Chapter 16

Light is an electromagnetic wave and travels at a speed of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} .\) The human eye is most sensitive to yellow-green light, which has a wavelength of \(5.45 \times 10^{-7} \mathrm{~m} .\) What is the frequency of this light?

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

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 \(750 \mathrm{~km}\) and traveled a distance of \(3700 \mathrm{~km}\) in \(5.3 \mathrm{~h}\). (a) What was the speed (in \(\mathrm{m} / \mathrm{s}\) ) of the wave? For reference, the speed of a 747 jetliner is about \(250 \mathrm{~m} / \mathrm{s}\). Find the wave's (b) frequency and (c) period.

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 \(20.0 \mathrm{~m}\) apart. Determine (a) the frequency and (b) the speed of the wave.

The right-most key on a piano produces a sound wave that has a frequency of \(4185.6 \mathrm{~Hz}\). Assuming that the speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\), find the corresponding wavelength.

A jetskier is moving at \(8.4 \mathrm{~m} / \mathrm{s}\) 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 \(1.2 \mathrm{~Hz}\), and the crests are separated by \(5.8 \mathrm{~m} .\) What is the wave speed?

The speed of a transverse wave on a string is \(450 \mathrm{~m} / \mathrm{s}\), and the wavelength is \(0.18 \mathrm{~m}\). The amplitude of the wave is \(2.0 \mathrm{~mm}\). How much time is required for a particle of the string to move through a total distance of \(1.0 \mathrm{~km}\) ?

A water-skier is moving at a speed of \(12.0 \mathrm{~m} / \mathrm{s}\). When she skis in the same direction as a traveling wave, she springs upward every \(0.600 \mathrm{~s}\) because of the wave crests. When she skis in the direction opposite to that in which the wave moves, she springs upward every \(0.500 \mathrm{~s}\) 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.

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 \(240 \mathrm{~m} / \mathrm{s}\). What would be the speed of the wave on the longer wire?

The middle \(C\) string on a piano is under a tension of \(944 \mathrm{~N}\). The period and wavelength of a wave on this string are \(3.82 \mathrm{~ms}\) and \(1.26 \mathrm{~m}\), respectively. Find the linear density of the string.

The mass of a string is \(5.0 \times 10^{-3} \mathrm{~kg}\), and it is stretched so that the tension in it is \(180 \mathrm{~N}\). A transverse wave traveling on this string has a frequency of \(260 \mathrm{~Hz}\) and a wavelength of \(0.60 \mathrm{~m}\). What is the length of the string?

Two wires are parallel, and one is directly above the other. Each has a length of \(50.0 \mathrm{~m}\) and a mass per unit length of \(0.020 \mathrm{~kg} / \mathrm{m}\). However, the tension in wire \(\mathrm{A}\) is \(6.00 \times 10^{2} \mathrm{~N}\), and the tension in wire \(\mathrm{B}\) is \(3.00 \times 10^{2} \mathrm{~N}\). 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 \(\underline{16.17}\) 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 \(0.055-\mathrm{kg}\) ball from the end of a wire. The wire has a length of \(0.95 \mathrm{~m}\) and a linear density of \(1.2 \times 10^{-4} \mathrm{~kg} / \mathrm{m}\). Using electronic equipment, the astronaut measures the time for a transverse pulse to travel the length of the wire and obtains a value of \(0.016 \mathrm{~s}\). The mass of the wire is negligible compared to the mass of the ball. Determine the acceleration due to gravity.

A steel cable of cross-sectional area \(2.83 \times 10^{-3} \mathrm{~m}^{2}\) is kept under a tension of \(1.00 \times 10^{4} \mathrm{~N}\). The density of steel is \(7860 \mathrm{~kg} / \mathrm{m}^{3}\) (this is not the linear density). At what speed does a transverse wave move along the cable?

Two blocks are connected by a wire that has a mass perunit length of \(8.50 \times 10^{-4} \mathrm{~kg} / \mathrm{m} .\) One block has a mass of \(19.0 \mathrm{~kg},\) and the other has a mass of 42.0 kg. These blocks are being pulled across a horizontal frictionless floor by a horizontal force \(\overrightarrow{\mathrm{P}}\) that is applied to the less massive block. A transverse wave travels on the wire between the blocks with a speed of \(352 \mathrm{~m} / \mathrm{s}\) (relative to the wire). The mass of the wire is negligible compared to the mass of the blocks. Find the magnitude of \(\overrightarrow{\mathbf{P}}\).

A copper wire, whose cross-sectional area is \(1.1 \times 10^{-6} \mathrm{~m}^{2},\) has a linear density of \(7.0 \times 10^{-3} \mathrm{~kg} / \mathrm{m}\) and is strung between two walls. At the ambient temperature, a transverse wave travels with a speed of \(46 \mathrm{~m} / \mathrm{s}\) on this wire. The coefficient of linear expansion for copper is \(17 \times 10^{-6}\left(\mathrm{C}^{\circ}\right)^{-1},\) and Young's modulus for copper is \(1.1 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\). What will be the speed of the wave when the temperature is lowered by \(14 \mathrm{C}^{\circ}\) ? Ignore any change in the linear density caused by the change in temperature.

The displacement (in meters) of a wave is given according to \(y=0.26 \sin (\pi t-3.7 \pi x)\) where \(t\) is in seconds and \(x\) is in meters, (a) Is the wave traveling in the \(+x\) or \(-x\) direction? (b) What is the displacement \(y\) when \(t=38 \mathrm{~s}\) and \(x=13 \mathrm{~m} ?\)

A wave traveling in the \(+x\) direction has an amplitude of \(0.35 \mathrm{~m},\) a speed of \(5.2 \mathrm{~m} /\) \(\mathrm{s},\) and a frequency of \(14 \mathrm{~Hz}\). 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 \(=0.37 \mathrm{~m}\), period \(=0.77 \mathrm{~s}\), wave speed \(=12 \mathrm{~m} / \mathrm{s} .\) 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 tension in a string is \(15 \mathrm{~N},\) and its linear density is \(0.85 \mathrm{~kg} / \mathrm{m} .\) A wave on the string travels toward the \(-x\) direction; it has an amplitude of \(3.6 \mathrm{~cm}\) and a frequency of \(12 \mathrm{~Hz}\) 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 \(\lambda\)

A transverse wave is traveling on a string. The displacement \(y\) of a particle from its equilibrium position is given by \(y=(0.021 \mathrm{~m}) \sin (25 t-2.0 x) .\) Note that the phase angle \(25 t-2.0 x\) is in radians, \(t\) is in seconds, and \(x\) is in meters. The linear density of the string is \(1.6 \times 10^{-2} \mathrm{~kg} / \mathrm{m} .\) What is the tension in the string?

The speed of a sound in a container of hydrogen at \(201 \mathrm{~K}\) is \(1220 \mathrm{~m} / \mathrm{s}\). What would be the speed of sound if the temperature were raised to \(405 \mathrm{~K}\) ? Assume that hydrogen behaves like an ideal gas.

A bat emits a sound whose frequency is \(91 \mathrm{kHz}\). The speed of sound in air at \(20.0{ }^{\circ} \mathrm{C}\) is \(343 \mathrm{~m} / \mathrm{s}\). However, the air temperature is \(35{ }^{\circ} \mathrm{C}\), so the speed of sound is not \(343 \mathrm{~m} / \mathrm{s} .\) Find the wavelength of the sound.

The distance between a loudspeaker and the left ear of a listener is \(2.70 \mathrm{~m}\). (a) Calculate the time required for sound to travel this distance if the air temperature is \(20^{\circ} \mathrm{C}\). (b) Assuming that the sound frequency is \(523 \mathrm{~Hz}\), 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 \(Y=2.0 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2},\) and the density of steel is \(\rho=7860 \mathrm{~kg} / \mathrm{m}^{3} .\) On a day when the temperature is \(20^{\circ} \mathrm{C}\), how many times greater is the speed of sound in the rail than in the air?

Argon (molecular mass \(=39.9 \mathrm{u}\) ) is a monatomic gas. As suming that it behaves like an ideal gas at \(298 \mathrm{~K}(\gamma=1.67),\) find (a) the rms speed of argon atoms and (b) the speed of sound in argon.

The wavelength of a sound wave in air is \(2.74 \mathrm{~m}\) at \(20{ }^{\circ} \mathrm{C}\). What is the wavelength of this sound wave in fresh water at \(20{ }^{\circ} \mathrm{C} ?\) (Hint: The frequency of the sound is the same in both media.)

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 \(5040 \mathrm{~m} / \mathrm{s},\) respectively. The sound travels a distance of \(125 \mathrm{~m}\) 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?

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 \(D=1.50 \mathrm{~m}\). A source of sound is located on the \(+x\) axis, its distances from microphones 1 and 2 being \(L_{1}\) and \(L_{2}\), respectively. The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). The sound reaches microphone 1 first, and then, 1.46 ms later, it reaches microphone 2 . Find the distances \(L_{1}\) and \(L_{2}\)

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 \(1.6 \mathrm{~s}\) after the first, and the third echo arrives \(1.1 \mathrm{~s}\) after the second. Assuming that the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\) and that there are no reflections of sound from the ground, find the distance between the cliffs.

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 \(20.2 \mathrm{u}\), and that of krypton is \(83.8 \mathrm{u}\). The temperature of the krypton is \(293 \mathrm{~K}\). What is the temperature of the neon?

When an earthquake occurs, two types of sound waves are generated and travel through the earth. The primary, or \(\mathrm{P}\), wave has a speed of about \(8.0 \mathrm{~km} / \mathrm{s}\) and the secondary, or \(\mathrm{S}\), wave has a speed of about \(4.5 \mathrm{~km} / \mathrm{s}\). A seismograph, located some distance away, records the arrival of the \(\mathrm{P}\) 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?

A monatomic ideal gas \((\gamma=1.67)\) is contained within a box whose volume is \(2.5 \mathrm{~m}^{3}\). The pressure of the gas is \(3.5 \times 10^{5} \mathrm{~Pa}\). The total mass of the gas is \(2.3 \mathrm{~kg}\). Find the speed of sound in the gas.

In a mixture of argon (atomic mass \(=39.9 \mathrm{u}\) ) and neon (atomic mass \(=20.2 \mathrm{u}\) ), the speed of sound is \(363 \mathrm{~m} / \mathrm{s}\) at \(3.00 \times 10^{2} \mathrm{~K}\). Assume that both monatomic gases behave as ideal gases. Find the percentage of the atoms that are argon and the percentage that are neon.

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 \(0.580 \mathrm{~s}\) on a day when the air temperature is \(293 \mathrm{~K},\) 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 \(298 \mathrm{~K}\), rather than the calibration temperature of \(293 \mathrm{~K}\), what percentage error would there be in the distance measured by the theodolite?

As a prank, someone drops a water-filled balloon out of a window. The balloon is released from rest at a height of \(10.0 \mathrm{~m}\) 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 \(20^{\circ} \mathrm{C}\) and ignoring the effect of air resistance on the balloon, determine how far above the man's ears this point is.

A typical adult ear has a surface area of \(2.1 \times 10^{-3} \mathrm{~m}^{2} .\) The sound intensity during a normal conversation is about \(3.2 \times 10^{-6} \mathrm{~W} / \mathrm{m}^{2}\) at the listener's ear. Assume that the sound strikes the surface of the ear perpendicularly. How much power is intercepted by the ear?

At a distance of \(3.8 \mathrm{~m}\) from a siren, the sound intensity is \(3.6 \times 10^{-2} \mathrm{~W} / \mathrm{m}^{2}\). Assuming that the siren radiates sound uniformly in all directions, find the total power radiated.

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 \(2.0 \times 10^{-6} \mathrm{~W} / \mathrm{m}^{2}\) at a distance of \(120 \mathrm{~m}\) from the explosion. Find the distance from the source at which the intensity is \(0.80 \times 10^{-6} \mathrm{~W} / \mathrm{m}^{2}\)

Suppose that sound is emitted uniformly in all directions by a public address system. The intensity at a location \(22 \mathrm{~m}\) away from the sound source is \(3.0 \times 10^{-4} \mathrm{~W} / \mathrm{m}^{2}\). What is the intensity at a spot that is \(78 \mathrm{~m}\) away?

A loudspeaker has a circular opening with a radius of \(0.0950 \mathrm{~m} .\) The electrical power needed to operate the speaker is \(25.0 \mathrm{~W}\). The average sound intensity at the opening is \(17.5 \mathrm{~W} / \mathrm{m}^{2}\). What percentage of the electrical power is converted by the speaker into sound power?

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 \(1.8 \mathrm{~cm}\), and it produces a sound intensity of \(5.9 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}\). How much time is required for the transducer to emit \(4800 \mathrm{~J}\) of sound energy?

Review 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 \(2.2 \times 10^{-2} \mathrm{~m}^{2}\). The mass of the lasagna is \(0.35 \mathrm{~kg}\), and its specific heat capacity is \(3200 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\). The temperature rises by \(72 \mathrm{C}^{\circ}\) in \(8.0\) minutes. What is the intensity of the microwaves in the oven?

Review Interactive Solution \(\underline{16.55}\) 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 \(2.2 \times 10^{-2} \mathrm{~m}^{2}\). The mass of the lasagna is \(0.35 \mathrm{~kg},\) and its specific heat capacity is \(3200 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\). The temperature rises by \(72 \mathrm{C}^{\circ}\) in 8.0 minutes. What is the intensity of the microwaves in the oven?

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 \(x=+123 \mathrm{~m}\). 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.

A rocket, starting from rest, travels straight up with an acceleration of \(58.0 \mathrm{~m} / \mathrm{s}^{2}\). When the rocket is at a height of \(562 \mathrm{~m}\), 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 \(\frac{1}{3} I\). Assuming that the speed of sound is \(343 \mathrm{~m} / \mathrm{s},\) find the time that has elapsed between the two measurements.

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 \(7.8 \mathrm{~dB}\) 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?

An amplified guitar has a sound intensity level that is \(14 \mathrm{~dB}\) greater than the same unamplified sound. What is the ratio of the amplified intensity to the unamplified intensity?

A recording engineer works in a soundproofed room that is \(44.0 \mathrm{~dB}\) quieter than the outside. If the sound intensity in the room is \(1.20 \times 10^{-10} \mathrm{~W} / \mathrm{m}^{2}\), what is the intensity outside?

When a person wears a hearing aid, the sound intensity level increases by \(30.0 \mathrm{~dB}\). By what factor does the sound intensity increase?