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 \(3.0 \times 10^{-2}\) 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 whyyour answer is much less than \(1.2 \times 10^{-8} \mathrm{m}\) .

A loud factory machine produces sound having a displace ment amplitude of \(1.00 \mu \mathrm{m},\) but the frequency of this sound can be adjusted. In order to prevent ear damage to the workers, the maximum pressure amplitude of the sound waves is limited to 10.0 \(\mathrm{Pa.}\) Under the conditions of this factory, the bulk modulus of air is \(1.42 \times 10^{5}\) Pa. What is the highest-frequency sound to which this machine can be adjusted without exceeding the prescribed limit? Is this frequency audible to the workers?

BIO UItrasound and Infrasound. (a) Whale communication. Blue whales apparently communicate with each other using sound of frequency 17 \(\mathrm{Hz}\) , which can be heard nearly 1000 \(\mathrm{km}\) away in the ocean. What is the wavelength of such a sound in seawater, where the speed of sound is 1531 \(\mathrm{m} / \mathrm{s} ?\) (b) Dolphin clicks. One type of sound that dolphins emit is a sharp click of wavelength 1.5 \(\mathrm{cm}\) in the ocean. What is the frequency of such clicks? (c) Dog whistles. One brand of dog whistles claims a frequency of 25 \(\mathrm{kH} z\) for its product. What is the wavelength of this sound? (d) Bats. While bats emit a wide variety of sounds, one type emits pulses of sound having a frequency between 39 \(\mathrm{kHz}\) and 78 \(\mathrm{kHz}\) . What is the range of wavelengths of this sound? (e) Sonograms. Ultrasound is used to view the interior of the body, much as \(\mathrm{x}\) rays are utilized. For sharp imagery, the wave length of the sound should be around one-fourth (or less) the size of the objects to be viewed. Approximately what frequency of sound is needed to produce a clear image of a tumor that is 1.0 \(\mathrm{mm}\) across if the speed of sound in the tissue is 1550 \(\mathrm{m} / \mathrm{s} ?\)

(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 oscillator produce sound of wavelength 28.5 \(\mathrm{cm}\) in this gas. What should the gas temperature be to achieve this wavelength?

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 \(2.18 \times 10^{9} \mathrm{Pa}\) . (b) If the pipe is filled with air 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 sound wave in air at \(20^{\circ} \mathrm{C}\) has a frequency of 150 \(\mathrm{Hz}\) and a displacement amplitude of \(5.00 \times 10^{-3} \mathrm{mm} .\) For this sound wave calculate the (a) pressure amplitude (in \(\mathrm{Pa} ) ;\) (b) intensity (in \(\mathrm{W} / \mathrm{m}^{2} ) ;(\mathrm{c})\) sound intensity level (in decibels).

You live on a busy street, but as a music lover, you want to reduce the traffic noise. (a) If you install special sound-reflecting windows that reduce the sound intensity level (in dB) by \(30 \mathrm{dB},\) by what fraction have you lowered the sound intensity (in \(\mathrm{W} / \mathrm{m}^{2} ) ?\) (b) If, instead, you reduce the intensity by half, what change (in dB) do you make in the sound intensity level?

BIO 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 \(6.0 \times 10^{-5}\) 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?

CP A baby's mouth is 30 \(\mathrm{cm}\) from her father's ear and 1.50 \(\mathrm{m}\) from her mother's ear. What is the difference between the sound intensity levels heard by the father and by the mother?

The Sacramento City Council 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?

CP At point \(A, 3.0\) m from a small source of sound that is emitting uniformly in all directions, the sound intensity level is 53 dB. (a) What is the intensity of the sound at \(A ?\) (b) How far from the source must you go so that the intensity is one-fourth of what it was at \(A ?\) (c) How far must you go so that the sound intensity level is one-fourth of what it was at \(A ?\) (d) Does intensity obey the inverse-square law? What about sound intensity level?

(a) If two sounds differ by 5.00 \(\mathrm{dB}\) , find the ratio of the intensity of the louder sound to that of the softer one. (b) If one sound is 100 times as intense as another, by how much do they differ in sound intensity level (in decibels)? (c) If you increase the volume of your stereo so that the intensity doubles, by how much does the sound intensity level increase?

Standing sound waves are produced in a pipe that is 1.20 \(\mathrm{m}\) long. For the fundamental and first two overtones, determine 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.

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

BIO The Human Voice. The human vocal tract is a pipe that extends about 17 \(\mathrm{cm}\) from the lips to the vocal folds (also called "vocal cords") near the middle of your throat. The vocal folds behave rather like the reed of a clarinet, and the vocal tract acts like a stopped pipe. Estimate the first three standing-wave frequencies of the vocal tract. Use \(v=344 \mathrm{m} / \mathrm{s} .\) The answers are only an estimate, since the position of lips and tongue affects the motion of air in the vocal tract.)

BIO The Vocal Tract. Many opera singers (and some pop singers) have a range of about 2\(\frac{1}{2}\) octaves or even greater. Suppose a soprano's range extends from A below middle \(\mathrm{C}\) (frequency 220 \(\mathrm{Hz}\) ) up to \(\mathrm{E}^{\mathrm{b}}\) -flat above high \(\mathrm{C}\) (frequency 1244 \(\mathrm{Hz}\) . Although the vocal tract is quite complicated, we can model it as a resonating air column, like an organ pipe, that is open at the top and closed at the bottom. The column extends from the mouth down to the diaphragm in the chest cavity, and we can also assume that the lowest note is the fundamental. How long is this column of air if \(v=354 \mathrm{m} / \mathrm{s} ?\) Does your result seem reasonable, on the basis of observations of your own body?

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 Shower. 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 emply 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 fundamental standing wave in the air column if the test tube is half filled with water?

CP You have a stopped pipe of adjustable length close to a taut \(85.0-\mathrm{cm}, 7.25-\mathrm{g}\) wire under a tension of 4110 \(\mathrm{N}\) . You want to adjust the length of the pipe so that, when it produces sound at its fundamental frequency, this sound causes the wire to vibrate in its second overtone with very large amplitude. How long should the pipe be?

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 172 Hz. You are 8.00 m from A. What is the closest you can be to \(B\) and be at a point of destructive interference?

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 Hz. Point \(P\) is 12.0 m from \(A\) and 13.4 \(\mathrm{m}\) from \(B\) . Is the interference at \(P\) constructive or destructive? Give the reasoning behind your answer.

Tuning a Violin. A violinist is tuning her instrument to concert \(\mathrm{A}(440 \mathrm{Hz}) .\) She plays the note while listening to an electronically generated tone of exactly that frequency and hears a beat of frequency 3 \(\mathrm{Hz}\) , which increases to 4 \(\mathrm{Hz}\) when she tightens her violin string slightly. (a) What was the frequency of the note played by her violin when she heard the 3 -Hz beat? (b) To get her violin perfectly tuned to concert \(\Lambda,\) should she tighten or loosen her string from what it was when she heard the 3 -Hz beat?

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

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

Adjusting Airplane Motors. The motors that drive airplane propellers are, in some cases, tuned by using beats. The whirring motor produces a sound wave having the same frequency as the propeller. (a) If one single-bladed propeller is turning at 575 \(\mathrm{rpm}\) and you hear a \(2.0-\mathrm{Hz}\) beat when you run the second propeller, what are the two possible frequencies (in rpm) of the second propeller? (b) Suppose you increase the speed of the second propeller slightly and find that the beat frequency changes to 2.1 \(\mathrm{Hz}\) . In part (a). which of the two answers was the correct one for the frequency of the second single-bladed propeller? How do you know?

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?

Two train whistles, \(A\) and \(B\) , each have a frequency of 392 \(\mathrm{Hz} . A\) is stationary and \(B\) is moving toward the right (away from \(A\) ) at a speed of 35.0 \(\mathrm{m} / \mathrm{s}\) . A listener is between the two whistles and is moving toward the right with a speed of 15.0 \(\mathrm{m} / \mathrm{s}\) (Fig. E16.45). No wind is blowing. (a) What is the frequency from A as heard by the listener? (b) What is the frequency from \(B\) as heard by the listener? (c) What is the beat frequency detected by the listener?

A swimming duck paddles the water with its feet once every 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's.

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 emitted 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 tirst at 18.0 \(\mathrm{m} / \mathrm{s}\) and (a) approaching the first and (b) receding from the first?

The siren of a fire engine that is driving northward at 30.0 \(\mathrm{m} / \mathrm{s}\) emits a sound of frequency 2000 \(\mathrm{Hz}\) . A truck in front of this fire engine is moving northward at 20.0 \(\mathrm{m} / \mathrm{s}\) . (a) What is the frequency of the siren's sound the fire engine's driver hears reflected from the back of the truck? (b) What wavelength would this driver measure for these reflected sound waves?

How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive 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.)

Extrasolar Planets. In the not-too-distant future, it should be possible to detect the presence of planets moving around other stars by measuring the Doppler shift in the infrared light they emit. If a planet is going around its star at 50.00 \(\mathrm{km} / \mathrm{s}\) while emitting infrared light of frequency \(3.330 \times 10^{14} \mathrm{Hz},\) what frequency light will be received from this planet when it is moving directly away from us? (Note: Infrared light is light having wavelengths longer than those of visible light.)

A jet plane flies 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.

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 in \(\mathrm{mi} / \mathrm{h} )\) 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 \(\mathrm{m} / \mathrm{s} ?\)

CP 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 \(f_{\text { beat }}=f_{0}(\Delta F / 2 F)\) . (b) Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of 440.0 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?

A soprano und a bass are singing a duet. While the soprano sings an \(A\) -sharp at 932 \(\mathrm{Hz}\) , the bass sings an \(A\) -sharp 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 sound intensity level, what must be (a) the ratio of the pressure 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 \(m\) and in nm \()\) does the soprano produce to sing her A-sharp at 72.0 \(\mathrm{dB} ?\)

CP The sound from a trumpet radiates uniformly in all directions in \(20^{\circ} \mathrm{C}\) air. At a distance of 5.00 \(\mathrm{m}\) from the trumpet the sound intensity level is 52.0 \(\mathrm{dB}\) . The frequency is 587 \(\mathrm{Hz}\) . (a) What is the pressure amplitude at this distance? (b) What is the displacement amplitude? (c) At what distance is the sound intensity level 30.0 \(\mathrm{dB}\) ?

CP A person is playing a small flute 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 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?

Ep A New Musical Instrument. 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 \(v_{\mathrm{s}}\) . (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 calculated 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 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 air 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?

The frequency of the note \(\mathrm{F}_{4}\) is 349 \(\mathrm{Hz} .\) (a) If an organ pipe is open at one end and closed at the other, what length must it have for its fundamental mode to produce this note at \(20.0^{\circ} \mathrm{C}\) (b) At what air temperature will the frequency be 370 \(\mathrm{Hz}\) , corresponding to a rise in pitch from F to F-sharp? (Ignore the change in length of the pipe due to the temperature change.)

Wagnerian Opera. A man marries a great Wagnerian soprano but, alas, he discovers he cannot stand Wagnerian opera. In order to save his eardrums, the unhappy man decides he must silence his larklike wife for good. His plan is to tie her to the front of his car and send car and soprano speeding toward a brick wall. This soprano is quite shrewd, however, having studied physics in her student days at the music conservatory. She realizes that this wall has a resonant frequency of 600 \(\mathrm{Hz}\) , which means that if a continuous sound wave of this frequency hits the wall, it will fall down, and she will be saved to sing more Isoldes. The car is heading toward the wall at a high speed of 30 \(\mathrm{m} / \mathrm{s}\) . (a) At what frequency must the soprano sing so that the will will crumble? (b) What frequency will the soprano hear reflected from the wall just before it crumbles?

A bat flies toward a wall, emitting a steady sound of frequency 1.70 \(\mathrm{kHz}\) . This bat hears its own sound plus the sound reflected by the wall. How fast should the bat fly in order to hear a beat frequency of 10.0 \(\mathrm{Hz}\) ?

BIO Ultrasound in Medicine. A \(2.00-\) MHz sound wave travels through a pregnant woman's abdomen and is reflected from the fetal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beals. The reflected sound is then mixed with the transmitted sound, and 72 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.