Solved: What did Fourier discover about complex periodic

Problem 1E Your friend says that frequency is a quantitative measure of pitch. Do you agree or disagree?

Problem 45E Make up a multiple-choice question that distinguishes between any of the terms listed in the Summary of Terms.

Problem 1P The highest frequency humans can hear is about 20,000 Hz. What is the wavelength of sound in air at this frequency? What is the wavelength of the lowest sounds we can hear, about 20 Hz?

Problem ?1R Consider three notes: A, 220 Hz; B, 440 Hz; and C, 660 Hz. Rank them from highest to lowest for a. pitch. b. frequency. c. wavelength.

Problem 1RQ Distinguish between noise and music.

Problem 2E As the pitch of sound increases, what happens to the frequency?

Problem 2P A violin string playing the note “A” oscillates at 440 Hz. What is the period of the string’s oscillation?

Problem 2R You blow across the mouths of identical bottles A, B, and C, each containing a different amount of water, as shown. From highest to lowest, rank the pitch of sound for each.

Problem 2RQ What are the three principal characteristics of musical tones?

Problem 3E Your classmate says that timbre refers to the quality of a steady musical sound composed of a mixture of harmonics of different intensities. Do you agree or disagree?

The string of a cello playing the note C oscillates at 264 Hz. What is the period of the string’s oscillation?

Problem 3RQ How does a high-pitch musical note compare with a low one in terms of frequency?

Problem 4E The yellow-green light emitted by streetlights matches the yellow-green color to which the human eye is most sensitive. Consequently, such a 100-W streetlight emits light that is better seen at night. Similarly, the monitored sound intensities of TV commercials are louder than the sound of regular programming, yet don’t exceed the regulated intensities. At what frequencies do advertisers concentrate the commercial’s sound?

Problem 4P How much more intense than the threshold of hearing is a sound of 10 dB? 30 dB? 60 dB?

Problem 4RQ How does the highest pitch one can hear vary with age?

Why should guitars be played offstage before they are brought onstage for a concert? (Hint: Think thermally.)

Problem 5P How much more intense is sound at 40 dB than sound at 0 dB?

Problem 5RQ What is a decibel, and how many decibels correspond to the lowest-intensity sound we can hear?

Problem 6E A guitar and a flute are in tune with each other. Explain how a change in temperature could alter this situation.

Problem 6P How much more intense is a sound of 40 dB than a sound of 30 dB?

Problem 6RQ Is the sound of 30 dB 30 times greater than the threshold of hearing, or 103 (a thousand) times greater?

Problem 7E If sound becomes louder, which wave characteristic is likely increasing—frequency, wavelength, amplitude, or speed?

Problem 7P A certain note has a frequency of 1000 Hz. What is the frequency of a note one octave above it? Two octaves above it? One octave below it? Two octaves below it?

Problem 7RQ Distinguish between sound intensity and loudness.

Problem 8E Explain how you can lower the pitch of a note on a guitar by altering the string’s (a) length, (b) tension, or (c) thickness or mass.

Problem 8P Starting with a fundamental tone, how many harmonics are between the first and second octaves? Between the second and third octaves? (See Figure 21.5 to get started.)

Problem 8RQ How do the loudest sounds we can tolerate compare with the faintest sounds?

Problem 9E Does the pitch of a note depend on sound frequency, loudness, quality, or on all of these?

Problem 9RQ What is it that determines the pitch of a note?

Problem 10E When a guitar string is struck, a standing wave is produced that causes the sounding board to oscillate with a large sustained amplitude, pushing back and forth against the surrounding air to generate sound. How does the frequency of the resulting sound compare with the frequency of the standing wave in the string?

Problem 10RQ If the fundamental frequency of a note is 200 Hz, what is the frequency of the second harmonic? The third harmonic?

Problem 11E The strings on a harp are of different lengths and produce different notes. How are different notes produced on a guitar, where all strings have the same length?

Problem 11RQ What exactly determines the musical quality of a note?

Problem 12E If a vibrating string is made shorter (as by holding a finger on it), how does this affect the frequency of vibration and pitch?

Problem 12RQ Why do the same notes plucked on a banjo and on a guitar have distinctly different sounds?

Problem 13E A nylon guitar string vibrates in a standing-wave pattern, as shown below. What is the wavelength of the wave?

What are the three principal classes of musical instruments?

Problem 14E Why do tuning forks with long tines vibrate at a lower frequency than short-tined forks? (? int:?This question could have been asked back in Chapter 8.)

Problem 14RQ Why do orchestras generally have a greater number of stringed instruments than wind instruments?

Problem 15E Why is the thickness greater for the bass strings of a guitar than for the treble strings?

Problem 15RQ What did Fourier discover about complex periodic wave patterns?

Problem 16E Will the thicker or thinner of two guitar strings of the same tension and length vibrate at the higher frequency?

A high-fidelity sound system may have a frequency range that extends up to or beyond 20,000 Hz. What is the purpose of this extended range?

Problem 17E Why does a vibrating guitar string not sound as loud when it is mounted on a work bench as it does when mounted on the guitar?

Problem 17RQ How was the sound signal captured on phonograph records of the 20th century? How is the sound signal captured on a CD?

Problem 18E Would a plucked guitar string vibrate for a longer time or a shorter time if the instrument had no sounding board? Why?

Problem 18RQ Why does the use of blue light allow more information on a DVD?

Problem 19E If you very lightly touch a guitar string at its midpoint, you can hear a tone that is one octave above the fundamental for that string. (An octave is a factor of 2 in frequency.) Explain.

Problem 20E If a guitar string vibrates in two segments, where can a tiny piece of folded paper be supported without flying off? How many pieces of folded paper could similarly be supported if the waveform were of three segments?

Problem 21E Your classmate says that a harmonic series of frequencies includes the fundamental frequency and integral multiples of the fundamental frequency. Do you agree or disagree?

Problem 22E Why do the same notes on a trumpet and on a saxophone sound different when both are played with the same pitch and loudness?

Problem 23E The amplitude of a transverse wave in a stretched string is the maximum displacement of the string from its equilibrium position. What does the amplitude of a longitudinal sound wave in air correspond to?

Problem 24E Which of the two musical notes displayed one at a time on an oscilloscope screen has the higher pitch?

Problem 25E In the oscilloscopes shown above, which screen displays the louder sound (assuming detection by equivalent microphones)?

Problem 26E Tom Senior makes music by setting small columns of air into vibration by blowing across the ends of drinking straws of various lengths. Which straws, the short ones or the long ones, produce lower pitch? What would you expect of the pitch produced by the much larger musical instrument behind Tom that uses resonant air columns excited by striking the ends of the tubes with paddles?

Which is a more objective measurement—sound intensity or loudness? Defend your answer.

Problem 28E One person has a threshold of hearing of 5 dB, and another of 10 dB. Which person has the more acute hearing?

Problem 29E How is an electronic organ able to imitate the sounds of various musical instruments?

Problem 30E After inhaling helium gas, a person talks with a high-pitched voice. One of the reasons for this is the higher speed of sound in helium than in air. Why does sound travel faster in helium?

Problem 31E Why does your voice sound fuller in the shower?

Problem 32E The frequency range for a telephone is between 500 Hz and 4000 Hz. Why does a telephone not do a very good job of transmitting music?

Problem 33E How many octaves does normal human hearing span? How many octaves are on a common piano keyboard? (If you’re not sure, look and see.)

The note middle C on a piano has a fundamental frequency of about 262 Hz. What is the frequency of the second harmonic of this note?

Problem 35E If the fundamental frequency of a guitar string is 220 Hz, what is the frequency of the second harmonic? Of the third harmonic?

Problem 36E If the fundamental frequency of a violin string is 440 Hz, what is the frequency of the second harmonic? Of the third harmonic?

Problem 37E How many nodes, not including the endpoints, are in a standing wave three wavelengths long? How many nodes are in a standing wave four wavelengths long?

Problem 38E How can you tune the note A3 on a piano to its proper frequency of 220 Hz with the aid of a tuning fork whose frequency is 440 Hz?

Problem 39E At an outdoor concert, the pitch of musical tones is ?not? affected on a windy day. Explain.

Problem 40E A trumpet has keys and valves that permit the trumpeter to change the length of the vibrating air column and the position of the nodes. A bugle has no such keys and valves, yet it can sound various notes. How do you think the bugler achieves different notes?

The human ear is sometimes called a Fourier analyzer. What does this mean, and why is it an appropriate description?

The width of a laser beam was and is significant in reading CDs and DVDs. The thinner the beam, the closer the series of pits can be. Why will blue laser light allow closer pits than red laser light?

Do all the people in a group hear the same music when they listen attentively as in Figure 21.14? Do all see the same sight when viewing a painting? Do all taste the same flavor when sipping the same wine? Do all perceive the same aroma when smelling the same perfume? Do all feel the same texture when touching the same fabric? Do all come to the same conclusion when listening to a logical presentation of ideas?

Problem 44E Why is it a safe prediction that you, presently rending this, will have a significantly greater toss of hearing in your later years than your grandparents experienced?

Distinguish between noise and music.

How does the highest pitch one can hear vary with age?

How does a high-pitch musical note relate to frequency?

What does a decibel measure?

How many decibels correspond to the lowest-intensity sound we can hear?

Is the sound of 30 dB 30 times greater than the threshold of hearing, or \(10^3\) (a thousand) times greater? Text Transcription: 10^3

Between sound intensity and loudness, which is more subjective?

How do the loudest sounds we can tolerate compare with the faintest sounds?

If the fundamental frequency of a note is 200 Hz, what is the frequency of the second harmonic?

What is meant by the fundamental frequency of a musical note?

If the fundamental frequency of a note is 200 Hz, what is the frequency of the third harmonic?

What determines the musical quality of a note?

Why do the same notes plucked on a banjo and on a guitar have distinctly different sounds?

What are the three principal classes of musical instruments?

Why do orchestras generally have a greater number of stringed instruments than wind instruments?

What did Fourier discover about complex periodic wave patterns?

A high-fidelity sound system may have a frequency range that extends up to or beyond 20,000 Hz. What is the purpose of this extended range?

Distinguish between analog and digital for how the sound signal is captured on phonograph records versus a CD.

Why does the use of blue light allow more information on a DVD?

The highest frequency humans can hear is about 20,000 Hz. What is the wavelength of sound in room temperature air at this frequency? What is the wavelength of the lowest sounds we can hear, about 20 Hz?

A violin string playing the note A oscillates at 440 Hz. What is the period of the string’s oscillation?

The string of a cello playing the note C oscillates at 264 Hz. What is the period of the string’s oscillation?

How much more intense than the threshold of hearing is a sound of 10 dB? 30 dB? 60 dB?

How much more intense is sound at 40 dB than sound at 0 dB?

How much more intense is a sound of 40 dB than a sound of 30 dB?

A certain note has a frequency of 1000 Hz. What is the frequency of a note one octave above it? Two octaves above it? One octave below it? Two octaves below it?

Starting with a fundamental tone, how many harmonics are between the first and second octaves? The second and third octaves? (See Figure 21.5 to get started.)

Consider three notes: 220 Hz, 440 Hz, and 660 Hz. Rank them from highest to lowest for a. pitch. b. frequency. c. wavelength.

Your friend says that frequency is a quantitative measure of pitch. Do you agree or disagree?

You blow across the mouths of identical bottles A, B, and C, each containing a different amount of water, as shown. From highest to lowest, rank the pitch of sound for each.

As the pitch of sound gets higher, what happens to the frequency?

Your classmate says that timbre refers to the quality of a steady musical sound composed of a mixture of harmonics of different intensities. Do you agree or disagree?

Why should guitars be played offstage before they are brought onstage for a concert? (Hint: Think thermally.)

If sound becomes louder, which wave characteristic is likely increasing: frequency, wavelength, amplitude, or speed?

Explain how you can lower the pitch of a note on a guitar by altering the string’s (a) length, (b) tension, or (c) thickness or mass.

Does the pitch of a note depend on sound frequency, loudness, quality, or all of these?

When a guitar string is struck, a standing wave is produced that causes the sounding board to oscillate with a large sustained amplitude, pushing back and forth against the surrounding air to generate sound. How does the frequency of the resulting sound compare with the frequency of the standing wave in the string?

If a vibrating string is made shorter (as by holding a finger on it), how does this affect the frequency of vibration and pitch?

The strings on a harp are of different lengths and produce different notes. How are different notes produced on a guitar, where all strings have the same length?

A nylon guitar string vibrates in a standing-wave pattern, as shown in the figure. What is the wavelength of the wave?

Why do tuning forks with long tines vibrate at a lower frequency than short-tined forks? (Hint: This question could have been asked back in Chapter 8.)

Why are the bass strings of a guitar thicker than the treble strings?

Will the thicker or thinner of two guitar strings of the same tension and length vibrate at the higher frequency?

Why doesn’t a vibrating guitar string sound as loud when it is mounted on a workbench as it does when mounted on the guitar?

Would a plucked guitar string vibrate for a longer time or a shorter time if the instrument had no sounding board? Why?

If you very lightly touch a vibrating guitar string at its midpoint, you can hear a tone that is one octave above the fundamental for that string. (An octave is a factor of 2 in frequency.) Explain.

If a guitar string vibrates in two segments, where can a tiny piece of folded paper be supported without flying off? How many pieces of folded paper could similarly be supported if the waveform were of three segments?

Why do the same notes on a trumpet and on a saxophone sound different when both are played with the same pitch and loudness?

The amplitude of a transverse wave in a stretched string is the maximum displacement of the string from its equilibrium position. What does the amplitude of a longitudinal sound wave in air correspond to?

Which of the two musical notes displayed one at a time on an oscilloscope screen has the higher pitch?

In the oscilloscopes shown in the preceding question, which screen displays the louder sound (assuming detection by equivalent microphones)?

Tom Senior makes music by setting small columns of air into vibration by blowing across the ends of drinking straws of various lengths. Which straws, the short ones or the long ones, produce a lower pitch? What would you expect of the pitch produced by the much larger musical instrument behind Tom that uses resonant air columns excited by striking the ends of the tubes with paddles?

Which is a more objective measurement: sound intensity or loudness? Defend your answer.

One person has a threshold of hearing of 5 dB, and another of 10 dB. Which person has the more acute hearing?

How is an electronic organ able to imitate the sounds of various musical instruments?

After inhaling helium gas, a person talks with a high-pitched voice. One of the reasons for this is the higher speed of sound in helium than in air. Why does sound travel faster in helium?

Why does your voice sound fuller in the shower?

The frequency range for a telephone is between 500 Hz and 3000 Hz. Why doesn’t a telephone do a very good job of transmitting music?

How many octaves does normal human hearing span? How many octaves are on a common piano keyboard? (If you’re not sure, look and see.)

The note middle C on a piano has a fundamental frequency of about 262 Hz. What is the frequency of the second harmonic of this note?

If the fundamental frequency of a guitar string is 220 Hz, what is the frequency of the second harmonic? Of the third harmonic?

If the fundamental frequency of a violin string is 440 Hz, what is the frequency of the second harmonic? Of the third harmonic?

How many nodes, not including the endpoints, are in a standing wave that is three wavelengths long? How many nodes are in a standing wave that is four wavelengths long?

At an outdoor concert, the pitch of musical tones is not affected on a windy day. Explain.

The human ear is sometimes called a Fourier analyzer. What does this mean, and why is it an appropriate description?

The width of a laser beam was and is significant in reading CDs and DVDs. The thinner the beam, the closer the series of pits can be. Why will blue laser light allow closer pits than red laser light?

Make up a multiple-choice question that distinguishes between any of the terms listed in the Summary of Terms.

The yellow-green light emitted by streetlights matches the yellow-green color to which the human eye is most sensitive. Consequently, such a 100-W streetlight emits light that is better seen at night. Similarly, the monitored sound intensities of TV commercials are louder than the sound of regular programming, yet they don’t exceed the regulated maximum intensities. Discuss the frequencies at which advertisers concentrate the commercials’ sound.

A guitar and a flute are in tune with each other. Discuss how a change in temperature could alter this situation.

Your classmate says that a harmonic series of frequencies includes the fundamental frequency and integral multiples of the fundamental frequency. Do you agree or disagree?

How can you tune the note A3 on a piano to its proper frequency of 220 Hz with the aid of a tuning fork whose frequency is 440 Hz? Discuss.

A trumpet has keys and valves that permit the trumpeter to change the length of the vibrating air column and the position of the nodes. A bugle has no such keys and valves, yet it can sound various notes. Discuss how the bugler achieves different notes.

Do all the people in a group hear the same music when they listen attentively, as in Figure 21.14? Do all see the same sight when viewing a painting? Do all taste the same flavor when sipping the same wine? Do all perceive the same aroma when smelling the same perfume? Do all feel the same texture when touching the same fabric? Do all come to the same conclusion when listening to a logical presentation of ideas?

Discuss why it is safe to say that you will have a significantly greater loss of hearing in your later years than your grandparents experienced.

What are the three principal characteristics of musical tones?