a. What are the three longest wavelengths for standing

Problem 1CQ Light can pass easily through water and through air, but light will reflect from the surface of a lake. What does this tell you about the speed of light in air and in water?

Problem 1P Figure P16.1 is a snapshot graph at t = 0 s of two waves on a taut string approaching each other at 1 m/s. Draw six snapshot graphs, stacked vertically, showing the string at 1 s intervals from t = 1 s to t = 6 s.

Problem 2CQ Ocean waves are partially reflected from the entrance to a harbor, where the depth of the water is suddenly less. What does this tell you about the speed of waves in water of different depths?

Problem 2P Figure P16.2 is a snapshot graph at t = 0 s of two waves approaching each other at 1 m/s. Draw four snapshot graphs, stacked vertically, showing the string at t = 2, 4, 6, and 8 s.

Problem 3CQ A string has an abrupt change in linear density at its midpoint so that the speed of a pulse on the left side is 2/3 of that on the right side. a. On which side is the linear density greater? Explain. b. From which side would you start a pulse so that its reflection from the midpoint would not be inverted? Explain.

Problem 3P Figure P16.3 a is a snapshot graph at t = 0 s of two waves on a string approaching each other at 1 m/s. At what time was the snapshot graph in Figure P16.3 b taken?

Problem 4CQ A guitarist finds that the frequency of one of her strings is too low by 1.4%. Should she increase or decrease the tension of the string? Explain.

Problem 4P Figure P16.6 is a snapshot graph at t = 0 s of a pulse on a string moving to the right at 1 m/s. The string is fixed at x = 5 m. Draw a history graph spanning the time interval t = 0 s to t = 10 s for the location x = 3 m on the string.

Problem 5CQ Certain illnesses inflame your vocal cords, causing them to swell. How does this affect the pitch of your voice? Explain.

Problem 5P At t = 0 s, a small “upward” (positive y ) pulse centered at x = 6.0 m is moving to the right on a string with fixed ends at x = 0.0 m and x = 10.0 m. The wave speed on the string is 4.0 m/s. At what time will the string next have the same appearance that it did at t = 0 s?

Problem 6CQ Figure Q16.6 shows a standing wave on a string that is oscillating at frequency . How many antinodes will there be if the frequency is doubled to ? Explain.

Problem 6P You are holding one end of an elastic cord that is fastened to a wall 3.0 m away. You begin shaking the end of the cord at 3.5 Hz, creating a continuous sinusoidal wave of wavelength 1.0 m. How much time will pass until a standing wave fills the entire length of the string?

Problem 7P A 2.0-m-long string is fixed at both ends and tightened until the wave speed is 40 m/s. What is the frequency of the standing wave shown in Figure P16.9 ?

Problem 7CQ Figure Q16.7 shows a standing sound wave in a tube of air that is open at both ends. a. Which mode (value of m ) standing wave is this? b. Is the air vibrating horizontally or vertically?

Problem 8CQ A typical flute is about 66 cm long. A piccolo is a very similar instrument, though it is smaller, with a length of about 32 cm. How does the pitch of a piccolo compare to that of a flute?

Problem 8P Figure P16.10 shows a standing wave oscillating at 100 Hz on a string. What is the wave speed?

Problem 9P A bass guitar string is 89 cm long with a fundamental frequency of 30 Hz. What is the wave speed on this string?

Problem 9CQ Some pipes on a pipe organ are open at both ends, others are closed at one end. For pipes that play low-frequency notes, there is an advantage to using pipes that are closed at one end. What is the advantage?

Problem 10CQ A flute player tunes her instrument when the air (and the flute) is cold. As she plays, the flute and the air inside it warm up. Both the changing speed of sound in the air inside and the thermal expansion of the flute affect the frequency of the sound wave produced by the flute. After some time, she finds that her playing is “ sharp” —the frequencies are too high. Which change produced this effect: the warming of the air or the warming of the body of the flute?

Problem 10P The fundamental frequency of a guitar string is 384 Hz. What is the fundamental frequency if the tension in the string is halved?

Problem 11CQ A friend’s voice sounds different over the telephone than it does in person. This is because telephones do not transmit frequencies over about 3000 Hz. 3000 Hz is well above the normal frequency of speech, so why does eliminating these high frequencies change the sound of a person’s voice?

Problem 11P 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 50.0 Hz, what is the frequency of the third-longest wavelength?

Problem 12CQ Suppose you were to play a trumpet after breathing helium, in which the speed of sound is much greater than in air. Would the pitch of the instrument be higher or lower than normal, or would it be unaffected by being played with helium inside the tube rather than air?

Problem 12P A 121-cm-long, 4.00 g string oscillates in its m = 3 mode with a frequency of 180 Hz and a maximum amplitude of 5.00 mm. What are (a) the wavelength and (b) the tension in the string?

Problem 13CQ If you pour liquid in 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?

Problem 13P A guitar string with a linear density of 2.0 g/m is stretched between supports that are 60 cm apart. The string is observed to form a standing wave with three antinodes when driven at a frequency of 420 Hz. What are (a) the frequency of the fifth harmonic of this string and (b) the tension in the string?

Problem 14CQ When you speak after breathing helium, in which the speed of sound is much greater than in air, your voice sounds quite different. The frequencies emitted by your vocal cords do not change since they are determined by the mass and tension of your vocal cords. So what does change when your vocal tract is filled with helium rather than air?

Problem 14P A violin string has a standard length of 32.8 cm. 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)?

Problem 15CQ Sopranos can sing notes at very high frequencies—over 1000 Hz. When they sing such high notes, it can be difficult to understand the words they are singing. Use the concepts of harmonics and formants to explain this.

Problem 15P 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?

Problem 16CQ When you hit a baseball with a bat, the bat flexes and then vibrates. We can model this vibration as a transverse standing wave. The modes of this standing wave are similar to the modes of a stretched string, but with one important difference: The ends of the bat are antinodes instead of nodes, because the ends of the bat are free to move. The modes thus look like the modes of a stretched string with antinodes replacing nodes and nodes replacing antinodes. If the ball hits the bat near an antinode of a standing-wave mode, the bat will start oscillating in this mode. The batter holds the bat at one end, which is also an antinode, so a large vibration of the bat causes an unpleasant vibration in the batter’s hands. This can be avoided if the ball hits the bat at what players call the “sweet spot,” which is a node of the standing-wave pattern. The first standing-wave mode of a vibrating bat is the m = 2mode. Sketch the appearance of this vibrational mode of the bat, then estimate the approximate distance of the sweet spot (as a fraction of the bat’s length) from the end of the bat.

Problem 16P The lowest frequency in the audible range is 20 Hz. What are the lengths of (a) the shortest open-open tube and (b) the shortest open-closed tube needed to produce this frequency?

Problem 17CQ If a cold gives you a stuffed-up nose, it changes the way your voice sounds, even if your vocal cords are not affected. Explain why this is so.

Problem 17P The contrabassoon is the wind instrument capable of sounding the lowest pitch in an orchestra. It is folded over several times to fit its impressive 18 ft length into a reasonable size instrument. a. If we model the instrument as an open-closed tube, what is its fundamental frequency? The sound speed inside is 350 m/s because the air is warmed by the player’s breath. b. The actual fundamental frequency of the contrabassoon is 27.5 Hz, which should be different from your answer in part a. This means the model of the instrument as an open-closed tube is a bit too simple. But if you insist on using that model, what is the “effective length” of the instrument?

Problem 18CQ A small boy and a grown woman both speak at approximately the same pitch. Nonetheless, it’s easy to tell which is which from listening to the sounds of their voices. How are you able to make this determination?

Problem 18P Figure P16.22 shows a standing sound wave in an 80-cmlong tube. The tube is filled with an unknown gas. What is the speed of sound in this gas?

Problem 19CQ Figure shows wave fronts of a circular wave. Are the displacements at the following pairs of positions in phase or out of phase? Explain. a. Aand B ________________ b. C and D ________________ c. E and F FIGURE

Problem 19P 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?

Problem 20MCQ Question refer to the snapshot graph of Figure Q16.18 . At t = 1 s, what is the displacement y of the string at x = 7 cm? A. -1.0 mm B. 0 mm C. 0.5 mm D. 1.0 mm E. 2.0 mm

Problem 20P The lowest pedal note on a large pipe organ has a fundamental frequency of 16 Hz. This extreme bass note is more felt as a rumble than heard with the ears. What is the length of the open-closed pipe that makes that note?

Problem 21MCQ Question refer to the snapshot graph of Figure Q16.18 . At x = 3 cm, what is the earliest time that y will equal 2 mm? A. 0.5 s B. 0.7 s C. 1.0 s D. 1.5 s E. 2.5 s

Problem 21P The fundamental frequency of an open-open tube is 1500 Hz when the tube is filled with 0°C helium. What is its frequency when filled with 0°C air?

Problem 22MCQ Question refer to the snapshot graph of Figure Q16.18 . At t = 1.5 s , what is the value of y at x = 10 cm ? A. -2.0 mm B. -1.0 mm C. -0.5 mm D. 0 mm E. 1.0 mm

Problem 22P 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. If you model the tube as an open-closed system, what are the first three resonant frequencies?

Problem 23MCQ Two sinusoidal waves with the same amplitude A and frequency f travel in opposite directions along a long string. You stand at one point and watch the string. The maximum displacement of the string at that point is A. A B. 2 A C. 0 D. There is not enough information to decide.

Problem 23P A drainage pipe running under a freeway is 30.0 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 m/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?

Problem 24MCQ A student in her physics lab measures the standing-wave modes of a tube. The lowest frequency that makes a resonance is 20 Hz. As the frequency is increased, the next resonance is at 60 Hz. What will be the next resonance after this? A. 80 Hz B. 100 Hz C. 120 Hz D. 180 Hz

Problem 24P 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 Hz? Assume a sound speed of 350 m/s. Does this result for the tube length seem reasonable, based on observations on your own body?

Problem 25MCQ An organ pipe is tuned to exactly 384 Hz when the temperature in the room is 20°C. Later, when the air has warmed up to 25°C, the frequency is A. Greater than 384 Hz. B. 384 Hz. C. Less than 384 Hz.

Problem 25P A child has an ear canal that is 1.3 cm long. At what sound frequencies in the audible range will the child have increased hearing sensitivity?

Problem 26MCQ Two guitar strings made of the same type of wire have the same length. String 1 has a higher pitch than string 2. Which of the following is true? A. The wave speed of string 1 is greater than that of string 2. ________________ B. The tension in string 2 is greater than that in string 1. ________________ C. The wavelength of the lowest standing-wave mode on string 2 is longer than that on string 1. ________________ D. The wavelength of the lowest standing-wave mode on string 1 is longer than that on string 2.

Problem 26P 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?

Problem 27MCQ The frequency of the lowest standing-wave mode on a 1.0-m-long string is 20 Hz. What is the wave speed on the string? A. 10 m/s B. 20 m/s C. 30 m/s D. 40 m/s

Problem 27P Deep-sea divers often breathe a mixture of helium and oxy-gen to avoid the complications of breathing high-pressure nitrogen. At great depths the mix is almost entirely helium, which has the side effect of making the divers’ voices sound very odd. Breathing helium doesn’t affect the frequency at which the vocal cords vibrate, but it does affect the frequencies of the formants. The text gives the frequencies of the first two formants for an “ee” vowel sound as 270 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?

Problem 28MCQ Suppose you pluck a string on a guitar and it produces the note A at a frequency of 440 Hz. Now you press your finger down on the string against one of the frets, making this point the new end of the string. The newly shortened string has 4/5 the length of the full string. When you pluck the string, its frequency will be A. 350 Hz B. 440 Hz C. 490 Hz D. 550 Hz

Problem 28P Two loudspeakers in a 20°C room emit 686 Hz sound waves along the x-axis. What is the smallest distance between the speakers for which the interference of the sound waves is destructive?

Problem 29P 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 30 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?

Problem 30P Two identical loudspeakers separated by distance d emit 170 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 m/s.

Problem 31P Figure P16.40 shows the circular wave fronts emitted by two sources. Make a table with rows labeled P, Q, and R and columns labeled , 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.

Problem 32P Two identical loudspeakers 2.0 m apart are emitting 1800 Hz sound waves into a room where the speed of sound is 340 m/s. Is the point 4.0 m directly 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?

Problem 33P 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?

Problem 35GP The fundamental frequency of a standing wave on a 1.0-m-long string is 440 Hz. What would be the wave speed of a pulse moving along this string?

Problem 34P 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?

Problem 36GP In addition to producing images, ultrasound can be used to heat tissues of the body for therapeutic purposes. When a sound wave hits the boundary between soft tissue and air, or between soft tissue and bone, most of the energy is reflected; only 0.11 %is transmitted. This means that standing waves can be set up in the body, creating excess thermal energy in the tissues at an antinode. Suppose 0.75 MHz ultrasound is directed through a layer of tissue with a bone 0.50 cm below the surface. Will standing waves be created? Explain.

Problem 37GP An 80-cm-long steel string with a linear density of 1.0 g/m is under 200 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°C room?

Problem 38GP 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-cm-long tendon with a cross-section area of . The density of tendon tissue is . For a reasonable tension of 500 N, what will be the resonant frequencies of her Achilles tendon?

Problem 39GP A string, stretched between two fixed posts, forms standingwave resonances at 325 Hz and 390 Hz. What is the largest possible value of its fundamental frequency?

Problem 40GP 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 mm, and spider silk has a density of . To give a resonance at 100 Hz, to what tension must a spider adjust a 12-cm-long strand of silk?

Problem 41GP 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 20°C room hears a note with a wavelength of 40 cm. What is the tension in the string?

Problem 42GP 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 string’s linear density is 0.600 g/m and the tension is 150 N, how long is the vibrating section of the violin string?

Problem 43GP A heavy piece of hanging sculpture is suspended by a 90-cm-long, 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?

Problem 44GP An experimenter finds that standing waves on a 0.80-m-long string, fixed at both ends, occur at 24 Hz and 32 Hz, but at no frequencies in between. a. What is the fundamental frequency? ________________ b. What is the wave speed on the string? ________________ c. Draw the standing-wave pattern for the string at 32 Hz.

Problem 45GP Astronauts visiting Planet X have a 2.5-m-long siring whose mass is 5.0 g. They tie the string to a support, stretch it horizontally over a pulley 2.0 m away, and hang a 1.0 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 Hz and 80 Hz, but at no frequencies in between. What is the value of g, the free-fall acceleration, on Planet X?

Problem 46P A 75 g bungee cord has an equilibrium length of 1.2 m. The cord is stretched to a length of 1.8 m, then vibrated at 20 Hz. This produces a standing wave with two antinodes. What is the spring constant of the bungee cord?

Problem 47GP A 2.5-cm-diameter steel cable (with density 7900 kg/m3) that is part of the suspension system for a footbridge stretches 14 m between the tower and the ground. After walking over the bridge, a hiker finds that the cable is vibrating in its fundamental mode with a period of 0.40 s. What is the tension in the cable?

Problem 48GP 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 P16.56 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?

Problem 49GP A steel wire is used to stretch a spring, as shown in Figure. 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? FIGURE

Problem 50GP Just as you are about to step into a nice hot bath, a small earthquake rattles your bathroom. Immediately afterward, you notice that the water in the tub is oscillating. The water in the center seems to be motionless while the water at the two ends alternately rises and falls, like a seesaw. You happen to know that your bathtub is 1.4 m long, and you count 10 complete oscillations of the water in 20 s. a. What is the wavelength of this standing wave? ________________ b. What is the speed of the waves that are reflecting back and forth inside the tub to create the standing wave?

Problem 51GP A microwave generator can produce microwaves at any frequency between 10 GHz and 20 GHz. As Figure P16.57 shows, the microwaves are aimed, through a small hole, into a “microwave cavity” that consists of a 10-cm-long cylinder with reflective ends. a. Which frequencies between 10 GHz and 20 GHz will create standing waves in the microwave cavity? b. For which of these frequencies is the cavity midpoint an antinode?

Problem 52GP 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?

Problem 53GP A carbon-dioxide laser emits infrared light with a wavelength of 10.6 mm. 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?

Problem 54GP 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 shows an experiment in which the tube is filled with pure oxygen and the piston is driven at 400 Hz. FIGURE a. Do the cork particles collect at standing-wave nodes or antinodes? Hint: Consider the appearance of the ends of the tube. ________________ b. What is the speed of sound in oxygen?

Problem 55GP A 40-cm-long tube has a 40-cmlong insert that can be pulled in and out, as shown in Figure P16.59. 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? The air temperature is 20°C.

Problem 57GP A 50-cm-long wire with a mass of 1.0 g and a tension of 440 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 the speed of sound is 340 m/s.

Problem 56GP A 1.0-m-tall vertical tube is filled with 20°C 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 sound wave in the air at the top of the tube?

Problem 58GP A 25-cm-long wire with a linear density of 20g/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 the speed of sound is 340 m/s.

Problem 59GP Two loudspeakers located along the x-axis as shown in Figure P16.61 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 m and 1.00 m, but at no points in between. What is the frequency of the sound? Assume the speed of sound is 340 m/s.

Problem 61GP FM station KCOM (“All commercials, all the time” ) transmits simultaneously, at a frequency of 99.9 MHz, from two broadcast towers placed precisely 31.5 m apart along a north-south line. a. What is the wavelength of KCOM’s transmissions? ________________ b. Suppose you stand 90.0 m due east of the point halfway between the two towers with your portable FM radio. Will you receive a strong or weak signal at this position? Why? ________________ c. You then stand 90.0 m due north of the northern tower with your radio. Will you receive a strong or weak signal at this position? Why?

Problem 60GP You are standing 2.50 m directly in front of one of the two loudspeakers shown in Figure P16.63. They are 3.00 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 20°C.

Problem 62GP Two loudspeakers, 4.0 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 m. a. What is the frequency of the sound? b. If the frequency is then increased while you remain 0.25 m from the center, what is the first frequency for which that location will be a maximum of sound intensity?

Problem 63GP 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?

Problem 64GP Certain birds produce vocalizations consisting of two distinct BIO frequencies that are not harmonically related—that is, the two frequencies are not harmonics of a common fundamental frequency. These two frequencies must be produced by two different vibrating structures in the bird’s vocal tract a. Wood ducks have been observed to make a call with approximately equal intensities at 850 Hz and 1200 Hz. The membranes that produce the vocalizations do not seem to vibrate at frequencies less than 500 Hz. Given this limitation, could these two frequencies be higher harmonics of a lower-frequency fundamental? ________________ b. If we model the duck’s vocal tract as an open-closed tube, what length has a fundamental frequency equal to the lower of the two frequencies in part a?

Problem 65GP 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 Hz and the note E should be at 659 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. 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? ________________ b. 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?

Problem 66GP 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?

Problem 67GP A student wailing at a stoplight notices that her turn signal, which has a period of 0.85 s, makes one blink exactly in sync with the turn signal of the car in front of her. The blinker of the car ahead men 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?

Problem 68GP Musicians can use beats to tune their instruments. One flute is properly tuned and plays the musical note A at exactly 440 Hz. A second player sounds the same note and hears that her instrument is slightly “flat” (i.e., at too low a frequency). Playing at the same time as the first flute, she hears two loud-soft-loud beats per second. Must she shorten or lengthen her flute, and by how much, to bring it into tune? Assume a speed of sound of 350 m/s.

Problem 69GP Police radars determine speed by measuring the Doppler 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?

Problem 70GP A Doppler blood flowmeter emits ultrasound at a frequency of 5.0 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 m/s?

Problem 71GP An ultrasound unit is being used to measure a patient’s heartbeat by combining the emitted 2.0 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 Hz. What is the maximum speed of the heart tissue?

Problem 72PP Harmonics and Harmony You know that certain musical notes sound good together— harmonious—whereas others do not. This harmony is related to the various harmonics of the notes. The musical notes C (262 Hz) and G (392 Hz) make a pleasant sound when played together; we call this consonance. As Figure P16.70 shows, the harmonics of the two notes are either far from each other or very close to each other (within a few Hz). This is the key to consonance: harmonics that are spaced either far apart or very close. The close harmonics have a beat frequency of a few Hz that is perceived as pleasant. If the harmonics of two notes are close but not too close, the rather high beat frequency between the two is quite unpleasant. This is what we hear as dissonance. Exactly how much a difference is maximally dissonant is a matter of opinion, but harmonic separations of 30 or 40 Hz seem to be quite unpleasant for most people. What is the beat frequency between the second harmonic of G and the third harmonic of C? A. 1 Hz B. 2 Hz C. 4 Hz D. 6 Hz

Problem 73PP Harmonics and Harmony You know that certain musical notes sound good together— harmonious—whereas others do not. This harmony is related to the various harmonics of the notes. The musical notes C (262 Hz) and G (392 Hz) make a pleasant sound when played together; we call this consonance. As Figure P16.70 shows, the harmonics of the two notes are either far from each other or very close to each other (within a few Hz). This is the key to consonance: harmonics that are spaced either far apart or very close. The close harmonics have a beat frequency of a few Hz that is perceived as pleasant. If the harmonics of two notes are close but not too close, the rather high beat frequency between the two is quite unpleasant. This is what we hear as dissonance. Exactly how much a difference is maximally dissonant is a matter of opinion, but harmonic separations of 30 or 40 Hz seem to be quite unpleasant for most people. Would a G-flat (frequency 370 Hz) and a C played together be consonant or dissonant? A. Consonant B. Dissonant

Problem 74PP Harmonics and Harmony You know that certain musical notes sound good together— harmonious—whereas others do not. This harmony is related to the various harmonics of the notes. The musical notes C (262 Hz) and G (392 Hz) make a pleasant sound when played together; we call this consonance. As Figure P16.70 shows, the harmonics of the two notes are either far from each other or very close to each other (within a few Hz). This is the key to consonance: harmonics that are spaced either far apart or very close. The close harmonics have a beat frequency of a few Hz that is perceived as pleasant. If the harmonics of two notes are close but not too close, the rather high beat frequency between the two is quite unpleasant. This is what we hear as dissonance. Exactly how much a difference is maximally dissonant is a matter of opinion, but harmonic separations of 30 or 40 Hz seem to be quite unpleasant for most people. An organ pipe open at both ends is tuned so that its fundamental frequency is a G. How long is the pipe? A. 43 cm B. 87 cm C. 130 cm D. 173 cm

Problem 75PP Harmonics and Harmony You know that certain musical notes sound good together— harmonious—whereas others do not. This harmony is related to the various harmonics of the notes. The musical notes C (262 Hz) and G (392 Hz) make a pleasant sound when played together; we call this consonance. As Figure P16.70 shows, the harmonics of the two notes are either far from each other or very close to each other (within a few Hz). This is the key to consonance: harmonics that are spaced either far apart or very close. The close harmonics have a beat frequency of a few Hz that is perceived as pleasant. If the harmonics of two notes are close but not too close, the rather high beat frequency between the two is quite unpleasant. This is what we hear as dissonance. Exactly how much a difference is maximally dissonant is a matter of opinion, but harmonic separations of 30 or 40 Hz seem to be quite unpleasant for most people. 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 P16.70 would be present? A. All of the harmonics in the figure would be present. B. 262, 786, and 1310 Hz C. 524, 1048, and 1572 Hz D. 262, 524, and 1048 Hz