Solved: A plane is traveling at Mach 2.0. An observer on

[Unless stated otherwise, assume \(T=20^\circ \mathrm C\) and \(v_{\mathrm{sound}}=343~\mathrm {m/s}\) in air.] Calculate the percent error made over one mile of distance by the “5-second rule” for estimating the distance from a lightning strike if the temperature is (a) \(30^\circ \mathrm C\), and (b) \(10^\circ \mathrm C\).

Problem 8P What is the intensity of a sound at the pain level of 120 dB? Compare it to that of a whisper at 20 dB.

Problem 9P What is the sound level of a sound whose intensity is 2.0 × 10?6 W/m2?

Problem 20P What would be the sound level (in dB) of a sound wave in air that corresponds to a displacement amplitude of vibrating air molecules of 0.13 mm at 300 Hz?

(I) A \(6000-H z \) tone must have what sound level to seem as loud as a \(100-H z\) tone that has a \(50-d B\) sound level? (See Fig. Equation Transcription: Text Transcription: 6000-Hz 100 - Hz 50 - dB

(I) What are the lowest and highest frequencies that an ear can detect when the sound level is \(30 \mathrm{~dB}\) ? (See Fig. 12-6.)

(II) Your auditory system can accommodate a huge range of sound levels. What is the ratio of highest to lowest intensity at (a) , (b) ? (See Fig. )

Problem 24P The A string on a violin has a fundamental frequency of 440 Hz. The length of the vibrating portion is 32 cm, and it has a mass of 0.35 g. Under what tension must the string be placed?

Problem 26P (a) What resonant frequency would you expect from blowing across the top of an empty soda bottle that is 18 cm deep, if you assumed it was a closed tube? (b) How would that change if it was one-third full of soda?

Problem 27P If you were to build a pipe organ with open-tube pipes spanning the range of human hearing (20 Hz to 20 kHz), what would be the range of the lengths of pipes required?

Problem 28P A tight guitar string has a frequency of 540 Hz as its third harmonic. What will be its fundamental frequency if it is fingered at a length of only 60% of its original length?

Problem 29P An unfingered guitar string is 0.73 m long and is tuned to play E above middle C (330 Hz). (a) How far from the end of this string must a fret (and your finger) be placed to play A above middle C (440 Hz)? (b) What is the wavelength on the string of this 440-Hz wave? (c) What are the frequency and wavelength of the sound wave produced in air at 20°C by this fingered string?

Problem 30P (a) Determine the length of an open organ pipe that emits middle C (262 Hz) when the temperature is 21 °C. (b) What are the wavelength and frequency of the fundamental standing wave in the tube? (c) What are ? and f in the traveling sound wave produced in the outside air?

Problem 31P An organ is in tune at 20°C. By what percent will the frequency be off at 5.0°C?

(II) How far from the mouthpiece of the flute in Example 12-10 should the hole be that must be uncovered to play D above middle C at 294 Hz ?

(II) (a) At \(T=20^{\circ} \mathrm{C}\), how long must an open organ pipe be to have a fundamental frequency of \(294 \mathrm{~Hz}\) ? (b) If this pipe is filled with helium, what is its fundamental frequency?

(II) A particular organ pipe can resonate at \(264 \mathrm{~Hz}\), \(440 \mathrm{~Hz}\), and \(616 \mathrm{~Hz}\), but not at any other frequencies in between. (a) Show why this is an open or a closed pipe. (b) What is the fundamental frequency of this pipe?

Problem 35P A uniform narrow tube 1.80 m long is open at both ends. It resonates at two successive harmonics of frequencies 275 Hz and 330 Hz. What is (a) the fundamental frequency, and (b) the speed of sound in the gas in the tube?

Problem 36P A pipe in air at 20°C is to be designed to produce two successive harmonics at 240 Hz and 280 Hz. How long must the pipe be, and is it open or closed?

Problem 37P How many overtones are present within the audible range for a 2.14-m-long organ pipe at 20°C (a) if it is open, and (b) if it is closed?

(III) The human ear canal is approximately long. It is open to the outside and is closed at the other end by the eardrum. Estimate the frequencies (in the audible range) of the standing waves in the ear canal. What is the relationship of your answer to the information in the graph of Fig. ?

Problem 39P A piano tuner hears one beat every 2.0s when trying to adjust two strings, one of which is sounding 440 Hz. How far off in frequency is the other string?

Problem 40P What is the beat frequency if middle C (262 Hz) and C# (277 Hz) are played together? What if each is played two octaves lower (each frequency reduced by a factor of 4)?

Problem 41P A certain dog whistle operates at 23.5 KHz. while another (brand X) operates at an unknown frequency. If humans can hear neither whistle when played separately, but a shrill whine of frequency 5000 Hz occurs when they are played simultaneously, estimate the operating frequency of brand X.

Problem 42P A guitar string produces 4 beats/s when sounded with a 350-Hz tuning fork and 9 beats/s when sounded with a 355-Hz tuning fork. What is the vibrational frequency of the string? Explain your reasoning.

(II) Two violin strings are tuned to the same frequency, . The tension in one string is then decreased by . What will be the beat frequency heard when the two strings are played together? [Hint: Recall Eq.

Problem 44P How many beats will be heard if two identical flutes each try to play middle C (262 Hz), but one is at 5.0°C and the other at 25.0°C?

Problem 45P You have three tuning forks, A, B, and C. Fork B has a frequency of 441 Hz; when A and B are sounded together, a beat frequency of 3 Hz is heard. When B and C are sounded together, the beat frequency is 4 Hz. What are the possible frequencies of A and C? What beat frequencies are possible when A and C are sounded together?

Problem 46P Two loudspeakers are 1.80m apart. A person stands 3.00m from one speaker and 3.50m from the other. (a) What is the lowest frequency at which destructive interference will occur at this point? (b) Calculate two other frequencies that also result in destructive interference at this point (give the next two highest). Let T = 20°C.

Problem 1Q What is the evidence that sound travels as a wave?

What is the evidence that sound is a form of energy?

Children sometimes play with a homemade "telephone" by attaching a string to the bottoms of two paper cups. When the string is stretched and a child speaks into one cup, the sound can be heard at the other cup (Fig. . Explain clearly how the sound wave travels from one cup to the other.

When a sound wave passes from air into water, do you expect the frequency or wavelength to change?

Problem 6Q The voice of a person who has inhaled helium sounds very high-pitched. Why?

Problem 5Q What evidence can you give that the speed of sound in air does not depend significantly on frequency?

Problem 7Q How will the air temperature in a room affect the pitch of organ pipes?

Problem 8Q Explain how a tube might be used as a filter to reduce the amplitude of sounds in various frequency ranges. (An example is a car muffler)

Why are the frets on a guitar (Fig. 12-30) spaced closer together as you move up the fingerboard toward the bridge?

Problem 10Q A noisy truck approaches you from behind a building. Initially you hear it but cannot see it. When it emerges and you do see it, its sound is suddenly “brighter”—you hear more of the high-frequency noise. Explain. [Hint: See Section 11-14 on diffraction ]

Problem 11Q Standing waves can be said to be due to “interference in space," whereas beats can be said to be due to "interference in time." Explain.

In Fig. , if the frequency of the speakers were lowered, would the points and (where destructive and constructive interference occur) move farther apart or closer together?

Problem 13Q Traditional methods of protecting the hearing of people who work in areas with very high noise levels have consisted mainly of efforts to block or reduce noise levels. With a relatively new technology, headphones are worn that do not block the ambient noise. Instead, a device is used which detects the noise, inverts it electronically, then feeds it to the headphones in addition to the ambient noise. How could adding more noise reduce the sound levels reaching the ears?

Consider the two waves shown in Fig. . Each wave can be thought of as a superposition of two sound waves with slightly different frequencies, as in Fig. In which of the waves, (a) or (b), are the two component frequencies farther apart? Explain.

Problem 15Q Is there a Doppler shirt if the source and observer move in the same direction, with the same velocity? Explain.

Problem 16Q If a wind is blowing, will this alter the frequency of the sound heard by a person at rest with respect to the source? Is the wavelength or velocity changed?

Figure shows various positions of a child in motion on a swing. monitor is blowing a whistle in front of the child on the ground. At which position, A through , will the child hear the highest frequency for the sound of the whistle? Explain your reasoning.

Problem 1P assume T = 20°C and vsound = 343 m/s in air. A hiker determines the length of a lake by listening for the echo of her shout reflected by a cliff at the far end of the lake. She hears the echo 2.0 s after shouting. Estimate the length of the lake.

(I) A sailor strikes the side of his ship just below the waterline. He hears the echo of the sound reflected from the ocean floor directly below \(2.5 \mathrm{~s}\) later. How deep is the ocean at this point? Assume the speed of sound in seawater is \(1560 \mathrm{~m} / \mathrm{s}\) (Table ) and does not vary significantly with depth. Equation Transcription: Text Transcription: 2.5 s 1560 m/s

Problem 3P assume T = 20°C and vsound = 343 m/s in air. (a) Calculate the wavelengths in air at 20°C for sounds in the maximum range of human hearing, 20 Hz to 20,000 Hz. (b) What is the wavelength of a 10-MHz ultrasonic wave?

(II) An ocean fishing boat is drifting just above a school of tuna on a foggy day. Without warning, an engine backfire occurs on another boat away (Fig. ). How much time elapses before the backfire is heard (a) by the fish, and by the fishermen?

Problem 5P assume T = 20°C and vsound = 343 m/s in air. A stone is dropped from the top of a cliff. The splash it makes when striking the water below is heard 3.5 s later. How high is the cliff?

(II) A person, with his ear to the ground, sees a huge stone strike the concrete pavement. A moment later two sounds are heard from the impact: one travels in the air and the other in the concrete, and they are apart. How far away did the impact occur? See Table

Problem 10P If two firecrackers produce a sound level of 95 dB when fired simultaneously at a certain place, what will be the sound level if only one is exploded? [Hint: Add intensities, not dB’s.]

(II) A person standing a certain distance from an airplane with four equally noisy jet engines is experiencing a sound level bordering on pain, 120 dB. What sound level would this person experience if the captain shut down all but one engine? [Hint: Add intensities, not dB’s.]

Problem 12P A cassette player is said to have a signal-to-noise ratio of 58 dB, whereas for a CD player it is 95 dB. What is the ratio of intensities of the signal and the background noise for each device?

A tuning fork is set into vibration above a vertical open tube filled with water (Fig. 12-35 ). The water level is allowed to drop slowly. As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is 0.125 m and again at 0.395 m. What is the frequency of the tuning fork?

(II) (a) Estimate the power output of sound from a person speaking in normal conversation. Use Table 12-2. Assume the sound spreads roughly uniformly over a sphere centered on the mouth. (b) How many people would it take to produce a total sound output of \(100 \mathrm{~W}\) of ordinary conversation? [Hint: Add intensities, not dB's.]

Problem 14P A 50-dB sound wave strikes an eardrum whose area is 5.0 × 10?5 m2. (a) How much energy is absorbed by the eardrum per second? (b) At this rate, how long would it take your eardrum to receive a total energy of 1.0 J?

Problem 15P Expensive amplifier A is rated at 250 W, while the more modest amplifier B is rated at 40 W. (a) Estimate the sound level in decibels you would expect at a point 3.5 m from a loudspeaker connected in turn to each amp. (b) Will the expensive amp sound twice as loud as the cheaper one?

At a rock concert, a \(\mathrm{dB}\) meter registered \(130 \mathrm{~dB}\) when placed \(2.8 \mathrm{~m}\) in front of a loudspeaker on the stage. (a) What was the power output of the speaker, assuming uniform spherical spreading of the sound and neglecting absorption in the air? (b) How far away would the sound level be a somewhat reasonable \(90 \mathrm{~dB}\)?

Problem 17P Human beings can typically detect a difference in sound level of 2.0 dB. What is the ratio of the amplitudes of two sounds whose levels differ by this amount? [Hint: See Section 11-9.]

Problem 87GP A bat emits a series of high-frequency sound pulses as it approaches a moth. The pulses are approximately 70.0 ms apart, and each is about 3.0 ms long. How far away can the moth be detected by the bat so that the echo from one pulse returns before the next pulse is emitted?

The "alpenhorn" (Fig. 12-38) was once used to send signals from one Alpine village to another. Since lower frequency sounds are less susceptible to intensity loss, long horns were used to create deep sounds. When played as a musical instrument, the alpenhorn must be blown in such a way that only one of the overtones is resonating. The most popular alpenhorn is about \(3.4 \mathrm{~m}\) long, and it is called the F sharp (or G flat) horn. What is the fundamental frequency of this horn, and which overtone is close to F sharp? (See Table 12-3.) Model as an open tube.

Problem 18P If the amplitude of a sound wave is tripled, (a) by what factor will the intensity increase? (b) By how many dB will the sound level increase?

Problem 19P Two sound waves have equal displacement amplitudes, but one has twice the frequency of the other. What is the ratio of their intensities?

Problem 91GP The intensity at the threshold of hearing for the human ear at a frequency of about 1000 Hz is /0 = 1.0 x 10 12 W/m2, for which /3. the sound level, is 0 dB. The threshold of pain at the same frequency is about 120 dB, or / = 1.0 W/m2, corresponding to an increase of intensity by a factor of 1012. By what factor does the displacement amplitude. A. vary?

(I) An organ pipe is 112 cm long. What are the fundamental and first three audible overtones if the pipe is (a) closed at one end, and (b) open at both ends?

Problem 47P Two piano strings are supposed to be vibrating at 132 Hz, but a piano tuner hears three beats every 2.0 s when they are played together. (a) If one is vibrating at 132 Hz, what must be the frequency of the other (is there only one answer)? (b) By how much (in percent) must the tension be increased or decreased to bring them in tune?

Problem 48P A source emits sound of wavelengths 2.64 m and 2.76 m in air. How many beats per second will be heard? (Assume T = 20°C.)

Problem 49P The predominant frequency of a certain fire engine’s siren is 1550 Hz when at rest. What frequency do you detect if you move with a speed of 30.0 m/s (a) toward the fire engine, and (b) away from it?

Problem 50P You are standing still. What frequency do you detect if a fire engine whose siren emits at 1550 Hz moves at a speed of 32 m/s (a) toward you, or (b) away from you?

Problem 51P (a) Compare the shift in frequency if a 2000-Hz source is moving toward you at 15 m/s, versus you moving toward it at 15 m/s. Are the two frequencies exactly the same? Are they close? (b) Repeat the calculation for 150 m/s and then again (c) for 300 m/s. What can you conclude about the asymmetry of the Doppler formulas?

Problem 52P Two automobiles are equipped with the same single frequency horn. When one is at rest and the other is moving toward the first at 15 m/s, the driver at rest hears a beat frequency of 5.5 Hz. What is the frequency the horns emit? Assume T = 20°C.

(II) A bat at rest sends out ultrasonic sound waves at 50.0 kHz and receives them returned from an object moving directly away from it at 25.0 m/s. What is the received sound frequency?

Problem 54P A bat flies toward a wall at a speed of 5.0 m/s. As it flies, the bat emits an ultrasonic sound wave with frequency 30.0 kHz. What frequency does the bat hear in the reflected wave?

Problem 55P In one of the original Doppler experiments, a tuba was played on a moving flat train car at a frequency of 75 Hz, and a second identical tuba played the same tone while at rest in the railway station. What beat frequency was heard if the train car approached the station at a speed of 10.0 m/s?

(II) A Doppler flow meter uses ultrasound waves to measure blood-flow speeds. Suppose the device emits sound at \(3.5 \mathrm{MHz}\), and the speed of sound in human tissue is taken to be \(1540 \mathrm{~m} / \mathrm{s}\). What is the expected beat frequency if blood is flowing in large leg arteries at \(2.0 \mathrm{~cm} / \mathrm{s}\) directly away from the sound source?

(III) The Doppler effect using ultrasonic waves of frequency \(2.25 \times 10^6 \mathrm{~Hz}\) is used to monitor the heartbeat of a fetus. A (maximum) beat frequency of \(500 \mathrm{~Hz}\) is observed. Assuming that the speed of sound in tissue is \(1.54 \times 10^3 \mathrm{~m} / \mathrm{s}\), calculate the maximum velocity of the surface of the beating heart.

Problem 58P A factory whistle emits sound of frequency 570 Hz. When the wind velocity is 12.0 m/s from the north, what frequency will observers hear who are located, at rest, (a) due north, (b) due south, (c) due east, and (d) due west, of the whistle? What frequency is heard by a cyclist heading (e) north or (f) west, toward the whistle at 15.0 m/s? Assume T = 20°C.

(a) How fast is an object moving on land if its speed at \(20^\circ C\) is Mach 0.33? (b) A high-flying jet cruising at 3000 km/h displays a Mach number of 3.2 on a screen. What is the speed of sound at that altitude?

Problem 60P An airplane travels at Mach 2.3 where the speed of sound is 310 m/s. (a) What is the angle the shock wave makes with the direction of the airplane’s motion? (b) If the plane is flying at a height of 7100 m, how long after it is directly overhead will a person on the ground hear the shock wave?

Problem 61P A space probe enters the thin atmosphere of a planet where the speed of sound is only about 35 m/s. (a) What is the probe’s Mach number if its initial speed is 15,000 km/h? (b) What is the angle of the shock wave relative to the direction of motion?

(II) A meteorite traveling 8500 m/s strikes the ocean. Determine the shock wave angle it produces (a) in the air just before entering the ocean, and (b) in the water just after entering. Assume \(\mathrm{T}=20^{\circ} \mathrm{C}\).

(II) Show that the angle \(\theta\) a sonic boom makes with the path of a supersonic object is given by Eq. 12-5.

(II) You look directly overhead and see a plane exactly 1.5 km above the ground flying faster than the speed of sound. By the time you hear the sonic boom, the plane has traveled a horizontal distance of 2.0 km. See Fig.12-14. Determine (a) the angle of the shock cone, \(\theta\), and (b) the speed of the plane (the Mach number). Assume the speed of sound is 330 m/s.

A fish finder uses a sonar device that sends \(20,000-\mathrm{Hz}\) sound pulses downward from the bottom of the boat, and then detects echoes. If the maximum depth for which it is designed to work is \(200 \mathrm{~m}\), what is the minimum time between pulses (in fresh water)?

Approximately how many octaves are there in the human audible range?

A science museum has a display called a sewer pipe symphony. It consists of many plastic pipes of various lengths, which are open on both ends. (a) If the pipes have lengths of \(3.0 \mathrm{~m}, 2.5 \mathrm{~m}, 2.0 \mathrm{~m}, 1.5 \mathrm{~m}\) and \(1.0 \mathrm{~m}\), what frequencies will be heard by a visitor's ear placed near the ends of the pipes? (b) Why does this display work better on a noisy day than on a quiet day?

Problem 68GP A single mosquito 5.0 m from a person makes a sound close to the threshold of human hearing (0 dB). What will be the sound level of 1000 such mosquitoes?

Problem 69GP What is the resultant sound level when an 82-dB sound and an 87-dB sound are heard simultaneously?

Problem 70GP The sound level 12.0 m from a loudspeaker, placed in the open, is 105 dB. What is the acoustic power output (W) of the speaker, assuming it radiates equally in all directions?

Problem 71GP A stereo amplifier is rated at 150 W output at 1000 Hz. The power output drops by 10 dB at 15 kHz. What is the power output in watts at 15 kHz?

Problem 72GP Workers around jet aircraft typically wear protective devices over their ears. Assume that the sound level of a jet airplane engine, at a distance of 30 m, is 140 dB, and that the average human ear has an effective radius of 2.0 cm. What would be the power intercepted by an unprotected ear at a distance of 30 m from a jet airplane engine?

In audio and communications systems, the gain, \(\beta\), in decibels is defined as \(\beta=10 \log \left(\frac{P_{\text {out }}}{P_{\text {in }}}\right)\), where \({P_{\text {in}}}\) is the power input to the system and \({P_{\text {out}}}\) is the power output. A particular stereo amplifier puts out 100 W of power for an input of 1 mW. What is its gain in dB?

Each string on a violin is tuned to a frequency \(1 \frac{1}{2}\) times that of its neighbor. The four equal-length strings are to be placed under the same tension; what must be the mass per unit length of each string relative to that of the lowest string? Equation Transcription: Text Transcription: 1 \frac{1}{2}

The A string of a violin is 32 cm long between fixed points with a fundamental frequency of 440 Hz and a mass per unit length of \(6.1 \times 10^{?4} \ \mathrm {kg/m}\). (a) What are the wave speed and tension in the string? (b) What is the length of the tube of a simple wind instrument (say, an organ pipe) closed at one end whose fundamental is also 440 Hz if the speed of sound is 343 m/s in air? (c) What is the frequency of the first overtone of each instrument?

A \(75-\mathrm{cm}\)-long guitar string of mass \(2.10 \mathrm{~g}\) is near a tube that is open at one end and also \(75 \mathrm{~cm}\) long. How much tension should be in the string if it is to produce resonance (in its fundamental mode) with the third harmonic in the tube?

(II) A highway overpass was observed to resonate as one full loop \(\left(\frac{1}{2} \lambda\right)\) when a small earthquake shook the ground vertically at 4.0 Hz. The highway department put a support at the center of the overpass, anchoring it to the ground as shown in Fig. 12-36. What resonant frequency would you now expect for the overpass? Earthquakes rarely do significant shaking above 5 or 6 Hz. Did the modifications do any good?

Problem 79GP A person hears a pure tone in the 500–1000-Hz range coming from two sources. The sound is loudest at points equidistant from the two sources. To determine exactly what the frequency is, the person moves about and finds that the sound level is minimal at a point 0.34 m farther from one source than the other. What is the frequency of the sound?

Problem 80GP Two trains emit 424-Hz whistles. One train is stationary. The conductor on the stationary train hears a 3.0-Hz beat frequency when the other train approaches. What is the speed of the moving train?

Problem 81GP The frequency of a steam train whistle as it approaches you is 538 Hz. After it passes you, its frequency is measured as 486 Hz. How fast was the train moving (assume constant velocity)?

Problem 82GP At a race track, you can estimate the speed of cars just by listening to the difference in pitch of the engine noise between approaching and receding cars. Suppose the sound of a certain car drops by a full octave (frequency halved) as it goes by on the straightaway. How fast is it going?

Two open organ pipes, sounding together, produce a beat frequency of \(11 \mathrm{~Hz}\). The shorter one is \(2.40 \mathrm{~m}\) long. How long is the other?

Two loudspeakers are at opposite ends of a railroad car as it moves past a stationary observer at 10.0m/s, as shown in Fig. 12-37. If the speakers have identical sound frequencies of 212 Hz, what is the beat frequency heard by the observer when (a) he listens from the position A, in front of the car, (b) he is between the speakers, at B, and (c) he hears the speakers after they have passed him, at C?

Problem 85GP If the velocity of blood flow in the aorta is normally about 0.32 m/s, what beat frequency would you expect if 5.50-MHz ultrasound waves were directed along the flow and reflected from the red blood cells? Assume that the waves travel with a speed of 1.54 × 103m/s.

Problem 86GP A bat flies toward a moth at speed 6.5 m/s while the moth is flying toward the bat at speed 5.0 m/s. The bat emits a sound wave of 51.35 kHz. What is the frequency of the wave detected by the bat after that wave reflects off the moth?

Problem 89GP Room acoustics for stereo listening can be compromised by the presence of standing waves, which can cause acoustic “dead spots” at the locations of the pressure nodes. Consider a living room 5.0 m long, 4.0 m wide, and 2.8 m high. Calculate the fundamental frequencies for the standing waves in this room.

A dramatic demonstration, called “singing rods,” involves a long, slender aluminum rod held in the hand near the rod’s midpoint. The rod is stroked with the other hand. With a little practice, the rod can be made to “sing,” or emit a clear, loud, ringing sound. For a 90-cm-long rod, (a) what is the fundamental frequency of the sound? (b) What is its wavelength in the rod, and (c) what is the traveling wavelength in air at \(20^\circ \mathrm C\)?

A plane is traveling at Mach 2.0. An observer on the ground hears the sonic boom \(1.5 \mathrm{~min}\) after the plane passes directly overhead. What is the plane's altitude?

Problem 93GP The wake of a speedboat is 15° in a lake where the speed of the water wave is 2.2 km/h. What is the speed of the boat?