Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as Wiley PLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
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Section 16.1 The Nature of Waves, Section 16.2 Periodic Waves

| 1 | ssm Light is an electromagnetic wave and travels at a speed of . The human eye is most sensitive to yellow-green light, which has a wavelength of . What is the frequency of this light?

| 2 | A woman is standing in the ocean, and she notices that after a wave crest passes, five more crests pass in a time of 50.0 s. The distance between two successive crests is 32 m. Determine, if possible, the wave’s (a) period, (b) frequency, (c) wavelength, (d) speed, and (e) amplitude. If it is not possible to determine any of these quantities, then so state.

| 3 | A longitudinal wave with a frequency of 3.0 Hz takes 1.7 s to travel the length of a 2.5-m Slinky (see Figure 16-3). Determine the wavelength of the wave.

| 4 | Tsunamis are fast-moving waves often generated by underwater earthquakes. In the deep ocean their amplitude is barely noticeable, but upon reaching shore, they can rise up to the astonishing height of a six-story building. One tsunami, generated off the Aleutian islands in Alaska, had a wavelength of 750 km and traveled a distance of 3700 km in 5.3 h. (a) What was the speed (in m/s) of the wave? For reference, the speed of a 747 jetliner is about 250 m/s. Find the wave’s (b) frequency and (c) period.

| 5 | ssm In Figure 16-2c the hand moves the end of the Slinky up and down through two complete cycles in one second. The wave moves along the Slinky at a speed of 0.50 m/s. Find the distance between two adjacent crests on the wave.

| 6 | A person lying on an air mattress in the ocean rises and falls through one complete cycle every five seconds. The crests of the wave causing the motion are 20.0 m apart. Determine (a) the frequency and (b) the speed of the wave.

| 7 | The right-most key on a piano produces a sound wave that has a frequency of 4185.6 Hz. Assuming that the speed of sound in air is 343 m/s, find the corresponding wavelength.

* | 8 | A jetskier is moving at 8.4 m/s in the direction in which the waves on a lake are moving. Each time he passes over a crest, he feels a bump. The bumping frequency is 1.2 Hz, and the crests are separated by 5.8 m. What is the wave speed?

* | 9 | The speed of a transverse wave on a string is 450 m/s, and the wavelength is 0.18 m. The amplitude of the wave is 2.0 mm. How much time is required for a particle of the string to move through a total distance of 1.0 km?

* | 10 | In Figure 16-3c the colored dot exhibits simple harmonic motion as the longitudinal wave passes. The wave has an amplitude of and a frequency of 4.0 Hz. Find the maximum acceleration of the dot.

** | 11 | A water-skier is moving at a speed of 12.0 m/s. When she skis in the same direction as a traveling wave, she springs upward every 0.600 s because of the wave crests. When she skis in the direction opposite to that in which the wave moves, she springs upward every 0.500 s in response to the crests. The speed of the skier is greater than the speed of the wave. Determine (a) the speed and (b) the wavelength of the wave.
Section 16.3 The Speed of a Wave on a String

| 12 | A wire is stretched between two posts. Another wire is stretched between two posts that are twice as far apart. The tension in the wires is the same, and they have the same mass. A transverse wave travels on the shorter wire with a speed of 240 m/s. What would be the speed of the wave on the longer wire?

| 13 | ssm www The middle C string on a piano is under a tension of 944 N. The period and wavelength of a wave on this string are 3.82 ms and 1.26 m, respectively. Find the linear density of the string.

| 14 | The mass of a string is , and it is stretched so that the tension in it is 180 N. A transverse wave traveling on this string has a frequency of 260 Hz and a wavelength of 0.60 m. What is the length of the string?

| 15 | Two wires are parallel, and one is directly above the other. Each has a length of 50.0 m and a mass per unit length of 0.020 kg/m. However, the tension in wire A is , and the tension in wire B is . Transverse wave pulses are generated simultaneously, one at the left end of wire A and one at the right end of wire B. The pulses travel toward each other. How much time does it take until the pulses pass each other?

| 16 | The drawing shows two transverse waves traveling on two strings. The linear density of each string is 0.065 kg/m, and the tension is provided by a 26-N block that is hanging from the string. Determine the speed of the wave in part (a) and part (b) of the drawing.

| 17 | Consult Interactive Solution 16.17 at in order to review a model for solving this problem. To measure the acceleration due to gravity on a distant planet, an astronaut hangs a 0.055-kg ball from the end of a wire. The wire has a length of 0.95 m and a linear density of . Using electronic equipment, the astronaut measures the time for a transverse pulse to travel the length of the wire and obtains a value of 0.016 s. The mass of the wire is negligible compared to the mass of the ball. Determine the acceleration due to gravity.

* | 18 | A steel cable of cross-sectional area is kept under a tension of . The density of steel is 7860 kg/m3 (this is not the linear density). At what speed does a transverse wave move along the cable?

* | 19 | ssm Two blocks are connected by a wire that has a mass perunit length of . One block has a mass of 19.0 kg, and the other has a mass of 42.0 kg. These blocks are being pulled across a horizontal frictionless floor by a horizontal force that is applied to the less massive block. A transverse wave travels on the wire between the blocks with a speed of 352 m/s (relative to the wire). The mass of the wire is negligible compared to the mass of the blocks. Find the magnitude of .

** | 20 | A copper wire, whose cross-sectional area is , has a linear density of and is strung between two walls. At the ambient temperature, a transverse wave travels with a speed of 46 m/s on this wire. The coefficient of linear expansion for copper is , and Young’s modulus for copper is . What will be the speed of the wave when the temperature is lowered by 14 C°? Ignore any change in the linear density caused by the change in temperature.

** | 21 | The drawing shows a 15.0-kg ball being whirled in a circular path on the end of a string. The motion occurs on a frictionless, horizontal table. The angular speed of the ball is . The string has a mass of 0.0230 kg. How much time does it take for a wave on the string to travel from the center of the circle to the ball?

Section 16.4 The Mathematical Description of a Wave
(Note: The phase angles and are measured in radians, not degrees.)

| 22 | The displacement (in meters) of a wave is given according to , where t is in seconds and x is in meters, (a) Is the wave traveling in the or direction? (b) What is the displacement y when and ?

| 23 | ssm A wave traveling in the direction has an amplitude of 0.35 m, a speed of 5.2 m/s, and a frequency of 14 Hz. Write the equation of the wave in the form given by either Equation 16.3 or 16.4.

| 24 | The drawing shows two graphs that represent a transverse wave on a string. The wave is moving in the direction. Using the information contained in these graphs, write the mathematical expression (similar to Equation 16.3 or 16.4) for the wave.

| 25 | A wave has the following properties: , , wave . The wave is traveling in the direction. What is the mathematical expression (similar to Equation 16.3 or 16.4) for the wave?

* | 26 | The tension in a string is 15 N, and its linear density is 0.85 kg/m. A wave on the string travels toward the direction; it has an amplitude of 3.6 cm and a frequency of 12 Hz. What are the (a) speed and (b) wavelength of the wave? (c) Write down a mathematical expression (like Equation 16.3 or 16.4) for the wave, substituting numbers for the variables A,f, and .

* | 27 | ssm A transverse wave is traveling on a string. The displacement y of a particle from its equilibrium position is given by . Note that the phase angle is in radians, t is in seconds, and x is in meters. The linear density of the string is . What is the tension in the string?

** | 28 | A transverse wave on a string has an amplitude of 0.20 m and a frequency of 175 Hz. Consider the particle of the string at . It begins with a displacement of when , according to Equation 16.3 or 16.4. How much time passes between the first two instants when this particle has a displacement of ?
Section 16.5 The Nature of Sound,
Section 16.6 The Speed of Sound

| 29 | ssm The speed of a sound in a container of hydrogen at 201 K is 1220 m/s. What would be the speed of sound if the temperature were raised to 405 K? Assume that hydrogen behaves like an ideal gas.

| 30 | A bat emits a sound whose frequency is 91 kHz. The speed of sound in air at 20.0 °C is 343 m/s. However, the air temperature is 35 °C, so the speed of sound is not 343 m/s. Find the wavelength of the sound.

| 31 | The distance between a loudspeaker and the left ear of a listener is 2.70 m. (a) Calculate the time required for sound to travel this distance if the air temperature is 20 °C. (b) Assuming that the sound frequency is 523 Hz, how many wavelengths of sound are contained in this distance?

| 32 | Have you ever listened for an approaching train by kneeling next to a railroad track and putting your ear to the rail? Young’s modulus for steel is , and the density of steel is . On a day when the temperature is 20 °C, how many times greater is the speed of sound in the rail than in the air?

| 33 | ssm Argon () is a monatomic gas. As suming that it behaves like an ideal gas at 298 K , find (a) the rms speed of argon atoms and (b) the speed of sound in argon.

| 34 | The wavelength of a sound wave in air is 2.74 m at 20 °C. What is the wavelength of this sound wave in fresh water at 20 °C? (Hint: The frequency of the sound is the same in both media.)

| 35 | As the drawing illustrates, a siren can be made by blowing a jet of air through 20 equally spaced holes in a rotating disk. The time it takes for succes sive holes to move past the air jet is the period of the sound. The siren is to produce a 2200-Hz tone. What must be the angular speed (in rad/s) of the disk?

| 36 | A sound wave is incident on a pool of fresh water. The sound enters the water perpendicularly and travels a distance of 0.45 m before striking a 0.15-m-thick copper block lying on the bottom. The sound passes through the block, reflects from the bottom surface of the block, and returns to the top of the water along the same path. How much time elapses between when the sound enters and when it leaves the water?

| 37 | An explosion occurs at the end of a pier. The sound reaches the other end of the pier by traveling through three media: air, fresh water, and a slender metal handrail. The speeds of sound in air, water, and the handrail are 343, 1482, and 5040 m/s, respectively. The sound travels a distance of 125 m in each medium. (a) Through which medium does the sound arrive first, second, and third? (b) After the first sound arrives, how much later do the second and third sounds arrive?

* | 38 | As the drawing shows, one microphone is located at the origin, and a second microphone is located on the axis. The microphones are separated by a distance of . A source of sound is located on the axis, its distances from microphones 1 and 2 being L1 and L2, respectively. The speed of sound is 343 m/s. The sound reaches microphone 1 first, and then, 1.46 ms later, it reaches microphone 2. Find the distances L1 and L2.

* | 39 | ssm A hunter is standing on flat ground between two vertical cliffs that are directly opposite one another. He is closer to one cliff than to the other. He fires a gun and, after a while, hears three echoes. The second echo arrives 1.6 s after the first, and the third echo arrives 1.1 s after the second. Assuming that the speed of sound is 343 m/s and that there are no reflections of sound from the ground, find the distance between the cliffs.

* | 40 | A sound wave travels twice as far in neon (Ne) as it does in krypton (Kr) in the same time interval. Both neon and krypton can be treated as monatomic ideal gases. The atomic mass of neon is 20.2 u, and that of krypton is 83.8 u. The temperature of the krypton is 293 K. What is the temperature of the neon?

* | 41 | ssm A long slender bar is made from an unknown material. The length of the bar is 0.83 m, its cross-sectional area is , and its mass is 2.1 kg. A sound wave travels from one end of the bar to the other end in . From which one of the materials listed in Table 10.1 is the bar most likely to be made?

* | 42 | When an earthquake occurs, two types of sound waves are generated and travel through the earth. The primary, or P, wave has a speed of about 8.0 km/s and the secondary, or S, wave has a speed of about 4.5 km/s. A seismograph, located some distance away, records the arrival of the P wave and then, 78 s later, records the arrival of the S wave. Assuming that the waves travel in a straight line, how far is the seismograph from the earthquake?

* | 43 | A monatomic ideal gas is contained within a box whose volume is 2.5 m3. The pressure of the gas is . The total mass of the gas is 2.3 kg. Find the speed of sound in the gas.

* | 44 | Consult Multiple-Concept Example 4 in order to review a model for solving this type of problem. Suppose you are standing by the side of a road in the Sahara desert where the temperature has reached a hot 56 °C (130 °F). A truck, traveling at a constant speed, passes by. After 4.00 s have elapsed, you use the ultrasonic ruler discussed in Example 4 to measure the distance to the truck. A sound pulse leaves the ultrasonic ruler and returns 0.120 s later. Assume that the average molecular mass of air is 28.9 u, air is an ideal diatomic gas (), and the truck moves a negligible distance in the time it takes for the sound pulse to reach it. Determine how fast the truck is moving.

** | 45 | In a mixture of argon () and neon (), the speed of sound is 363 m/s at . Assume that both monatomic gases behave as ideal gases. Find the percentage of the atoms that are argon and the percentage that are neon.

** | 46 | Multiple-Concept Example 4 presents one method for modeling this type of problem. Civil engineers use a transit theodolite when surveying. A modern version of this device determines distance by measuring the time required for an ultrasonic pulse to reach a target, reflect from it, and return. Effectively, such a theodolite is calibrated properly when it is programmed with the speed of sound appropriate for the ambient air temperature. (a) Suppose the round-trip time for the pulse is 0.580 s on a day when the air temperature is 293 K, the temperature for which the instrument is calibrated. How far is the target from the theodolite? (b) Assume that air behaves as an ideal gas. If the air temperature were 298 K, rather than the calibration temperature of 293 K, what percentage error would there be in the distance measured by the theodolite?

** | 47 | ssm www As a prank, someone drops a water-filled balloon out of a window. The balloon is released from rest at a height of 10.0 m above the ears of a man who is the target. Then, because of a guilty conscience, the prankster shouts a warning after the balloon is released. The warning will do no good, however, if shouted after the balloon reaches a certain point, even if the man could react infinitely quickly. Assuming that the air temperature is 20 °C and ignoring the effect of air resistance on the balloon, determine how far above the man’s ears this point is.

| 48 | A typical adult ear has a surface area of . The sound intensity during a normal conversation is about at the listener’s ear. Assume that the sound strikes the surface of the ear perpendicularly. How much power is intercepted by the ear?

| 49 | ssm www At a distance of 3.8 m from a siren, the sound intensity is . Assuming that the siren radiates sound uniformly in all directions, find the total power radiated.

| 50 | A rocket in a fireworks display explodes high in the air. The sound spreads out uniformly in all directions. The intensity of the sound is at a distance of 120 m from the explosion. Find the distance from the source at which the intensity is .

| 51 | Suppose that sound is emitted uniformly in all directions by a public address system. The intensity at a location 22 m away from the sound source is . What is the intensity at a spot that is 78 m away?

| 52 | Suppose in Conceptuat Example 8 (see Figure 16-24) that the person is producing 1.1 mW of sound power. Some of the sound is reflected from the floor and ceiling. The intensity of this reflected sound at a distance of 3.0 m from the source is . What is the total sound intensity due to both the direct and reflected sounds, at this point?

| 53 | ssm A loudspeaker has a circular opening with a radius of 0.0950 m. The electrical power needed to operate the speaker is 25.0 W. The average sound intensity at the opening is 17.5 W/m2. What percentage of the electrical power is converted by the speaker into sound power?

* | 54 | Deep ultrasonic heating is used to promote healing of torn tendons. It is produced by applying ultrasonic sound over the affected area of the body. The sound transducer (generator) is circular with a radius of 1.8 cm, and it produces a sound intensity of . How much time is required for the transducer to emit 4800 J of sound energy?

* | 55 | Review Interactive Solution 16.55 at for one approach to this problem. A dish of lasagna is being heated in a microwave oven. The effective area of the lasagna that is exposed to the microwaves is . The mass of the lasagna is 0.35 kg, and its specific heat capacity is 3200 J/(kg · C°). The temperature rises by 72 C° in 8.0 minutes. What is the intensity of the microwaves in the oven?

* | 56 | Two sources of sound are located on the x axis, and each emits power uniformly in all directions. There are no reflections. One source is positioned at the origin and the other at . The source at the origin emits four times as much power as the other source. Where on the x axis are the two sounds equal in intensity? Note that there are two answers.

** | 57 | ssm A rocket, starting from rest, travels straight up with an acceleration of 58.0 m/s2. When the rocket is at a height of 562 m, it produces sound that eventually reaches a ground-based monitoring station directly below. The sound is emitted uniformly in all directions. The monitoring station measures a sound intensity I. Later, the station measures an intensity . Assuming that the speed of sound is 343 m/s, find the time that has elapsed between the two measurements.
Section 16.8 Decibels

| 58 | A middle-aged man typically has poorer hearing than a middle-aged woman. In one case a woman can just begin to hear a musical tone, while a man can just begin to hear the tone only when its intensity level is increased by 7.8 dB relative to that for the woman. What is the ratio of the sound intensity just detected by the man to that just detected by the woman?

| 59 | An amplified guitar has a sound intensity level that is 14 dB greater than the same unamplified sound. What is the ratio of the amplified intensity to the unamplified intensity?

| 60 | A recording engineer works in a soundproofed room that is 44.0 dB quieter than the outside. If the sound intensity in the room is , what is the intensity outside?

| 61 | ssm When a person wears a hearing aid, the sound intensity level increases by 30.0 dB. By what factor does the sound intensity increase?

| 62 | The sound intensity level at a rock concert is 115 dB, while that at a jazz fest is 95 dB. Determine the ratio of the sound intensity at the rock concert to that at the jazz fest.

| 63 | The equation , which defines the decibel, can be written in terms of power P (in watts) rather than intensity I (in watts/meter2). The form can be used to compare two power levels in terms of decibels. Suppose that stereo amplifier A is rated at watts per channel, and amplifier B has a rating of watts per channel, (a) Expressed in decibels, how much more powerful is A compared to B? (b) Will A sound more than twice as loud as B? Justify your answer.

* | 64 | ssm Review Conceptuat Example 8 as background for this problem. A loudspeaker is generating sound in a room. At a certain point, the sound waves coming directly from the speaker (without reflecting from the walls) create an intensity level of 75.0 dB. The waves reflected from the walls create, by themselves, an intensity level of 72.0 dB at the same point. What is the total intensity level?(Hint: The answer is not 147.0 dB.)

* | 65 | Two identical rifles are shot at the same time, and the sound intensity level is 80.0 dB. What would be the sound intensity level if only one rifle were shot? (Hint: The answer is not 40.0 dB.)

* | 66 | Hearing damage may occur when a person is exposed to a sound intensity level of 90.0 dB (relative to the threshold of hearing) for a period of 9.0 hours. An eardrum has an area of . How much sound energy is incident on the eardrum during this time?

* | 67 | A portable radio is sitting at the edge of a balcony 5.1 m above the ground. The unit is emitting sound uniformly in all directions. By accident, it falls from rest off the balcony and continues to play on the way down. A gardener is working in a flower bed directly below the falling unit. From the instant the unit begins to fall, how much time is required for the sound intensity level heard by the gardener to increase by 10.0 dB?

** | 68 | A source emits sound uniformly in all directions. A radial line is drawn from this source. On this line, determine the positions of two points, 1.00 m apart, such that the intensity level at one point is 2.00 dB greater than that at the other.

** | 69 | ssmSuppose that when a certain sound intensity level (in dB) triples, the sound intensity (in W/m2) also triples. Determine this sound intensity level. Section 16.9 The Doppler Effect

| 70 | The security alarm on a parked car goes off and produces a frequency of 960 Hz. The speed of sound is 343 m/s. As you drive toward this parked car, pass it, and drive away, you observe the frequency to change by 95 Hz. At what speed are you driving?

| 71 | You are riding your bicycle directly away from a stationary source of sound and hear a frequency that is 1.0% lower than the emitted frequency. The speed of sound is 343 m/s. What is your speed?

| 72 | You are flying in an ultra-light aircraft at a speed of 39 m/s. An eagle, whose speed is 18 m/s, is flying directly toward you. Each of the given speeds is relative to the ground. The eagle emits a shrill cry whose frequency is 3400 Hz. The speed of sound is 330 m/s. What frequency do you hear?

| 73 | ssm From a vantage point very close to the track at a stock car race, you hear the sound emitted by a moving car. You detect a frequency that is 0.86 times as small as that emitted by the car when it is stationary. The speed of sound is 343 m/s. What is the speed of the car?

| 74 | Interactive LearningWare 16.2 at provides some pertinent background for this problem. A convertible moves toward you and then passes you; all the while, its loudspeakers are producing a sound. The speed of the car is a constant 9.00 m/s, and the speed of sound is 343 m/s. What is the ratio of the frequency you hear while the car is approaching to the frequency you hear while the car is moving away?

* | 75 | ssm An aircraft carrier has a speed of 13.0 m/s relative to the water. A jet is catapulted from the deck and has a speed of 67.0 m/s relative to the water. The engines produce a 1550-Hz whine, and the speed of sound is 343 m/s. What is the frequency of the sound heard by the crew on the ship?

* | 76 | Multiple-Concept Example 11 presents a model for solving this type of problem. A bungee jumper jumps from rest and screams with a frequency of 589 Hz. The air temperature is 20 °C. What is the frequency heard by the people on the ground below when she has fallen a distance of 11.0 m? Assume that the bungee cord has not yet taken effect, so she is in free-fall.

* | 77 | Refer to Interactive Solution 16.77 at for one approach to this type of problem. Two trucks travel at the same speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.14 times the frequency he hears when the trucks are stationary. The speed of sound is 343 m/s. At what speed is each truck moving?

* | 78 | Consult Multiple-Concept Example 11 in order to review a model for solving this type of problem. A car is accelerating while its horn is blowing. Just after the car passes a stationary person, the person hears a frequency of 966.0 Hz. Fourteen seconds later, the frequency heard by the person has decreased to 912.0 Hz. When the car is stationary, its horn emits a sound whose frequency is . The speed of sound is 343 m/s. What is the acceleration of the car?

* | 79 | ssm Two submarines are underwater and approaching each other head-on. Sub A has a speed of 12 m/s and sub B has a speed of 8 m/s. Sub A sends out a 1550-Hz sonar wave that travels at a speed of 1522 m/s. (a) What is the frequency detected by sub B? (b) Part of the sonar wave is reflected from B and returns to A. What frequency does A detect for this reflected wave?

** | 80 | A microphone is attached to a spring that is suspended from the ceiling, as the drawing indicates. Directly below on the floor is a stationary 440-Hz source of sound. The microphone vibrates up and down in simple harmonic motion with a period of 2.0 s. The difference between the maximum and minimum sound frequencies detected by the microphone is 2.1 Hz. Ignoring any reflections of sound in the room and using 343 m/s for the speed of sound, determine the amplitude of the simple harmonic motion.

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