathrm{m}$ from $t=0 \mathrm{s}$ to $t=6 \mathrm{s}$
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 5

FIGURE EX21.5 is a snapshot graph at $t=0$ s of two waves moving to the right at $1.0 \mathrm{m} / \mathrm{s}$. The string is fixed at $x=8.0 \mathrm{m}$ Draw four snapshot graphs, stacked vertically, showing the string at $t=2,4,6,$ and $8 \mathrm{s}$.
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 6

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 EX21.6?
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 7

FIGURE EX21.7 shows a standing wave oscillating at $100 \mathrm{Hz}$ on a string. What is the wave speed?

Donald Albin

Numerade Educator

Problem 8

FIGURE EX.21.8 shows a standing wave that is oscillating at frequency $f_{0}$
a. How many antinodes will there be if the frequency is doubled to $2 / 8 ?$ Explain.
b. If the tension in the string is increased by a factor of four, for what frequency, in terms of $f_{0},$ will the string continue to oscillate as a standing wave with three antinodes?
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 9

Standing waves on a $1.0-$ m-long string that is fixed at both ends are seen at successive frequencies of $24 \mathrm{Hz}$ and $36 \mathrm{Hz}$
a. What are the fundamental frequency and the wave speed?
b. Draw the standing-wave pattern when the string oscillates at
$36 \mathrm{Hz}$

Donald Albin

Numerade Educator

Problem 10

a. What are the three longest wavelengths for standing waves on a $240-\mathrm{cm}$ -long string that is fixed at both ends?
b. If the frequency of the second-longest wavelength is $50 \mathrm{Hz}$, what is the frequency of the thind-longest wavelength?

Donald Albin

Numerade Educator

Problem 11

A 121 -cm-long, $4.0 \mathrm{g}$ string oscillates in its $m=3$ mode with a frequency of $180 \mathrm{Hz}$ and a maximum amplitude of $5.0 \mathrm{mm}$ What are (a) the wavelength and (b) the tension in the string?

Donald Albin

Numerade Educator

Problem 12

A heavy piece of 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 of $80 \mathrm{Hz}$. What is the mass of the sculpture?

Donald Albin

Numerade Educator

Problem 13

A carbon dioxide laser is an infrared laser. A $\mathrm{CO}_{2}$ laser with a cavity length of 53.00 cm oscillates in the $m=100,000$ mode. What are the wavelength and frequency of the laser beam?

Donald Albin

Numerade Educator

Problem 14

What are the three longest wavelengths for standing sound waves in a 121 -cm-long tube that is (a) open at both ends and
(b) open at one end, closed at the other?

Donald Albin

Numerade Educator

Problem 15

FIGURE EX21.15 shows a standing sound wave in an $80-\mathrm{cm}$ -long tube. The tube is filled with an unknown gas. What is the speed of sound in this gas?
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 16

The fundamental frequency of an open-open tube is $1500 \mathrm{Hz}$ when the tube is filled with $0^{\circ} \mathrm{C}$ helium. What is its frequency when filled with $0^{\circ} \mathrm{C}$ air?

Donald Albin

Numerade Educator

Problem 17

The lowest pedal note on a large pipe organ has a fundamental frequency of 16.4 Hz. This extreme bass note, four octaves below middle $C$, is more felt as a rumble than heard with the ears. What is the length of pipe between the sounding hole and the open end?

Donald Albin

Numerade Educator

Problem 18

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 19

A violin string is $30 \mathrm{cm}$ long. It sounds the musical note
A $(440 \mathrm{Hz})$ when played without fingering. How far from the end of the string should you place your finger to play the note $\mathbf{C}(523 \mathrm{Hz}) ?$

Donald Albin

Numerade Educator

Problem 20

Two loudspeakers emit sound waves along the $x$ -axis. The sound 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 $60 \mathrm{cm}$
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?

Donald Albin

Numerade Educator

Problem 21

Two loudspeakers in a $20^{\circ} \mathrm{C}$ room emit $686 \mathrm{Hz}$ sound waves along the $x$ -axis.
a. If the speakers are in phase, what is the smallest distance between the speakers for which the interference of the sound waves is perfectly destructive?
b. If the speakers are out of phase, what is the smallest distance between the speakers for which the interference of the sound waves is maximum constructive?

Donald Albin

Numerade Educator

Problem 22

Two in-phase 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 $340 \mathrm{m} / \mathrm{s}$.

Donald Albin

Numerade Educator

Problem 23

What is the thinnest film of $\mathrm{MgF}_{2}(n=1.39)$ on glass that produces a strong reflection for orange light with a wavelength of $600 \mathrm{nm} ?$

Donald Albin

Numerade Educator

Problem 24

A very thin oil film $(n=1.25)$ floats on water $(n=1.33)$ What is the thinnest film that produces a strong reflection for green light with a wavelength of $500 \mathrm{nm} ?$

Donald Albin

Numerade Educator

Problem 25

FIGURE EX21.25 shows the circular wave fronts emitted by two wave sources.
a. Are these sources in phase or out of phase? Explain.
b. Make a table with rows labeled $P$, $Q$, and $R$ and columns Labeled $r_{1}, r_{2}, \Delta r,$ and $\mathrm{C} / \mathrm{D} .$ Fill in the table for points $\mathrm{P}$, $\mathrm{Q}$ and $R,$ 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.
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 26

FIGURE ex21.26 shows the circular wave fronts emitted by two
wave sources.
a. Are these sources in phase or out of phase? Explain.
b. Make a table with rows labeled $P, Q,$ and $R$ and columns labeled $r_{1}, r_{2}, \Delta r,$ and $\mathrm{C} / \mathrm{D} .$ Fill in the table for points $\mathrm{P}, \mathrm{Q},$ and
R, giving the distances as multiples of $\lambda$ and indicating, with a
Cor a D, whether the interference at that point is constructive
or destructive.

Donald Albin

Numerade Educator

Problem 27

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

Donald Albin

Numerade Educator

Problem 28

Two out-of-phase radio antennas at $x=\pm 300 \mathrm{m}$ on the $x$ axis are emitting $3.0 \mathrm{MHz}$ radio waves. Is the point $(x, y)=$ $(300 \mathrm{m}, 800 \mathrm{m})$ a point of maximum constructive interference, perfect destructive interference, or something in between?

Donald Albin

Numerade Educator

Problem 29

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 30

A flute player hears four beats per second when she compares her note to a 523 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 31

Two lasers with very nearly the same wavelength can generate a beat frequency if both laser beams illuminate a photodetector with a very fast response. In an experiment, one laser's wavelength has been stabilized at $780.54510 \mathrm{nm} .$ The second laser starts with a longer wavelength that is slowly decreased until the beat frequency between the two lasers is $98.5 \mathrm{MHz}$. What is the second laser"s wavelength?

Donald Albin

Numerade Educator

Problem 32

Two waves on a string travel in opposite directions at $100 \mathrm{m} / \mathrm{s}$ FIGURE EX21.32 shows a snapshot graph of the string at $t=0 \mathrm{s}$ when the two waves are overlapped, and a snapshot graph of the left-traveling wave at $t=0.050 \mathrm{s} .$ Draw a
snapshot graph of the right-traveling wave at $t=0.050 \mathrm{s}$.
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 33

A 2.0 -m-long string vibrates at its second-harmonic frequency with a maximum amplitude of $2.0 \mathrm{cm} .$ One end of the string is at $x=0 \mathrm{cm} .$ Find the oscillation amplitude at $x=10,20,30,40$ and $50 \mathrm{cm}$.

Donald Albin

Numerade Educator

Problem 34

A string vibrates at its third-harmonic frequency. The amplitude at a point $30 \mathrm{cm}$ from one end is half the maximum amplitude. How long is the string?

Donald Albin

Numerade Educator

Problem 35

A string of length $L$ vibrates at its fundamental frequency. The amplitude at a point $\frac{1}{4} L$ from one end is $2.0 \mathrm{cm} .$ What is the amplitude of each of the traveling waves that form this standing wave?

Donald Albin

Numerade Educator

Problem 36

Two sinusoidal waves with equal wavelengths travel along a string in opposite directions at $3.0 \mathrm{m} / \mathrm{s}$. The time between two successive instants when the antinodes are at maximum height is $0.25 \mathrm{s} .$ What is the wavelength?

Donald Albin

Numerade Educator

Problem 37

A particularly beautiful note reaching your car 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?

Donald Albin

Numerade Educator

Problem 38

A violinist places her finger so that the vibrating section of a $1.0 \mathrm{g} / \mathrm{m}$ string has a length of $30 \mathrm{cm},$ then she draws her bow across it. A listener nearby in a $20^{\circ} \mathrm{C}$ room hears a note with a wavelength of $40 \mathrm{cm} .$ What is the tension in the string?

Donald Albin

Numerade Educator

Problem 39

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 40

When mass $M$ is tied to the bottom of a long, thin wire suspended from the ceiling, the wire's second-harmonic frequency is $200 \mathrm{Hz}$. Adding an additional $1.0 \mathrm{kg}$ to the hanging mass increases the second-harmonic frequency to $245 \mathrm{Hz}$. What is $M ?$

Donald Albin

Numerade Educator

Problem 41

Astronauts visiting Planet X have a 2.5 -m-long string whose mass is $5.0 \mathrm{g}$. They tie the string to a support, stretch it horizontally over a pulley $2.0 \mathrm{m}$ away, and hang a $1.0 \mathrm{kg}$ mass on the free end. Then the astronauts begin to excite standing waves on the string. Their data show that standing waves exist at frequencies of $64 \mathrm{Hz}$ and $80 \mathrm{Hz}$, but at no frequencies in between. What is the value of $g$, the free-fall acceleration on Planet X?

Donald Albin

Numerade Educator

Problem 42

A 75 g bungee cord has an equilibrium length of $1.20 \mathrm{m}$. The cord is stretched to a length of $1.80 \mathrm{m},$ then vibrated at $20 \mathrm{Hz}$ This produces a standing wave with two antinodes. What is the spring constant of the bungee cord?

Donald Albin

Numerade Educator

Problem 43

A 22 -cm-long, 1.0 -mm-diameter copper wire is joined smoothly to a 60 -cm-long, 1.0 -mm-diameter aluminum wire. The resulting wire is stretched with $20 \mathrm{N}$ of tension between fixed supports $82 \mathrm{cm}$ apart. The densities of copper and aluminum are $8920 \mathrm{kg} / \mathrm{m}^{3}$ and $2700 \mathrm{kg} / \mathrm{m}^{3},$ respectively.
a. What is the lowest-frequency standing wave for which there is a node at the junction between the two metals?
b. At that frequency, how many antinodes are on the aluminum wire?

Donald Albin

Numerade Educator

Problem 44

In a laboratory experiment, one end of a horizontal string is tied to a support while the other end passes over a frictionless pulley and is tied to a $1.5 \mathrm{kg}$ sphere. Students determine the frequencies of standing waves on the horizontal segment of the string, then they raise a beaker of water until the hanging $1.5 \mathrm{kg}$ sphere is completely submerged. The frequency of the fifth harmonic with the sphere submerged cxactly matches the frequency of the third harmonic before the sphere was submerged. What is the diameter of the sphere?

Donald Albin

Numerade Educator

Problem 45

Il What is the fundamental frequency of the steel wire in FIGURE P21.45?
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 46

Il The two strings in FIGURE $\mathrm{P} 21.46$ are of equal length and are being driven at equal frequencies. The linear density of the left string is $2.0 \mathrm{g} / \mathrm{m}$. What is the linear density of the right string?
(FIGURE CAN'T COPY)

Paul A.

California State Polytechnic University, Pomona

Problem 47

The microwave generator in FIGURE P21.47 can produce microwaves at any frequency between $10 \mathrm{GHz}$ and $20 \mathrm{GHz}$ The microwaves are aimed, through a small hole, into a "microwave cavity" that consists of a $10-\mathrm{cm}-$ long cylinder with reflective ends.
a. Which frequencies will create standing waves in the microwave cavity?
b. For which of these frequencies is the cavity midpoint an antinode?

Donald Albin

Numerade Educator

Problem 48

An open-open organ pipe is $78.0 \mathrm{cm}$ long. An open-closed pipe has a fundamental frequency equal to the third harmonic of the open-open pipe. How long is the open-closed pipe?

Donald Albin

Numerade Educator

Problem 49

A narrow column of $20^{\circ} \mathrm{C}$ air is found to have standing waves at frequencies of $390 \mathrm{Hz}, 520 \mathrm{Hz},$ and $650 \mathrm{Hz}$ and at no frequencies in between these. The behavior of the tube at frequencies less than 390 Hz or greater than $650 \mathrm{Hz}$ is not known.
Is this an open-open tube or an open-closed tube? Explain.
b. How long is the tube?
c. Draw a displacement graph of the 520 Hz standing wave in the tube.
d. The air in the tube is replaced with carbon dioxide, which has a sound speed of $280 \mathrm{m} / \mathrm{s} .$ What are the new frequencies of these three modes?

Donald Albin

Numerade Educator

Problem 50

In $1866,$ the German scientist Adolph Kundt developed a technique for accurately measuring the speed of sound in various gases. A long glass tube, known today as a Kundt's tube, has a vibrating piston at one end and is closed at the other. Very finely ground particles of cork are sprinkled in the bottom of the tube before the piston is inserted. As the vibrating piston is slowly moved forward, there are a few positions that cause the cork particles to collect in small, regularly spaced piles along the bottom. FIGURE P21.50 shows an experiment in which the tube is filled with pure oxygen and the piston is driven at $400 \mathrm{Hz}$. What is the speed of sound in oxygen?
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 51

Il A $40-\mathrm{cm}-$ long tube has a $40-\mathrm{cm}-$ long insert that can be pulled in and out. 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? Assume $v_{\text {sound }}=343 \mathrm{m} / \mathrm{s}$.

Donald Albin

Numerade Educator

Problem 52

A $1.0-$ m-tall vertical tube is filled with $20^{\circ} \mathrm{C}$ water. A tuning fork vibrating at $580 \mathrm{Hz}$ is held just over the top of the tube as the water is slowly drained from the bottom. At what water heights, measured from the bottom of the tube, will there be a standing wave in the tube above the water?

Donald Albin

Numerade Educator

Problem 53

A $50-\mathrm{cm}$ -long wire with a mass of $1.0 \mathrm{g}$ and a tension of $440 \mathrm{N}$ passes across the open end of an open-closed tube of air. The wire, which is fixed at both ends, is bowed at the center so as to vibrate at its fundamental frequency and generate a sound wave. Then the tube length is adjusted until the fundamental frequency of the tube is heard. What is the length of the tube? Assume $v_{\text {sound }}=340 \mathrm{m} / \mathrm{s}$.

Donald Albin

Numerade Educator

Problem 54

If $A$ 25-cm-long wire with a linear density of $20 \mathrm{g} / \mathrm{m}$ passes across the open end of an $85-$ cm-long open-closed tube of air. If the wire, which is fixed at both ends, vibrates at its fundamental frequency, the sound wave it generates excites the second vibrational mode of the tube of air. What is the tension in the wire? Assume $v_{\text {sound }}=340 \mathrm{m} / \mathrm{s}$.

Donald Albin

Numerade Educator

Problem 55

A vertical tube, open at both ends, is lowered into a tank of water until it is partially filled. The top portion of the tube, above the water, is filled with a gas that, because it is denser than air, remains in the tube. A 50.0 -cm-long, 1.00 g horizontal wire is stretched just above the top of the tube with $440 \mathrm{N}$ of tension. Bowing the wire at its center causes the wire to vibrate at its fundamental frequency. The water level in the tube is adjusted until the sound from the vibrating wire sets up a standing sound wave in the gas. The water is then lowered another $30.5 \mathrm{cm}$ until the next standing sound wave is detected. Use this information to determine the speed of sound in the gas.

Donald Albin

Numerade Educator

Problem 56

A longitudinal standing wave can be created in a long, thin aluminum rod by stroking the rod with very dry fingers. This is often done as a physics demonstration, creating a high-pitched, very annoying whine. From a wave perspective, the standing wave is equivalent to a sound standing wave in an open-open tube. In particular, both ends of the rod are anti-nodes. What is the fundamental frequency of a $2.0-\mathrm{m}$ -long aluminum rod?

Donald Albin

Numerade Educator

Problem 57

An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but it's too dangerous to go inside. Recalling your recent physics class, you decide to try setting up standing-wave resonances inside the tunnel. Using your subsonic amplifier and loudspeaker, you find resonances at $4.5 \mathrm{Hz}$ and 6.3 Hz, and at no frequencies between these. It's rather chilly inside the tunnel, so you estimate the sound speed to be $335 \mathrm{m} / \mathrm{s}$. Based on your measurements, how far is it to the end of the tunnel?

Donald Albin

Numerade Educator

Problem 58

Analyze the standing sound waves in an open-closed tube to show that the possible wavelengths and frequencies are given by Equation 21.18.

Donald Albin

Numerade Educator

Problem 59

Two in-phase loudspeakers emit identical $1000 \mathrm{Hz}$ sound waves along the $x$ -axis. What distance should one speaker be placed behind the other for the sound to have an amplitude 1.5 times that of each speaker alone?

Donald Albin

Numerade Educator

Problem 60

Two loudspeakers emit sound waves of the same frequency along the $x$ -axis. The amplitude of each wave is $a$. The sound intensity is minimum when speaker 2 is $10 \mathrm{cm}$ behind speaker $1 .$ The intensity increases as speaker 2 is moved forward and first reaches maximum, with amplitude $2 a$, when it is $30 \mathrm{cm}$ in front of speaker $1 .$ What is
a. The wavelength of the sound?
b. The phase difference between the two loudspeakers?
c. The amplitude of the sound (as a multiple of $a$ ) if the speak-
ers are placed side by side?

Donald Albin

Numerade Educator

Problem 61

Two loudspeakers emit sound waves along the $x$ -axis. A listener in front of both speakers hears a maximum sound intensity when speaker 2 is at the origin and speaker 1 is at $x=0.50 \mathrm{m}$. If speaker 1 is slowly moved forward, the sound intensity decreases and then increases, reaching another maximum when speaker 1 is
at $x=0.90 \mathrm{m}$
a. What is the frequency of the sound? Assume $v_{\text {tanad }}=340 \mathrm{m} / \mathrm{s}$
b. What is the phase difference between the speakers?

Donald Albin

Numerade Educator

Problem 62

Two loudspeakers emit sound waves along the $x$ -axis. Speaker
2 is 2.0 m behind speaker $1 .$ Both loudspeakers are connected to the same signal generator, which is oscillating at $340 \mathrm{Hz}$, but the wire to speaker 1 passes through a box that delays the signal by $1.47 \mathrm{ms} .$ Is the interference along the $x$ -axis maximum constructive interference, perfect destructive interference, or something in between? Assume $v_{\text {sund }}=340 \mathrm{m} / \mathrm{s}$.

Donald Albin

Numerade Educator

Problem 63

A sheet of glass is coated with a 500 -nm-thick layer of oil $(n=1.42)$
a. For what visible wavelengths of light do the reflected waves interfere constructively?
b. For what visible wavelengths of light do the reflected waves interfere destructively?
c. What is the color of reflected light? What is the color of transmitted light?

Donald Albin

Numerade Educator

Problem 64

Il Example 21.10 showed that a 92 -nm-thick coating of $\mathrm{MgF}_{2}$ $(n=1.39)$ on glass acts as an antireflection coating for light with a wavelength of $510 \mathrm{nm}$. Without the coating, the intensity of reflected light is $I_{0}=c a^{2},$ where $a$ is the amplitude of the reflected light wave and $c$ is an unknown proportionality constant.
a. Let $I_{\lambda}$ be the intensity of light reflected from the coated glass
at wavelength $\lambda$. Find an expression for the ratio $I_{\lambda} / I_{0}$ as a function of the wavelength $\lambda$. This ratio is the reflection intensity from the coated glass relative to the reflection intensity from uncoated glass. A ratio less than 1 indicates that the coating is reducing the reflection intensity. Hint: The amplitude of the superposition of two waves depends on the phase difference between the waves. Although not entirely accurate, assume that both reflected waves have amplitude $a$.
b. Evaluate $I_{A} / I_{0}$ at $\lambda=400,450,500,550,600,650,$ and
$700 \mathrm{nm} .$ This spans the range of visible light.
c. Draw a graph of $I_{\lambda} / I_{0}$ versus $\lambda$.

Donald Albin

Numerade Educator

Problem 65

A manufacturing firm has hired your company, Acoustical Consulting, to help with a problem. Their employees are complaining about the annoying hum from a piece of machinery. Using a frequency meter, you quickly determine that the machine emits a rather loud sound at $1200 \mathrm{Hz}$. After investigating, you tell the owner that you cannot solve the problem entirely, but you can at least improve the situation by eliminating reflections of this sound from the walls. You propose to do this by installing mesh screens in front of the walls. A portion of the sound will reflect from the mesh; the rest will pass through the mesh and reflect from the wall. How far should the mesh be placed in front of the wall for this scheme to work?

Donald Albin

Numerade Educator

Problem 66

A soap bubble is essentially a very thin film of water $(n=$ 1.33) surrounded by air. The colors that you see in soap bubbles are produced by interference, much like the colors of dichroic glass.
a. Derive an expression for the wavelengths $\lambda_{\mathrm{c}}$ for which constructive interference causes a strong reflection from a soap bubble of thickness $d$ Hint: Think about the reflection phase shifts at both boundaries.
b. What visible wavelengths of light are strongly reflected from a 390 -nm-thick soap bubble? What color would such a soap bubble appear to be?

Donald Albin

Numerade Educator

Problem 67

II Two radio antennas are separated by $2.0 \mathrm{~m}$. Both broadcast identical $750 \mathrm{MHz}$ waves. If you walk around the antennas in a circle of radius $10 \mathrm{~m}$, how many maxima will you detect?

Donald Albin

Numerade Educator

Problem 68

You are standing $2.5 \mathrm{m}$ directly in front of one of the two loudspeakers shown in FIGURE P21.68. They are 3.0 $\mathrm{m}$ 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 $20^{\circ} \mathrm{C}$.
(FIGURE CAN'T COPY)

Donald Albin

Numerade Educator

Problem 69

Two loudspeakers in a plane, 5.0 apart, are playing the same frequency. If you stand $12.0 \mathrm{m}$ in front of the plane of the speakers, centered between them, you hear a sound of maximum intensity. As you walk parallel to the plane of the speakers, staying $12.0 \mathrm{m}$ in front of them, you first hear a minimum of sound intensity when you are directly in front of one of the speakers.
a. What is the frequency of the sound? Assume a sound speed of $340 \mathrm{m} / \mathrm{s}$
b. If you stay $12.0 \mathrm{m}$ directly in front of one of the speakers, for what other frequencies between $100 \mathrm{Hz}$ and $1000 \mathrm{Hz}$ is there
a minimum sound intensity at this point?

Donald Albin

Numerade Educator

Problem 70

Two in-phase loudspeakers are located at $(x, y)$ coordinates $(-3.0 \mathrm{m},+2.0 \mathrm{m})$ and $(-3.0 \mathrm{m},-2.0 \mathrm{m}) .$ They emit identical
sound waves with a $2.0 \mathrm{m}$ wavelength and amplitude $a$. Determine the amplitude of the sound at the five positions on the $y$ axis $(x=0)$ with $y=0.0 \mathrm{m}, 0.5 \mathrm{m}, 1.0 \mathrm{m}, 1.5 \mathrm{m},$ and $2.0 \mathrm{m}$.

Donald Albin

Numerade Educator

Problem 71

Your firm has been hired to design a system that allows airplane pilots to make instrument landings in rain or fog. You've decided to place two radio transmitters $50 \mathrm{m}$ apart on either side of the runway. These two transmitters will broadcast the same frequency, but out of phase with each other. This will cause a nodal line to extend straight off the end of the runway (see Figure $21.30 b$ ). As long as the airplane's receiver is silent, the pilot knows she's directly in line with the runway. If she drifts to one side or the other, the radio will pick up a signal and sound a warning beep. To have sufficient accuracy, the first intensity maxima need to be 60 m on either side of the nodal line at a distance of 3.0 km. What frequency should you specify for the transmitters?

Donald Albin

Numerade Educator

Problem 72

Two radio antennas are 100 m apart along a north-south line. They broadcast identical radio waves at a frequency of $3.0 \mathrm{MHz}$ Your job is to monitor the signal strength with a handheld receiver. To get to your first measuring point, you walk $800 \mathrm{m}$ east from the midpoint between the antennas, then 600 m north.
a. What is the phase difference between the waves at this point?
b. Is the interference at this point maximum constructive, perfect destructive, or somewhere in between? Explain.
c. If you now begin to walk farther north, does the signal strength increase, decrease, or stay the same? Explain.

Donald Albin

Numerade Educator

Problem 73

Il The three identical Loudspeakers in FIGURE $\mathrm{P} 21.73$ play a $170 \mathrm{Hz}$ tone in a room where the speed of sound is $340 \mathrm{m} / \mathrm{s} .$ You are standing $4.0 \mathrm{m}$ in front of the middle speaker. At this point, the amplitude of the wave from cach speaker is $a$. a. What is the amplitude at this
point?
b. How far must speaker 2 be moved to the left to produce a maximum amplitude at the point where you are standing?
c. When the amplitude is maximum, by what factor is the sound intensity greater than the sound intensity from a single speaker?

Donald Albin

Numerade Educator

Problem 74

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note $A$ should have a frequency of $440 \mathrm{Hz}$ and the note $\mathrm{E}$ should be at $659 \mathrm{Hz}$
a. What is the frequency difference between the third harmonic of the A and the second harmonic of the E?
b. 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 E strings simultancously and listens for beats between the harmonics. What beat frequency indicates that the B string is properly tuned?
c. The tuner starts with the tension in the B string a little low, then tightens it. What is the frequency of the B string when she hears four beats per second?

Donald Albin

Numerade Educator

Problem 75

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 76

Two loudspeakers face each other from opposite walls of a room. Both are playing exactly the same frequency, thus setting up a standing wave with distance $\lambda / 2$ between antinodes. Assume that
$\lambda$ is much less than the room width, so there are many antinodes.
a. Yvette starts at one speaker and runs toward the other at speed $v_{Y} .$ As the does so, she hears a loud-soft-loud modulation of the sound intensity. From your perspective, as you sit at rest in the room, Yvette is running through the nodes and antinodes of the standing wave. Find an expression for the number of sound maxima she hears per second.
b. From Yvette's perspective, the two sound waves are Doppler shifted. They're not the same frequency, so they don't create
a standing wave. Instead, she hears a loud-soft-loud modulation of the sound intensity because of beats. Find an expression for the beat frequency that Yvette hears.
c. Are your answers to parts a and b the same or different? Should they be the same or different?

Donald Albin

Numerade Educator

Problem 77

Two loudspeakers emit $400 \mathrm{Hz}$ notes. One speaker sits on the ground. The other speaker is in the back of a pickup truck. You hear eight beats per second as the truck drives away from you. What is the truck's speed?

Donald Albin

Numerade Educator

Problem 78

a. The frequency of a standing wave on a string is $f$ when the string's tension is $T$. If the tension is changed by the small amount $\Delta T$, without changing the length, show that the frequency changes by an amount $\Delta f$ such that
1
$$
\frac{\Delta f}{f}=\frac{1}{2} \frac{\Delta T}{T}
$$
b. Two identical strings vibrate at $500 \mathrm{Hz}$ when stretched with the same tension. What percentage increase in the tension of one of the strings will cause five beats per second when both strings vibrate simultancously?

Donald Albin

Numerade Educator

Problem 79

A 280 Hz sound wave is directed into one end of a trombone slide and a microphone is placed at the other end to record the intensity of sound waves that are transmitted through the tube. The straight sides of the slide are $80 \mathrm{cm}$ in length and $10 \mathrm{cm}$ apart with a semicircular bend at the end. For what slide extensions $s$ will the microphone detect a maximum of sound intensity?

Donald Albin

Numerade Educator

Problem 80

As the captain of the scientific team sent to Planet. Physics, one of your tasks is to measure $g$. You have a long, thin wire labeled $1.00 \mathrm{g} / \mathrm{m}$ and a $1.25 \mathrm{kg}$ weight. You have your accurate space cadet chronometer but, unfortunately, you seem to have forgotten a meter stick. Undeterred, you first find the midpoint of the wire by folding it in half. You then attach one end of the wire to the wall of your laboratory, stretch it horizontally to pass over a pulley at the midpoint of the wire, then tie the $1.25 \mathrm{kg}$ weight to the end hanging over the pulley. By vibrating the wire, and measuring time with your chronometer, you find that the wire's second harmonic frequency is $100 \mathrm{Hz}$. Next, with the $1.25 \mathrm{kg}$ weight still tied to one end of the wire, you attach the other end to the ceiling to make a pendulum. You find that the pendulum requires 314 s to complete 100 oscillations. Pulling out your trusty calculator, you get to work. What value of $g$ will you report back to headquarters?

Donald Albin

Numerade Educator

Problem 81

A steel wire is used to stretch a spring. An oscillating magnetic field drives the steel wire back and forth. A standing wave with three antinodes is created when the spring is stretched $8.0 \mathrm{cm} .$ What stretch of the spring produces a standing wave with two antinodes?

Donald Albin

Numerade Educator

Problem 82

Ultrasound has many medical applications, one of which is to monitor fetal heartbeats by reflecting ultrasound off a fetus in the womb.
a. Consider an object moving at speed $v_{\mathrm{o}}$ toward an at-rest source that is emitting sound waves of frequency $f_{0} .$ Show that the reflected wave (i.c., the echo) that returns to the source has a Doppler-shifted frequency $$f_{\text {echo }}=\left(\frac{v+v_{\mathrm{o}}}{v-v_{\mathrm{o}}}\right) f_{0}$$ where $v$ is the speed of sound in the medium.
b. Suppose the object's speed is much less than the wave speed: $v_{\mathrm{o}} \ll v .$ Then $f_{\text {echo }} \approx f_{0},$ and a microphone that is sensitive to these frequencies will detect a beat frequency if it listens to $f_{0}$ and $f_{\text {echo }}$ simultaneously. Use the binomial approximation and other appropriate approximations to show that the beat frequency is $f_{\text {beat }} \approx\left(2 v_{0} / v\right) f_{0}$
c. The reflection of $2.40 \mathrm{MHz}$ ultrasound waves from the sur-
face of a fetus's beating heart is combined with the $2.40 \mathrm{MHz}$ wave to produce a beat frequency that reaches a maximum of $65 \mathrm{~Hz}$. What is the maximum speed of the surface of the heart? The speed of ultrasound waves within the body is $1540 \mathrm{~m} / \mathrm{s}$
d. Suppose the surface of the heart moves in simple harmonic motion at 90 beats/min. What is the amplitude in $\mathrm{mm}$ of the heartbeat?

Donald Albin

Numerade Educator

Problem 83

A water wave is called a deep-water wave if the water's depth is more than one-quarter of the wavelength. Unlike the waves we've considered in this chapter, the speed of a deep-water wave depends on its wavelength:
$$
v=\sqrt{\frac{g \lambda}{2 \pi}}
$$
Longer wavelengths travel faster. Let's apply this to standing waves. Consider a diving pool that is $5.0 \mathrm{m}$ decp and $10.0 \mathrm{m}$ wide. Standing water waves can set up across the width of the pool. Because water sloshes up and down at the sides of the pool, the boundary conditions require antinodes at $x=0$ and $x=L$. Thus a standing water wave resembles a standing sound wave in an open-open tube.
a. What are the wavelengths of the first three standing-wave modes for water in the pool? Do they satisfy the condition for being deep-water waves? Draw a graph of each.
b. What are the wave speeds for each of these waves?
c. Derive a general expression for the frequencies $f_{m}$ of the possible standing waves. Your expression should be in terms of $m, g,$ and $L$
d. What are the oscillation periods of the first three standing wave modes?

Donald Albin

Numerade Educator

Problem 84

The broadcast antenna of an AM radio station is located at the edge of town. The station owners would like to beam all of the energy into town and none into the countryside, but a single antenna radiates energy equally in all directions. FIGURE $\mathrm{CP} 21.84$ shows two parallel antennas separated by distance $L$. Both antennas broadcast a signal at wavelength $\lambda$, but antenna 2 can delay its broadcast relative to antenna 1 by a time interval $\Delta t$ in order to create a phase difference $\Delta \phi_{0}$ between the sources. Your task is to find values of $L$ and $\Delta t$ such that the waves interfere constructively on the town side and destructively on the country side.
(FIGURE CAN'T COPY)
Let antenna 1 be at $x=0 .$ The wave that travels to the right is $a \sin [2 \pi(x / \lambda-t / T)] .$ The left wave is $a \sin [2 \pi(-x / \lambda-u / T)]$
(It must be this, rather than $a \sin [2 \pi(x / \lambda+u / T)]$, so that the two waves match at $x=0 .$ ) Antenna 2 is at $x=L$. It broadcasts wave $\left.a \sin [2 \pi(x-L) / \lambda-t / T)+\phi_{20}\right]$ to the right and wave
$a \sin \left[2 \pi(-(x-L) / \lambda-u / T)+\phi_{20}\right]$ to the left.
a. What is the smallest value of $L$ for which you can create perfect constructive interference on the town side and perfect destructive interference on the country side? Your answer will be a multiple or fraction of the wavelength $\lambda$.
b. What phase constant $\phi_{20}$ of antenna 2 is needed?
c. What fraction of the oscillation period $T$ must $\Delta t$ be to produce the proper value of $\phi_{20} ?$
d. Evaluate both $L$ and $\Delta t$ for the realistic AM radio frequency of $1000 \mathrm{KHz}$
Comment: This is a simple example of what is called a phased array, where phase differences between identical emitters are used to "steer" the radiation in a particular direction. Phased arrays are widely used in radar technology.

Mayukh Banik

Numerade Educator