Waves/Acoustics

Question: Energy Delivered to the Ear. Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate. Typically, the diameter of this membrane is about 8.4 mm in humans. (a) How much energy is delivered to the eardrum each second when someone whispers (20 dB) a secret in your ear? (b) To comprehend how sensitive the ear is to very small amounts of energy, calculate how fast a typical 2.0-mg mosquito would have to fly (in mm/s) to have this amount of kinetic energy.

Question: A 60.0-m-long brass rod is struck at one end. A person at the other end hears two sounds as a result of two longitudinal waves, one traveling in the metal rod and the other traveling in air. What is the time interval between the two sounds? (The speed of sound in air is 344 m/s; see Tables 11.1 and 12.1 for relevant information about brass.)

Question: What must be the stress (F/A) in a stretched wire of a material whose Young’s modulus is Y for the speed of longitudinal waves to equal 30 times the speed of transverse waves?

Question: (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.

Question: (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.

Eavesdropping! You are trying to overhear a juicy conversation, but from your distance of 15.0 m, it sounds like only an average whisper of 20.0 db. How close should you move to the chatterboxes for the sound level to be 60.0 dB?

BIO Human Hearing. A fan at a rock concert is 30 m from the stage, and at this point the sound intensity level is 110 dB (a) How much energy is transferred to her eardrums each second? (b) How fast would a 2.0-mg mosquito have to fly (in mm/s) to have this much kinetic energy? Compare the mosquito’s speed with that found for the whisper in part (a) of Exercise 16.13.

A sound wave in air at 20°C has a frequency of 320 Hz and a displacement amplitude of $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{mm}$ For this sound wave calculate the (a) pressure amplitude (in Pa); (b) intensity $\mathbf{W}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$(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 dB, by what fraction have you lowered the sound intensity $\mathbf{(}\mathbf{W}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}\mathbf{)}\mathbf{?}$ (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 Hz has a pressure amplitude of about $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{Pa}$. Calculate the (a) intensity; (b) sound intensity level; (c) displacement amplitude of this sound wave at 20°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?

A baby’s mouth is 30 cm from her father’s ear and 1.50 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 dB to 70 db. With the new law, what is the ratio of the new allowed intensity to the previously allowed intensity?

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 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?

Question: Standing sound waves are produced in a pipe that is 1.20 m long. For the fundamental and first two overtones, determine the locations along the pipe (measured from the left end) of the displacement 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 524 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 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 m/s. (The answers are only an estimate, since the position of lips and tongue affects the motion of air in the vocal tract.)

The Vocal Tract. Many opera singers (and some pop singers) have a range of about ${\mathbf{2}}^{\mathbf{1}\mathbf{/}\mathbf{2}}$ octaves or even greater. Suppose a soprano’s range extends from A below middle C (frequency 220 Hz) up to E-flat above high C (frequency 1244 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 m/s? Does your result seem reasonable, on the basis of observations of your own body?

The longest pipe found in most medium-size pipe organs is 4.88 m (16 ft) long. What is the frequency of the note corresponding to the fundamental mode if the pipe is (a) open at both ends, (b) open at one end and closed at the other?

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 ${\mathbf{\lambda }}_{\mathbf{0}}\mathbf{=}\mathbf{2}\mathbf{L}\mathbf{/}\mathbf{n}$ and the frequencies are given by ${\mathbf{f}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{nv}}{\mathbf{4}\mathbf{L}}{\mathbf{nf}}_{\mathbf{1}}$, where n = 1, 2, 3, c.(b) Modelling it as a pipe, find the frequency of fundamental and the first two overtones for a shower 2.50 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 m/s and the test tube act as a stopped pipe. (a) If the length of the air column in the test tube is 14.0 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?

You have a stopped pipe of adjustable length close to a taut 62.0-cm, 7.25-g wire under a tension of 4110 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?

A 75.0-cm-long wire of mass 5.625 g is tied at both ends and adjusted to a tension of 35.0 N. When it is vibrating in its second overtone, find (a) the frequency and wavelength at which it is vibrating and (b) the frequency and wavelength of the sound waves it is producing.

Small speakers A and B are driven in phase at 725 Hz by the same audio oscillator. Both speakers start out 4.50 m from the listener, but speaker A is slowly moved away (Fig. E16.34). (a) At what distance d will the sound from the speakers first produce destructive interference at the listener’s location? (b) If A is moved even farther away than in part (a), at what distance d will the speakers next produce destructive interference at the listener’s

Two loudspeakers, A and B (Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q. What is the lowest frequency for which (a) constructive interference occurs at point Q; (b) destructive interference occurs at point Q?

Two loudspeakers, A and B (see Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. The frequency of the sound waves produced by the loudspeakers is 206 Hz. Consider a point P between the speakers and along the line connecting them, a distance x to the right of A. Both speakers emit sound waves that travel directly from the speaker to point P. For what values of x will (a) destructive interference occur at P; (b) constructive interference occur at P? (c) 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. Speaker B is 12.0 m to the right of speaker A. The frequency of the waves emitted by each speaker is 688 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?

 Two loudspeakers, $\mathit{A}$ and  $\mathbf{B}$, are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is $\mathbf{172}\mathbf{Hz}$. You are $\mathbf{8}\mathbf{.}\mathbf{00}\text{\hspace{0.17em}}\mathbf{m}$ from $\mathit{A}$ . What is the closest you can be to $\mathit{B}$ and be at a point of destructive interference?

Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone. For what frequencies does their sound at the speakers produce (a) constructive interference and (b) destructive interference?

Two guitarists attempt to play the same note of wavelength $\mathbf{64}\mathbf{.}\mathbf{8}\mathbf{cm}$   at the same time, but one of the instruments is slightly out of tune and plays a note of wavelength  $\mathbf{65}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{cm}$ instead. What is the frequency of the beats these musicians hear when they play together?

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

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 575rpm and you hear 2Hz beats 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 2.1Hz. 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?

Two organ pipes, open at one end but closed at the other, are each $\mathbf{1}\mathbf{.}\mathbf{14}\mathbf{m}$ long. One is now lengthened by  $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{cm}$ . Find the beat frequency that they produce when playing together in their fundamentals.

In Example $\mathbf{16}\mathbf{.}\mathbf{18}$  (Section $\mathbf{16}\mathbf{.}\mathbf{8}$ ), suppose the police car is moving away from the warehouse at  $\mathbf{20}\mathbf{m}/\mathbf{s}$ . What frequency does the driver of the police car hear reflected from the warehouse?

On the planet Arrakis a male ornithoid is flying toward his mate at $\mathbf{25}\mathbf{m}/\mathbf{s}$ while singing at a frequency of  $\mathbf{1200}\mathbf{Hz}$ . If the stationary female hears a tone of  $\mathbf{1240}\mathbf{Hz}$, what is the speed of sound in the atmosphere of Arrakis?

A railroad train is traveling at 25m/s in still air. The frequency of the note emitted by the locomotive whistle is 400Hz. What is the wavelength of the sound waves (a) in front of the locomotive and (b) behind the locomotive? What is the frequency of the sound heard by a stationary listener (c) in front of the locomotive and (d) behind the locomotive?

Two train whistles, A and B, each have a frequency of 393Hz. A is stationary and B is moving toward the right (away from A) at a speed of 35m/s. A listener is between the two whistles and is moving toward the right with a speed of 15m/s. 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) A sound source producing 1.00-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 swimming duck paddles the water with its feet once every 1.6 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 m/s, and the crests of the waves ahead of the duck are spaced 0.12 m apart. (a) What is the duck’s speed? (b) How far apart are the crests behind the duck?

A railroad train is traveling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 352 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m/s and (a) approaching the first and (b) receding from the first?

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

While sitting in your car by the side of a country road, you are approached by your friend, who happens to be in an identical car. You blow your car’s horn, which has a frequency of 260 Hz. Your friend blows his car’s horn, which is identical to yours, and you hear a beat frequency of 6.0 Hz. How fast is your friend approaching you?

Two swift canaries fly toward each other, each moving at 15.0 m/s relative to the ground, each warbling a note of frequency 1750 Hz. (a) What frequency note does each bird hear from the other one? (b) What wavelength will each canary measure for the note from the other one?

The siren of a fire engine that is driving northward at 30.0 m/s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m/s. (a) What is the frequency of the siren’s sound that 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?

A stationary police car emits a sound of frequency 1200 Hz that bounces off a car on the highway and returns with a frequency of 1250 Hz. The police car is right next to the highway, so the moving car is traveling directly toward or away from it. (a) How fast was the moving car going? Was it moving toward or away from the police car? (b) What frequency would the police car have received if it had been traveling toward the other car at 20.0 m/s?

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.)

A jet plane flies overhead at Mach 1.70 and at a constant altitude of 1250 m. (a) What is the angle $\mathit{\sigma }$ 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 a space shuttle at one instant during its re-entry into the atmosphere makes an angle of 58.0° with its direction of motion. The speed of sound at this altitude is 331 m/s. (a) What is the Mach number of the shuttle at this instant, and (b) how fast (in m/s and in mi/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 m/s?

A soprano and a bass are singing a duet. While the soprano sings an A-sharp at 932 Hz, the bass sings an A-sharp but three octaves lower. In this concert hall, the density of air is 1.20 kg/m³ and its bulk modulus is $\mathbf{1}\mathbf{.}\mathbf{42}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{ }\mathbf{Pa}$. 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 dB?