A 44-cm-diameter water tank is filled with 35 cm of water.

What is superposition?

What is a standing wave?

How are standing waves related to music?

What is interference?

What are beats?

Why is superposition important?

Two pulses on a string approach each other at speeds of 1 m/s. What is the shape of the string at t = 6 s?

A very long string has a linear density of 5.0 g/m and is stretched with a tension of 8.0 N. 100 Hz waves with amplitudes of 2.0 mm are generated at the ends of the string. a. What is the node spacing along the resulting standing wave? b. What is the maximum displacement of the string?

Standing-wave frequencies can be measured very accurately. Consequently, standing waves are often used in experiments to make accurate measurements of other quantities. One such experiment, shown in FIGURE 17.10, uses standing waves to measure the free-fall acceleration g. A heavy mass is suspended from a 1.65-m-long, 5.85 g steel wire; then an oscillating magnetic field (because steel is magnetic) is used to excite the m = 3 standing wave on the wire. Measuring the frequency for different masses yields the data given in the table. Analyze these data to determine the local value of g

A standing wave on a string vibrates as shown at the right. Suppose the string tension is quadrupled while the frequency and the length of the string are held constant. Which standing-wave pattern is produced?

Helium-neon lasers emit the red laser light commonly used in classroom demonstrations and supermarket checkout scanners. A helium-neon laser operates at a wavelength of precisely 632.9924 nm when the spacing between the mirrors is 310.372 mm. a. In which mode does this laser operate? b. What is the next longest wavelength that could form a standing wave in this laser cavity?

An open-open tube of air supports standing waves at frequencies of 300 Hz and 400 Hz and at no frequencies between these two. The second harmonic of this tube has frequency a. 100 Hz b. 200 Hz c. 400 Hz d. 600 Hz e. 800 Hz

A shower stall is 2.45 m (8 ft) tall. For what frequencies less than 500 Hz are there standing sound waves in the shower stall?

Two loudspeakers emit waves with l = 2.0 m. Speaker 2 is 1.0 m in front of speaker 1. What, if anything, can be done to cause constructive interference between the two waves? a. Move speaker 1 forward (to the right) 1.0 m. b. Move speaker 1 forward (to the right) 0.5 m. c. Move speaker 1 backward (to the left) 0.5 m. d. Move speaker 1 backward (to the left) 1.0 m. e. Nothing. The situation shown already causes constructive interference. f. Constructive interference is not possible for any placement of the speakers.

The eardrum, which transmits sound vibrations to the sensory organs of the inner ear, lies at the end of the ear canal. For adults, the ear canal is about 2.5 cm in length. What frequency standing waves can occur in the ear canal that are within the range of human hearing? The speed of sound in the warm air of the ear canal is 350 m/s

The interference at point C in Figure 17.26 is a. Maximum constructive. b. Constructive, but less than maximum. c. Maximum destructive. d. Destructive, but less than maximum. e. There is no interference at point C

A clarinet is 66.0 cm long. A flute is nearly the same length, with 63.6 cm between the hole the player blows across and the end of the flute. What are the frequencies of the lowest note and the next higher harmonic on a flute and on a clarinet? The speed of sound in warm air is 350 m/s

These two loudspeakers are in phase. They emit equalamplitude sound waves with a wavelength of 1.0 m. At the point indicated, is the interference maximum constructive, maximum destructive, or something in between?

You are standing in front of two side-by-side loudspeakers playing sounds of the same frequency. Initially there is almost no sound at all. Then one of the speakers is moved slowly away from you. The sound intensity increases as the separation between the speakers increases, reaching a maximum when the speakers are 0.75 m apart. Then, as the speaker continues to move, the intensity starts to decrease. What is the distance between the speakers when the sound intensity is again a minimum?

You hear three beats per second when two sound tones are generated. The frequency of one tone is 610 Hz. The frequency of the other is a. 604 Hz b. 607 Hz c. 613 Hz d. 616 Hz e. Either a or d. f. Either b or c.

Two loudspeakers emit 500 Hz sound waves with an amplitude of 0.10 mm. Speaker 2 is 1.00 m behind speaker 1, and the phase difference between the speakers is 90. What is the amplitude of the sound wave at a point 2.00 m in front of speaker 1?

Magnesium fluoride (MgF2) is used as an antireflection coating on lenses. The index of refraction of MgF2 is 1.39. What is the thinnest film of MgF2 that works as an antireflection coating at l = 510 nm, near the center of the visible spectrum?

Two loudspeakers in a plane are 2.0 m apart and in phase with each other. Both emit 700 Hz sound waves into a room where the speed of sound is 341 m/s. A listener stands 5.0 m in front of the loudspeakers and 2.0 m to one side of the center. Is the interference at this point maximum constructive, maximum destructive, or in between? How will the situation differ if the loudspeakers are out of phase?

The little brown bat is a common species in North America. It emits echolocation pulses at a frequency of 40 kHz, well above the range of human hearing. To allow researchers to hear these bats, the bat detector shown in FIGURE 17.30 combines the bats sound wave at frequency f1 with a wave of frequency f2 from a tunable oscillator. The resulting beat frequency is then amplified and sent to a loudspeaker. To what frequency should the tunable oscillator be set to produce an audible beat frequency of 3 kHz?

FIGURE Q17.1 shows a standing wave oscillating on a string at frequency f0. a. What mode (m-value) is this? b. How many antinodes will there be if the frequency is doubled to 2f0?

If you take snapshots of a standing wave on a string, there are certain instants when the string is totally flat. What has happened to the energy of the wave at those instants?

FIGURE Q17.3 shows the displacement of a standing sound wave in a 32-cm-long horizontal tube of air open at both ends. a. What mode (m-value) is this? b. Are the air molecules moving horizontally or vertically? Explain. c. At what distances from the left end of the tube do the molecules oscillate with maximum amplitude? d. At what distances from the left end of the tube does the air pressure oscillate with maximum amplitude?

An organ pipe is tuned to exactly 384 Hz when the room temperature is 20C. If the room temperature later increases to 22C, does the pipes frequency increase, decrease, or stay the same? Explain.

If you pour liquid into a tall, narrow glass, you may hear sound with a steadily rising pitch. What is the source of the sound? And why does the pitch rise as the glass fills?

A flute filled with helium will, until the helium escapes, play notes at a much higher pitch than normal. Why?

In music, two notes are said to be an octave apart when one note is exactly twice the frequency of the other. Suppose you have a guitar string playing frequency f0. To increase the frequency by an octave, to 2f0, by what factor would you have to (a) increase the tension or (b) decrease the length?

FIGURE Q17.8 is a snapshot graph of two plane waves passing through a region of space. Each wave has a 2.0 mm amplitude and the same wavelength. What is the net displacement of the medium at points a, b, and c?

FIGURE Q17.9 shows the circular waves emitted by two in-phase sources. Are a, b, and c points of maximum constructive interference, maximum destructive interference, or in between?

A trumpet player hears 5 beats per second when she plays a note and simultaneously sounds a 440 Hz tuning fork. After pulling her tuning valve out to slightly increase the length of her trumpet, she hears 3 beats per second against the tuning fork. Was her initial frequency 435 Hz or 445 Hz? Explain.

FIGURE EX17.1 is a snapshot graph at t = 0 s of two waves approaching each other at 1.0 m/s. Draw six snapshot graphs, stacked vertically, showing the string at 1 s intervals from t = 1 s to t = 6 s.

FIGURE EX17.2 is a snapshot graph at t = 0 s of two waves approaching each other at 1.0 m/s. Draw six snapshot graphs, stacked vertically, showing the string at 1 s intervals from t = 1 s to t = 6 s.

FIGURE EX17.3a is a snapshot graph at t = 0 s of two waves approaching each other at 1.0 m/s. At what time was the snapshot graph in FIGURE EX17.3b taken?

FIGURE EX17.4 is a snapshot graph at t = 0 s of two waves moving to the right at 1.0 m/s. The string is fixed at x = 8.0 m. Draw four snapshot graphs, stacked vertically, showing the string at t = 2, 4, 6, and 8 s.

FIGURE EX17.5 shows a standing wave oscillating at 100 Hz on a string. What is the wave speed?

FIGURE EX17.6 shows a standing wave on a 2.0-m-long string that has been fixed at both ends and tightened until the wave speed is 40 m/s. What is the frequency?

wave on a string that is oscillating at 100 Hz. a. How many antinodes will there be if the frequency is increased to 200 Hz? b. If the tension is increased by a factor of 4, at what frequency will the string continue to oscillate as a standing wave that looks like the one in the figure?

a. 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 Hz, what is the frequency of the third-longest wavelength?

Standing waves on a 1.0-m-long string that is fixed at both ends are seen at successive frequencies of 36 Hz and 48 Hz. a. What are the fundamental frequency and the wave speed? b. Draw the standing-wave pattern when the string oscillates at 48 Hz.

The two highest-pitch strings on a violin are tuned to 440 Hz (the A string) and 659 Hz (the E string). What is the ratio of the mass of the A string to that of the E string? Violin strings are all the same length and under essentially the same tension.

A heavy piece of hanging sculpture is suspended by a 90-cmlong, 5.0 g steel wire. When the wind blows hard, the wire hums at its fundamental frequency of 80 Hz. What is the mass of the sculpture

A carbon dioxide laser is an infrared laser. A CO2 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?

Microwaves pass through a small hole into the microwave cavity of FIGURE EX17.13. What frequencies between 10 GHz and 20 GHz will create standing waves in the cavity?

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?

FIGURE EX17.15 shows a standing sound wave in an 80-cm-long tube. The tube is filled with an unknown gas. What is the speed of sound in this gas?

The fundamental frequency of an open-open tube is 1500 Hz when the tube is filled with 0C helium. What is its frequency when filled with 0C air?

We can make a simple model of the human vocal tract as an open-closed tube extending from the opening of the mouth to the diaphragm. What is the length of this tube if its fundamental frequency equals a typical speech frequency of 250 Hz? The speed of sound in the warm air is 350 m/s

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

A bass clarinet can be modeled as a 120-cm-long open-closed tube. A bass clarinet player starts playing in a 20C room, but soon the air inside the clarinet warms to where the speed of sound is 352 m/s. Does the fundamental frequency increase or decrease? By how much?

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

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. As FIGURE EX17.21 shows, both ends of the rod are antinodes. What is the fundamental frequency of a 2.0-m-long aluminum rod?

Two loudspeakers emit sound waves along the x-axis. The sound has maximum intensity when the speakers are 20 cm apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of 60 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?

Two loudspeakers in a 20C room emit 686 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 maximum 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?

Noise-canceling headphones are an application of destructive interference. Each side of the headphones uses a microphone to pick up noise, delays it slightly, then rebroadcasts the noise next to your ear where it can interfere with the incoming sound wave of the noise. Suppose you are sitting 1.8 m from an annoying, 110 Hz buzzing sound. What is the minimum headphone delay, in ms, that will cancel this noise?

What is the thinnest film of MgF2 1n = 1.392 on glass that produces a strong reflection for orange light with a wavelength of 600 nm?

A very thin oil film 1n = 1.252 floats on water 1n = 1.332. What is the thinnest film that produces a strong reflection for green light with a wavelength of 500 nm?

FIGURE EX17.27 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 r1, r2, r, and C/D. Fill in the table for points P, Q, and R, giving the distances as multiples of l and indicating, with a C or a D, whether the interference at that point is constructive or destructive.

FIGURE EX17.28 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 r1, r2, r, and C/D. Fill in the table for points P, Q, and R, giving the distances as multiples of l and indicating, with a C or a D, whether the interference at that point is constructive or destructive.

Two in-phase loudspeakers, which emit sound in all directions, are sitting side by side. One of them is moved sideways by 3.0 m, then forward by 4.0 m. Afterward, constructive interference is observed 1 4 and 3 4 of the distance between the speakers along the line that joins them. What is the maximum possible wavelength of the sound waves?

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

Two out-of-phase radio antennas at x = {300 m on the x-axis are emitting 3.0 MHz radio waves. Is the point 1x, y2 = 1300 m, 800 m2 a point of maximum constructive interference, maximum destructive interference, or something in between?

Two strings are adjusted to vibrate at exactly 200 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?

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?

Traditional Indonesian music uses an ensemble called a gamelan that is based on tuned percussion instruments somewhat like gongs. In Bali, the gongs are often grouped in pairs that are slightly out of tune with each other. When both are played at once, the beat frequency lends a distinctive vibrating quality to the music. Suppose a pair of gongs are tuned to produce notes at 151 Hz and 155 Hz. How many beats are heard if the gongs are struck together and both ring for 2.5 s?

Two microwave signals of nearly equal wavelengths can generate a beat frequency if both are directed onto the same microwave detector. In an experiment, the beat frequency is 100 MHz. One microwave generator is set to emit microwaves with a wavelength of 1.250 cm. If the second generator emits the longer wavelength, what is that wavelength?

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

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

Tendons are, essentially, elastic cords stretched between two fixed ends. As such, they can support standing waves. A woman has a 20-cm-long Achilles tendonconnecting the heel to a muscle in the calfwith a cross-section area of 90 mm2 . The density of tendon tissue is 1100 kg/m3 . For a reasonable tension of 500 N, what will be the fundamental frequency of her Achilles tendon

Biologists think that some spiders tune strands of their web to give enhanced response at frequencies corresponding to those at which desirable prey might struggle. Orb spider web silk has a typical diameter of 20 mm, and spider silk has a density of 1300 kg/m3 . To have a fundamental frequency at 100 Hz, to what tension must a spider adjust a 12-cm-long strand of silk?

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

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

A steel wire is used to stretch the spring of FIGURE P17.42. 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 cm. What stretch of the spring produces a standing wave with two antinodes?

Astronauts visiting Planet X have a 250-cm-long string whose mass is 5.00 g. They tie the string to a support, stretch it horizontally over a pulley 2.00 m away, and hang a 4.00 kg mass on the free end. Then the astronauts begin to excite standing waves on the horizontal portion of the string. Their data are as follows: m Frequency (Hz) 1 31 2 66 3 95 4 130 5 162 Use the best-fit line of an appropriate graph to determine the value of g, the free-fall acceleration on Planet X

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

A metal wire under tension T0 vibrates at its fundamental frequency. For what tension will the second-harmonic frequency be the same as the fundamental frequency at tension T0?

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 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 kg sphere is completely submerged. The frequency of the fifth harmonic with the sphere submerged exactly matches the frequency of the third harmonic before the sphere was submerged. What is the diameter of the sphere?

A vibrating standing wave on a string radiates a sound wave with intensity proportional to the square of the standing-wave amplitude. When a piano key is struck and held down, so that the string continues to vibrate, the sound level decreases by 8.0 dB in 1.0 s. What is the strings damping time constant t?

What is the fundamental frequency of the steel wire i

The two strings in FIGURE P17.49 are of equal length and are being driven at equal frequencies. The linear density of the left string is 5.0 g/m. What is the linear density of the right string?

Western music uses a musical scale with equal temperament tuning, which means that any two adjacent notes have the same frequency ratio r. That is, notes n and n + 1 are related by fn+1 = rfn for all n. In this system, the frequency doubles every 12 notesan interval called an octave. a. What is the value of r? b. Orchestras tune to the note A, which has a frequency of 440 Hz. What is the frequency of the next note of the scale (called A-sharp)?

An open-open organ pipe is 78.0 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?

Deep-sea divers often breathe a mixture of helium and oxygen to avoid getting the bends from breathing high-pressure nitrogen. The helium has the side effect of making the divers voices sound odd. Although your vocal tract can be roughly described as an open-closed tube, the way you hold your mouth and position your lips greatly affects the standing-wave frequencies of the vocal tract. This is what allows different vowels to sound different. The ee sound is made by shaping your vocal tract to have standing-wave frequencies at, normally, 270 Hz and 2300 Hz. What will these frequencies be for a helium-oxygen mixture in which the speed of sound at body temperature is 750 m/s? The speed of sound in air at body temperature is 350 m/s

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 Kundts 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 P17.53 shows an experiment in which the tube is filled with pure oxygen and the piston is driven at 400 Hz. What is the speed of sound in oxygen?

The 40-cm-long tube of FIGURE P17.54 has a 40-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 cm, 56.7 cm, and 70.9 cm. What is the frequency of the tuning fork? Assume vsound = 343 m/s

A 1.0-m-tall vertical tube is filled with 20C water. A tuning fork vibrating at 580 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?

A 44-cm-diameter water tank is filled with 35 cm of water. A 3.0-mm-diameter spigot at the very bottom of the tank is opened and water begins to flow out. The water falls into a 2.0-cm-diameter, 40-cm-tall glass cylinder. As the water falls and creates noise, resonance causes the column of air in the cylinder to produce a tone at the columns fundamental frequency. What are (a) the frequency and (b) the rate at which the frequency is changing (Hz/s) when the cylinder has been filling for 4.0 s? You can assume that the height of the water in the tank does not appreciably change in 4.0 s

A 25-cm-long wire with a linear density of 20 g/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 vsound = 340 m/s

An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but its 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 Hz and 6.3 Hz, and at no frequencies between these. Its rather chilly inside the tunnel, so you estimate the sound speed to be 335 m/s. Based on your measurements, how far is it to the end of the tunnel?

Two in-phase loudspeakers emit identical 1000 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?

Analyze the standing sound waves in an open-closed tube to show that the possible wavelengths and frequencies are given by Equation 17.18.

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 cm behind speaker 1. The intensity increases as speaker 2 is moved forward and first reaches maximum, with amplitude 2a, when it is 30 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 speakers are placed side by side?

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 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 m. a. What is the frequency of the sound? Assume vsound = 340 m/s. b. What is the phase difference between the speakers?

A sheet of glass is coated with a 500-nm-thick layer of oil 1n = 1.422. 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?

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 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?

A soap bubble is essentially a very thin film of water 1n = 1.332 surrounded by air. The colors that you see in soap bubbles are produced by interference. a. Derive an expression for the wavelengths lC 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?

Engineers are testing a new thin-film coating whose index of refraction is less than that of glass. They deposit a 560-nm-thick layer on glass, then shine lasers on it. A red laser with a wavelength of 640 nm has no reflection at all, but a violet laser with a wavelength of 400 nm has a maximum reflection. How the coating behaves at other wavelengths is unknown. What is the coatings index of refraction?

Scientists are testing a transparent material whose index of refraction for visible light varies with wavelength as n = 30.0 nm1/2 /l1/2, where l is in nm. If a 295-nm-thick coating is placed on glass 1n = 1.502 for what visible wavelengths will the reflected light have maximum constructive interference?

You are standing 2.5 m directly in front of one of the two loudspeakers shown in FIGURE P17.68. They are 3.0 m apart and both are playing a 686 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 20C

Two loudspeakers in a plane, 5.0 m apart, are playing the same frequency. If you stand 12.0 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 m in front of them, you first hear a minimum of sound intensity when you are directly in front of one of the speakers. What is the frequency of the sound? Assume a sound speed of 340 m/s.

Two identical loudspeakers separated by distance x each emit sound waves of wavelength l and amplitude a along the x-axis. What is the minimum value of the ratio x/l for which the amplitude of their superposition is also a?

The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is 340 m/s. You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each 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?

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 Hz and the note E should be at 659 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 Hz tuning fork. She then strikes the A and E strings simultaneously and listens for beats between the harmonics. What beat frequency indicates that the E string is properly tuned? c. The tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

A flutist assembles her flute in a room where the speed of sound is 342 m/s. When she plays the note A, it is in perfect tune with a 440 Hz tuning fork. After a few minutes, the air inside her flute has warmed to where the speed of sound is 346 m/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?

You have two small, identical boxes that generate 440 Hz notes. While holding one, you drop the other from a 20-m-high balcony. How many beats will you hear before the falling box hits the ground? You can ignore air resistance

Two loudspeakers emit 400 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 trucks speed?

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

A 280 Hz sound wave is directed into one end of the trombone slide seen in FIGURE CP17.77. 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 cm in length and 10 cm apart with a semicircular bend at the end. For what slide extensions s will the microphone detect a maximum of sound intensity?

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 g/m and a 1.25 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 kg weight to the end hanging over the pulley. By vibrating the wire, and measuring time with your chronometer, you find that the wires secondharmonic frequency is 100 Hz. Next, with the 1.25 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?

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

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 vo toward an at-rest source that is emitting sound waves of frequency f0. Show that the reflected wave (i.e., the echo) that returns to the source has a Doppler-shifted frequency fecho = 1 v + vo v - vo 2 f0 where v is the speed of sound in the medium. b. Suppose the objects speed is much less than the wave speed: vo V v. Then fecho f0, and a microphone that is sensitive to these frequencies will detect a beat frequency if it listens to f0 and fecho simultaneously. Use the binomial approximation and other appropriate approximations to show that the beat frequency is fbeat 12vo/v2 f0. c. The reflection of 2.40 MHz ultrasound waves from the surface of a fetuss beating heart is combined with the 2.40 MHz wave to produce a beat frequency that reaches a maximum of 65 Hz. What is the maximum speed of the surface of the heart? The speed of ultrasound waves within the body is 1540 m/s. d. Suppose the surface of the heart moves in simple harmonic motion at 90 beats/min. What is the amplitude in mm of the heartbeat?

A water wave is called a deep-water wave if the waters depth is more than one-quarter of the wavelength. Unlike the waves weve considered in this chapter, the speed of a deep-water wave depends on its wavelength: v = B gl 2p Longer wavelengths travel faster. Lets apply this to standing waves. Consider a diving pool that is 5.0 m deep and 10.0 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? b. What are the wave speeds for each of these waves? c. Derive a general expression for the frequencies fm 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?

What are the overarching findings of Part IV?

What laws of physics govern oscillations and waves?

What are the most important models of Part IV?

What are the most important tools introduced in Part IV?