In audio and communications systems, the gain, /3,

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.

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 s later. How deep is the ocean at this point? Assume the speed of sound in sea water is 1560 m/s (Table 16-1) and does not vary significantly with depth.

(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 15-MHz ultrasonic wave?

On a warm summer day (27C), it takes 4.70 s for an echo to return from a cliff across a lake. On a winter day, it takes 5.20 s. What is the temperature on the winter day?

A motion sensor can accurately measure the distance d to an object repeatedly via the sonar technique used in Example 16-2. A short ultrasonic pulse is emitted and reflects from any objects it encounters, creating echo pulses upon their arrival back at the sensor. The sensor measures the time interval t between the emission of the original pulse and the arrival of the first echo, (a) The smallest time interval t that can be measured with high precision is 1.0 ms. What is the smallest distance (at 20C) that can be measured with the motion sensor? (b) If the motion sensor makes 15 distance measurements every second (that is, it emits 15 sound pulses per second at evenly spaced time intervals), the measurement of t must be completed within the time interval between the emissions of successive pulses. What is the largest distance (at 20C) that can be measured with the motion sensor? (c) Assume that during a lab period the rooms temperature increases from 20C to 23C. What percent error will this introduce into the motion sensors distance measurements?

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 1.35 km away (Fig. 16-32). How much time elapses before the backfire is heard (a) by the fish, and (b) by the fishermen?FIGURE 16-32 ^ Problem 6.

A stone is dropped from the top of a cliff. The splash it makes when striking the water below is heard 3.0 s later. How high is the cliff?

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 0.75 s apart. How far away did the impact occur? See Table 16-1.

Calculate the percent error made over one mile of distance bv the 5-second rule fnr estimating the distance from a

The pressure amplitude of a sound wave in air (p = 1.29 kg/m3) at 0C is 3.0 X 10_3Pa. What is the displacement amplitude if the frequency is (a) 150 Hz and (b) 15 kHz?

What must be the pressure amplitude in a sound wave in air (0C) if the air molecules undergo a maximum displacement equal to the diameter of an oxygen molecule, about 3.0 X 10-10m? Assume a sound-wave frequency of (a) 55 Hz and (b) 5.5 kHz.

Write an expression that describes the pressure variation as a function of x and t for the waves described in Problem 11.

The pressure variation in a sound wave is given by AP = 0.0035 sin(0.3877jc - 135077?), where AP is in pascals, x in meters, and t in seconds. Determine (a) the wavelength, (b) the frequency, (c) the speed, and (d) the displacement amplitude of the wave. Assume the density of the medium to be p = 2.3 X 103 kg/m3.

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

What is the sound level of a sound whose intensity is 2.0 X 10-6 W/m2?

What are the lowest and highest frequencies that an ear can detect when the sound level is 40 dB? (See Fig. 16-6.)

Your auditory system can accommodate a huge range of sound levels. What is the ratio of highest to lowest intensity at (a) 100 Hz, (b) 5000 Hz? (See Fig. 16-6.)

You are trying to decide between two new stereo amplifiers. One is rated at 100 W per channel and the other is rated at 150 W per channel. In terms of dB, how much louder will the more powerful amplifier be when both are producing sound at their maximum levels?

At a painfully loud concert, a 120-dB sound wave travels away from a loudspeaker at 343 m/s. How much sound wave energy is contained in each 1.0-cm3 volume of air in the region near this loudspeaker?

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?

A person standing a certain distance from an airplane with four equally noisy jet engines is experiencing a sound level of 130 dB. What sound level would this person experience if the captain shut down all but one engine?

A cassette player is said to have a signal-to-noise ratio of 62 dB, whereas for a CD player it is 98 dB. What is the ratio of intensities of the signal and the background noise for each device?

(a) Estimate the power output of sound from a person speaking in normal conversation. Use Table 16-2. 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 75 W of ordinary conversation? [Hint: Add intensities, not dBs.]

A 50-dB sound wave strikes an eardrum whose area is 5.0 X 10-5m2. (a) How much energy is received by the eardrum ner second? (h \ At this rate, how Iona would it

Expensive amplifier A is rated at 250 W, while the more modest amplifier B is rated at 45 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 dB meter registered 130 dB when placed 2.2 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 85 dB?

A fireworks shell explodes 100 m above the ground, creating a colorful display of sparks. How much greater is the sound level of the explosion for a person standing at a point directly below the explosion than for a person a horizontal distance of 200 m away (Fig. 16-33)?FIGURE 16-33 Problem 27.

If the amplitude of a sound wave is made 2.5 times greater, (a) by what factor will the intensity increase? (b) By how many dB will the sound level increase?

Two sound waves have equal displacement amplitudes, but one has 2.6 times the frequency of the other, (a) Which has the greater pressure amplitude and by what factor is it greater? (b) What is the ratio of their intensities?

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 380 Hz?

(a) Calculate the maximum displacement of air molecules when a 330-Hz sound wave passes whose intensity is at the threshold of pain (120 dB). (b) What is the pressure amplitude in this wave?

A jet plane emits 5.0 X 105J of sound energy per second, (a) What is the sound level 25 m away? Air absorbs sound at a rate of about 7.0dB/km; calculate what the sound level will be (b) 1.00 km and (c) 7.50 km away from this jet plane, taking into account air absorption.

What would you estimate for the length of a bass clarinet, assuming that it is modeled as a closed tube and that the lowest note that it can play is a whose frequency is 69.3 Hz?

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?

An organ pipe is 124 cm long. Determine the fundamental and first three audible overtones if the pipe is (a) closed at one end, and (b) open at both ends.

(a) What resonant frequency would you expect from blowing across the top of an empty soda bottle that is 21 cm deep, if you assumed it was a closed tube? (b) How would that change if it was one-third full of soda?

If you were to build a pipe organ with open-tube pipes snannino the. ranae of human hearino (7.0 H z to ?.0kH7Y

Estimate the frequency of the sound of the ocean when you put your ear very near a 20-cm-diameter seashell (Fig. 16-34).FIGURE 16-34 Problem 38.

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 25C by this fingered string?

(a) Determine the length of an open organ pipe that emits middle C (262 Hz) when the temperature is 15C. (b) What are the wavelength and frequency of the fundamental standing wave in the tube? (c) What are A and / in the traveling sound wave produced in the outside air?

An organ is in tune at 22.0C. By what percent will the frequency be off at 5.0C?

How far from the mouthpiece of the flute in Example 16-11 should the hole be that must be uncovered to play F above middle C at 349 Hz?

A bugle is simply a tube of fixed length that behaves as if it is open at both ends. A bugler, by adjusting his lips correctly and blowing with proper air pressure, can cause a harmonic (usually other than the fundamental) of the air column within the tube to sound loudly. Standard military tunes like Taps and Reveille require only four musical notes: G4 (392 Hz), C5 (523 Hz), E5 (659 Hz), and G5 (784 Hz). (a) For a certain length , a bugle will have a sequence of four consecutive harmonics whose frequencies very nearly equal those associated with the notes G4, C5, E5, and G5. Determine this t (b) Which harmonic is each of the (approximate) notes G4, C5, E5, and G5 for the bugle?

A particular organ pipe can resonate at 264 Hz, 440 Hz, and 616 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?

When a players finger presses a guitar string down onto a fret, the length of the vibrating portion of the string is shortened, thereby increasing the strings fundamental frequency (see Fig. 16-35). The strings tension and mass per unit length remain unchanged. If the unfingered length of the string is = 65.0 cm, determine the positions x of the first six frets, if each fret raises the pitch of the fundamental by one musical note in comparison to the neighboring fret. On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is 21/12.

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?

A pipe in air at 23.0C 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?

How many overtones are present within the audible range for a 2.48-m-long organ pipe at 20C (a) if it is open, and (b) if it is closed?

Determine the fundamental and first overtone frequencies for an 8.0-m-long hallway with all doors closed. Model the hallway as a tube closed at both ends.

In a quartz oscillator, used as a stable clock in electronic devices, a transverse (shear) standing sound wave is excited across the thickness d of a quartz disk and its frequency / is detected electronically. The parallel faces of the disk are unsupported and so behave as free ends when the sound wave reflects from them (see Fig. 16-36). If the oscillator is designed to operate with the first harmonic, determine the required disk thickness if / = 12.0 MHz. The density and shear modulus of quartz are p = 2650 kg/m3 and G = 2.95 X 1010N/m2.FIGURE 16-36 Problem 50.

The human ear canal is approximately 2.5 cm 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. 16-6?

Approximately what are the intensities of the first two overtones of a violin compared to the fundamental? How many decibels softer than the fundamental are the first and second overtones? (See Fig. 16-14.)

A piano tuner hears one beat every 2.0 s when trying to adjust two strings, one of which is sounding 370 Hz. How far off in frequency is the other string?

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)?

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.

The two sources of sound in Fig. 16-15 face each other and emit sounds of equal amplitude and equal frequency (294 Hz) but 180 out of phase. For what minimum separation of the two speakers will there be some point at which (n\ mrrmlp.te romtrnrtivp interferpnrp nrrnrs and (h\ comnletp

How many beats will be heard if two identical flutes, each 0.66 m long, try to play middle C (262 Hz), but one is at 5.0C and the other at 28C?

Two loudspeakers are placed 3.00 m apart, as shown in Fig. 16-37. They emit 494-Hz sounds, in phase. A microphone is placed 3.20 m distant from a point midway between the two speakers, where an intensity maximum is recorded, (a) How far must the microphone be moved to the right to find the first intensity minimum? (b) Suppose the speakers are reconnected so that the 494-Hz sounds they emit are exactly out of phase. At what positions are the intensity maximum and minimum now?FIGURE 16-37 Problem 58.

Two piano strings are supposed to be vibrating at 220 Hz, but a piano tuner hears three beats every 2.0 s when they are played together, (a) If one is vibrating at 220.0 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?

A source emits sound of wavelengths 2.64 m and 2.72 m in air. (a) How many beats per second will be heard? (Assume T = 20C.) (b) How far apart in space are the regions of maximum intensity?

The predominant frequency of a certain fire trucks siren is 1350 Hz when at rest. What frequency do you detect if you move with a speed of 30.0 m/s (a) toward the fire truck, and (b) away from it?

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 30.0 m/s. What is the received sound frequency?

(a) Compare the shift in frequency if a 2300-Hz source is moving toward you at 18 m/s, versus you moving toward it at 18 m/s. Are the two frequencies exactly the same? Are they close? (b) Repeat the calculation for 160 m/s and then again (c) for 320 m/s. What can you conclude about the asymmetry of the Doppler formulas? (d) Show that at low speeds (relative to the speed of sound), the two formulas source approaching and detector approachingyield the same result.

Two automobiles are equipped with the same singlefrequency 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 4.5 Hz. What is the frequency the horns emit? Assume T = 20C.

A police car sounding a siren with a frequency of 1280 Hz is traveling at 120.0 km/h. (a) What frequencies does an observer standing next to the road hear as the car approaches and as it recedes? (b) What frequencies are heard in a car traveling at 90.0 km/h in the opposite direction before and after passing the police car? (c) The police car nasses a car traveling in the same direction at 80.0 km /h.

A bat flies toward a wall at a speed of 7.0 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?

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 in the station if the train car approached the station at a speed of 12.0 m/s?

If a speaker mounted on an automobile broadcasts a song, with what speed (km/h) does the automobile have to move toward a stationary listener so that the listener hears the song with each musical note shifted up by one note in comparison to the song heard by the automobiles driver? On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is 21/12.

A wave on the surface of the ocean with wavelength 44 m is traveling east at a speed of 18 m/s relative to the ocean floor. If, on this stretch of ocean surface, a powerboat is moving at 15 m/s (relative to the ocean floor), how often does the boat encounter a wave crest, if the boat is traveling (a) west, and (b) east?

A factory whistle emits sound of frequency 720 Hz. When the wind velocity is 15.0 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 ( /) west, toward the whistle at 12.0 m/s? Assume T = 20C

The Doppler effect using ultrasonic waves of frequency 2.25 X 106 Hz is used to monitor the heartbeat of a fetus. A (maximum) beat frequency of 260 Hz is observed. Assuming that the speed of sound in tissue is 1.54 X 103 m/s, calculate the maximum velocity of the surface of the beating heart.

An airplane travels at Mach 2.0 where the speed of sound is 310 m/s. (a) What is the angle the shock wave makes with the direction of the airplanes motion? (b) If the plane is flying at a height of 6500 m, how long after it is directly overhead will a person on the ground hear the shock wave?

A space probe enters the thin atmosphere of a planet where the speed of sound is only about 45 m/s. (a) What is the probes 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?

A meteorite traveling 8800 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 T = 20C.

Show that the angle 6 a sonic boom makes with the path of a supersonic object is given by Eq. 16-12.

You look directly overhead and see a plane exactly 1.25 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. 16-38. Determine (a) the angle of the shock cone, 6, and (b) the speed of the plane (the Mach number). Assume the speed of sound is 330 m/s.FIGURE 16-38 Problem 76.

A supersonic jet traveling at Mach 2.2 at an altitude of 9500 m passes directly over an observer on the ground. Where will the plane be relative to the observer when the latter hears the sonic boom? (See Fig. 16-39.)FIGURE 16-39 Problem 77.

A fish finder uses a sonar device that sends 20,000-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 75 m, what is the minimum time between pulses (in fresh water)?

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 m, 2.5 m, 2.0 m, 1.5 m and 1.0 m, what frequencies will be heard by a visitors ear placed near the ends of the pipes? (b) Why does this display work better on a noisy day than on a quiet day?

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 100 such mosquitoes?

What is the resultant sound level when an 82-dB sound and an 89-dB sound are heard simultaneously?

The sound level 9.00 m from a loudspeaker, placed in the nnen is 11 5 dR What is the

A stereo amplifier is rated at 175 W output at 1000 Hz. The power output drops by 12 dB at 15 kHz. What is the power output in watts at 15 kHz?

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 130 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, /3, in decibels is defined as - where P[n is the power input to the system and Pout is the power output. A particular stereo amplifier puts out 125 W nf nnwer fnr an innut nf 1.0mW. What is its pain

For large concerts, loudspeakers are sometimes used to amplify a singers sound. The human brain interprets sounds that arrive within 50 ms of the original sound as if they came from the same source. Thus if the sound from a loudspeaker reaches a listener first, it would sound as if the loudspeaker is the source of the sound. Conversely, if the singer is heard first and the loudspeaker adds to the sound within 50 ms, the sound would seem to come from the singer, who would now seem to be singing louder. The second situation is desired. Because the signal to the loudspeaker travels at the speed of light (3 X 108m/s), which is much faster than the speed of sound, a delay is added to the signal sent to the loudspeaker. How much delay must be added if the loudspeaker is 3.0 m behind the singer and we want its sound to arrive 30 ms after the singers?

Manufacturers typically offer a particular guitar string in a choice of diameters so that players can tune their instruments with a preferred string tension. For example, a nylon high-E string is available in a low- and high-tension model with diameter 0.699 mm and 0.724 mm, respectively. Assuming the density p of nylon is the same for each model, compare (as a ratio) the tension in a tuned high- and low-tension string.

The high-E string on a guitar is fixed at both ends with length I = 65.0 cm and fundamental frequency fi = 329.6 Hz. On an acoustic guitar, this string typically has a diameter of 0.33 mm and is commonly made of brass (7760 kg/m3), while on an electric guitar it has a diameter of 0.25 mm and is made of nickel-coated steel (7990 kg/m3). Compare (as a ratio) the high-E string tension on an acoustic versus an electric guitar.

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 7.2 X 10-4 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 tuning fork is set into vibration above a vertical open tube filled with water (Fig. 16-40). 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?FIGURE 16-40 Problem 90.

Two identical tubes, each closed at one end, have a fundamental frequency of 349 Hz at 25.0C. The air temperature is increased to 30.0C in one tube. If the two pipes are sounded together now, what beat frequency results?

Each string on a violin is tuned to a frequency 1 \ times that of its neighbor. The four equal-length strings are to be placed under the. same tension* what must be the mass tier unit

The diameter D of a tube does affect the node at the open end of a tube. The end correction can be roughly approximated as adding D/3 to the effective length of the tube. For a closed tube of length 0.60 m and diameter 3.0 cm, what are the first four harmonics, taking the end correction into consideration?

A person hears a pure tone in the 500 to 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 0.28 m farther from one source than the other. What is the frequency of the sound?

The frequency of a steam train whistle as it approaches you is 552 Hz. After it passes you, its frequency is measured as 486 Hz. How fast was the train moving (assume constant velocity)?

Two trains emit 516-Hz whistles. One train is stationary. The conductor on the stationary train hears a 3.5-Hz beat frequency when the other train approaches. What is the speed of the moving train?

Two loudspeakers are at opposite ends of a railroad car as it moves past a stationary observer at 10.0 m/s, as shown in Fig. 16-41. If the speakers have identical sound frequencies of 348 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?FIGURE 16-41 Problem 97.

Two open organ pipes, sounding together, produce a beat frequency of 8.0 Hz. The shorter one is 2.40 m long. How long is the other?

A bat flies toward a moth at speed 7.5 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?

If the velocity of blood flow in the aorta is normally about 0.32 m/s, what beat frequency would you expect if 3.80-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 X 103 m/s.

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?

(a) Use the binomial expansion to show that Eqs. 16-9a and 16-10a become essentially the same for small relative velocity between source and observer. (b) What percent error would result if Eq. 16-10a were used instead of Eq. 16-9a for a relative velocity of 18.0 m/s?

Two loudspeakers face each other at opposite ends of a long corridor. They are connected to the same source which produces a pure tone of 282 Hz. A person walks frnm one sneaker toward the other at a sneed o f 1.4 m /s.

A Doppler flow meter is used to measure the speed of blood flow. Transmitting and receiving elements are placed on the skin, as shown in Fig. 16-42. Typical sound-wave frequencies of about 5.0 MHz are used, which have a reasonable chance of being reflected from red blood cells. By measuring the frequency of the reflected waves, which are Doppler-shifted because the red blood cells are moving, the speed of the blood flow can be deduced. Normal blood flow speed is about 0.1 m/s. Suppose that an artery is partly constricted, so that the speed of the blood flow is increased, and the flow meter measures a Doppler shift of 780 Hz. What is the speed of blood flow in the constricted region? The effective angle between the sound waves (both transmitted and reflected) and the direction of blood flow is 45. Assume the velocity of sound in tissue is 1540 m/s.FIGURE 16-42 Problem 104.

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?

A source of sound waves (wavelength A) is a distance I from a detector. Sound reaches the detector directly, and also by reflecting off an obstacle, as shown in Fig. 16-43. The obstacle is equidistant from source and detector. When the obstacle is a distance d to the right of the line of sight between source and detector, as shown, the two waves arrive in phase. How much farther to Detector the right must the obstacle be moved if the two waves are to be out of phase by \ wavelength, so destructive interference occurs? (Assume A I, d.) FIGURE 16-43 Problem 106.

A dramatic demonstration, called singing rods, involves a long, slender aluminum rod held in the hand near the rods 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 75-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 wavelength of the sound in air at 20C?

Assuming that the maximum displacement of the air molecules in a sound wave is about the same as that of the speaker cone that produces the sound (Fig. 16-44), estimate by how much a loudspeaker cone moves for a fairly loud (105 dB) sound of (a) 8.0 kHz, and (b) 35 Hz. FIGURE 16-44 Problem 108.

(Ill) The manner in which a string is plucked determines the mixture of harmonic amplitudes in the resulting wave. Consider a string exactly |-m long that is fixed at both its ends located at x = 0.0 and x = \ m. The first five harmonics of this string have wavelengths X\ = 1.0 m, A2 = |m, A3 = \ m, A4 = \ m, and A5 = | m. According to Fouriers theorem, any shape of this string can be formed by a sum of its harmonics, with each harmonic having its own unique amplitude A. We limit the sum to the first five harmonics in the expression . (2tt \ . (2tt D(x) = AiSinl x j + v42sinl x A . (2ir \ A . (2tt \ A . (2ir \ + A 3 sin I xJ + A 4 sin I xJ + A 5 sin I xJ, and D is the displacement of the string at a time t = 0. Imagine plucking this string at its midpoint (Fig. 16-45a) or at a point two-thirds from the left end (Fig. 16-45b). Using a graphing calculator or computer program, show that the above expression can fairly accurately represent the shape in: (a) Fig. 16-45a, if A \ = 1.00, A 2 = 0.00, A 3 = -0.11, A 4 = 0.00, and , . A s = 0.040; and in (b) Fig. 16-45b, if A 1 = 0.87, A 2 = -0.22, A 3 = 0.00, A 4 = 0.054, and A s = -0.035. FIGURE 16-45 Problem 109