Figure 1440shows a wave on a string moving to the right.

Awave travels along a stretched horizontal rope. The vertical distance from crest to trough for this wave is 13 cm and the horizontal distance from crest to trough is 28 cm. What are (a) the wavelength and (b) the amplitude of this wave?

A surfer oating beyond the breakers notes 14 waves per minute passing her position. If the wavelength of these waves is 34 m, what is their speed?

The speed of surface waves in water decreases as the water becomes shallower. Suppose waves travel across the surface of a lake with a speed of 2.0 m/s and a wavelength of 1.5 m. When these waves move into a shallower part of the lake, their speed decreases to 1.6 m/s, though their frequency remains the same. Find the wavelength of the waves in the shallower water.

A tsunami traveling across deep water can have a speed of 750 km/h and a wavelength of 310 km. What is the frequency of such a wave?

A4.5-Hz wave with an amplitude of 12 cm and a wavelength of 27 cm travels along a stretched horizontal string. (a) How far does a given peak on the wave travel in a time interval of 0.50 s? (b) How far does a knot on the string travel in the same time interval? (c)How would your answers to parts (a) and (b) change if the amplitude of the wave were halved? Explain.

The speed of a deepwater wave with a wavelength is given approximately by . Find the speed and frequency of a deepwater wave with a wavelength of

In shallow water of depth d the speed of waves is approximately . Find the speed and frequency of a wave with wavelength 0.75 cm in water that is 2.6 cm deep.

Consider a wave on a string with constant tension. If the frequency of the wave is doubled, by what multiplicative factor does (a) the speed and (b) the wavelength change?

Suppose you would like to double the speed of a wave on a string. By what multiplicative factor must you increase the tension?

Predict/Explain Two strings are made of the same material and have equal tensions. String 1 is thick; string 2 is thin. (a) Is the speed of waves on string 1 greater than, less than, or equal to the speed of waves on string 2? (b) Choose the best explanation from among the following: I. Since the strings are made of the same material, the wave speeds will also be the same. II. A thick string implies a large mass per length and a slow wave speed. III. Athick string has a greater force constant, and therefore a greater wave speed.

Predict/Explain Two strings are made of the same material and have waves of equal speed. String 1 is thick; string 2 is thin.(a)Is the tension in string 1 greater than, less than, or equal to the tension in string 2? (b) Choose the best explanation from among the following: I. String 1 must have a greater tension to compensate for its greater mass per length. II. String 2 will have a greater tension because it is thinner than string 1. III. Equal wave speeds implies equal tensions.

The three waves, A, B and C, shown in Figure 1432propagate on strings with equal tensions and equal mass per length. Rank the waves in order of increasing (a) frequency, (b) wavelength, and (c) speed. Indicate ties where appropriate.

Waves on a particular string travel with a speed of 16 m/s. By what factor should the tension in this string be changed to produce waves with a speed of 32 m/s?

A brother and sister try to communicate with a string tied between two tin cans (Figure 1433). If the string is 9.5 m long, has a mass of 32 g, and is pulled taut with a tension of 8.6 N, how much time does it take for a wave to travel from one end of the string to the other?

(a)Suppose the tension is increased in the previous problem. Does a wave take more, less, or the same time to travel from one end to the other? Calculate the time of travel for tensions of (b) 9.0 N and (c) 10.0 N.

A5.2-m wire with a mass of 87 g is attached to the mast of a sailboat. If the wire is given a thunk at one end, it takes 0.094 s for the resulting wave to reach the other end. (a) What is the tension in the wire? (b)Would the tension found in part (a) be larger or smaller if the mass of the wire is greater than 87 g? (c) Calculate the tension for a 97-g wire.

Two steel guitar strings have the same length. String Ahas a diameter of 0.50 mm and is under 410.0 N of tension. String B has a diameter of 1.0 mm and is under a tension of 820.0 N. Find the ratio of the wave speeds, , in these two strings.

Use dimensional analysis to show how the speed vof a wave on a string of circular cross section depends on the tension in the string, T, the radius of the string, R, and its mass per volume, . r

Write an expression for a harmonic wave with an amplitude of 0.16 m, a wavelength of 2.1 m, and a period of 1.8 s. The wave is transverse, travels to the right, and has a displacement of 0.16 m at and .

Write an expression for a transverse harmonic wave that has a wavelength of 2.6 m and propagates to the right with a speed of 14.3 m/s. The amplitude of the wave is 0.11 m, and its displacement at and

The vertical displacement of a wave on a string is described by the equation , in which A, B, and C are positive constants. (a) Does this wave propagate in the positive or negative xdirection? (b)What is the wavelength of this wave? (c) What is the frequency of this wave? (d) What is the smallest positive value of x where the displacement of this wave is zero at

The vertical displacement of a wave on a string is described by the equation , in which A, B, and C are positive constants. (a) Does this wave propagate in the positive or negative x direction? (b) What is the physical meaning of the constant A? (c) What is the speed of this wave? (d) What is the smallest positive time t for which the wave has zero displacement at the point

Awave on a string is described by the following equation: (a) What is the amplitude of this wave? (b) What is its wavelength? (c) What is its period? (d) What is its speed? (e) In which direction does the wave travel?

Consider the wave function given in the previous problem. Sketch this wave from to for the following times: (a) ; (b) ; (c) . (d) What is the least amount of time required for a given point on this wave to move from to ? Verify your answer by referring to the sketches for parts (a), (b), and (c).

Four waves are described by the following equations, in which all distances are measured in centimeters and all times are measured in seconds: (a) Which of these waves travel in the direction? (b) Which of these waves travel in the direction? (c) Which wave has the highest frequency? (d) Which wave has the greatest wavelength? (e) Which wave has the greatest speed?

At Zion National Park a loud shout produces an echo 1.80 s later from a colorful sandstone cliff. How far away is the cliff?

Dolphins of the open ocean are classied as Type II Odontocetes (toothed whales). These animals use ultrasonic clicks with a frequency of about 55 kHz to navigate and nd prey. (a)Suppose a dolphin sends out a series of clicks that are reected back from the bottom of the ocean 75 m below. How much time elapses before the dolphin hears the echoes of the clicks? (The speed of sound in seawater is approximately 1530 m/s.) (b) What is the wavelength of 55-kHz sound in the ocean?

The lowest note on a piano is A, four octaves below the A given in Table 143. The highest note on a piano is C, four octaves above middle C. Find the frequencies and wavelengths (in air) of these notes.

Asound wave in air has a frequency of 425 Hz. (a) What is its wavelength? (b)If the frequency of the sound is increased, does its wavelength increase, decrease, or stay the same? Explain.(c)Calculate the wavelength for a sound wave with a frequency of 475 Hz.

When you drop a rock into a well, you hear the splash 1.5 seconds later. (a) How deep is the well? (b) If the depth of the well were doubled, would the time required to hear the splash be greater than, less than, or equal to 3.0 seconds? Explain.

Arock is thrown downward into a well that is 8.85 m deep. If the splash is heard 1.20 seconds later, what was the initial speed of the rock?

If the distance to a point source of sound is doubled, by what multiplicative factor does the intensity change?

The intensity level of sound in a truck is 92 dB. What is the intensity of this sound?

The distance to a point source is decreased by a factor of three. (a) By what multiplicative factor does the intensity increase? (b) By what additive amount does the intensity level increase?

Sound 1 has an intensity of . Sound 2 has an intensity level that is 2.5 dB greater than the intensity level of sound 1. What is the intensity of sound 2?

Abird-watcher is hoping to add the white-throated sparrow to her life list of species. How far could she be from the bird described in Example 143 and still be able to hear it? Assume no reections or absorption of the sparrows sound.

Residents of Hawaii are warned of the approach of a tsunami by sirens mounted on the tops of towers. Suppose a siren produces a sound that has an intensity level of 120 dB at a distance of 2.0 m. Treating the siren as a point source of sound, and ignoring reections and absorption, nd the intensity level heard by an observer at a distance of (a)12 m and (b)21 m from the siren. (c) How far away can the siren be heard?

In a pig-calling contest, a caller produces a sound with an intensity level of 110 dB. How many such callers would be required to reach the pain level of 120 dB?

Twenty violins playing simultaneously with the same intensity combine to give an intensity level of 82.5 dB. (a)What is the intensity level of each violin? (b) If the number of violins is increased to 40, will the combined intensity level be more than, less than, or equal to 165 dB? Explain.

The radius of a typical human eardrum is about 4.0 mm. Find the energy per second received by an eardrum when it listens to sound that is (a) at the threshold of hearing and (b) at the threshold of pain.

A point source of sound that emits uniformly in all directions is located in the middle of a large, open eld. The intensity at Brittanys location directly north of the source is twice that at Phillips position due east of the source. What is the distance between Brittany and Phillip if Brittany is 12.5 m from the source?

Predict/Explain Ahorn produces sound with frequency . Let the frequency you hear when you are at rest and the horn moves toward you with a speed u be ; let the frequency you hear when the horn is at rest and you move toward it with a speed u be . (a) Is greater than, less than, or equal to ? (b) Choose the best explanation from among the following: I. Amoving observer encounters wave crests more often than a stationary observer, leading to a higher frequency. II. The relative speeds are the same in either case. Therefore, the frequencies will be the same as well. III. A moving source causes the wave crests to bunch up, leading to a higher frequency than for a moving observer.

You are heading toward an island in your speedboat when you see a friend standing on shore at the base of a cliff. You sound the boats horn to get your friends attention. Let the wavelength of the sound produced by the horn be , the wavelength as heard by your friend be , and the wavelength of the echo as heard on the boat be . Rank these wavelengths in order of increasing length. Indicate ties where appropriate.

Aperson with perfect pitch sits on a bus bench listening to the 450-Hz horn of an approaching car. If the person detects a frequency of 470 Hz, how fast is the car moving?

A train moving with a speed of 31.8 m/s sounds a 136-Hz horn. What frequency is heard by an observer standing near the tracks as the train approaches?

In the previous problem, suppose the stationary observer sounds a horn that is identical to the one on the train. What frequency is heard from this horn by a passenger in the train?

A bat moving with a speed of 3.25 m/s and emitting sound of 35.0 kHz approaches a moth at rest on a tree trunk. (a) What frequency is heard by the moth? (b) If the speed of the bat is increased, is the frequency heard by the moth higher or lower? (c) Calculate the frequency heard by the moth when the speed of the bat is 4.25 m/s.

Amotorcycle and a police car are moving toward one another. The police car emits sound with a frequency of 502 Hz and has a speed of 27.0 m/s. The motorcycle has a speed of 13.0 m/s. What frequency does the motorcyclist hear?

In the previous problem, suppose that the motorcycle and the police car are moving in the same direction, with the motorcycle in the lead. What frequency does the motorcyclist hear in this case?

Hearing the siren of an approaching re truck, you pull over to the side of the road and stop. As the truck approaches, you hear a tone of 460 Hz; as the truck recedes, you hear a tone of 410 Hz. How much time will it take for the truck to get from your position to the re 5.0 km away, assuming it maintains a constant speed?

With what speed must you approach a source of sound to observe a 15% change in frequency?

A particular jet engine produces a tone of 495 Hz. Suppose that one jet is at rest on the tarmac while a second identical jet ies overhead at 82.5% of the speed of sound. The pilot of each jet listens to the sound produced by the engine of the other jet. (a) Which pilot hears a greater Doppler shift? Explain. (b) Calculate the frequency heard by the pilot in the moving jet. (c)Calculate the frequency heard by the pilot in the stationary jet.

(a) If bicyclist Abeeps a 315-Hz horn, what frequency is heard by bicyclist B? (b) Which of the following would cause the greater increase in the frequency heard by bicyclist B: (i) bicyclist Aspeeds up by 1.50 m/s, or (ii) bicyclist B speeds up by 1.50 m/s? Explain.

Atrain on one track moves in the same direction as a second train on the adjacent track. The rst train, which is ahead of the second train and moves with a speed of 36.8 m/s, blows a horn whose frequency is 124 Hz. If the frequency heard on the second train is 135 Hz, what is its speed?

Two cars traveling with the same speed move directly away from one another. One car sounds a horn whose frequency is 205 Hz and a person in the other car hears a frequency of 192 Hz. What is the speed of the cars?

The Shinkansen, the Japanese bullet train, runs at high speed from Tokyo to Nagoya. Riding on the Shinkansen, you notice that the frequency of a crossing signal changes markedly as you pass the crossing. As you approach the crossing, the frequency you hear is f; as you recede from the crossing the frequency you hear is . What is the speed of the train?

Two wave pulses on a string approach one another at the time , as shown in Figure 1434. Each pulse moves with a speed of 1.0 m/s. Make a careful sketch of the resultant wave at the times , 2.0 s, 2.5 s, 3.0 s, and 4.0 s, assuming that the superposition principle holds for these waves.

Suppose pulse 2 in Problem 57 is inverted, so that it is a downward deection of the string rather than an upward deection. Repeat Problem 57 in this case.

Two wave pulses on a string approach one another at the time , as shown in Figure 1435. Each pulse moves with a speed of 1.0 m/s. Make a careful sketch of the resultant wave at the times , 2.0 s, 2.5 s, 3.0 s, and 4.0 s, assuming that the superposition principle holds for these waves.

Suppose pulse 2 in Problem 59 is inverted, so that it is a downward deection of the string rather than an upward deection. Repeat Problem 59 in this case.

A pair of in-phase stereo speakers is placed side by side, 0.85 m apart. You stand directly in front of one of the speakers,1.1 m from the speaker. What is the lowest frequency that will produce constructive interference at your location?

Two violinists, one directly behind the other, play for a listener directly in front of them. Both violinists sound concert A(440 Hz). (a)What is the smallest separation between the violinists that will produce destructive interference for the listener? (b) Does this smallest separation increase or decrease if the violinists produce a note with a higher frequency? (c) Repeat part (a) for violinists who produce sounds of 540 Hz.

Two loudspeakers are placed at either end of a gymnasium, both pointing toward the center of the gym and equidistant from it. The speakers emit 266-Hz sound that is in phase. An observer at the center of the gym experiences constructive interference. How far toward either speaker must the observer walk to rst experience destructive interference?

(a) In the previous problem, does the required distance increase, decrease, or stay the same if the frequency of the speakers is lowered? (b) Calculate the distance to the rst position of destructive interference if the frequency emitted by the speakers is lowered to 238 Hz.

Two speakers with opposite phase are positioned 3.5 m apart, both pointing toward a wall 5.0 m in front of them (Figure 1436). An observer standing against the wall midway between the speakers hears destructive interference. If the observer hears constructive interference after moving 0.84 m to one side along the wall, what is the frequency of the sound emitted by the speakers?

Suppose, in Example 147, that the speakers have opposite phase. What is the lowest frequency that gives destructive interference in this case?

When you blow across the opening of a partially lled 2-L soda pop bottle you hear a tone. (a) If you take a sip of the pop and blow across the opening again, does the tone you hear have a higher frequency, a lower frequency, or the same frequency as before? (b)Choose the bestexplanation from among the following: I. The same pop bottle will give the same frequency regardless of the amount of pop it contains. II. The greater distance from the top of the bottle to the level of the pop results in a higher frequency. III. Alower level of pop results in a longer column of air, and hence a lower frequency.

An organ pipe that is open at both ends is 3.5 m long. What is its fundamental frequency?

Astring 1.5 m long with a mass of 2.6 g is stretched between two xed points with a tension of 93 N. Find the frequency of the fundamental on this string.

Astring is tied down at both ends. Some of the standing waves on this string have the following frequencies: 100 Hz, 200 Hz, 250 Hz, and 300 Hz. It is also known that there are no standing waves with frequencies between 250 Hz and 300 Hz. (a) What is the fundamental frequency of this string? (b) What is the frequency of the third harmonic?

Standing Waves in the Human Ear The human ear canal is much like an organ pipe that is closed at one end (at the tympanic membrane or eardrum) and open at the other (Figure 1437). A typical ear canal has a length of about 2.4 cm. (a) What are the fundamental frequency and wavelength of the ear canal? (b) Find the frequency and wavelength of the ear canals third harmonic. (Recall that the third harmonic in this case is the standing wave with the second-lowest frequency.) (c) Suppose a person has an ear canal that is shorter than 2.4 cm. Is the fundamental frequency of that persons ear canal greater than, less than, or the same as the value found in part (a)? Explain. [Note that the frequencies found in parts (a) and (b) correspond closely to the frequencies of enhanced sensitivity in Figure 1428.]

A guitar string 66 cm long vibrates with a standing wave that has three antinodes. (a) Which harmonic is this? (b) What is the wavelength of this wave?

A12.5-g clothesline is stretched with a tension of 22.1 N between two poles 7.66 m apart. What is the frequency of (a)the fundamental and (b) the second harmonic? (c) If the tension in the clothesline is increased, do the frequencies in parts (a) and (b) increase, decrease, or stay the same? Explain.

(a)In the previous problem, will the frequencies increase, decrease, or stay the same if a more massive rope is used? (b) Repeat Problem 73 for a clothesline with a mass of 15.0 g.

The organ pipe in Figure 1438 is 2.75 m long. (a) What is the frequency of the standing wave shown in the pipe? (b) What is the fundamental frequency of this pipe?

The frequency of the standing wave shown in Figure 1439 is 202 Hz. (a) What is the fundamental frequency of this pipe? (b) What is the length of the pipe?

An organ pipe open at both ends has a harmonic with a frequency of 440 Hz. The next higher harmonic in the pipe has a frequency of 495 Hz. Find (a) the frequency of the fundamental and (b) the length of the pipe.

When guitar strings Aand B are plucked at the same time, a beat frequency of 2 Hz is heard. If string A is tightened, the beat frequency increases to 3 Hz. Which of the two strings had the lower frequency initially?

(a)Is the beat frequency produced when a 245-Hz tone and a 240-Hz tone are played together greater than, less than, or equal to the beat frequency produced when a 140-Hz tone and a 145-Hz tone are played together? (b)Choose the best explanation from among the following: I. The beat frequency is determined by the difference in frequencies and is independent of their actual values. II. The higher frequencies will produce a higher beat frequency. III. The percentage change in frequency for 240 and 245 Hz is less than for 140 and 145 Hz, resulting in a lower beat frequency

Two tuning forks have frequencies of 278 Hz and 292 Hz. What is the beat frequency if both tuning forks are sounded simultaneously?

Tuning a PianoTo tune middle C on a piano, a tuner hits the key and at the same time sounds a 261-Hz tuning fork. If the tuner hears 3 beats per second, what are the possible frequencies of the piano key?

Two musicians are comparing their clarinets. The rst clarinet produces a tone that is known to be 441 Hz. When the two clarinets play together they produce eight beats every 2.00 seconds. If the second clarinet produces a higher pitched tone than the rst clarinet, what is the second clarinets frequency?

Two strings that are xed at each end are identical, except that one is 0.560 cm longer than the other. Waves on these strings propagate with a speed of 34.2 m/s, and the fundamental frequency of the shorter string is 212 Hz. (a)What beat frequency is produced if each string is vibrating with its fundamental frequency? (b) Does the beat frequency in part (a) increase or decrease if the longer string is increased in length? (c) Repeat part (a), assuming that the longer string is 0.761 cm longer than the shorter string.

Atuning fork with a frequency of 320.0 Hz and a tuning fork of unknown frequency produce beats with a frequency of 4.5 Hz. If the frequency of the 320.0-Hz fork is lowered slightly by placing a bit of putty on one of its tines, the new beat frequency is 7.5 Hz. (a) Which tuning fork has the lower frequency? Explain. (b)What is the nal frequency of the 320.0-Hz tuning fork? (c) What is the frequency of the other tuning fork?

Identical cellos are being tested. One is producing a fundamental frequency of 130.9 Hz on a string that is 1.25 m long and has a mass of 109 g. On the second cello the same string isngered to reduce the length that can vibrate. If the beat frequency produced by these two strings is 4.33 Hz, what is the vibrating length of the second string?

A friend in another city tells you that she has two organ pipes of different lengths, one open at both ends, the other open at one end only. In addition, she has determined that the beat frequency caused by the second-lowest frequency of each pipe is equal to the beat frequency caused by the third-lowest frequency of each pipe. Her challenge to you is to calculate the length of the organ pipe that is open at both ends, given that the length of the other pipe is 1.00 m.

A harmonic wave travels along a string. (a) At a point where the displacement of the string is greatest, is the kinetic energy of the string a maximum or a minimum? Explain. (b)At a point where the displacement of the string is zero, is the kinetic energy of the string a maximum or a minimum? Explain.

A harmonic wave travels along a string. (a) At a point where the displacement of the string is greatest, is the potential energy of the string a maximum or a minimum? Explain. (b) At a point where the displacement of the string is zero, is the potential energy of the string a maximum or a minimum? Explain.

Figure 1440shows a wave on a string moving to the right. For each of the points indicated on the string, AF, state whether it is (I, moving upward; II, moving downward; or III, instantaneously at rest) at the instant pictured.

You stand near the tracks as a train approaches with constant speed. The train is operating its horn continuously, and you listen carefully to the sound it makes. For each of the following properties of the sound, state whether it increases, decreases, or stays the same as the train gets closer: (a) the intensity; (b) the frequency; (c) the wavelength; (d) the speed.

Sitting peacefully in your living room one stormy day, you see a ash of lightning through the windows. Eight and a half seconds later thunder shakes the house. Estimate the distance from your house to the bolt of lightning.

The fundamental of an organ pipe that is closed at one end and open at the other end is 261.6 Hz (middle C). The second harmonic of an organ pipe that is open at both ends has the same frequency. What are the lengths of these two pipes?

The Loudest Animal The loudest sound produced by a living organism on Earth is made by the bowhead whale (Balaena mysticetus). These whales can produce a sound that, if heard in air at a distance of 3.00 m, would have an intensity level of 127 dB. This is roughly the equivalent of 5000 trumpeting elephants. How far away can you be from a 127-dB sound and still just barely hear it?(Assume a point source, and ignore reections and absorption.)

Hearing a Good Hit Physicist Robert Adair, once appointed the ofcial physicist to the National League by the commissioner of baseball, believes that the crack of the bat can tell an outelder how well the ball has been hit. According to Adair, a good hit makes a sound of 510 Hz, while a poor hit produces a sound of 170 Hz. What is the difference in wavelength of these sounds?

A standing wave of 603 Hz is produced on a string that is 1.33 m long and xed on both ends. If the speed of waves on this string is 402 m/s, how many antinodes are there in the standing wave?

Measuring Hearing LossTo determine the amount of temporary hearing loss loud music can cause in humans, researchers studied a group of 20 adult females who were exposed to 110-dB music for 60 minutes. Eleven of the 20 subjects showed a 20.0-dB reduction in hearing sensitivity at 4000 Hz. What is the intensity corresponding to the threshold of hearing for these subjects?

Hearing a Pin Drop The ability to hear a pin drop is the sign of sensitive hearing. Suppose a 0.55-g pin is dropped from a height of 28 cm, and that the pin emits sound for 1.5 s when it lands. Assuming all of the mechanical energy of the pin is converted to sound energy, and that the sound radiates uniformly in all directions, nd the maximum distance from which a person can hear the pin drop. (This is the ideal maximum distance, but atmospheric absorption and other factors will make the actual maximum distance considerably smaller.)

A machine shop has 120 equally noisy machines that together produce an intensity level of 92 dB. If the intensity level must be reduced to 82 dB, how many machines must be turned off?

When you blow across the top of a soda pop bottle you hear a fundamental frequency of 206 Hz. Suppose the bottle is now lled with helium. (a)Does the fundamental frequency increase, decrease, or stay the same? Explain. (b) Find the new fundamental frequency. (Assume that the speed of sound in helium is three times that in air.)

Speed of a Tsunami Tsunamis can have wavelengths between 100 and 400 km. Since this is much greater than the average depth of the oceans (about 4.3 km), the ocean can be considered as shallow water for these waves. Using the speed of waves in shallow water of depth d given in Problem 7, nd the typical speed for a tsunami. (Note: In the open ocean, tsunamis generally have an amplitude of less than a meter, allowing them to pass ships unnoticed. As they approach shore, however, the water depth decreases and the waves slow down. This can result in an increase of amplitude to as much as 37 m or more.)

Two trains with 124-Hz horns approach one another. The slower of the two trains has a speed of 26 m/s. What is the speed of the fast train if an observer standing near the tracks between the trains hears a beat frequency of 4.4 Hz?

Jim is speeding toward James Island with a speed of 24 m/s when he sees Betsy standing on shore at the base of a cliff (Figure 1441). Jim sounds his 330-Hz horn. (a) What frequency does Betsy hear? (b)Jim can hear the echo of his horn reected back to him by the cliff. Is the frequency of this echo greater than or equal to the frequency heard by Betsy? Explain. (c) Calculate the frequency Jim hears in the echo from the cliff.

Two ships in a heavy fog are blowing their horns, both of which produce sound with a frequency of 175.0 Hz (Figure 1442). One ship is at rest; the other moves on a straight line that passes through the one at rest. If people on the stationary ship hear a beat frequency of 3.5 Hz, what are the two possible speeds and directions of motion of the moving ship?

Cracking Your Knuckles When you crack a knuckle, you cause the knuckle cavity to widen rapidly. This, in turn, allows the synovial uid to expand into a larger volume. If this expansion is sufciently rapid, it causes a gas bubble to form in the uid in a process known as cavitation. This is the mechanism responsible for the cracking sound. (Cavitation can also cause pits in rapidly rotating ships propellers.) If a crack produces a sound with an intensity level of 57 dB at your ear, which is 18 cm from the knuckle, how far from your knuckle can the crack be heard? Assume the sound propagates uniformly in all directions, with no reections or absorption.

Asteel guitar string has a tension T, length L, and diameter D. Give the multiplicative factor by which the fundamental frequency of the string changes under the following conditions: (a) The tension in the string is increased by a factor of 4. The diameter is D and the length is L. (b) The diameter of the string is increased by a factor of 3. The tension is T and the length is L.(c)The length of the string is halved. The tension is T and the diameter is D.

ASlinky has a mass of 0.28 kg and negligible length. When it is stretched 1.5 m, it is found that transverse waves travel the length of the Slinky in 0.75 s. (a)What is the force constant, k, of the Slinky? (b)If the Slinky is stretched farther, will the time required for a wave to travel the length of the Slinky increase, decrease, or stay the same? Explain. (c) If the Slinky is stretched 3.0 m, how much time does it take a wave to travel the length of the Slinky? (The Slinky stretches like an ideal spring, and propagates transverse waves like a rope with variable tension.)

OSHA Noise StandardsOSHA, the Occupational Safety and Health Administration, has established standards for workplace exposure to noise. According to OSHAs Hearing Conservation Standard, the permissible noise exposure per day is 95.0 dB for 4 hours or 105 dB for 1 hour. Assuming the eardrum is 9.5 mm in diameter, nd the energy absorbed by the eardrum (a) with 95.0 dB for 4 hours and (b) with 105 dB for 1 hour. (c)Is OSHAs safety standard simply a measure of the amount of energy absorbed by the ear? Explain.

Thundersticks at Ball Games Thundersticks are a popular noisemaking device at many sporting events. Atypical thunderstick is a hollow plastic tube about 82 cm long and 8.5 cm in diameter. When two thundersticks are hit sharply together, they produce a copious amount of noise. (a) Which dimension, the length or diameter, is more important in determining the frequency of the sound emitted by the thundersticks? Explain. (b)Estimate the characteristic frequency of the thundersticks sound. (c)Suppose a single pair of thundersticks produces sound with an intensity level of 95 dB. What is the intensity level of 1200 pairs of thundersticks clapping simultaneously?

An organ pipe 2.5 m long is open at one end and closed at the other end. What is the linear distance between a node and the adjacent antinode for the third harmonic in this pipe?

Two identical strings with the same tension vibrate at 631 Hz. If the tension in one of the strings is increased by 2.25%, what is the resulting beat frequency?

The Sound of a Black HoleAstronomers using the Chandra X-ray Observatory have discovered that the Perseus Black Hole, some 250 million light years away, produces sound waves in the gaseous halo that surrounds it. The frequency of this sound is the same as the frequency of the 59th B-at below the B-at given in Table 143. How long does it take for this sound wave to complete one cycle? Give your answer in years.

The Love Song of the Midshipman Fish When the plainn midshipman sh (Porichthys notatus) migrates from deep Pacic waters to the west coast of North America each summer, the males begin to sing their love song, which some describe as sounding like a low-pitched motorboat. Houseboat residents and shore dwellers are kept awake for nights on end by the amorous sh. The love song consists of a single note, the second Aat below middle C. (a) If the speed of sound in seawater is 1531 m/s, what is the wavelength of the midshipmans song? (b) What is the wavelength of the sound after it emerges into the air? (Information on the musical scale is given in Table 143.)

Arope of length Land mass Mhangs vertically from a ceiling. The tension in the rope is only that due to its own weight. (a) Suppose a wave starts near the bottom of the rope and propagates upward. Does the speed of the wave increase, decrease, or stay the same as it moves up the rope? Explain. (b)Show that the speed of waves a height yabove the bottom of the rope is

Experiments on water waves show that the speed of waves in shallow water is independent of their wavelength (see Problem 7). Using this observation and dimensional analysis, determine how the speed v of shallow-water waves depends on the depth of the water, d, the mass per volume of water, , and the acceleration of gravity, g.

A deepwater wave of wavelength has a speed given approximately by . Find an expression for the period of a deepwater wave in terms of its wavelength. (Note the similarity of your result to the period of a pendulum.)

Beats and Standing Waves In Problem 63, suppose the observer walks toward one speaker with a speed of 1.35 m/s. (a)What frequency does the observer hear from each speaker? (b) What beat frequency does the observer hear? (c) How far must the observer walk to go from one point of constructive interference to the next? (d) How many times per second does the observer hear maximum loudness from the speakers? Compare your result with the beat frequency from part (b).

Suppose the air passages in a certain P. walkeri crest produce a bent tube 2.7 m long. What is the fundamental frequency of this tube, assuming the bend has no effect on the frequency? (For comparison, a typical human hearing range is 20 Hz to 20 kHz.) A. 0.0039 Hz B. 32 Hz C. 64 Hz D. 130 Hz

Paleontologists believe the crest of a female P. walkeri was probably shorter than the crest of a male. If this was the case, would the fundamental frequency of a female be greater than, less than, or equal to the fundamental frequency of a male?

Suppose the fundamental frequency of a particular female was 74 Hz. What was the length of the air passages in this females crest? A. 1.2 m B. 2.3 m C. 2.7 m D. 4.6 m

As a young P. walkeri matured, the air passages in its crest might increase in length from 1.5 m to 2.7 m, causing a decrease in the standing wave frequencies. Referring to Figure 1443, do you expect the change in the fundamental frequency to be greater than, less than, or equal to the change in the second harmonic frequency?

Referring to Example 146 Suppose the engineer adjusts the speed of the train until the sound he hears reected from the cliff is 775 Hz. The trains whistle still produces a tone of 650.0 Hz. (a) Is the new speed of the train greater than, less than, or equal to 21.2 m/s? Explain. (b) Find the new speed of the train.

Referring to Example 146 Suppose the train is backing away from the cliff with a speed of 18.5 m/s and is sounding its 650.0-Hz whistle. (a)What is the frequency heard by the observer standing near the tunnel entrance? (b) What is the frequency heard by the engineer?

Referring to Example 149 Suppose we add more water to the soda pop bottle. (a) Does the fundamental frequency increase, decrease, or stay the same? Explain. (b) Find the fundamental frequency if the height of water in the bottle is increased to 7.5 cm. The height of the bottle is still 26.0 cm.

Referring to Example 149 The speed of sound increases slightly with temperature. (a) Does the fundamental frequency of the bottle increase, decrease, or stay the same as the air heats up on a warm day? Explain. (b) Find the fundamental frequency if the speed of sound in air increases to 348 m/s. Assume the bottle is 26.0 cm tall, and that it contains water to a depth of 6.5 cm.