SeCTIon 3 SECTION 3 Objectives Harmonics · 2018-04-04 · ©RubberBall/Getty Images harmonic...

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Key Termsfundamental frequency timbreharmonic series beat

Standing Waves on a Vibrating StringAs discussed in the chapter “Vibrations and Waves,” a variety of standing waves can occur when a string is fixed at both ends and set into vibration. The vibrations on the string of a musical instrument, such as the violin in Figure 3.1, usually consist of many standing waves together at the same time, each of which has a different wavelength and frequency. So, the sounds you hear from a stringed instrument, even those that sound like a single pitch, actually consist of multiple frequencies.

Figure 3.2, on the next page, shows several possible vibrations on an idealized string. The ends of the string, which cannot vibrate, must always be nodes (N ). The simplest vibration that can occur is shown in the first row of Figure 3.2. In this case, the center of the string experiences the most displacement, and so it is an antinode (A). Because the distance from one node to the next is always half a wavelength, the string length (L) must equal λ1/2. Thus, the wavelength is twice the string length (λ1 = 2L).

As described in the chapter on waves, the speed of a wave equals the frequency times the wavelength, which can be rearranged as shown.

v = f λ, so f = v _ λ

By substituting the value for wavelength found above into this equation for frequency, we see that the frequency of this vibration is equal to the speed of the wave divided by twice the string length.

fundamental frequency = f1 = v _ λ1

= v _ 2L

This frequency of vibration is called the fundamental frequency of the vibrating string. Because frequency is inversely proportional to wave-length and because we are considering the greatest possible wavelength, the fundamental frequency is the lowest possible frequency of a standing wave on this string.

Harmonics are integral multiples of the fundamental frequency.The next possible standing wave for a string is shown in the second row of Figure 3.2. In this case, there are three nodes instead of two, so the string length is equal to one wavelength. Because this wavelength is half the previous wavelength, the frequency of this wave is twice that of the fundamental frequency.

f2 = 2f1

HarmonicsObjectives

Differentiate between the harmonic series of open and closed pipes.

Calculate the harmonics of a vibrating string and of open and closed pipes.

Relate harmonics and timbre.

Relate the frequency difference between two waves to the number of beats heard per second.

fundamental frequency the lowest frequency of vibration of a standing wave

Stringed Instruments The vibrating strings of a violin produce standing waves whose frequencies depend on the string lengths.

FIGURE 3.1

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Differentiated Instruction

Below levelStudents may be confused as to why the addition of one node equals half the wave-length and double the frequency. It may help them to visualize folding the string in half with the second node as the center.

Preview vocabularylatin word origins Often terms used in physics take their meaning from their Latin roots. Fundamental, for instance, comes from the Latin fundamentum, meaning “foundation.” Apply this root to the definition of fundamental frequency (i.e., the foundation is the lowest point of a structure, just as fundamental frequency is the lowest frequency of a standing wave).

SeeIng SounDSPurpose Observe sound waves from a variety of sources.

Materials oscilloscope, microphone, small amplifier (if needed), assorted sound sources

Caution Consult the oscilloscope’s user’s manual for instructions on the proper use of the oscilloscope.

Procedure Connect the microphone or the output from the microphone and amplifier to the input of the oscillo-scope. Display the sound pattern of the human voice by having several students speak or sing into the microphone. Have the students note the patterns on the screen of the oscilloscope. Show the characteristic of a single frequency sound by inputting the sound from an oscillator, sine wave generator, tuning fork, or a student’s voice at a single pitch.

Then have several students play the same note on various instruments one at a time. Have the students note the characteristic harmonics of each instrument as they are displayed on the oscilloscope screen.

� Plan and Prepare

� Teach

Demonstration

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harmonic series a series of frequen-cies that includes the fundamental frequency and integral multiples of the fundamental frequency

This pattern continues, and the frequency of the standing wave shown in the third row of Figure 3.2 is three times the fundamental frequency. More generally, the frequencies of the standing wave patterns are all integral multiples of the fundamental frequency. These frequencies form what is called a harmonic series. The fundamental frequency (f1) corresponds to the first harmonic, the next frequency (f2) corresponds to the second harmonic, and so on.

Because each harmonic is an integral multiple of the fundamental frequency, the equation for the fundamental frequency can be general-ized to include the entire harmonic series. Thus, fn = nf1, where f1 is the fundamental frequency (f1 = v __

2L ) and fn is the frequency of the nth

harmonic. The general form of the equation is written as follows:

Harmonic Series of Standing Waves on a Vibrating String

fn = n v _ 2L

n = 1, 2, 3, . . .

frequency = harmonic number × (speed of waves on the string)

___ (2)(length of vibrating string)

Note that v in this equation is the speed of waves on the vibrating string and not the speed of the resultant sound waves in air. If the string vibrates at one of these frequencies, the sound waves produced in the surrounding air will have the same frequency. However, the speed of these waves will be the speed of sound waves in air, and the wavelength of these waves will be that speed divided by the frequency.

Did YOU Know?

When a guitar player presses down on a guitar string at any point, that point becomes a node and only a portion of the string vibrates. As a result, a single string can be used to create a variety of fundamental frequencies. In the equation on this page, L refers to the portion of the string that is vibrating.

Figure 3.2

The harmonic SerieS

λ1 = 2L f1fundamental frequency, or first harmonic

λ2 = L f2 = 2f1 second harmonic

λ3 = 2 __ 3 L f3 = 3f1 third harmonic

λ4 = 1 __ 2 L f4 = 4f1 fourth harmonic

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Teaching TipHave students visualize the harmonics to help them recall the appropriate equations. Both ends of a string fixed at each end must be nodes. For the first harmonic (one loop), the length L of the string must equal 1 _ 2 λ. Thus, 1 _ 2 λ = L and λ = 2L. For other harmonics, students can use this same technique with the appropriate multiples of 1 _ 2 λ.

FIguRe 3.2 Be sure students understand that each loop corresponds to half a wavelength.

Ask Find the wavelength (λ5) and

frequency (f5) for the next possible case

in the harmonic series.

Answer: λ5 = 2 _

5 L, f

5 = 5f

1

TEACH FROM VISUALS

PRe-APPoint out to advanced students that frequency depends on both string length and wave speed, as shown by the equation for the harmonic series. Thus, two strings of the same length will not necessarily have the same fundamental frequency. The string’s tension and mass per unit length affect the speed of waves on the string, so the fundamental frequency can be changed by varying either of these factors. To test their understanding of frequency’s

dependence on string length and speed, supply students with multiple scenarios that encourage them to substitute values into the harmonic series equation as well as solve for unknowns.

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Standing Waves in an Air ColumnStanding waves can also be set up in a tube of air, such as the inside of a trumpet, the column of a saxophone, or the pipes of an organ like those shown in Figure 3.3. While some waves travel down the tube, others are reflected back upward. These waves traveling in opposite directions combine to produce standing waves. Many brass instruments and woodwinds produce sound by means of these vibrating air columns.

If both ends of a pipe are open, all harmonics are present.The harmonic series present in an organ pipe depends on whether the reflecting end of the pipe is open or closed. When the reflecting end of the pipe is open, as is illustrated in Figure 3.4, the air molecules have complete freedom of motion, so an antinode (of displacement) exists at this end. If a pipe is open at both ends, each end is an antinode. This situation is the exact opposite of a string fixed at both ends, where both ends are nodes.

Because the distance from one node to the next ( 1 __ 2 λ) equals the distance from one antinode to the next, the pattern of standing waves that can occur in a pipe open at both ends is the same as that of a vibrating string. Thus, the entire harmonic series is present in this case, as shown in Figure 3.4, and our earlier equation for the harmonic series of a vibrating string can be used.

Harmonic Series of a Pipe Open at Both Ends

fn = n v _ 2L

n = 1, 2, 3, . . .

frequency = harmonic number × (speed of sound in the pipe)

____ (2)(length of vibrating air column)

In this equation, L represents the length of the vibrating air column. Just as the fundamental frequency of a string instrument can be varied by changing the string length, the fundamental frequency of many wood-wind and brass instruments can be varied by changing the length of the vibrating air column.

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Waves in a Pipe The harmonic series present in each of these organ pipes depends on whether the end of the pipe is open or closed.

FIGURE 3.3

Did YOU Know?

A flute is similar to a pipe open at both ends. When all keys of a flute are closed, the length of the vibrating air column is approximately equal to the length of the flute. As the keys are opened one by one, the length of the vibrating air column decreases, and the fundamental frequency increases.

(a) First harmonic (b) Second harmonic (c) Third harmonic

Harmonics in an Open-Ended Pipe In a pipe open at both ends, each end is an antinode of displacement, and all harmonics are present.

FIGURE 3.4

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Problem Solving

Teaching TipThe memory device used for a vibrating string can be used for open and closed pipes as well. Ask students to visualize the standing wave, remembering that closed ends must be nodes and open ends must be antinodes. Sketching diagrams for each case will show that pipes closed at one end must have half a loop ( 1 __ 4 λ) more than pipes open at both ends.

� Teach continued

TAke IT FuRTheRAllow students to take a concrete look at how speed affects frequency by comparing the frequency of an open pipe filled with water to one filled with air. The speed of sound is 346 m/s in air and 1490 m/s in water.

The frequency in water is 4.3 times the frequency in air.

420 Chapter 12

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Harmonics in a Pipe Closed at One End In a pipe closed at one end, the closed end is a node of displacement and the open end is an anti-node of displacement. In this case, only the odd harmonics are present.

FIGURE 3.5

(a) First harmonic (b) Second harmonic (c) Third harmonic

If one end of a pipe is closed, only odd harmonics are present.When one end of an organ pipe is closed, as is illustrated in Figure 3.5, the movement of air molecules is restricted at this end, making this end a node. In this case, one end of the pipe is a node and the other is an antinode. As a result, a different set of standing waves can occur.

As shown in Figure 3.5(a), the simplest possible standing wave that can exist in this pipe is one for which the length of the pipe is equal to one-fourth of a wavelength. Hence, the wavelength of this standing wave equals four times the length of the pipe. Thus, in this case, the fundamen-tal frequency equals the velocity divided by four times the pipe length.

f1 = v _ λ1

= v _ 4L

For the case shown in Figure 3.5(b), the length of the pipe is equal to three-fourths of a wavelength, so the wavelength is four-thirds the length of the pipe (λ3 = 4 __ 3 L). Substituting this value into the equation for fre-quency gives the frequency of this harmonic.

f3 = v _ λ3

= v _ 4 __ 3 L

= 3v _ 4L

= 3f1

The frequency of this harmonic is three times the fundamental frequency. Repeating this calculation for the case shown in Figure 3.5(c) gives a frequency equal to five times the fundamental frequency. Thus, only the odd-numbered harmonics vibrate in a pipe closed at one end. We can generalize the equation for the harmonic series of a pipe closed at one end as follows:

Harmonic Series of a Pipe Closed at One End

fn = n v _ 4L

n = 1, 3, 5, . . .

frequency = harmonic number × (speed of sound in the pipe)

____ (4)(length of vibrating air column)

A PIPE CLOSED AT ONE ENDSnip off the corners of one end of the straw so that the end tapers to a point, as shown below. Chew on this end to flatten it, and you create a double-reed instrument! Put your lips around the tapered end of the straw, press them together tightly, and blow through the straw. When you hear a steady tone, slowly snip off pieces of the straw at the other end. Be careful to keep about the same amount of pressure with your lips. How does the pitch change as the straw becomes shorter? How can you account for this change in pitch? You may be able to produce more than one tone for any given length of the straw. How is this possible?

MATERIALSstraw•

scissors•

SAFETY Always use caution when working with scissors.

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TeACheR’S noTeSThis lab demonstrates the effect of pipe length on pitch. The experiment works best with narrow paper straws.

Students should find that the shorter the straw is, the higher the fundamental frequency is. One length of straw can produce more than one possible tone because there are several possible wavelengths of standing waves for a given length of straw.

The sound frequencies can be demonstrated visually by blowing into a microphone connected to an oscillo-scope and watching the results on the oscilloscope screen.homework options This QuickLab can easily be performed outside of the physics lab room.

QuickLab

Below levelReview with students the difference between pitch and loudness. Explain that pitch is how frequency is perceived by the human ear. Pitch is described on a continuum of low and high, because pitch is determined by frequency. Loudness, on the other hand, is determined by amplitude, which is the maximum displace-ment of a periodic wave.

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Harmonics

Sample Problem B What are the first three harmonics in a 2.45 m long pipe that is open at both ends? What are the first three harmonics of this pipe when one end of the pipe is closed? Assume that the speed of sound in air is 345 m/s.

ANALYZE Given: L = 2.45 m v = 345 m/s

Unknown: Pipe open at both ends: f1 f2 f3

Pipe closed at one end: f1 f3 f5

PLAN Choose an equation or situation:When the pipe is open at both ends, the fundamental frequency can be found by using the equation for the entire harmonic series:

fn = n v _ 2L

, n = 1, 2, 3, . . .

When the pipe is closed at one end, use the following equation:

fn = n v _ 4L

, n = 1, 3, 5, . . .

In both cases, the second two harmonics can be found by multiplying the harmonic numbers by the fundamental frequency.

SOLVE Substitute the values into the equations and solve:For a pipe open at both ends:

f1 = n v _ 2L

= (1) ( 345m/s __ (2)(2.45m)

) = 70.4 Hz

The next two harmonics are the second and the third:

f2 = 2f1 = (2)(70.4 Hz) = 141 Hz

f3 = 3f1 = (3)(70.4 Hz) = 211 Hz

For a pipe closed at one end:

f1 = n v _ 4L

= (1) ( 345m/s __ (2)(2.45m)

) = 35.2 Hz

The next possible harmonics are the third and the fifth:

f3 = 3f1 = (3)(35.2 Hz) = 106 Hz

f5 = 5f1 = (5)(35.2 Hz) = 176 Hz

Continued

Interactive DemoHMDScience.com

PREMIUM CONTENT

Tips and TricksBe sure to use the correct harmonic numbers for each situation. For a pipe open at both ends, n = 1, 2, 3, etc. For a pipe closed at one end, only odd harmonics are pres ent, so n = 1, 3, 5, etc.

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Problem Solving

Classroom PracticehARMonICSOne string on a toy guitar is 34.5 cm long.

a. What is the wavelength of its first harmonic?

b. The string is plucked, and the speed of waves on the string is 410 m/s. What are the frequencies of the first three harmonics?

Answers: a. 69.0 cm b. 590 Hz, 1200 Hz, 1800 Hz

� Teach continued

AlTeRnATIve APPRoACheSFor the open pipe, both open ends must be antinodes; thus L = 1 _ 2 λ

1, or λ

1 = 2L =

2(2.45 m) = 4.90 m. The first harmonic can now be found with the equation

f = v _ λ

, as follows:

f1 = v _

λ1

= 345 m/s

_ 4.90 m

= 70.4 Hz

For the closed pipe, one end is a node and one is an antinode; hence, L = 1 __ 4 λ

1, or λ

1 = 4L

= 4(2.45m) = 9.80 m.

Thus, the first harmonic is as follows:

f1 = v _

λ1

= 345 m/s

_ 9.80 m

= 35.2 Hz

The other harmonics can be found by multiplying the harmonic number by the funda mental frequency, as in Sample Problem B.

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Trumpets, saxophones, and clarinets are similar to a pipe closed at one end. For example, although the trumpet shown in Figure 3.6 has two open ends, the player’s mouth effectively closes one end of the instru-ment. In a saxophone or a clarinet, the reed closes one end.

Despite the similarity between these instruments and a pipe closed at one end, our equation for the harmonic series of pipes does not directly apply to such instruments. One reason the equation does not apply is that any deviation from the cylindrical shape of a pipe affects the harmonic series of an instrument. Another reason is that the open holes in many instruments affect the harmonics. For example, a clarinet is primar-ily cylindrical, but there are some even harmonics in a clarinet’s tone at relatively small intensities. The shape of a saxophone is such that the harmonic series in a saxophone is similar to that in a cylindrical pipe open at both ends even though only one end of the saxo-phone is open. These deviations are in part responsible for the variety of sounds that can be produced by different instruments.

Shape and Harmonic Series Variations in shape give each instrument a different harmonic series.

FIGURE 3.6

Harmonics (continued)

CHECK YOUR WORK

In a pipe open at both ends, the first possible wavelength is 2L; in a pipe closed at one end, the first possible wavelength is 4L. Because frequency and wavelength are inversely proportional, the fundamental frequency of the open pipe should be twice that of the closed pipe, that is, 70.4 = (2)(35.2).

1. What is the fundamental frequency of a 0.20 m long organ pipe that is closed at one end, when the speed of sound in the pipe is 352 m/s?

2. A flute is essentially a pipe open at both ends. The length of a flute is approximately 66.0 cm. What are the first three harmonics of a flute when all keys are closed, making the vibrating air column approximately equal to the length of the flute? The speed of sound in the flute is 340 m/s.

3. What is the fundamental frequency of a guitar string when the speed of waves on the string is 115 m/s and the effective string lengths are as follows?

a. 70.0 cm b. 50.0 cm c. 40.0 cm

4. A violin string that is 50.0 cm long has a fundamental frequency of 440 Hz. What is the speed of the waves on this string?

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PRoBleM guIDe BUse this guide to assign problems.

Se = Student Edition Textbook

Pw = Sample Problem Set I (online)

PB = Sample Problem Set II (online)

Solving for:

fn Se Sample, 1–3; Ch. Rvw. 34–35, 36b, 40

Pw 7–9

PB 5–7

v Se 4

Pw Sample, 1–3

PB 8–10

L Se Ch. Rvw. 36a, 41*

Pw 4–6

PB Sample, 1–4

*Challenging Problem

AnswersPractice B 1. 440 Hz

2. 260 Hz, 520 Hz, 780 Hz

3. a. 82.1 Hz

b. 115 Hz

c. 144 Hz

4. 440 m/s

DeConSTRuCTIng PRoBleMSWhen a pipe is open at one end, the first possible wavelength is 2L. When a pipe is open at both ends, it is 4L. Students should use this tip to check their equation writing. Emphasize than an incorrect value for wavelength will affect their final answer.

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Harmonics account for sound quality, or timbre.Figure 3.7 shows the harmonics present in a tuning fork, a cla rinet, and a viola when each sounds the musical note A-natural. Each instrument has its own characteristic mixture of harmonics at varying intensities.

The harmonics shown in the second column of Figure 3.7 add together according to the principle of superposition to give the resultant waveform shown in the third column. Since a tuning fork vibrates at only its funda-mental frequency, its waveform is simply a sine wave. (Some tuning forks also vibrate at higher frequencies when they are struck hard enough.) The waveforms of the other instruments are more complex because they consist of many harmonics, each at different intensities. Each individual harmonic waveform is a sine wave, but the resultant wave is more com-plex than a sine wave because each individual waveform has a different frequency.

In music, the mixture of harmonics that produces the characteristic sound of an instrument is referred to as the spectrum of the sound. From the perspective of the listener, this spectrum results in sound quality, or timbre. A clarinet sounds different from a viola because of differences in timbre, even when both instruments are sounding the same note at the same volume. The rich harmonics of most instruments provide a much fuller sound than that of a tuning fork.

The intensity of each harmonic varies within a particular instrument, depending on frequency, amplitude of vibration, and a variety of other factors. With a violin, for example, the intensity of each harmonic

timbre the musical quality of a tone resulting from the combination of harmonics present at different intensities

Figure 3.7

Harmonics of a Tuning fork, a clarineT, and a Viola aT THe same PiTcH

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Teaching TipExplain the distinction sometimes made between musical sounds and noise. When individual waveforms are regular, the resultant waveform, although complex, is composed of repeating patterns. On the other hand, when the individual waveforms are not periodic, the resultant waveform is nonrepeating. Sounds that are considered to be musical typically have waveforms with repeating patterns, while those that are considered to be noise do not.

FIguRe 3.7 Point out that the second column in Figure 3.7 represents the spectrum of wave amplitudes of harmonics produced by each instrument.

Ask The harmonics shown in Figure 3.7 are for the musical note A-natural ( f

1 = 220 Hz). List the frequencies

produced by the clarinet in order of intensity, from greatest to least.

Answer: 660 Hz, 220 Hz, 440 Hz, 1540 Hz, 880 Hz, 1100 Hz

� Teach continued

TEACH FROM VISUALS

Below levelStudents may have heard the word quality only in its everyday use (for example, “The quality of this fabric is poor.”) Emphasize that the word quality is not being used here to imply whether or not sound is good or bad. It is being used to mean “trait” or “distinguish-ing attribute.” Use the word in a sentence such as this one: “Joe possesses the quality of patience.” The sound quality of a flute is different from that of a shrill whistle. This does

not mean one is better or worse than the other; it means that each has its own attri-butes and the two can be distinguished from one another.

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A uditoriums, churches, concert halls, libraries, and music rooms are designed with specific functions in mind. One auditorium may be made

for rock concerts, while another is constructed for use as a lecture hall. Your school’s auditorium, for instance, may allow you to hear a speaker well but make a band sound damped and muffled.

Rooms are often constructed so that sounds made by a speaker or a musical instrument bounce back and forth against the ceiling, walls, floor, and other surfaces. This repetitive echo is called reverberation. The reverberation time is the amount of time it takes for a sound’s intensity to decrease by 60 dB.

For speech, the auditorium should be designed so that the reverberation time is relatively short. A repeated echo of each word could become confusing to listeners.

Music halls may differ in construction depending on the type of music usually played there. For example, rock

music is generally less pleasing with a large amount of reverberation, but more reverberation is sometimes desired for orchestral and choral music.

For these reasons, you may notice a difference in the way ceilings, walls, and furnishings are designed in different rooms. Ceilings designed for a lot of reverberation are flat and hard. Ceilings in libraries and other quiet places are often made of soft or textured material to muffle sounds. Padded furnishings and plants can also be strategically arranged to absorb sound. All of these different factors are considered and combined to accommodate the audi tory function of a room.

Reverberation

depends on where the string is bowed, the speed of the bow on the string, and the force the bow exerts on the string. Because there are so many factors involved, most instruments can produce a wide variety of tones.

Even though the waveforms of a clarinet and a viola are more com-plex than those of a tuning fork, note that each consists of repeating patterns. Such waveforms are said to be periodic. These repeating patterns occur because each frequency is an integral multiple of the fundamental frequency.

Fundamental frequency determines pitch.The frequency of a sound determines its pitch. In musical instruments, the fundamental frequency of a vibration typically determines pitch. Other harmonics are sometimes referred to as overtones. In the chromatic (half-step) musical scale, there are 12 notes, each of which has a charac-teristic frequency. The frequency of the thirteenth note is exactly twice that of the first note, and together the 13 notes constitute an octave. For stringed instruments and open-ended wind instruments, the frequency of the second har monic of a note corresponds to the frequency of the octave above that note.

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why It MattersReveRBeRATIonThe discussion of harmonics in this section considers the interference of the sound waves from a single instru-ment. This feature discusses the interference that occurs when echoes interfere with the original sound waves and with one another.

extensionStudents may be surprised to learn that reverberation is an important consider-ation when buildings are being designed. This opportunity can be used to discuss acoustical engineering as a possible career choice for students who are interested in acoustics.

englISh leARneRSEnglish learners may be confused by the term octave. If their native language contains many Latinate roots, or if they have been exposed to the religious or literary meaning of the word, they may associate octave with eight rather than with twelve. Clarify that the word is not being used to imply a number; it is being used to signify the interval between one musical pitch and another with half or double its frequency.

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Destructiveinterference

Destructiveinterference

Constructiveinterference

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t1 t2 t3

Conceptual Challenge

FIGURE 3.8

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BeatsSo far, we have considered the superposition of waves in a harmonic series, where each frequency is an integral multiple of the fundamental frequency. When two waves of slightly different frequencies interfere, the interference pattern varies in such a way that a listener hears an alternation between loudness and softness. The variation from soft to loud and back to soft is called a beat.

Sound waves at slightly different frequencies produce beats.Figure 3.8 shows how beats occur. In Figure 3.8(a), the waves produced by two tuning forks of different frequencies start exactly opposite one another. These waves combine according to the superposition principle, as shown in Figure 3.8(b). When the two waves are exactly opposite one another, they are said to be out of phase, and complete destructive interference occurs. For this reason, no sound is heard at t1.

Because these waves have different frequencies, after a few more cycles, the crest of the blue wave matches up with the crest of the red wave, as at t2. At this point, the waves are said to be in phase.

beat the periodic variation in the amplitude of a wave that is the superposition of two waves of slightly different frequencies

Superposition and BeatsBeats are formed by the interference of two waves of slightly different frequencies traveling in the same direction. In this case, constructive interference is greatest at t2, when the two waves are in phase.

Concert Violins Before a performance, musicians tune their instruments to match their fundamental frequencies. If a conductor hears the number of beats decreasing as two violin players are tuning, are the fundamental frequen-cies of these violins becoming closer together or farther apart? Explain.

Tuning Flutes How could two flute players use beats to ensure that their instruments are in tune with each other?

Sounds from a Guitar Will the speed of waves on a vibrating guitar string be the same as the speed of the sound waves in the air that are generated by this vibra-tion? How will the frequency and wavelength of the waves on the string compare with the frequency and wavelength of the sound waves in the air?

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FIguRe 3.8 Be sure students under-stand that this figure depicts two waves at a particular point in space as time passes (rather than an expanse of space at an instant of time, as in previous wave representations).

Ask At what time(s) are the two waves exactly out of phase? At what time(s) are the two waves exactly in phase?

Answer: t1 and t

3 ; t

2

AnswersConceptual Challenge 1. The fundamental frequencies are

getting closer together because if the number of beats heard each second is decreasing, the two waves are closer to being completely in phase at all points.

2. When the two flutists play the same note, the number of beats heard each second is the frequency difference between the two flutes. If one of the flutes is adjusted until no beats are heard, the two flutes will be in tune with each other.

3. The speed of the sound waves in air will not be the same as the speed of waves on the string. There is no frequency change because the vibrations are still occurring at the same rate, so the wavelength must change with the wave speed (because v = λf ).

� Teach continued

TEACH FROM VISUALS

Below levelStudents can remember that destructive interference means waves are out of phase by remembering that the Latin prefix de- typically indicates “reversal,” “removal,” or “lack.” There is a lack of sound when there is destructive interference.

426 Chapter 12

Reviewing Main Ideas

1. On a piano, the note middle C has a fundamental frequency of 262 Hz. What is the second harmonic of this note?

2. If the piano wire in item 1 is 66.0 cm long, what is the speed of waves on this wire?

3. A piano tuner using a 392 Hz tuning fork to tune the wire for G-natural hears four beats per second. What are the two possible frequencies of vibration of this piano wire?

4. In a clarinet, the reed end of the instrument acts as a node and the first open hole acts as an antinode. Because the shape of the clarinet is nearly cylindrical, its harmonic series approximately follows that of a pipe closed at one end. What harmonic series is predominant in a clarinet?

Critical Thinking

5. Which of the following are different for a trumpet and a banjo when both play notes at the same fundamental frequency?a. wavelength in air of the first harmonicb. which harmonics are presentc. intensity of each harmonicd. speed of sound in air

Now constructive interference occurs, and the sound is louder. Because the blue wave has a higher frequency than the red wave, the waves are out of phase again at t3, and no sound is heard.

As time passes, the waves continue to be in and out of phase, the interference constantly shifts between constructive interference and destructive interference, and the listener hears the sound getting softer and louder and then softer again. You may have noticed a similar phe-nomenon on a playground swing set. If two people are swinging next to one another at different frequencies, the two swings may alternate between being in phase and being out of phase.

The number of beats per second corresponds to the difference between frequencies.In our previous example, there is one beat, which occurs at t2. One beat corresponds to the blue wave gaining one entire cycle on the red wave. This is because to go from one destructive interference to the next, the red wave must lag one entire cycle behind the blue wave. If the time that lapses from t1 to t3 is one second, then the blue wave completes one more cycle per second than the red wave. In other words, its frequency is greater by 1 Hz. By generalizing this, you can see that the frequency difference between two sounds can be found by the number of beats heard per second.

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SECTION 3 FORMATIVE ASSESSMENT

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Answers to Section Assessment

Teaching TipPoint out that the number of beats is always the absolute value of the difference between two frequencies.

Measuring the number of beats does not tell you which of two instruments has a higher frequency. If two waves are exactly in phase, there are no beats and the sounds are exactly in tune.

Assess Use the Formative Assessment on this page to evaluate student mastery of the section.

Reteach For students who need additional instruction, download the Section Study Guide.

Response to Intervention To reassess students’ mastery, use the Section Quiz, available to print or to take directly online at hMDScience.com.

� Assess and Reteach

1. 524 Hz

2. 348 m/s

3. 388 Hz and 396 Hz

4. the odd harmonics

5. b, c

Sound 427

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