Chapter 16

Example 16.1 (Section 16.1\()\) showed that for sound waves in air with frequency 1000 \(\mathrm{Hz}\) , a displacement amplitude of\( 1.2 \times 10^{-8} \mathrm{m}\) produces a pressure amplitude of \(3.0 \times 10^{-2} \mathrm{Pa}\) . Water at \(20^{\circ} \mathrm{C}\) has a bulk modulus of \(2.2 \times 10^{9} \mathrm{Pa}\) , and the speed of sound in water at this temperature is 1480 \(\mathrm{m} / \mathrm{s}\) . For \(1000-\mathrm{Hz}\) sound waves in \(20^{\circ} \mathrm{C}\) water, what displacement amplitude is produced if the pressure amplitude is \(3.0 \times 10^{-2}\) Pa? Explain why your answer is much less than \(1.2 \times 10^{-8} \mathrm{Pa}\) .

(a) In a liquid with density 1300 \(\mathrm{kg} / \mathrm{m}^{3}\) , longitudinal waves with frequency 400 \(\mathrm{Hz}\) are found to have wavelength 8.00 \(\mathrm{m}\) . Calculate the bulk modulus of the liquid. (b) A metal bar with a length of 1.50 \(\mathrm{m}\) has density 6400 \(\mathrm{kg} / \mathrm{m}^{3}\) . Longitudinal sound waves take \(3.90 \times 10^{-4} \mathrm{s}\) to travel from one end of the bar to the other. What is Young's modulus for this metal?

An oscillator vibrating at 1250 Hz produces a sound wave that travels through an ideal gas at 325 \(\mathrm{m} / \mathrm{s}\) when the gas temperature is \(22.0^{\circ} \mathrm{C}\) . For a certain experiment, you need to have the same uscillator produce sound of waveleugth 28.5 \(\mathrm{cm}\) in this gas. What should the gas temperature be to achieve this wavelength?

(a) Show that the fractional change in the speed of sound \((d v / v)\) due to a very small temperature change \(d T\) is given by \(d v / v=\frac{1}{2} d T / T .\) (Hint: Start with Eq. \(16.10 .\) (b) The speed of sound in air at \(20^{\circ} \mathrm{C}\) is found to be 344 \(\mathrm{m} / \mathrm{s}\) . Use the result in part (a) to find the change in the speed of sound for a \(1.0^{\circ} \mathrm{C}\) change in air temperature.

Longitudinal Waves in Different Fluids. (a) A longitudinal wave propagating in a water-filled pipe has intensity \(3.00 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\) and frequency 3400 \(\mathrm{Hz}\) . Find the amplitude \(A\) and wavelength \(\lambda\) of the wave. Water has density 1000 \(\mathrm{kg} / \mathrm{m}^{3}\) and bulk modulus \(2.18 \times 10^{9} \mathrm{Pa}\) . Water has density 1000 \(\mathrm{kg} / \mathrm{m}^{3}\) at pressure \(1.00 \times 10^{5} \mathrm{Pa}\) and density 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) , what will be the amplitude \(A\) and wavelength \(\lambda\) of a longitudinal wave with the same intensity and frequency as in part (a)? (c) In which fluid is the amplitude larger, water or air? What is the ratio of the two amplitudes? Why is this ratio so different from 1.00\(?\)

(a) What is the sound intensity level in a car when the sound intensity is 0.500\(\mu \mathrm{W} / \mathrm{m}^{2} 7\) (b) What is the sound intensity level in the air near a jackhammer when the pressure amplitude of the sound is 0.150 \(\mathrm{Pa}\) and the temperature is \(20.0^{\circ} \mathrm{C}\) ?

For a person with normal hearing, the faintest sound that can be heard at a frequency of 400 \(\mathrm{Hz}\) has a pressure amplitude of about \(6.0 \times 10^{-5} \mathrm{Pa}\) . Calculate the (a) intensity; (b) sound intensity level; (c) displacement amplitude of this sound wave at \(20^{\circ} \mathrm{C}\) .

The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity level than when a single one cries? (b) To increase the sound intensity level again by the same number of decibels as in part (a), how many more crying babies are required?

The Sacramento City Council recently adopted a law to reduce the allowed sound intensity level of the much despised leaf blowers from their current level of about 95 \(\mathrm{dB}\) to 70 \(\mathrm{dB}\) . With the new law, what is the ratio of the new allowed intensity to the previously allowed intensity?

(a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0 dB? (b) Explain why you don't need to know the original sound intensity.

The fundamental frequency of a pipe that is open at both ends is 594 Hz (a) How long is this pipe? If one end is now closed, find (b) the wavelength and (c) the frequency of the new fundamental.

Standing sound waves are produced in a pipe that is 1.20 \(\mathrm{m}\) long. For the fundamental and first two overtones, determine the the locations along the pipe (measured from the left end) of the dis- placement nodes and the pressure nodes if (a) the pipe is open at both ends and (b) the pipe is closed at the left end and open at the right end.

Find the fundamental frequency and the frequency of the first three overtones of a pipe 45.0 \(\mathrm{cm}\) long (a) if the pipe is open at both ends and (b) if the pipe is closed at one end. Use \(v=344 \mathrm{m} / \mathrm{s} .\) (c) For each of these cases, what is the number of the highest harmonic that may be heard by a person who can hear frequencies from 20 \(\mathrm{Hz}\) to \(20,000 \mathrm{Hz}\) ?

A certain pipe produces a fundamental frequency of 262 \(\mathrm{Hz}\) in air. (a) If the pipe is filled with helium at the same temperature, what fundamental frequency does it produce? (The molar mass of air is 28.8 \(\mathrm{g} / \mathrm{mol}\) , and the molar mass of helium is 4.00 \(\mathrm{g} / \mathrm{mol}\) . (b) Does your answer to part (a) depend on whether the pipe is open or stopped? Why or why not?

Singing in the Showen. A pipe closed at both ends can have standing waves inside of it, but you normally don't hear them because little of the sound can get out. But you can hear them if you are inside the pipe, such as someone singing in the shower. (a) Show that the wavelengths of standing waves in a pipe of length \(L\) that is closed at both ends are \(\lambda_{n}=2 L / n\) and the frequencies are given by \(f_{n}=n v / 2 L=n f_{1},\) where \(n=1,2,3, \ldots\) (b) Modeling it as a pipe, find the frequency of the fundamental and the first two overtones for a shower 2.50 \(\mathrm{m}\) tall. Are these frequencies audible?

You blow across the open mouth of an empty test tube and produce the fundamental standing wave of the air column inside the test tube. The speed of sound in air is 344 \(\mathrm{m} / \mathrm{s}\) and the test tube acts as a stopped pipe. (a) If the length of the air column in the test tube is 14.0 \(\mathrm{cm}\) , what is the frequency of this standing wave? (b) What is the frequency of the nfundamental standing wave in the air column if the test tube is half filled with water?

Two loudspeakers, \(A\) and \(B(\mathrm{Fig} .16 .40),\) are driven by the same amplifier and emit sinusoidal waves in phase. Speaker \(B\) is 2.00 \(\mathrm{m}\) to the right of speaker \(A\) . Consider point \(Q\) along the exten- sion of the line connecting the speakers, 1.00 \(\mathrm{m}\) to the right of speaker \(B .\) Both speakers emit sound waves that travel directly from the speaker to point \(Q .\) (a) What is the lowest frequency for which constructive interference occurs at point \(Q ?\) (b) What is the lowest frequency for which destructive interference occurs at point \(Q ?\)

Two loudspeakers, \(A\) and \(B,\) are driven by the same amplifier and emit sinusoidal waves in phase. Speaker \(B\) is 12.0 \(\mathrm{m}\) to the right of speaker \(A\) . The frequency of the waves emitted by each speaker is 688 \(\mathrm{Hz}\) You are standing between the speakers, along the line connecting them, and are at a point of constructive interference. How far must you walk toward speaker \(B\) to move to a point of destructive interference? effects like those in parts (a) and (b) are almost never a factor in listening to home stereo equipment. Why not?

Two loudspeakers, \(A\) and \(B,\) are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is 860 \(\mathrm{Hz}\) Point \(P\) is 12.0 \(\mathrm{m}\) from \(A\) and 13.4 \(\mathrm{m}\) from \(B .\) Is the interference at \(P\) constructive or destructive? Give the reasoning behind your answer.

Two organ pipes, open at one end but closed at the other, are each 1.14 \(\mathrm{m}\) long. One is now lengthened by 2.00 \(\mathrm{cm}\) . Find the frequency of the beat they produce when playing together in theifundamcntal.

Two identical taut strings under the same tension \(F\) produce a note of the same fundamental frequency \(f_{0}\) . The tension in one of them is now increased by a very small amount \(\Delta F\) . (a) If they are played together in their fundamental, show that the frequency of the beat produced is \(f_{\text { keet }}=f_{0}(\Delta F / 2 F)\) . (b) Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of 440.0 \(\mathrm{Hz}\) . One of the strings is retuned by increasing its tension. When this is done, 1.5 beats per second are heard when both strings are plucked simultaneously at their centers. By what percentage was the string tension changed?

On the planet Arrakis a male ornithoid is flying toward his mate at 25.0 \(\mathrm{m} / \mathrm{s}\) while singing at a frequency of 1200 \(\mathrm{Hz}\) . If the stationary female hears a tone of 1240 \(\mathrm{Hz}\) , what is the speed of sound in the atmosphere of Arrakis?

A swimming duck paddles the water with its feet once cvery 1.6 \(\mathrm{s}\) , producing surface waves with this period. The duck is moving at constant speed in a pond where the speed of surface waves is 0.32 \(\mathrm{m} / \mathrm{s}\) , and the crests of the waves ahead of the duck are spaced 0.12 \(\mathrm{m}\) apart. (a) What is the duck's speed? (b) How far apart are the crests behind the duck?

Moving Source vs. Moving Listener. (a) A sound source producing \(1.00-\mathrm{kHz}\) waves moves toward a stationary listener at one-half the speed of sound. What frequency will the listener hear? (b) Suppose instead that the source is stationary and the listener moves toward the source at one- half the speed of sound. What frequency does the listener hear? How does your answer compare to that in part (a)? Explain on physical grounds why the two answers differ.

A car alarm is emitting sound waves of frequency 520 \(\mathrm{Hz}\) You are on a motorcycle, traveling directly away from the car. How fast must you be traveling if you detect a frequency of 490 \(\mathrm{Hz}\) ?

A railroad train is traveling at 30.0 \(\mathrm{m} / \mathrm{s}\) in still air. The frequency of the note cmitted by the train whistle is 262 \(\mathrm{Hz}\) . What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 \(\mathrm{m} / \mathrm{s}\) and (a) approaching the first; and (b) receding from the first?

Doppler Radar. A giant thunderstorm is moving toward a weather station at \(45.0 \mathrm{mi} / \mathrm{h}(20.1 \mathrm{m} / \mathrm{s}) .\) If the station sends a radar beam of frequency 200.0 \(\mathrm{MHz}\) toward the storm, what is the difference in frequency between the emitted beam and the beam reflected back from the storm? Be careful to carry plenty of significant figures! (Hint The storm reflects the same frequency that it receives.)

How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we recelve from it is 10.0\(\%\) higher than the frequency of the light it is emitting? Would it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)

The shock-wave cone created by the space shuttle at one instant during its reentry into the atmosphere makes an angle of \(58.0^{\circ}\) with its direction of motion. The speed of sound at this altitude is 331 \(\mathrm{m} / \mathrm{s}\) (a) What is the Mach number of the shuttle at this instant, and (b) how fast (in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{mi} / \mathrm{h} )\) is it traveling relative to the atmosphere? (c) What would be its Mach number and the angle of its shock-wave cone if it flew at the same speed but at low altitude where the speed of sound is 344 \(\mathrm{m} / \mathrm{s} ?\)

A jet plane flics overhead at Mach 1.70 and at a constant altitude of 950 \(\mathrm{m}\) (a) What is the angle \(\alpha\) of the shock-wave cone? (b) How much time after the plane passes directly overhead do you hear the sonic boom? Neglect the variation of the speed of sound with altitude.

A soprano and a bass are singing a duet. While the soprano \(\operatorname{sings}\) an \(A^{\prime \prime}\) at 932 . Hz, the bass sings an \(A^{\prime \prime}\) but three octaves lower. In this concert hall, the density of air is 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) and its bulk modulus is \(1.42 \times 10^{5} \mathrm{Pa}\) . In order for their notes to have the same Rnund intensity level, what must he (a) the ratio of the pressur amplitude of the bass to that of the soprano, and (b) the ratio of the displacement amplitude of the bass to that of the soprano? (c) What displacement amplitude (in \(\mathrm{m}\) and \(\mathrm{nm}\) ) does the soprano produce to sing her \(A^{\prime \prime}\) at 72.0 \(\mathrm{dB} ?\)

A person is playing a small fute 10.75 \(\mathrm{cm}\) long, open at one end and closed at the other, near a taut string having a fundamental frequency of 600.0 \(\mathrm{Hz}\) . If the speed of sound is 344.0 \(\mathrm{m} / \mathrm{s}\) , for which harmonics of the flute will the string resonate? In each case, which harmonic of the string is in resonance?

A New Musical Tnstrument. You have designed a new musical instrument of very simple construction. Your design consists of a metal tube with length \(L\) and diameter \(L / 10\) . You have stretched a string of mass per unit length \(\mu\) across the open end of the tube. The other end of the tube is closed. To produce the musical effect you're looking for, you want the frequency of the third-harmonic standing wave on the string to be the same as the fundamental frequency for sound waves in the air column in the tube. The speed of sound waves in this air column is \(v_{v}\) (a) What must be the tension of the string to produce the desired effect? (b) What happens to the sound produced by the instrument if the tension is changed to twice the value calculatod in part (a)?(c) For the tension calculated in part (a), what other harmonics of the string, if any, are in resonance with standing waves in the air column?

An organ pipe has two successive harmonics with frequencies 1372 and 1764 \(\mathrm{Hz}\) (a) Is this an open or a stopped pipe? Explain. (b) What two harmonics are these? (c) What is the length of the pipe?

(a) Determine the first three normal-mode frequencies for a pipe of length \(L\) that is closed at both ends. Explain your reasoning. (b) Use the results of part (a) to estimate the normal-mode frequencies of a shower stall. Explain the connectiva between these frequencies and the observation that your singing voice probably sounds better in the shower, especially when you sing at certain frequencies.

One type of steel has a density of \(7.8 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}\) and a breaking stress of \(7.0 \times 10^{3} \mathrm{N} / \mathrm{m}^{2}\) . A cylindrical guitar string is to be made of 4.00 \(\mathrm{g}\) of this steel. (a) What are the length and radius of the longest and thinnest string that can be placed under a tension of 900 \(\mathrm{N}\) without breaking? (b) What is the highest fundamental frequency that this string could have?

A long tube contains air at a pressure of 1.00 atm and a temperature of \(77.0^{\circ} \mathrm{C}\) . The tube is open at one end and closed at the other by a movable piston. A tuning fork near the open end is vibrating with a frequency of 500 \(\mathrm{Hz}\) . Resonance is produced when the piston is at distances \(18.0,55.5,\) and 93.0 \(\mathrm{cm}\) from the open end. (a) From these measurements, what is the speed of sound in alr at \(77.0^{\circ} \mathrm{C} ?\) (b) From the result of part (a), what is the value of \(\gamma ?\) (c) These data show that a displacement antinode is slightly outside of the open end of the tube. How far outside is it?

A standing wave with a frequency of 1100 \(\mathrm{Hz}\) in a column of methane \(\left(\mathrm{CH}_{4}\right)\) at \(20.0^{\circ} \mathrm{C}\) produces nodes that are 0.200 \(\mathrm{m}\) apart. What is the value of \(\gamma\) for methane? (The molar mass of methane is 16.0 \(\mathrm{g} / \mathrm{mol}\) )

Two identical loudspcakers are located at points \(A\) and \(B\) , 2.00 \(\mathrm{m}\) apart. The loudspeakers are driven by the same amplifier and produce sound waves with a frequency of 784 \(\mathrm{Hz}\) . Take the speed of sound in air to be 344 \(\mathrm{m} / \mathrm{s}\) . A small microphone is moved out from point \(B\) along a line perpendicular to the line connecting \(A\) and \(B(\text { line } B C \text { in Fig. } 16.44)\) (a) At what distances from \(B\) will there be destructive interference? (b) At what distances from \(B\) will there be constructive interference? (c) If the frequency is made low enough, there will be no positions along the line \(B C\) at which destructive interference occurs. How low must the frequency be for this to be the case?

A small sphere of radius \(R\) is arranged to pulsate so that its radius varies in simple harmonic motion between a minimum of \(R-\Delta R\) and a maximum of \(R+\Delta R\) with frequency \(f .\) This produces sound waves in the surrounding air of density \(\rho\) and bulk modulus \(B\) (a) Find the intensity of sound waves at the surface of the sphere. (The amplitude of oscillation of the sphere is the same as that of the air at the surface of the sphere.) (b) Find the total acoustic powcr radiated by the sphcrc. (c) At a distance \(d \gg R\) from the center of the sphere, find the amplitude, pressure amplitude, and intensity of the sound wave.

Ultrasound in Medicine. \(\quad\) A \(2.00-\mathrm{MHz}\) sound wave travels through a pregnant woman's abdomen and is retiected from the fotal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and 85 beats per second are detected. The speed of sound in body tissue is 1500 \(\mathrm{m} / \mathrm{s}\) . Calculate the speed of the fetal heart wall at the instant this measurement is made.

The sound source of a ship's sonar system operates at a frequency of 22.0 \(\mathrm{kH}_{2}\) . The speed of sound in water (assumed to be at a uniform \(20^{\circ} \mathrm{C} )\) is 1482 \(\mathrm{m} / \mathrm{s}\) (a) What is the wavelength of the waves cmitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling directly toward the slip at 4.95 \(\mathrm{m} / \mathrm{s} ?\) The ship is at rest in the water.

A police siren of frequency firm is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude \(A_{p}\) and frequency \(f_{p}\) (a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.

Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey's speed. (The "horseshoe" that gives the hat its name is a depression around the nostrils that acts like a focusing mirror, so that the hat emits sound in a narrow beam like a flashlight.) A Rhinolophus flying at speed \(v_{\text { tot }}\) emits sound of fre-quency \(f_{\text { but }}\) ; the sound it hears reflected from an insect flying toward it has a higher frequency \(f_{\text { rent }}(\text { a) Show that the speed of the insect is }\) where \(v\) is the speed of sound. (b) If \(f_{\mathrm{bat}}=80.7 \mathrm{kHz}, \quad f_{\mathrm{rell}}=\) \(83.5 \mathrm{kHz},\) and \(v_{\mathrm{bat}}=3.9 \mathrm{m} / \mathrm{s},\) calculate the speed of the insect.

A woman stands at rest in front of a large, smooth wall. She holds a vibrating tuning fork of frequency \(f_{0}\) directly in front of her (between her and the wall). (a) The woman now runs toward the wall with speed \(v_{\mathrm{W}}\) . She detects beats due to the interference between the sound waves reaching her directly from the fork and those reaching her after being reflected from the wall. How many beats per second will she detect? (Note: If the beat frequency is too large, the woman may have to use some instrumentation other than her ears to detect and count the beats.) (b) If the woman instead runs away from the wall, holding the tuning fork at her back so it is between her and the wall, how many beats per second will she detect?

Two loudspeakers, \(A\) and \(B,\) radiate sound uniformly in all directions in air at \(20^{\circ} \mathrm{C}\) . The acoustic power output from \(A\) is \(8.00 \times 10^{-4} \mathrm{W}\) , and from \(B\) it is \(6.00 \times 10^{-5} \mathrm{W}\) . Both loudspeakcrs are vibrating in phase at a frequency of 172 \(\mathrm{Hz}\) (a) Determine the difference in phase of the two signals at a point \(C\) along the line joining \(A\) and \(B, 3.00 \mathrm{m}\) from \(B\) and 4.00 \(\mathrm{m}\) from \(A(\mathrm{Fig} .16 .46)\) . (b) Determine the intensity and sound intensity level at \(C\) from speaker \(A\) if speaker \(B\) is turned off and the intensity and sound intensity level at point \(C\) from speaker \(B\) if speaker \(A\) is turned off. (c) With both speakers on, what are the intensity and sound intensity level at \(C ?\)