Problems that are not marked with a star are considered the easiest to solve. Problems that are marked with a single star (*) are more difficult, while those marked with a double star (**) are the most difficult.

Section 16.1 The Nature of Waves, Section 16.2 Periodic Waves


A person standing in the ocean notices that after a wave crest passes by, ten more crests pass in a time of 120 s. What is the frequency of the wave?

Example 1


Light is an electromagnetic wave and travels at a speed of 3.00×108 m/s. The human eye is most sensitive to yellow-green light, which has a wavelength of 5.45×10–7 m. What is the frequency of this light?


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.

Example 1


Consider the freight train in Figure 16.7. Suppose 15 boxcars pass by in a time of 12.0 s and each has a length of 14.0 m. (a) What is the frequency at which each boxcar passes? (b) What is the speed of the train?


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.

Example 1


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.


Suppose the amplitude and frequency of the transverse wave in Figure 16.2c are, respectively, 1.3 cm and 5.0 Hz. Find the total vertical distance (in cm) through which the colored dot moves in 3.0 s.


 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 *  

A 3.49-rad/s (33 rpm) record has a 5.00-kHz tone cut in the groove. If the groove is located 0.100 m from the center of the record (see drawing), what is the wavelength in the groove?


 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


The mass of a string is 5.0×10–3 kg, 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?


The linear density of the A string on a violin is 7.8×10–4 kg/m. A wave on the string has a frequency of 440 Hz and a wavelength of 65 cm. What is the tension in the string?

Example 2


A vibrator moves one end of a rope up and down to generate a wave. The tension in the rope is 58 N. The frequency is then doubled. To what value must the tension be adjusted, so the new wave has the same wavelength as the old one?


A transverse wave is traveling with a speed of 300 m/s on a horizontal string. If the tension in the string is increased by a factor of four, what is the speed of the wave?



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 6.00×102 N, and the tension in wire B is 3.00×102 N. 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?


Consult Interactive Solution 16.17 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 1.2×10–4 kg/m. 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 *  

Review Conceptual Example 3 before starting this problem. The amplitude of a transverse wave on a string is 4.5 cm. The ratio of the maximum particle speed to the speed of the wave is 3.1. What is the wavelength (in cm) of the wave?

 19 *  

The drawing at right shows a frictionless incline and pulley. The two blocks are connected by a wire (mass per unit length=0.0250 kg/m) and remain stationary. A transverse wave on the wire has a speed of 75.0 m/s. Neglecting the weight of the wire relative to the tension in the wire, find the masses m1 and m2 of the blocks.



 20 **  

A copper wire, whose cross-sectional area is 1.1×10–6 m2, has a linear density of 7.0×10–3 kg/m 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 17×10–6(C°)–1, and Young’s modulus for copper is 1.1×1011 N/m2. 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 w=12.0 rad/s. 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 (2pft2px/l) and (2pft+2px/l) are measured in radians, not degrees.)


A wave traveling in the +x 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.


A wave has the following properties: amplitude=0.37 m, period=0.77 s, wave speed=12 m/s. The wave is traveling in the –x direction. What is the mathematical expression (similar to Equation 16.3 or 16.4) for the wave?



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



A wave causes a displacement y that is given in meters according to y=(0.45) sin (8.0pt+px), where t and x are expressed in seconds and meters, respectively. (a) Find the amplitude, the frequency, the wavelength, and the speed of the wave. (b) Is this wave traveling in the +x or –x direction?


 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 –x 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 l.

 27 *  

A transverse wave is traveling on a string. The displacement y of a particle from its equilibrium position is given by y=(0.021 m) sin (25t2.0x). Note that the phase angle 25t2.0x is in radians, t is in seconds, and x is in meters. The linear density of the string is 1.6×10–2 kg/m. 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 x=0 m. It begins with a displacement of y=0 m when t=0 s, according to Equation 16.3 or 16.4. How much time passes between the first two instants when this particle has a displacement of y=0.10 m?

Section 16.5 The Nature of Sound, Section 16.6 The Speed of Sound


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.

Example 4


For research purposes a sonic buoy is tethered to the ocean floor and emits an infrasonic pulse of sound. The period of this sound is 71 ms. Determine the wavelength of the sound.


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?



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 Y=2.0×1011 N/m2, and the density of steel is r=7860 kg/m3. 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?


At 20 °C the densities of fresh water and ethyl alcohol are, respectively, 998 and 789 kg/m3. Find the ratio of the adiabatic bulk modulus of fresh water to the adiabatic bulk modulus of ethyl alcohol at 20 °C.



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?


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?



Consult Interactive LearningWare 16.1 for insight into this problem. At what temperature is the speed of sound in helium (ideal gas, g=1.67, atomic mass=4.003 u) the same as its speed in oxygen at 0 °C?

Example 4


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 successive 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 w (in rad/s) of the disk?


 38 *  

As the drawing shows, one microphone is located at the origin, and a second microphone is located on the +y axis. The microphones are separated by a distance of D=1.50 m. A source of sound is located on the +x 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 *  

A long slender bar is made from an unknown material. The length of the bar is 0.83 m, its cross-sectional area is 1.3×10–4 m2, and its mass is 2.1 kg. A sound wave travels from one end of the bar to the other end in 1.9×10–4 s. From which one of the materials listed in Table 10.1 is the bar most likely to be made?


 40 *  

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?

 41 *  

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.


 42 *  

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?

 43 *  

A monatomic ideal gas (g=1.67) is contained within a box whose volume is 2.5 m3. The pressure of the gas is 3.5×105 Pa. The total mass of the gas is 2.3 kg. Find the speed of sound in the gas.

Example 4

 44 *  

At a height of ten meters above the surface of a freshwater lake, a sound pulse is generated. The echo from the bottom of the lake returns to the point of origin 0.140 s later. The air and water temperatures are 20 °C. How deep is the lake?

 45 **  

A jet is flying horizontally, as the drawing shows. When the plane is directly overhead at B, a person on the ground hears the sound coming from A in the drawing. The average temperature of the air is 20 °C. If the speed of the plane at A is 164 m/s, what is its speed at B, assuming that it has a constant acceleration?


 46 **  

The sonar unit on a boat is designed to measure the depth of fresh water (r=1.00×103 kg/m3, Bad=2.20×109 Pa). When the boat moves into salt water (r=1025 kg/m3, Bad=2.37×109 Pa), the sonar unit is no longer calibrated properly. In salt water, the sonar unit indicates the water depth to be 10.0 m. What is the actual depth of the water?

 47 **  

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.


Section 16.7 Sound Intensity


A typical adult ear has a surface area of 2.1×10–3 m2. The sound intensity during a normal conversation is about 3.2×10–6 W/m2 at the listener’s ear. Assume that the sound strikes the surface of the ear perpendicularly. How much power is intercepted by the ear?


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?



The average sound intensity inside a busy restaurant is 3.2×10–5 W/m2. How much energy goes into each ear (area=2.1×10–3 m2) during a one-hour meal?


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 3.0×10–4 W/m2. What is the intensity at a spot that is 78 m away?



Suppose in Conceptual Example 8 (see Figure 16.25) 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 4.4×10–6 W/m2. What is the total sound intensity due to both the direct and reflected sounds, at this point?


At a distance of 3.8 m from a siren, the sound intensity is 3.6×10–2W/m2. Assuming that the siren radiates sound uniformly in all directions, find the total power radiated.

Example 6

 54 *  

Deep ultrasonic heating is used to promote healing of torn tendons. It is produced by applying ultrasonic sound to the body. The sound transducer (generator) is circular with a radius of 1.8 cm, and it produces a sound intensity of 5.9×103 W/m2. How much time is required for the transducer to emit 4800 J of sound energy?

 55 *  

Review Interactive Solution 16.55 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 2.2×10–2 m2. 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 *  

When a helicopter is hovering 1450 m directly overhead, an observer on the ground measures a sound intensity I. Assume that sound is radiated uniformly from the helicopter and that ground reflections are negligible. How far must the helicopter fly in a straight line parallel to the ground before the observer measures a sound intensity of I?

 57 **  

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


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?


The bellow of a territorial bull hippopotamus has been measured at 115 dB above the threshold of hearing. What is the sound intensity?



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 1.20×10–10 W/m2, what is the intensity outside?


Humans can detect a difference in sound intensity levels as small as 1.0 dB. What is the ratio of the sound intensities?

Example 9


The equation b=(10 dB) log (I/I0), which defines the decibel, is sometimes written in terms of power P (in watts) rather than intensity I (in watts/meter2). The form b=(10 dB) log (P/P0) can be used to compare two power levels in terms of decibels. Suppose that stereo amplifier A is rated at P=250 watts per channel, and amplifier B has a rating of P0=45 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.


For information, read Problem 62 before working this problem. Stereo manufacturers express the power output of a stereo amplifier using the decibel, abbreviated as dBW, where the “W” indicates that a reference power level of P0=1.00 W has been used. If an amplifier has a power rated at 17.5 dBW, how many watts of power can this amplifier deliver?


 64 *  

Review Conceptual 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?


 65 *  

In a discussion person A is talking 1.5 dB louder than person B, and person C is talking 2.7 dB louder than person A. What is the ratio of the sound intensity of person C to the sound intensity of person B?


 66 *  

The sound intensity level of a person speaking normally is about 65 dB above the threshold of hearing. What is the minimum number of people speaking simultaneously, each with this intensity level, that is necessary to produce a sound intensity level at least 78 dB above the threshold of hearing?

 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?

Example 9

 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 **  

Suppose 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


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?


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?



Suppose you are stopped for a traffic light, and an ambulance approaches you from behind with a speed of 18 m/s. The siren on the ambulance produces sound with a frequency of 955 Hz. The speed of sound in air is 343 m/s. What is the wavelength of the sound reaching your ears?


A speeder looks in his rearview mirror. He notices that a police car has pulled behind him and is matching his speed of 38 m/s. The siren on the police car has a frequency of 860 Hz when the police car and the listener are stationary. The speed of sound is 343 m/s. What frequency does the speeder hear when the siren is turned on in the moving police car?



A bird is flying directly toward a stationary bird-watcher and emits a frequency of 1250 Hz. The bird-watcher, however, hears a frequency of 1290 Hz. What is the speed of the bird, expressed as a percentage of the speed of sound?

 75 *  

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?

Example 10

 76 *  

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 Solutions 16.77 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 *  

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?

 79 *  

A motorcycle starts from rest and accelerates along a straight line at 2.81 m/s2. The speed of sound is 343 m/s. A siren at the starting point remains stationary. How far has the motorcycle gone when the driver hears the frequency of the siren at 90.0% of the value it has when the motorcycle is stationary?


 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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