543543 543

Superposition and Standing Waves

CHAPTE R OUTL I N E

18.1 Superposition andInterference

18.2 Standing Waves

18.3 Standing Waves in a StringFixed at Both Ends

18.4 Resonance

18.5 Standing Waves in AirColumns

18.6 Standing Waves in Rods andMembranes

18.7 Beats: Interference in Time

18.8 Nonsinusoidal Wave Patterns

! Guitarist Carlos Santana takes advantage of standing waves on strings. He changes to ahigher note on the guitar by pushing the strings against the frets on the fingerboard, shorten-ing the lengths of the portions of the strings that vibrate. (Bettmann/Corbis)

Chapter 18

544

In the previous two chapters, we introduced the wave model. We have seen thatwaves are very different from particles. A particle is of zero size, while a wave has acharacteristic size—the wavelength. Another important difference between wavesand particles is that we can explore the possibility of two or more waves combining atone point in the same medium. We can combine particles to form extended objects,but the particles must be at different locations. In contrast, two waves can both be present at the same location, and the ramifications of this possibility are explored inthis chapter.

When waves are combined, only certain allowed frequencies can exist on systemswith boundary conditions—the frequencies are quantized. Quantization is a notion thatis at the heart of quantum mechanics, a subject that we introduce formally in Chapter40. There we show that waves under boundary conditions explain many of the quan-tum phenomena. For our present purposes in this chapter, quantization enables us tounderstand the behavior of the wide array of musical instruments that are based onstrings and air columns.

We also consider the combination of waves having different frequencies and wave-lengths. When two sound waves having nearly the same frequency interfere, we hearvariations in the loudness called beats. The beat frequency corresponds to the rate ofalternation between constructive and destructive interference. Finally, we discuss howany nonsinusoidal periodic wave can be described as a sum of sine and cosinefunctions.

18.1 Superposition and Interference

Many interesting wave phenomena in nature cannot be described by a single travelingwave. Instead, one must analyze complex waves in terms of a combination of travelingwaves. To analyze such wave combinations, one can make use of the superpositionprinciple:

Waves that obey this principle are called linear waves. In the case of mechanical waves,linear waves are generally characterized by having amplitudes much smaller than theirwavelengths. Waves that violate the superposition principle are called nonlinear wavesand are often characterized by large amplitudes. In this book, we deal only with linearwaves.

One consequence of the superposition principle is that two traveling waves canpass through each other without being destroyed or even altered. For instance,

If two or more traveling waves are moving through a medium, the resultant value ofthe wave function at any point is the algebraic sum of the values of the wave func-tions of the individual waves.

Superposition principle

when two pebbles are thrown into a pond and hit the surface at different places, theexpanding circular surface waves do not destroy each other but rather pass througheach other. The complex pattern that is observed can be viewed as two independentsets of expanding circles. Likewise, when sound waves from two sources move throughair, they pass through each other.

Figure 18.1 is a pictorial representation of the superposition of two pulses. Thewave function for the pulse moving to the right is y1, and the wave function for thepulse moving to the left is y2. The pulses have the same speed but different shapes, andthe displacement of the elements of the medium is in the positive y direction for bothpulses. When the waves begin to overlap (Fig. 18.1b), the wave function for the result-ing complex wave is given by y1 ! y2. When the crests of the pulses coincide(Fig. 18.1c), the resulting wave given by y1 ! y2 has a larger amplitude than that of theindividual pulses. The two pulses finally separate and continue moving in their originaldirections (Fig. 18.1d). Note that the pulse shapes remain unchanged after the interac-tion, as if the two pulses had never met!

The combination of separate waves in the same region of space to produce a resultantwave is called interference. For the two pulses shown in Figure 18.1, the displacement ofthe elements of the medium is in the positive y direction for both pulses, and the resultantpulse (created when the individual pulses overlap) exhibits an amplitude greater thanthat of either individual pulse. Because the displacements caused by the two pulses are inthe same direction, we refer to their superposition as constructive interference.

Now consider two pulses traveling in opposite directions on a taut string where onepulse is inverted relative to the other, as illustrated in Figure 18.2. In this case, whenthe pulses begin to overlap, the resultant pulse is given by y1 ! y2, but the values of thefunction y2 are negative. Again, the two pulses pass through each other; however, be-cause the displacements caused by the two pulses are in opposite directions, we refer totheir superposition as destructive interference.

S ECT I O N 18 . 1 • Superposition and Interference 545

(c)

(d)

(b)

(a)

y2 y 1

y 1+ y2

y 1+ y2

y2y 1

Active Figure 18.1 (a–d) Two pulses traveling on a stretched string in opposite direc-tions pass through each other. When the pulses overlap, as shown in (b) and (c), thenet displacement of the string equals the sum of the displacements produced by eachpulse. Because each pulse produces positive displacements of the string, we refer totheir superposition as constructive interference. (e) Photograph of the superposition oftwo equal, symmetric pulses traveling in opposite directions on a stretched spring.

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! PITFALL PREVENTION18.1 Do Waves Really

Interfere?In popular usage, the term inter-fere implies that an agent affects asituation in some way so as to pre-clude something from happen-ing. For example, in Americanfootball, pass interference meansthat a defending player has af-fected the receiver so that he isunable to catch the ball. This isvery different from its use inphysics, where waves pass througheach other and interfere, but donot affect each other in any way.In physics, interference is similarto the notion of combination as de-scribed in this chapter.

Constructive interference

Destructive interference

(e)

At the Active Figures link at http://www.pse6.com, you can choose theamplitude and orientation of each of the pulses and study the interferencebetween them as they pass each other.

546 C H A P T E R 18 • Superposition and Standing Waves

(a)

(b)

(d)

(e)

y 1

y 2

y 1

y 2

y 2

y 1

y 2

y 1

(c)y 1+ y 2

Active Figure 18.2 (a–e) Two pulses traveling in opposite directions and having dis-placements that are inverted relative to each other. When the two overlap in (c), theirdisplacements partially cancel each other. (f ) Photograph of the superposition of twosymmetric pulses traveling in opposite directions, where one is inverted relative to theother.

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Quick Quiz 18.1 Two pulses are traveling toward each other, each at 10 cm/son a long string, as shown in Figure 18.3. Sketch the shape of the string at t " 0.6 s.

Quick Quiz 18.2 Two pulses move in opposite directions on a string and areidentical in shape except that one has positive displacements of the elements of thestring and the other has negative displacements. At the moment that the two pulsescompletely overlap on the string, (a) the energy associated with the pulses has disap-peared (b) the string is not moving (c) the string forms a straight line (d) the pulseshave vanished and will not reappear.

1 cm

At the Active Figures linkat http://www.pse6.com, youcan choose the amplitude andorientation of each of thepulses and watch the interfer-ence as they pass each other.

Figure 18.3 (Quick Quiz 18.1) The pulses on this string are traveling at 10 cm/s.

(f)

Superposition of Sinusoidal Waves

Let us now apply the principle of superposition to two sinusoidal waves traveling in thesame direction in a linear medium. If the two waves are traveling to the right and havethe same frequency, wavelength, and amplitude but differ in phase, we can expresstheir individual wave functions as

y1 " A sin(kx # $t) y2 " A sin(kx # $t ! %)

where, as usual, k " 2&/', $ " 2&f, and % is the phase constant, which we discussed inSection 16.2. Hence, the resultant wave function y is

y " y1 ! y2 " A[sin(kx # $t) ! sin(kx # $t ! %)]

To simplify this expression, we use the trigonometric identity

If we let a " kx # $t and b " kx # $t ! %, we find that the resultant wave functiony reduces to

This result has several important features. The resultant wave function y also is sinus-oidal and has the same frequency and wavelength as the individual waves because thesine function incorporates the same values of k and $ that appear in the original wavefunctions. The amplitude of the resultant wave is 2A cos(%/2), and its phase is %/2. Ifthe phase constant % equals 0, then cos (%/2) " cos 0 " 1, and the amplitude of theresultant wave is 2A—twice the amplitude of either individual wave. In this case thewaves are said to be everywhere in phase and thus interfere constructively. That is, thecrests and troughs of the individual waves y1 and y2 occur at the same positions andcombine to form the red curve y of amplitude 2A shown in Figure 18.4a. Because the

y " 2A cos ! %

2 " sin !kx # $t !%

2 "

sin a ! sin b " 2 cos ! a # b2 " sin ! a ! b

2 "

S ECT I O N 18 . 1 • Superposition and Interference 547

y

= 0°

y1 and y2 are identical

x

yy1 y2 y

x

x

y

(a)

(b)

(c)

φ

yy1

y2

= 180°φ

= 60°φ

y

Active Figure 18.4 The superpositionof two identical waves y1 and y2 (blueand green) to yield a resultant wave(red). (a) When y1 and y2 are in phase,the result is constructive interference.(b) When y1 and y2 are & rad out ofphase, the result is destructive interfer-ence. (c) When the phase angle has avalue other than 0 or & rad, the resul-tant wave y falls somewhere between theextremes shown in (a) and (b).

At the Active Figures linkat http://www.pse6.com, youcan change the phase relation-ship between the waves andobserve the wave representingthe superposition.

Resultant of two travelingsinusoidal waves

individual waves are in phase, they are indistinguishable in Figure 18.4a, in which theyappear as a single blue curve. In general, constructive interference occurs whencos(%/2) " ( 1. This is true, for example, when % " 0, 2&, 4&, . . . rad—that is, when% is an even multiple of &.

When % is equal to & rad or to any odd multiple of &, then cos(%/2) " cos(&/2) "0, and the crests of one wave occur at the same positions as the troughs of the secondwave (Fig. 18.4b). Thus, the resultant wave has zero amplitude everywhere, as a conse-quence of destructive interference. Finally, when the phase constant has an arbitraryvalue other than 0 or an integer multiple of & rad (Fig. 18.4c), the resultant wave hasan amplitude whose value is somewhere between 0 and 2A.

Interference of Sound Waves

One simple device for demonstrating interference of sound waves is illustrated inFigure 18.5. Sound from a loudspeaker S is sent into a tube at point P, where there is aT-shaped junction. Half of the sound energy travels in one direction, and half travels inthe opposite direction. Thus, the sound waves that reach the receiver R can travelalong either of the two paths. The distance along any path from speaker to receiver iscalled the path length r. The lower path length r1 is fixed, but the upper path lengthr2 can be varied by sliding the U-shaped tube, which is similar to that on a slide trom-bone. When the difference in the path lengths )r " #r2 # r1 # is either zero or some in-teger multiple of the wavelength ' (that is )r " n', where n " 0, 1, 2, 3, . . .), the twowaves reaching the receiver at any instant are in phase and interfere constructively, asshown in Figure 18.4a. For this case, a maximum in the sound intensity is detected atthe receiver. If the path length r2 is adjusted such that the path difference )r " '/2,3'/2, . . . , n'/2 (for n odd), the two waves are exactly & rad, or 180°, out of phase atthe receiver and hence cancel each other. In this case of destructive interference, nosound is detected at the receiver. This simple experiment demonstrates that a phasedifference may arise between two waves generated by the same source when they travelalong paths of unequal lengths. This important phenomenon will be indispensable inour investigation of the interference of light waves in Chapter 37.

It is often useful to express the path difference in terms of the phase angle % be-tween the two waves. Because a path difference of one wavelength corresponds to aphase angle of 2& rad, we obtain the ratio %/2& " )r/' or

(18.1)

Using the notion of path difference, we can express our conditions for constructiveand destructive interference in a different way. If the path difference is any even multi-ple of '/2, then the phase angle % " 2n&, where n " 0, 1, 2, 3, . . . , and the interfer-ence is constructive. For path differences of odd multiples of '/2, % " (2n ! 1)&,where n " 0, 1, 2, 3, . . . , and the interference is destructive. Thus, we have theconditions

)r "%

2& '

548 C H A P T E R 18 • Superposition and Standing Waves

r 1

r 2

R

Speaker

S

PReceiver

Figure 18.5 An acoustical system for demonstrat-ing interference of sound waves. A sound wavefrom the speaker (S) propagates into the tubeand splits into two parts at point P. The two waves,which combine at the opposite side, are detectedat the receiver (R). The upper path length r2 canbe varied by sliding the upper section.

Relationship between pathdifference and phase angle

for constructive interference

and (18.2)

for destructive interference

This discussion enables us to understand why the speaker wires in a stereo systemshould be connected properly. When connected the wrong way—that is, when the posi-tive (or red) wire is connected to the negative (or black) terminal on one of the speakersand the other is correctly wired—the speakers are said to be “out of phase”—one speakercone moves outward while the other moves inward. As a consequence, the sound wavecoming from one speaker destructively interferes with the wave coming from the other—along a line midway between the two, a rarefaction region due to one speaker issuperposed on a compression region from the other speaker. Although the two soundsprobably do not completely cancel each other (because the left and right stereo signalsare usually not identical), a substantial loss of sound quality occurs at points alongthis line.

)r " (2n ! 1) '

2

)r " (2n) '

2

S E C T I O N 18 . 2 • Standing Waves 549

Example 18.1 Two Speakers Driven by the Same Source

3.00 m

8.00 m

r 2

r 1

O

0.350 m

1.85 m

P1.15 m

Figure 18.6 (Example 18.1) Two speakers emit sound waves toa listener at P.

18.2 Standing Waves

The sound waves from the speakers in Example 18.1 leave the speakers in the forwarddirection, and we considered interference at a point in front of the speakers. Supposethat we turn the speakers so that they face each other and then have them emit soundof the same frequency and amplitude. In this situation, two identical waves travel in

Figure 18.6 shows the physical arrangement of thespeakers, along with two shaded right triangles that can bedrawn on the basis of the lengths described in the problem.From these triangles, we find that the path lengths are

and

Hence, the path difference is r2 # r1 " 0.13 m. Because werequire that this path difference be equal to '/2 for the firstminimum, we find that ' " 0.26 m.

To obtain the oscillator frequency, we use Equation16.12, v " 'f, where v is the speed of sound in air, 343 m/s:

What If? What if the speakers were connected out ofphase? What happens at point P in Figure 18.6?

Answer In this situation, the path difference of '/2 com-bines with a phase difference of '/2 due to the incorrectwiring to give a full phase difference of '. As a result, thewaves are in phase and there is a maximum intensity atpoint P.

1.3 kHzf "v'

"343 m/s0.26 m

"

r2 " √(8.00 m)2 ! (1.85 m)2 " 8.21 m

r1 " √(8.00 m)2 ! (1.15 m)2 " 8.08 m

A pair of speakers placed 3.00 m apart are driven by thesame oscillator (Fig. 18.6). A listener is originally at point O,which is located 8.00 m from the center of the line connect-ing the two speakers. The listener then walks to point P,which is a perpendicular distance 0.350 m from O, beforereaching the first minimum in sound intensity. What is thefrequency of the oscillator?

Solution To find the frequency, we must know the wave-length of the sound coming from the speakers. With this in-formation, combined with our knowledge of the speed ofsound, we can calculate the frequency. The wavelength canbe determined from the interference information given. Thefirst minimum occurs when the two waves reaching the lis-tener at point P are 180° out of phase—in other words, whentheir path difference )r equals '/2. To calculate the path dif-ference, we must first find the path lengths r1 and r2 .

opposite directions in the same medium, as in Figure 18.7. These waves combine in ac-cordance with the superposition principle.

We can analyze such a situation by considering wave functions for two transverse si-nusoidal waves having the same amplitude, frequency, and wavelength but traveling inopposite directions in the same medium:

y1 " A sin(kx # $t) y2 " A sin(kx ! $t)

where y1 represents a wave traveling in the ! x direction and y2 represents one travelingin the # x direction. Adding these two functions gives the resultant wave function y:

y " y1 ! y2 " A sin(kx # $t) ! A sin(kx ! $t)

When we use the trigonometric identity sin(a ( b) " sin(a) cos(b) ( cos(a) sin(b),this expression reduces to

(18.3)

Equation 18.3 represents the wave function of a standing wave. A standing wave,such as the one shown in Figure 18.8, is an oscillation pattern with a stationary outlinethat results from the superposition of two identical waves traveling in opposite direc-tions.

Notice that Equation 18.3 does not contain a function of kx # $t. Thus, it is not anexpression for a traveling wave. If we observe a standing wave, we have no sense of mo-tion in the direction of propagation of either of the original waves. If we compare thisequation with Equation 15.6, we see that Equation 18.3 describes a special kind of sim-ple harmonic motion. Every element of the medium oscillates in simple harmonic mo-tion with the same frequency $ (according to the cos $t factor in the equation). How-ever, the amplitude of the simple harmonic motion of a given element (given by thefactor 2A sin kx, the coefficient of the cosine function) depends on the location x ofthe element in the medium.

The maximum amplitude of an element of the medium has a minimum value ofzero when x satisfies the condition sin kx " 0, that is, when

kx " &, 2&, 3&, . . .

Because k " 2&/', these values for kx give

(18.4)

These points of zero amplitude are called nodes.

x "'

2, ',

3'

2, * * * "

n'

2 n " 0, 1, 2, 3, * * *

y " (2A sin kx) cos $t

550 C H A P T E R 18 • Superposition and Standing Waves

v

v

Figure 18.7 Two speakers emitsound waves toward each other.When they overlap, identical wavestraveling in opposite directions willcombine to form standing waves.

Antinode Antinode

Node

2A sin kx

Node

Figure 18.8 Multiflash photograph of a standing wave on a string. The time behaviorof the vertical displacement from equilibrium of an individual element of the string isgiven by cos $t. That is, each element vibrates at an angular frequency $. The ampli-tude of the vertical oscillation of any elements of the string depends on the horizontalposition of the element. Each element vibrates within the confines of the envelopefunction 2A sin kx.

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! PITFALL PREVENTION18.2 Three Types of

AmplitudeWe need to distinguish carefullyhere between the amplitude ofthe individual waves, which is A,and the amplitude of the simpleharmonic motion of the elementsof the medium, which is 2A sin kx.A given element in a standingwave vibrates within the con-straints of the envelope function 2A sin kx, where x is that ele-ment’s position in the medium.This is in contrast to traveling sinusoidal waves, in which all ele-ments oscillate with the same am-plitude and the same frequency,and the amplitude A of the waveis the same as the amplitude A ofthe simple harmonic motion ofthe elements. Furthermore, wecan identify the amplitude of thestanding wave as 2A.

The element with the greatest possible displacement from equilibrium has an ampli-tude of 2A, and we define this as the amplitude of the standing wave. The positions inthe medium at which this maximum displacement occurs are called antinodes. Theantinodes are located at positions for which the coordinate x satisfies the conditionsin kx " ( 1, that is, when

Thus, the positions of the antinodes are given by

(18.5)

In examining Equations 18.4 and 18.5, we note the following important features ofthe locations of nodes and antinodes:

x "'

4,

3'

4,

5'

4, * * * "

n'

4 n " 1, 3, 5, * * *

kx "&

2,

3&

2,

5&

2, * * *

S E C T I O N 18 . 2 • Standing Waves 551

The distance between adjacent antinodes is equal to '/2.

The distance between adjacent nodes is equal to '/2.

The distance between a node and an adjacent antinode is '/4.

Wave patterns of the elements of the medium produced at various times bytwo waves traveling in opposite directions are shown in Figure 18.9. The blue andgreen curves are the wave patterns for the individual traveling waves, and the redcurves are the wave patterns for the resultant standing wave. At t " 0 (Fig. 18.9a),the two traveling waves are in phase, giving a wave pattern in which each element ofthe medium is experiencing its maximum displacement from equilibrium. Onequarter of a period later, at t " T/4 (Fig. 18.9b), the traveling waves have movedone quarter of a wavelength (one to the right and the other to the left). At thistime, the traveling waves are out of phase, and each element of the medium is pass-ing through the equilibrium position in its simple harmonic motion. The result iszero displacement for elements at all values of x—that is, the wave pattern is astraight line. At t " T/2 (Fig. 18.9c), the traveling waves are again in phase, produc-ing a wave pattern that is inverted relative to the t " 0 pattern. In the standing wave,the elements of the medium alternate in time between the extremes shown inFigure 18.9a and c.

(a) t = 0

y1

y2

yN N N N N

AA

(b) t = T/4

y2

y1

y

(c) t = T/2

y1

A A

y2

yN N N N N

A A

A A

Active Figure 18.9 Standing-wave patterns produced at various times by two waves ofequal amplitude traveling in opposite directions. For the resultant wave y, the nodes(N) are points of zero displacement, and the antinodes (A) are points of maximum dis-placement.

At the Active Figures linkat http://www.pse6.com, youcan choose the wavelength ofthe waves and see the standingwave that results.

Position of antinodes

552 C H A P T E R 18 • Superposition and Standing Waves

Quick Quiz 18.3 Consider a standing wave on a string as shown in Figure18.9. Define the velocity of elements of the string as positive if they are moving upwardin the figure. At the moment the string has the shape shown by the red curve in Figure18.9a, the instantaneous velocity of elements along the string (a) is zero for allelements (b) is positive for all elements (c) is negative for all elements (d) varies withthe position of the element.

Quick Quiz 18.4 Continuing with the scenario in Quick Quiz 18.3, at themoment the string has the shape shown by the red curve in Figure 18.9b, the instanta-neous velocity of elements along the string (a) is zero for all elements (b) is positive forall elements (c) is negative for all elements (d) varies with the position of the element.

Example 18.2 Formation of a Standing Wave

18.3 Standing Waves in a StringFixed at Both Ends

Consider a string of length L fixed at both ends, as shown in Figure 18.10. Standingwaves are set up in the string by a continuous superposition of waves incident on andreflected from the ends. Note that there is a boundary condition for the waves on the

Two waves traveling in opposite directions produce a stand-ing wave. The individual wave functions are

y1 " (4.0 cm) sin(3.0x # 2.0t)

y2 " (4.0 cm) sin(3.0x ! 2.0t)

where x and y are measured in centimeters.

(A) Find the amplitude of the simple harmonic motion ofthe element of the medium located at x " 2.3 cm.

Solution The standing wave is described by Equation 18.3;in this problem, we have A " 4.0 cm, k " 3.0 rad/cm, and$ " 2.0 rad/s. Thus,

y " (2A sin kx) cos $t " [(8.0 cm)sin 3.0x] cos 2.0t

Thus, we obtain the amplitude of the simple harmonic mo-tion of the element at the position x " 2.3 cm by evaluatingthe coefficient of the cosine function at this position:

(B) Find the positions of the nodes and antinodes if oneend of the string is at x " 0.

Solution With k " 2&/' " 3.0 rad/cm, we see that thewavelength is ' " (2&/3.0) cm. Therefore, from Equation18.4 we find that the nodes are located at

0, 1, 2, 3, . . .x " n '

2" n ! &

3 " cm n "

4.6 cm" (8.0 cm) sin (6.9 rad) "

y max " (8.0 cm)sin 3.0x #x"2.3

and from Equation 18.5 we find that the antinodes are lo-cated at

(C) What is the maximum value of the position in thesimple harmonic motion of an element located at anantinode?

Solution According to Equation 18.3, the maximum posi-tion of an element at an antinode is the amplitude of thestanding wave, which is twice the amplitude of the individualtraveling waves:

ymax " 2A(sin kx)max " 2(4.0 cm)(( 1) "

where we have used the fact that the maximum value ofsin kx is ( 1. Let us check this result by evaluating the coeffi-cient of our standing-wave function at the positions wefound for the antinodes:

In evaluating this expression, we have used the fact that nis an odd integer; thus, the sine function is equal to ( 1,depending on the value of n.

" (8.0 cm) sin $n ! &

2 " rad % " ( 8.0 cm

" (8.0 cm) sin $3.0n ! &

6 " rad %y max " (8.0 cm)sin 3.0x # x"n(&/6)

(8.0 cm

1, 3, 5, . . .x " n '

4" n ! &

6 " cm n "

string. The ends of the string, because they are fixed, must necessarily have zerodisplacement and are, therefore, nodes by definition. The boundary condition resultsin the string having a number of natural patterns of oscillation, called normal modes,each of which has a characteristic frequency that is easily calculated. This situation inwhich only certain frequencies of oscillation are allowed is called quantization. Quan-tization is a common occurrence when waves are subject to boundary conditions andwill be a central feature in our discussions of quantum physics in the extended versionof this text.

Figure 18.11 shows one of the normal modes of oscillation of a string fixed atboth ends. Except for the nodes, which are always stationary, all elements of thestring oscillate vertically with the same frequency but with different amplitudes ofsimple harmonic motion. Figure 18.11 represents snapshots of the standing wave atvarious times over one half of a period. The red arrows show the velocities of variouselements of the string at various times. As we found in Quick Quizzes 18.3 and 18.4,

S E C T I O N 18 . 3 • Standing Waves in a String Fixed at Both Ends 553

L

(a) (c)

(b) (d)

n = 2

n = 3

L = λ2

L = – λ332

n = 1 L = – λ112

f1 f3

f2

N

A

N

λ

λλ

Active Figure 18.10 (a) A string of length L fixed at both ends. The normal modes ofvibration form a harmonic series: (b) the fundamental, or first harmonic; (c) the sec-ond harmonic; (d) the third harmonic.

At the Active Figures linkat http://www.pse6.com, youcan choose the mode numberand see the correspondingstanding wave.

NN Nt = 0

(a)

(b) t = T/ 8

t = T/4(c)

t = 3T/ 8(d)

(e) t = T/ 2

Figure 18.11 A standing-wave pattern in a taut string. The five“snapshots” were taken at intervals of one eighth of the period.(a) At t " 0, the string is momentarily at rest. (b) At t " T/8,the string is in motion, as indicated by the red arrows, and dif-ferent parts of the string move in different directions with dif-ferent speeds. (c) At t " T/4, the string is moving but horizon-tal (undeformed). (d) The motion continues as indicated.(e) At t " T/2, the string is again momentarily at rest, but thecrests and troughs of (a) are reversed. The cycle continues until ultimately, when a time interval equal to T has passed, theconfiguration shown in (a) is repeated.

all elements of the string have zero velocity at the extreme positions (Figs. 18.11aand 18.11e) and elements have varying velocities at other positions (Figs. 18.11bthrough 18.11d).

The normal modes of oscillation for the string can be described by imposing therequirements that the ends be nodes and that the nodes and antinodes be separatedby one fourth of a wavelength. The first normal mode that is consistent with theboundary conditions, shown in Figure 18.10b, has nodes at its ends and one antinodein the middle. This is the longest-wavelength mode that is consistent with our require-ments. This first normal mode occurs when the length of the string is half the wave-length '1, as indicated in Figure 18.10b, or '1 " 2L. The next normal mode (see Fig.18.10c) of wavelength '2 occurs when the wavelength equals the length of the string,that is, when '2 " L. The third normal mode (see Fig. 18.10d) corresponds to the casein which '3 " 2L/3. In general, the wavelengths of the various normal modes for astring of length L fixed at both ends are

(18.6)

where the index n refers to the nth normal mode of oscillation. These are the possiblemodes of oscillation for the string. The actual modes that are excited on a string arediscussed shortly.

The natural frequencies associated with these modes are obtained from the rela-tionship f " v/', where the wave speed v is the same for all frequencies. Using Equa-tion 18.6, we find that the natural frequencies fn of the normal modes are

(18.7)

These natural frequencies are also called the quantized frequencies associated with the vi-brating string fixed at both ends.

Because (see Eq. 16.18), where T is the tension in the string and + is itslinear mass density, we can also express the natural frequencies of a taut string as

(18.8)

The lowest frequency f1, which corresponds to n " 1, is called either the fundamentalor the fundamental frequency and is given by

(18.9)

The frequencies of the remaining normal modes are integer multiples of the fun-damental frequency. Frequencies of normal modes that exhibit an integer-multiple re-lationship such as this form a harmonic series, and the normal modes are called

f1 "1

2L √ T

+

fn "n

2L √ T

+ n " 1, 2, 3, * * *

v " √T/+

fn " v'n

" n v

2L n " 1, 2, 3, * * *

'n "2Ln

n " 1, 2, 3, . . .

554 C H A P T E R 18 • Superposition and Standing Waves

Multiflash photographs of standing-wave patterns in a cord driven by a vibrator at itsleft end. The single-loop pattern represents the first normal mode (the fundamental),n " 1). The double-loop pattern represents the second normal mode (n " 2), and thetriple-loop pattern represents the third normal mode (n " 3).

Fundamental frequency of ataut string

Frequencies of normal modesas functions of string tensionand linear mass density

Frequencies of normal modesas functions of wave speed andlength of string

Wavelengths of normal modes

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harmonics. The fundamental frequency f1 is the frequency of the first harmonic; thefrequency f2 " 2f1 is the frequency of the second harmonic; and the frequency fn " nf1is the frequency of the nth harmonic. Other oscillating systems, such as a drumhead,exhibit normal modes, but the frequencies are not related as integer multiples ofa fundamental. Thus, we do not use the term harmonic in association with these typesof systems.

In obtaining Equation 18.6, we used a technique based on the separation distancebetween nodes and antinodes. We can obtain this equation in an alternative manner.Because we require that the string be fixed at x " 0 and x " L, the wave functiony(x, t) given by Equation 18.3 must be zero at these points for all times. That is, theboundary conditions require that y(0, t) " 0 and y(L, t) " 0 for all values of t. Becausethe standing wave is described by y " 2A(sin kx) cos $t, the first boundary condition,y(0, t) " 0, is automatically satisfied because sin kx " 0 at x " 0. To meet the secondboundary condition, y(L, t) " 0, we require that sin kL " 0. This condition is satisfiedwhen the angle kL equals an integer multiple of & rad. Therefore, the allowed valuesof k are given by1

knL " n& n " 1, 2, 3, . . . (18.10)

Because kn " 2&/'n , we find that

which is identical to Equation 18.6.Let us examine further how these various harmonics are created in a string. If we

wish to excite just a single harmonic, we must distort the string in such a way that itsdistorted shape corresponds to that of the desired harmonic. After being released,the string vibrates at the frequency of that harmonic. This maneuver is difficult toperform, however, and it is not how we excite a string of a musical instrument. If thestring is distorted such that its distorted shape is not that of just one harmonic, theresulting vibration includes various harmonics. Such a distortion occurs in musicalinstruments when the string is plucked (as in a guitar), bowed (as in a cello), orstruck (as in a piano). When the string is distorted into a nonsinusoidal shape, onlywaves that satisfy the boundary conditions can persist on the string. These are theharmonics.

The frequency of a string that defines the musical note that it plays is that of thefundamental. The frequency of the string can be varied by changing either the tensionor the string’s length. For example, the tension in guitar and violin strings is varied bya screw adjustment mechanism or by tuning pegs located on the neck of the instru-ment. As the tension is increased, the frequency of the normal modes increases inaccordance with Equation 18.8. Once the instrument is “tuned,” players vary the fre-quency by moving their fingers along the neck, thereby changing the length of theoscillating portion of the string. As the length is shortened, the frequency increasesbecause, as Equation 18.8 specifies, the normal-mode frequencies are inversely propor-tional to string length.

! 2&

'n" L " n& or 'n "

2Ln

S E C T I O N 18 . 3 • Standing Waves in a String Fixed at Both Ends 555

1 We exclude n " 0 because this value corresponds to the trivial case in which no wave exists (k " 0).

Quick Quiz 18.5 When a standing wave is set up on a string fixed at bothends, (a) the number of nodes is equal to the number of antinodes (b) the wavelengthis equal to the length of the string divided by an integer (c) the frequency is equal tothe number of nodes times the fundamental frequency (d) the shape of the string atany time is symmetric about the midpoint of the string.

556 C H A P T E R 18 • Superposition and Standing Waves

Example 18.3 Give Me a C Note!

Middle C on a piano has a fundamental frequency of 262 Hz,and the first A above middle C has a fundamental frequencyof 440 Hz.

(A) Calculate the frequencies of the next two harmonics ofthe C string.

Solution Knowing that the frequencies of higher harmon-ics are integer multiples of the fundamental frequencyf1 " 262 Hz, we find that

f2 " 2f1 "

f3 " 3f1 "

(B) If the A and C strings have the same linear mass density +and length L, determine the ratio of tensions in the two strings.

Solution Using Equation 18.9 for the two strings vibratingat their fundamental frequencies gives

f1A "1

2L √ TA

+ and f1C "

12L

√ TC

+

786 Hz

524 Hz

Setting up the ratio of these frequencies, we find that

What If? What if we look inside a real piano? In this case,the assumption we made in part (B) is only partially true. Thestring densities are equal, but the length of the A string isonly 64 percent of the length of the C string. What is the ratioof their tensions?

Answer Using Equation 18.8 again, we set up the ratio offrequencies:

TA

TC" (0.64)2 ! 440

262 "2

" 1.16

f1A

f1C"

L C

L A √ TA

TC" ! 100

64 " √ TA

TC

2.82TA

TC" ! f1A

f1C"2

" ! 440262 "

2"

f1A

f1C" √ TA

TC

Example 18.4 Guitar Basics

The high E string on a guitar measures 64.0 cm in lengthand has a fundamental frequency of 330 Hz. By pressingdown so that the string is in contact with the first fret (Fig.18.12), the string is shortened so that it plays an F note thathas a frequency of 350 Hz. How far is the fret from the neckend of the string?

Solution Equation 18.7 relates the string’s length to thefundamental frequency. With n " 1, we can solve for thespeed of the wave on the string,

Because we have not adjusted the tuning peg, the tension inthe string, and hence the wave speed, remain constant. Wecan again use Equation 18.7, this time solving for L and

v "2Ln

fn "2(0.640 m)

1 (330 Hz) " 422 m/s

substituting the new frequency to find the shortened stringlength:

The difference between this length and the measuredlength of 64.0 cm is the distance from the fret to the neckend of the string, or

What If? What if we wish to play an F sharp, which we doby pressing down on the second fret from the neck in Figure18.12? The frequency of F sharp is 370 Hz. Is this fret another3.7 cm from the neck?

Answer If you inspect a guitar fingerboard, you will findthat the frets are not equally spaced. They are far apart nearthe neck and close together near the opposite end. Conse-quently, from this observation, we would not expect the Fsharp fret to be another 3.7 cm from the end.

Let us repeat the calculation of the string length, thistime for the frequency of F sharp:

This gives a distance of 0.640 m # 0.571 m " 0.069 m "6.9 cm from the neck. Subtracting the distance from theneck to the first fret, the separation distance between thefirst and second frets is 6.9 cm # 3.7 cm " 3.2 cm.

L " n v

2fn" (1)

422 m/s2(370 Hz)

" 0.571 m

3.7 cm.

L " n v

2fn" (1)

422 m/s2(350 Hz)

" 0.603 m " 60.3 cm

Figure 18.12 (Example 18.4) Playing an F note on a guitar.

Char

les D

. Win

ters

Explore this situation at the Interactive Worked Example link at http://www.pse6.com.

Interactive

S E C T I O N 18 . 3 • Standing Waves in a String Fixed at Both Ends 557

Example 18.5 Changing String Vibration with Water Interactive

One end of a horizontal string is attached to a vibratingblade and the other end passes over a pulley as in Figure18.13a. A sphere of mass 2.00 kg hangs on the end of thestring. The string is vibrating in its second harmonic. A con-tainer of water is raised under the sphere so that the sphereis completely submerged. After this is done, the string vi-brates in its fifth harmonic, as shown in Figure 18.13b. Whatis the radius of the sphere?

Solution To conceptualize the problem, imagine what hap-pens when the sphere is immersed in the water. The buoy-ant force acts upward on the sphere, reducing the tension inthe string. The change in tension causes a change in thespeed of waves on the string, which in turn causes a changein the wavelength. This altered wavelength results in thestring vibrating in its fifth normal mode rather than the sec-ond. We categorize the problem as one in which we willneed to combine our understanding of Newton’s secondlaw, buoyant forces, and standing waves on strings. We beginto analyze the problem by studying Figure 18.13a. Newton’ssecond law applied to the sphere tells us that the tension inthe string is equal to the weight of the sphere:

&F " T1 # mg " 0

T1 " mg " (2.00 kg)(9.80 m/s2) " 19.6 N

where the subscript 1 is used to indicate initial variablesbefore we immerse the sphere in water. Once the sphere is

immersed in water, the tension in the string decreases toT2. Applying Newton’s second law to the sphere again in thissituation, we have

T2 ! B # mg " 0

(1) B " mg # T2

The desired quantity, the radius of the sphere, will appear inthe expression for the buoyant force B. Before proceedingin this direction, however, we must evaluate T2. We do thisfrom the standing wave information. We write the equationfor the frequency of a standing wave on a string (Equation18.8) twice, once before we immerse the sphere and onceafter, and divide the equations:

where the frequency f is the same in both cases, because itis determined by the vibrating blade. In addition, the linearmass density + and the length L of the vibrating portion ofthe string are the same in both cases. Solving for T2, wehave

Substituting this into Equation (1), we can evaluate thebuoyant force on the sphere:

B " mg # T2 " 19.6 N # 3.14 N " 16.5 N

Finally, expressing the buoyant force (Eq. 14.5) in termsof the radius of the sphere, we solve for the radius:

B " ,watergVsphere " ,waterg ( &r3)

To finalize this problem, note that only certain radii of thesphere will result in the string vibrating in a normal mode.This is because the speed of waves on the string must bechanged to a value such that the length of the string is an in-teger multiple of half wavelengths. This is a feature of thequantization that we introduced earlier in this chapter—thesphere radii that cause the string to vibrate in a normalmode are quantized.

7.38 cm" 7.38 - 10#2 m "

r " √3 3B4&,waterg

" √3 3(16.5 N)4&(1 000 kg/m3)(9.80 m/s2)

43

T2 " ! n1

n 2"2

T1 " ! 25 "

2(19.6 N) " 3.14 N

f "n1

2L √ T1

+

f "n2

2L √ T2

+

9: 1 "n1

n2 √ T1

T2

You can adjust the mass at the Interactive Worked Example link at http://www.pse6.com.

(b)

(a)

Figure 18.13 (Example 18.5) When the sphere hangs in air,the string vibrates in its second harmonic. When the sphere isimmersed in water, the string vibrates in its fifth harmonic.

18.4 Resonance

We have seen that a system such as a taut string is capable of oscillating in one ormore normal modes of oscillation. If a periodic force is applied to such a system,the amplitude of the resulting motion is greatest when the frequency of the ap-plied force is equal to one of the natural frequencies of the system. We discussedthis phenomenon, known as resonance, briefly in Section 15.7. Although ablock–spring system or a simple pendulum has only one natural frequency, standing-wave systems have a whole set of natural frequencies, such as that given by Equation18.7 for a string. Because an oscillating system exhibits a large amplitude when drivenat any of its natural frequencies, these frequencies are often referred to as resonancefrequencies.

Figure 18.14 shows the response of an oscillating system to various driving fre-quencies, where one of the resonance frequencies of the system is denoted by f0 . Notethat the amplitude of oscillation of the system is greatest when the frequency of thedriving force equals the resonance frequency. The maximum amplitude is limited byfriction in the system. If a driving force does work on an oscillating system that is ini-tially at rest, the input energy is used both to increase the amplitude of the oscillationand to overcome the friction force. Once maximum amplitude is reached, the workdone by the driving force is used only to compensate for mechanical energy loss dueto friction.

Examples of Resonance

A playground swing is a pendulum having a natural frequency that depends on itslength. Whenever we use a series of regular impulses to push a child in a swing, theswing goes higher if the frequency of the periodic force equals the natural frequencyof the swing. We can demonstrate a similar effect by suspending pendulums of differ-ent lengths from a horizontal support, as shown in Figure 18.15. If pendulum A is setinto oscillation, the other pendulums begin to oscillate as a result of waves transmittedalong the beam. However, pendulum C, the length of which is close to the length of A,oscillates with a much greater amplitude than pendulums B and D, the lengths ofwhich are much different from that of pendulum A. Pendulum C moves the way it doesbecause its natural frequency is nearly the same as the driving frequency associatedwith pendulum A.

Next, consider a taut string fixed at one end and connected at the opposite end toan oscillating blade, as illustrated in Figure 18.16. The fixed end is a node, and theend connected to the blade is very nearly a node because the amplitude of the blade’smotion is small compared with that of the elements of the string. As the blade oscil-lates, transverse waves sent down the string are reflected from the fixed end. As welearned in Section 18.3, the string has natural frequencies that are determined by itslength, tension, and linear mass density (see Eq. 18.8). When the frequency of theblade equals one of the natural frequencies of the string, standing waves are pro-duced and the string oscillates with a large amplitude. In this resonance case, thewave generated by the oscillating blade is in phase with the reflected wave, and thestring absorbs energy from the blade. If the string is driven at a frequency that is notone of its natural frequencies, then the oscillations are of low amplitude and exhibitno stable pattern.

Once the amplitude of the standing-wave oscillations is a maximum, the mechani-cal energy delivered by the blade and absorbed by the system is transformed to internalenergy because of the damping forces caused by friction in the system. If the appliedfrequency differs from one of the natural frequencies, energy is transferred to thestring at first, but later the phase of the wave becomes such that it forces the blade toreceive energy from the string, thereby reducing the energy in the string.

Resonance is very important in the excitation of musical instruments based on aircolumns. We shall discuss this application of resonance in Section 18.5.

558 C H A P T E R 18 • Superposition and Standing WavesA

mpl

itude

f0Frequency of driving force

Figure 18.14 Graph of the ampli-tude (response) versus driving fre-quency for an oscillating system.The amplitude is a maximum atthe resonance frequency f0.

A

B

C

D

Figure 18.15 An example of reso-nance. If pendulum A is set into os-cillation, only pendulum C, whoselength matches that of A, eventu-ally oscillates with large amplitude,or resonates. The arrows indicatemotion in a plane perpendicular tothe page.

Vibratingblade

Figure 18.16 Standing waves areset up in a string when one end isconnected to a vibrating blade.When the blade vibrates at one ofthe natural frequencies of thestring, large-amplitude standingwaves are created.

18.5 Standing Waves in Air Columns

Standing waves can be set up in a tube of air, such as that inside an organ pipe, as theresult of interference between longitudinal sound waves traveling in opposite direc-tions. The phase relationship between the incident wave and the wave reflected fromone end of the pipe depends on whether that end is open or closed. This relationshipis analogous to the phase relationships between incident and reflected transverse wavesat the end of a string when the end is either fixed or free to move (see Figs. 16.14 and16.15).

In a pipe closed at one end, the closed end is a displacement node because thewall at this end does not allow longitudinal motion of the air. As a result, at aclosed end of a pipe, the reflected sound wave is 180° out of phase with the incidentwave. Furthermore, because the pressure wave is 90° out of phase with the displace-ment wave (see Section 17.2), the closed end of an air column corresponds to apressure antinode (that is, a point of maximum pressure variation).

The open end of an air column is approximately a displacement antinode2

and a pressure node. We can understand why no pressure variation occurs at an openend by noting that the end of the air column is open to the atmosphere; thus, the pres-sure at this end must remain constant at atmospheric pressure.

You may wonder how a sound wave can reflect from an open end, as there may notappear to be a change in the medium at this point. It is indeed true that the medium

S E C T I O N 18 . 5 • Standing Waves in Air Columns 559

Quick Quiz 18.6 A wine glass can be shattered through resonance by main-taining a certain frequency of a high-intensity sound wave. Figure 18.17a shows a sideview of a wine glass vibrating in response to such a sound wave. Sketch the standing-wave pattern in the rim of the glass as seen from above. If an integral number of waves“fit” around the circumference of the vibrating rim, how many wavelengths fit aroundthe rim in Figure 18.17a?

(a) (b)

Figure 18.17 (Quick Quiz 18.6) (a) Standing-wave pattern in a vibrating wine glass.The glass shatters if the amplitude of vibration becomes too great. (b) A wine glass shat-tered by the amplified sound of a human voice.

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2 Strictly speaking, the open end of an air column is not exactly a displacement antinode. A com-pression reaching an open end does not reflect until it passes beyond the end. For a tube of circularcross section, an end correction equal to approximately 0.6R, where R is the tube’s radius, must beadded to the length of the air column. Hence, the effective length of the air column is longer than thetrue length L. We ignore this end correction in this discussion.

through which the sound wave moves is air both inside and outside the pipe. However,sound is a pressure wave, and a compression region of the sound wave is constrainedby the sides of the pipe as long as the region is inside the pipe. As the compression re-gion exits at the open end of the pipe, the constraint of the pipe is removed and thecompressed air is free to expand into the atmosphere. Thus, there is a change in thecharacter of the medium between the inside of the pipe and the outside even thoughthere is no change in the material of the medium. This change in character is sufficientto allow some reflection.

With the boundary conditions of nodes or antinodes at the ends of the air column,we have a set of normal modes of oscillation, as we do for the string fixed at both ends.Thus, the air column has quantized frequencies.

The first three normal modes of oscillation of a pipe open at both ends are shownin Figure 18.18a. Note that both ends are displacement antinodes (approximately). Inthe first normal mode, the standing wave extends between two adjacent antinodes,which is a distance of half a wavelength. Thus, the wavelength is twice the length of thepipe, and the fundamental frequency is f1 " v/2L. As Figure 18.18a shows, the fre-quencies of the higher harmonics are 2f1 , 3f1 , . . . . Thus, we can say that

560 C H A P T E R 18 • Superposition and Standing Waves

L

λ1 = 2Lf1 = — = —v

λ1

v2L

λ2 = Lf2 = — = 2f1

vL

λ3 = — L

f3 = — = 3f13v2L

23

(a) Open at both ends

λ1 = 4Lf1 = — = —v

λ1

v4L

λ3 = — L

f3 = — = 3f13v4L

λ5 = — L

f5 = — = 5f15v4L

45

43

First harmonic

Second harmonic

Third harmonic

First harmonic

Third harmonic

Fifth harmonic

(b) Closed at one end, open at the other

A AN

A A

A

AN N

AA A ANN N

N

A NAN

AA NNAN

λ

λ

λ

λ

λ

λ

λ

λ

Figure 18.18 Motion of elements of air in standing longitudinal waves in a pipe,along with schematic representations of the waves. In the schematic representations,the structure at the left end has the purpose of exciting the air column into a normalmode. The hole in the upper edge of the column assures that the left end acts as anopen end. The graphs represent the displacement amplitudes, not the pressure ampli-tudes. (a) In a pipe open at both ends, the harmonic series created consists of all inte-ger multiples of the fundamental frequency: f1 , 2f1 , 3f1 , . . . . (b) In a pipe closed atone end and open at the other, the harmonic series created consists of only odd-integermultiples of the fundamental frequency: f1 , 3f1 , 5f1 , . . . .

In a pipe open at both ends, the natural frequencies of oscillation form a harmonicseries that includes all integral multiples of the fundamental frequency.

! PITFALL PREVENTION18.3 Sound Waves in Air

Are Longitudinal, notTransverse

Note that the standing longitudi-nal waves are drawn as transversewaves in Figure 18.18. This is be-cause it is difficult to draw longitu-dinal displacements—they are inthe same direction as the propaga-tion. Thus, it is best to interpretthe curves in Figure 18.18 as agraphical representation of thewaves (our diagrams of stringwaves are pictorial representa-tions), with the vertical axis repre-senting horizontal displacementof the elements of the medium.

Because all harmonics are present, and because the fundamental frequency is given bythe same expression as that for a string (see Eq. 18.7), we can express the natural fre-quencies of oscillation as

(18.11)

Despite the similarity between Equations 18.7 and 18.11, you must remember that v inEquation 18.7 is the speed of waves on the string, whereas v in Equation 18.11 is thespeed of sound in air.

If a pipe is closed at one end and open at the other, the closed end is a displace-ment node (see Fig. 18.18b). In this case, the standing wave for the fundamental modeextends from an antinode to the adjacent node, which is one fourth of a wavelength.Hence, the wavelength for the first normal mode is 4L, and the fundamental frequencyis f1 " v/4L. As Figure 18.18b shows, the higher-frequency waves that satisfy our condi-tions are those that have a node at the closed end and an antinode at the open end;this means that the higher harmonics have frequencies 3f1 , 5f1 , . . . .

fn " n v

2L n " 1, 2, 3, . . .

S E C T I O N 18 . 5 • Standing Waves in Air Columns 561

In a pipe closed at one end, the natural frequencies of oscillation form a harmonicseries that includes only odd integral multiples of the fundamental frequency.

We express this result mathematically as

(18.12)

It is interesting to investigate what happens to the frequencies of instruments basedon air columns and strings during a concert as the temperature rises. The soundemitted by a flute, for example, becomes sharp (increases in frequency) as it warms upbecause the speed of sound increases in the increasingly warmer air inside the flute(consider Eq. 18.11). The sound produced by a violin becomes flat (decreases in fre-quency) as the strings thermally expand because the expansion causes their tension todecrease (see Eq. 18.8).

Musical instruments based on air columns are generally excited by resonance. Theair column is presented with a sound wave that is rich in many frequencies. The air col-umn then responds with a large-amplitude oscillation to the frequencies that matchthe quantized frequencies in its set of harmonics. In many woodwind instruments, theinitial rich sound is provided by a vibrating reed. In the brasses, this excitation is pro-vided by the sound coming from the vibration of the player’s lips. In a flute, the initialexcitation comes from blowing over an edge at the mouthpiece of the instrument. Thisis similar to blowing across the opening of a bottle with a narrow neck. The sound ofthe air rushing across the edge has many frequencies, including one that sets the aircavity in the bottle into resonance.

fn " n v

4L n " 1, 3, 5, . . .

Quick Quiz 18.7 A pipe open at both ends resonates at a fundamentalfrequency fopen. When one end is covered and the pipe is again made to resonate,the fundamental frequency is fclosed. Which of the following expressions describeshow these two resonant frequencies compare? (a) fclosed " fopen (b) fclosed "

fopen (c) fclosed " 2 fopen (d) fclosed " fopen

Quick Quiz 18.8 Balboa Park in San Diego has an outdoor organ. Whenthe air temperature increases, the fundamental frequency of one of the organ pipes(a) stays the same (b) goes down (c) goes up (d) is impossible to determine.

32

12

Natural frequencies of a pipeopen at both ends

Natural frequencies of a pipeclosed at one end and open atthe other

562 C H A P T E R 18 • Superposition and Standing Waves

Example 18.6 Wind in a Culvert

A section of drainage culvert 1.23 m in length makes ahowling noise when the wind blows.

(A) Determine the frequencies of the first three harmonicsof the culvert if it is cylindrical in shape and open at bothends. Take v " 343 m/s as the speed of sound in air.

Solution The frequency of the first harmonic of a pipeopen at both ends is

Because both ends are open, all harmonics are present;thus,

f2 " 2f1 " and f3 " 3f1 "

(B) What are the three lowest natural frequencies of theculvert if it is blocked at one end?

Solution The fundamental frequency of a pipe closed atone end is

417 Hz278 Hz

139 Hzf1 "v

2L"

343 m/s2(1.23 m)

"

In this case, only odd harmonics are present; hence, the

next two harmonics have frequencies f3 " 3f1 "

and f5 " 5f1 "

(C) For the culvert open at both ends, how many of the har-monics present fall within the normal human hearing range(20 to 20 000 Hz)?

Solution Because all harmonics are present for a pipe openat both ends, we can express the frequency of the highestharmonic heard as fn " nf1 where n is the number of har-monics that we can hear. For fn " 20 000 Hz, we find that thenumber of harmonics present in the audible range is

Only the first few harmonics are of sufficient amplitude tobe heard.

143n " 20 000 Hz

139 Hz "

349 Hz.

209 Hz

69.7 Hzf1 "v

4L"

343 m/s4(1.23 m)

"

Example 18.7 Measuring the Frequency of a Tuning Fork

A simple apparatus for demonstrating resonance in an aircolumn is depicted in Figure 18.19. A vertical pipe open atboth ends is partially submerged in water, and a tuning forkvibrating at an unknown frequency is placed near the top ofthe pipe. The length L of the air column can be adjusted bymoving the pipe vertically. The sound waves generated bythe fork are reinforced when L corresponds to one of theresonance frequencies of the pipe.

For a certain pipe, the smallest value of L forwhich a peak occurs in the sound intensity is 9.00 cm. Whatare

(A) the frequency of the tuning fork

(B) the values of L for the next two resonance frequen-cies?

Solution

(A) Although the pipe is open at its lower end to allow thewater to enter, the water’s surface acts like a wall at one end.Therefore, this setup can be modeled as an air columnclosed at one end, and so the fundamental frequency isgiven by f1 " v/4L. Taking v " 343 m/s for the speed ofsound in air and L " 0.090 0 m, we obtain

Because the tuning fork causes the air column to resonate atthis frequency, this must also be the frequency of the tuningfork.

(B) Because the pipe is closed at one end, we know fromFigure 18.18b that the wavelength of the fundamentalmode is ' " 4L " 4(0.090 0 m) " 0.360 m. Because thefrequency of the tuning fork is constant, the next twonormal modes (see Fig. 18.19b) correspond to lengths of

L " 3'/4 " and L " 5'/4 " 0.450 m.0.270 m

953 Hzf1 "v

4L"

343 m/s4(0.090 0 m)

"

L

Water

f = ?

Firstresonance

Secondresonance

(thirdharmonic) Third

resonance(fifth

harmonic)

(b)

(a)

λ/4 3λ/4

5λ/4

λ λ

λ

Figure 18.19 (Example 18.7) (a) Apparatus for demon-strating the resonance of sound waves in a pipe closed atone end. The length L of the air column is varied bymoving the pipe vertically while it is partially submerged inwater. (b) The first three normal modes of the system shownin part (a).

S E C T I O N 18 . 6 • Standing Waves in Rods and Membranes 563

18.6 Standing Waves in Rods and Membranes

Standing waves can also be set up in rods and membranes. A rod clamped in the mid-dle and stroked parallel to the rod at one end oscillates, as depicted in Figure 18.20a.The oscillations of the elements of the rod are longitudinal, and so the broken lines inFigure 18.20 represent longitudinal displacements of various parts of the rod. For clar-ity, we have drawn them in the transverse direction, just as we did for air columns. Themidpoint is a displacement node because it is fixed by the clamp, whereas the ends aredisplacement antinodes because they are free to oscillate. The oscillations in this setupare analogous to those in a pipe open at both ends. The broken lines in Figure 18.20arepresent the first normal mode, for which the wavelength is 2L and the frequency isf " v/2L, where v is the speed of longitudinal waves in the rod. Other normal modesmay be excited by clamping the rod at different points. For example, the secondnormal mode (Fig. 18.20b) is excited by clamping the rod a distance L/4 away fromone end.

Musical instruments that depend on standing waves in rods include triangles,marimbas, xylophones, glockenspiels, chimes, and vibraphones. Other devices thatmake sounds from bars include music boxes and wind chimes.

Two-dimensional oscillations can be set up in a flexible membrane stretched over acircular hoop, such as that in a drumhead. As the membrane is struck at some point,waves that arrive at the fixed boundary are reflected many times. The resulting soundis not harmonic because the standing waves have frequencies that are not related by in-teger multiples. Without this relationship, the sound may be more correctly describedas noise than as music. This is in contrast to the situation in wind and stringed instru-ments, which produce sounds that we describe as musical.

Some possible normal modes of oscillation for a two-dimensional circular mem-brane are shown in Figure 18.21. While nodes are points in one-dimensional standing

A N A

λ1 = 2L

(a)

L

f1 = – = –vλ1

v2L

λ

λ

Figure 18.20 Normal-mode longi-tudinal vibrations of a rod oflength L (a) clamped at the middleto produce the first normal modeand (b) clamped at a distance L/4from one end to produce the sec-ond normal mode. Note that thebroken lines represent oscillationsparallel to the rod (longitudinalwaves).01 11 21 02 31 12

1 1.59 2.14 2.30 2.65 2.92

41 22 03 51 32 61

3.16 3.50 3.60 3.65 4.06 4.15

Elements of the medium moving outof the page at an instant of time.

Elements of the medium moving intothe page at an instant of time.

Figure 18.21 Representation of some of the normal modes possible in a circular mem-brane fixed at its perimeter. The pair of numbers above each pattern corresponds tothe number of radial nodes and the number of circular nodes. Below each pattern is afactor by which the frequency of the mode is larger than that of the 01 mode. The fre-quencies of oscillation do not form a harmonic series because these factors are not inte-gers. In each diagram, elements of the membrane on either side of a nodal line movein opposite directions, as indicated by the colors. (Adapted from T. D. Rossing, The Sci-ence of Sound, 2nd ed, Reading, Massachusetts, Addison-Wesley Publishing Co., 1990)

NA A

(b)

L4–

A N

λ2 = L

f2 = – = 2f1vL

λ

564 C H A P T E R 18 • Superposition and Standing Waves

waves on strings and in air columns, a two-dimensional oscillator has curves alongwhich there is no displacement of the elements of the medium. The lowest normalmode, which has a frequency f1 , contains only one nodal curve; this curve runsaround the outer edge of the membrane. The other possible normal modes show ad-ditional nodal curves that are circles and straight lines across the diameter of themembrane.

18.7 Beats: Interference in Time

The interference phenomena with which we have been dealing so far involve the su-perposition of two or more waves having the same frequency. Because the amplitude ofthe oscillation of elements of the medium varies with the position in space of the ele-ment, we refer to the phenomenon as spatial interference. Standing waves in strings andpipes are common examples of spatial interference.

We now consider another type of interference, one that results from the superposi-tion of two waves having slightly different frequencies. In this case, when the two wavesare observed at the point of superposition, they are periodically in and out of phase.That is, there is a temporal (time) alternation between constructive and destructive in-terference. As a consequence, we refer to this phenomenon as interference in time or tem-poral interference. For example, if two tuning forks of slightly different frequencies arestruck, one hears a sound of periodically varying amplitude. This phenomenon iscalled beating:

Beating is the periodic variation in amplitude at a given point due to the superposi-tion of two waves having slightly different frequencies.

The number of amplitude maxima one hears per second, or the beat frequency,equals the difference in frequency between the two sources, as we shall show below.The maximum beat frequency that the human ear can detect is about 20 beats/s.When the beat frequency exceeds this value, the beats blend indistinguishably with thesounds producing them.

A piano tuner can use beats to tune a stringed instrument by “beating” a noteagainst a reference tone of known frequency. The tuner can then adjust the string ten-sion until the frequency of the sound it emits equals the frequency of the referencetone. The tuner does this by tightening or loosening the string until the beats pro-duced by it and the reference source become too infrequent to notice.

Consider two sound waves of equal amplitude traveling through a medium withslightly different frequencies f1 and f2 . We use equations similar to Equation 16.10to represent the wave functions for these two waves at a point that we choose asx " 0:

y1 " A cos $1t " A cos 2&f1t

y2 " A cos $2t " A cos 2&f2t

Using the superposition principle, we find that the resultant wave function at this point is

y " y1 ! y2 " A(cos 2&f1t ! cos 2&f2t)

The trigonometric identity

cos a ! cos b " 2 cos ! a # b2 " cos ! a ! b

2 "

Definition of beating

allows us to write the expression for y as

(18.13)

Graphs of the individual waves and the resultant wave are shown in Figure 18.22. Fromthe factors in Equation 18.13, we see that the resultant sound for a listener standing atany given point has an effective frequency equal to the average frequency ( f1 ! f2)/2and an amplitude given by the expression in the square brackets:

(18.14)

That is, the amplitude and therefore the intensity of the resultant sound vary intime. The broken blue line in Figure 18.22b is a graphical representation of Equation18.14 and is a sine wave varying with frequency ( f1 # f2)/2.

Note that a maximum in the amplitude of the resultant sound wave is detectedwhenever

This means there are two maxima in each period of the resultant wave. Because the am-plitude varies with frequency as ( f1 # f2)/2, the number of beats per second, or thebeat frequency f beat, is twice this value. That is,

(18.15)

For instance, if one tuning fork vibrates at 438 Hz and a second one vibrates at442 Hz, the resultant sound wave of the combination has a frequency of 440 Hz (themusical note A) and a beat frequency of 4 Hz. A listener would hear a 440-Hz soundwave go through an intensity maximum four times every second.

f beat " # f1 # f 2 #

cos 2& ! f 1 # f 2

2 "t " ( 1

A resultant " 2A cos 2& ! f1 # f2

2 " t

y " $2A cos 2& ! f1 # f 2

2 " t% cos 2& ! f1 ! f 2

2 " t

S E C T I O N 18 . 7 • Beats: Interference in Time 565

y

(a)

(b)

y

t

t

Active Figure 18.22 Beats are formed by the combination of two waves of slightly dif-ferent frequencies. (a) The individual waves. (b) The combined wave has an amplitude(broken line) that oscillates in time.

At the Active Figures linkat http://www.pse6.com, youcan choose the two frequenciesand see the correspondingbeats.

Resultant of two waves ofdifferent frequencies but equalamplitude

Beat frequency

Quick Quiz 18.9 You are tuning a guitar by comparing the sound of thestring with that of a standard tuning fork. You notice a beat frequency of 5 Hz whenboth sounds are present. You tighten the guitar string and the beat frequency rises to8 Hz. In order to tune the string exactly to the tuning fork, you should (a) continue totighten the string (b) loosen the string (c) impossible to determine.

18.8 Nonsinusoidal Wave Patterns

The sound wave patterns produced by the majority of musical instruments are nonsi-nusoidal. Characteristic patterns produced by a tuning fork, a flute, and a clarinet,each playing the same note, are shown in Figure 18.23. Each instrument has its owncharacteristic pattern. Note, however, that despite the differences in the patterns, eachpattern is periodic. This point is important for our analysis of these waves.

It is relatively easy to distinguish the sounds coming from a violin and a saxophoneeven when they are both playing the same note. On the other hand, an individual un-trained in music may have difficulty distinguishing a note played on a clarinet from thesame note played on an oboe. We can use the pattern of the sound waves from varioussources to explain these effects.

This is in contrast to a musical instrument that makes a noise, such as the drum, inwhich the combination of frequencies do not form a harmonic series. When frequen-cies that are integer multiples of a fundamental frequency are combined, the result is amusical sound. A listener can assign a pitch to the sound, based on the fundamentalfrequency. Pitch is a psychological reaction to a sound that allows the listener to placethe sound on a scale of low to high (bass to treble). Combinations of frequencies thatare not integer multiples of a fundamental result in a noise, rather than a musicalsound. It is much harder for a listener to assign a pitch to a noise than to a musicalsound.

The wave patterns produced by a musical instrument are the result of the superpo-sition of various harmonics. This superposition results in the corresponding richnessof musical tones. The human perceptive response associated with various mixtures ofharmonics is the quality or timbre of the sound. For instance, the sound of the trumpetis perceived to have a “brassy” quality (that is, we have learned to associate the adjec-tive brassy with that sound); this quality enables us to distinguish the sound of thetrumpet from that of the saxophone, whose quality is perceived as “reedy.” The clar-inet and oboe, however, both contain air columns excited by reeds; because of thissimilarity, it is more difficult for the ear to distinguish them on the basis of theirsound quality.

The problem of analyzing nonsinusoidal wave patterns appears at first sight to be aformidable task. However, if the wave pattern is periodic, it can be represented asclosely as desired by the combination of a sufficiently large number of sinusoidal wavesthat form a harmonic series. In fact, we can represent any periodic function as a seriesof sine and cosine terms by using a mathematical technique based on Fourier’stheorem.3 The corresponding sum of terms that represents the periodic wave pattern

566 C H A P T E R 18 • Superposition and Standing Waves

Example 18.8 The Mistuned Piano Strings

Tuning fork

Flute

Clarinet

(a)

(b)

(c)

t

t

t

Figure 18.23 Sound wave patternsproduced by (a) a tuning fork,(b) a flute, and (c) a clarinet, eachat approximately the same fre-quency. (Adapted from C. A. Culver,Musical Acoustics, 4th ed., NewYork, McGraw-Hill Book Company,1956, p. 128.)

3 Developed by Jean Baptiste Joseph Fourier (1786–1830).

Two identical piano strings of length 0.750 m are eachtuned exactly to 440 Hz. The tension in one of the strings isthen increased by 1.0%. If they are now struck, what is thebeat frequency between the fundamentals of the two strings?

Solution We find the ratio of frequencies if the tension inone string is 1.0% larger than the other:

" 1.005

f2

f1"

(v 2/2L)(v1/2L)

"v2

v1"

√T2/+

√T1/+" √ T2

T1" √ 1.010T1

T1

Thus, the frequency of the tightened string is

f2 " 1.005f1 " 1.005(440 Hz) " 442 Hz

and the beat frequency is

f beat " 442 Hz # 440 Hz " 2 Hz.

is called a Fourier series. Let y(t) be any function that is periodic in time with periodT, such that y(t ! T ) " y(t). Fourier’s theorem states that this function can bewritten as

(18.16)

where the lowest frequency is f1 " 1/T. The higher frequencies are integer multiplesof the fundamental, fn " nf1 , and the coefficients An and Bn represent the amplitudesof the various waves. Figure 18.24 represents a harmonic analysis of the wave patternsshown in Figure 18.23. Note that a struck tuning fork produces only one harmonic(the first), whereas the flute and clarinet produce the first harmonic and many higherones.

Note the variation in relative intensity of the various harmonics for the fluteand the clarinet. In general, any musical sound consists of a fundamental frequencyf plus other frequencies that are integer multiples of f, all having differentintensities.

y(t) " &n

(An sin 2&fnt ! Bn cos 2&fnt)

S E C T I O N 18 . 8 • Nonsinusoidal Wave Patterns 567

Rel

ativ

e in

tens

ity

Tuningfork

1 2 3 4 5 6

Rel

ativ

e in

tens

ity

1 2 3 4 5 6 7

Flute

Rel

ativ

e in

tens

ity

1 2 3 4 5 6 7 8 9

Clarinet

Harmonics

(c)

Harmonics

(b)

Harmonics

(a)

Figure 18.24 Harmonics of the wave patterns shown in Figure 18.23. Note the varia-tions in intensity of the various harmonics. (Adapted from C. A. Culver, Musical Acoustics,4th ed., New York, McGraw-Hill Book Company, 1956.)

(b)(a) (c)

Each musical instrument has its own characteristic sound and mixture of harmonics.Instruments shown are (a) the violin, (b) the saxophone, and (c) the trumpet.

Fourier’s theorem

! PITFALL PREVENTION18.4 Pitch vs. FrequencyDo not confuse the term pitch withfrequency. Frequency is the physicalmeasurement of the number ofoscillations per second. Pitch is apsychological reaction to soundthat enables a person to place thesound on a scale from high to low,or from treble to bass. Thus, fre-quency is the stimulus and pitch isthe response. Although pitch is re-lated mostly (but not completely)to frequency, they are not thesame. A phrase such as “the pitchof the sound” is incorrect becausepitch is not a physical property ofthe sound.

Phot

ogra

phs c

ourte

sy o

f (a)

©19

89 G

ary B

uss/

FPG;

(b

) and

(c) ©

1989

Rich

ard

Laird

/FPG

We have discussed the analysis of a wave pattern using Fourier’s theorem. Theanalysis involves determining the coefficients of the harmonics in Equation 18.16 froma knowledge of the wave pattern. The reverse process, called Fourier synthesis, can alsobe performed. In this process, the various harmonics are added together to form a re-sultant wave pattern. As an example of Fourier synthesis, consider the building of asquare wave, as shown in Figure 18.25. The symmetry of the square wave results in onlyodd multiples of the fundamental frequency combining in its synthesis. In Figure18.25a, the orange curve shows the combination of f and 3f. In Figure 18.25b, we haveadded 5f to the combination and obtained the green curve. Notice how the generalshape of the square wave is approximated, even though the upper and lower portionsare not flat as they should be.

Figure 18.25c shows the result of adding odd frequencies up to 9f. This ap-proximation (purple curve) to the square wave is better than the approximationsin parts a and b. To approximate the square wave as closely as possible, we would need to add all odd multiples of the fundamental frequency, up to infinite frequency.

Using modern technology, we can generate musical sounds electronically by mixing different amplitudes of any number of harmonics. These widely used elec-tronic music synthesizers are capable of producing an infinite variety of musical tones.

568 C H A P T E R 18 • Superposition and Standing Waves

(c)

f + 3f + 5f + 7f + 9f

Square wavef + 3f + 5f + 7f + 9f + ...

(b)

f + 3f + 5f

5f

f

3f

(a)

ff + 3f

3f

Active Figure 18.25 Fourier synthesis of a square wave, which is represented by the sum of oddmultiples of the first harmonic, which has frequency f. (a) Waves of frequency f and 3f are added.(b) One more odd harmonic of frequency 5f is added. (c) The synthesis curve approaches closerto the square wave when odd frequencies up to 9f are added.

At the Active Figures linkat http://www.pse6.com, youcan add in harmonics withfrequencies higher than 9f totry to synthesize a square wave.

Questions 569

The superposition principle specifies that when two or more waves move througha medium, the value of the resultant wave function equals the algebraic sum of the val-ues of the individual wave functions.

When two traveling waves having equal amplitudes and frequencies superimpose,the resultant wave has an amplitude that depends on the phase angle % between thetwo waves. Constructive interference occurs when the two waves are in phase, corre-sponding to % " 0, 2&, 4&, . . . rad. Destructive interference occurs when the twowaves are 180° out of phase, corresponding to % " &, 3&, 5&, . . . rad.

Standing waves are formed from the superposition of two sinusoidal waves havingthe same frequency, amplitude, and wavelength but traveling in opposite directions.The resultant standing wave is described by the wave function

y " (2A sin kx) cos $t (18.3)

Hence, the amplitude of the standing wave is 2A, and the amplitude of the simple har-monic motion of any particle of the medium varies according to its position as 2A sin kx.The points of zero amplitude (called nodes) occur at x " n'/2 (n " 0, 1, 2, 3, . . .).The maximum amplitude points (called antinodes) occur at x " n'/4 (n " 1, 3,5, . . .). Adjacent antinodes are separated by a distance '/2. Adjacent nodes also areseparated by a distance '/2.

The natural frequencies of vibration of a taut string of length L and fixed at bothends are quantized and are given by

n " 1, 2, 3, . . . (18.8)

where T is the tension in the string and + is its linear mass density. The natural fre-quencies of vibration f1, 2f1, 3f1, . . . form a harmonic series.

An oscillating system is in resonance with some driving force whenever the fre-quency of the driving force matches one of the natural frequencies of the system. Whenthe system is resonating, it responds by oscillating with a relatively large amplitude.

Standing waves can be produced in a column of air inside a pipe. If the pipe is openat both ends, all harmonics are present and the natural frequencies of oscillation are

n " 1, 2, 3, . . . (18.11)

If the pipe is open at one end and closed at the other, only the odd harmonics are present, and the natural frequencies of oscillation are

n " 1, 3, 5, . . . (18.12)

The phenomenon of beating is the periodic variation in intensity at a given point dueto the superposition of two waves having slightly different frequencies.

fn " nv

4L

fn " nv

2L

fn "n

2L √ T+

S U M M A RY

Take a practice test forthis chapter by clicking onthe Practice Test link athttp://www.pse6.com.

Does the phenomenon of wave interference apply only tosinusoidal waves?

2. As oppositely moving pulses of the same shape (one up-ward, one downward) on a string pass through each other,there is one instant at which the string shows no displace-ment from the equilibrium position at any point. Has theenergy carried by the pulses disappeared at this instant oftime? If not, where is it?

1. 3. Can two pulses traveling in opposite directions on thesame string reflect from each other? Explain.

When two waves interfere, can the amplitude of the resul-tant wave be greater than either of the two original waves?Under what conditions?

5. For certain positions of the movable section shown in Fig-ure 18.5, no sound is detected at the receiver—a situation

4.

Q U E S T I O N S

570 C H A P T E R 18 • Superposition and Standing Waves

corresponding to destructive interference. This suggeststhat energy is somehow lost. What happens to the energytransmitted by the speaker?

When two waves interfere constructively or destructively, isthere any gain or loss in energy? Explain.

7. A standing wave is set up on a string, as shown in Figure18.10. Explain why no energy is transmitted along thestring.

8. What limits the amplitude of motion of a real vibratingsystem that is driven at one of its resonant frequencies?

9. Explain why your voice seems to sound better than usualwhen you sing in the shower.

10. What is the purpose of the slide on a trombone or of thevalves on a trumpet?

11. Explain why all harmonics are present in an organ pipeopen at both ends, but only odd harmonics are present ina pipe closed at one end.

12. Explain how a musical instrument such as a piano may betuned by using the phenomenon of beats.

13. To keep animals away from their cars, some people mountshort, thin pipes on the fenders. The pipes give out a high-pitched wail when the cars are moving. How do they createthe sound?

14. When a bell is rung, standing waves are set up around thebell’s circumference. What boundary conditions mustbe satisfied by the resonant wavelengths? How does a crackin the bell, such as in the Liberty Bell, affect the satisfyingof the boundary conditions and the sound emanatingfrom the bell?

15. An archer shoots an arrow from a bow. Does the string of thebow exhibit standing waves after the arrow leaves? If so, andif the bow is perfectly symmetric so that the arrow leavesfrom the center of the string, what harmonics are excited?

6.

16. Despite a reasonably steady hand, a person often spills hiscoffee when carrying it to his seat. Discuss resonance as apossible cause of this difficulty, and devise a means for solv-ing the problem.

An airplane mechanic notices that the sound from a twin-engine aircraft rapidly varies in loudness when both en-gines are running. What could be causing this variationfrom loud to soft?

18. When the base of a vibrating tuning fork is placed againsta chalkboard, the sound that it emits becomes louder. Thisis because the vibrations of the tuning fork are transmittedto the chalkboard. Because it has a larger area than thetuning fork, the vibrating chalkboard sets more air into vi-bration. Thus, the chalkboard is a better radiator of soundthan the tuning fork. How does this affect the length oftime during which the fork vibrates? Does this agree withthe principle of conservation of energy?

19. If you wet your finger and lightly run it around the rim ofa fine wineglass, a high-frequency sound is heard. Why?How could you produce various musical notes with a set ofwineglasses, each of which contains a different amount ofwater?

20. If you inhale helium from a balloon and do your best tospeak normally, your voice will have a comical quacky qual-ity. Explain why this “Donald Duck effect” happens. Cau-tion : Helium is an asphyxiating gas and asphyxiation cancause panic. Helium can contain poisonous contaminants.

21. You have a standard tuning fork whose frequency is 262 Hzand a second tuning fork with an unknown frequency.When you tap both of them on the heel of one of yoursneakers, you hear beats with a frequency of 4 per second.Thoughtfully chewing your gum, you wonder whether theunknown frequency is 258 Hz or 266 Hz. How can youdecide?

17.

Section 18.1 Superposition and Interference

1. Two waves in one string are described by the wave functions

y1 " 3.0 cos(4.0x # 1.6t)

and

y2 " 4.0 sin(5.0x # 2.0t)

where y and x are in centimeters and t is in seconds. Find thesuperposition of the waves y1 ! y2 at the points (a) x " 1.00,t " 1.00, (b) x " 1.00, t " 0.500, and (c) x " 0.500, t " 0.(Remember that the arguments of the trigonometric func-tions are in radians.)

2. Two pulses A and B are moving in opposite directionsalong a taut string with a speed of 2.00 cm/s. The ampli-

tude of A is twice the amplitude of B. The pulses areshown in Figure P18.2 at t " 0. Sketch the shape of thestring at t " 1, 1.5, 2, 2.5, and 3 s.

1, 2, 3 " straightforward, intermediate, challenging " full solution available in the Student Solutions Manual and Study Guide

" coached solution with hints available at http://www.pse6.com " computer useful in solving problem

" paired numerical and symbolic problems

P R O B L E M S

4

y(cm)

2.00 cm/s

–2.00 cm/s

x(cm)

2

2 4 6 8 10 12 14 16 18 20

AB

Figure P18.2

Problems 571

Two pulses traveling on the same string are described by

(a) In which direction does each pulse travel? (b) At whattime do the two cancel everywhere? (c) At what point dothe two pulses always cancel?

4. Two waves are traveling in the same direction along astretched string. The waves are 90.0° out of phase. Eachwave has an amplitude of 4.00 cm. Find the amplitude ofthe resultant wave.

Two traveling sinusoidal waves are described by thewave functions

y1 " (5.00 m) sin[&(4.00x # 1 200t)]

and

y2 " (5.00 m) sin[&(4.00x # 1 200t # 0.250)]

where x, y1 , and y2 are in meters and t is in seconds.(a) What is the amplitude of the resultant wave? (b) Whatis the frequency of the resultant wave?

6. Two identical sinusoidal waves with wavelengths of 3.00 mtravel in the same direction at a speed of 2.00 m/s. Thesecond wave originates from the same point as the first,but at a later time. Determine the minimum possible timeinterval between the starting moments of the two waves ifthe amplitude of the resultant wave is the same as that ofeach of the two initial waves.

7. Review problem. A series of pulses, each of amplitude0.150 m, is sent down a string that is attached to a post atone end. The pulses are reflected at the post and travelback along the string without loss of amplitude. What isthe net displacement at a point on the string where twopulses are crossing, (a) if the string is rigidly attached tothe post? (b) if the end at which reflection occurs is free toslide up and down?

8. Two loudspeakers are placed on a wall 2.00 m apart. A lis-tener stands 3.00 m from the wall directly in front of oneof the speakers. A single oscillator is driving the speakersat a frequency of 300 Hz. (a) What is the phase differencebetween the two waves when they reach the observer?(b) What If? What is the frequency closest to 300 Hz towhich the oscillator may be adjusted such that the observerhears minimal sound?

9. Two speakers are driven by the same oscillator whose fre-quency is 200 Hz. They are located on a vertical pole adistance of 4.00 m from each other. A man walks straighttoward the lower speaker in a direction perpendicular tothe pole as shown in Figure P18.9. (a) How many timeswill he hear a minimum in sound intensity, and (b) howfar is he from the pole at these moments? Take the speedof sound to be 330 m/s and ignore any sound reflectionscoming off the ground.

10. Two speakers are driven by the same oscillator whose fre-quency is f. They are located a distance d from each otheron a vertical pole. A man walks straight toward the lower

5.

y1 "5

(3x # 4t)2 ! 2 and y2 "

#5(3x ! 4t # 6)2 ! 2

3.

speaker in a direction perpendicular to the pole, as shownin Figure P18.9. (a) How many times will he hear a mini-mum in sound intensity? (b) How far is he from the poleat these moments? Let v represent the speed of sound, andassume that the ground does not reflect sound.

Two sinusoidal waves in a string are defined by thefunctions

y1 " (2.00 cm) sin(20.0x # 32.0t)

and

y2 " (2.00 cm) sin(25.0x # 40.0t)

where y1, y2, and x are in centimeters and t is in seconds.(a) What is the phase difference between these two wavesat the point x " 5.00 cm at t " 2.00 s? (b) What is the posi-tive x value closest to the origin for which the two phasesdiffer by ( & at t " 2.00 s? (This is where the two wavesadd to zero.)

12. Two identical speakers 10.0 m apart are driven by the sameoscillator with a frequency of f " 21.5 Hz (Fig. P18.12).(a) Explain why a receiver at point A records a minimum insound intensity from the two speakers. (b) If the receiver ismoved in the plane of the speakers, what path should it takeso that the intensity remains at a minimum? That is, deter-

11.

dL

Figure P18.9 Problems 9 and 10.

9.00 m

10.0 m

y

(x,y)

Ax

Figure P18.12

572 C H A P T E R 18 • Superposition and Standing Waves

mine the relationship between x and y (the coordinates ofthe receiver) that causes the receiver to record a minimumin sound intensity. Take the speed of sound to be 344 m/s.

Section 18.2 Standing Waves

13. Two sinusoidal waves traveling in opposite directions inter-fere to produce a standing wave with the wave function

y " (1.50 m) sin(0.400x) cos(200t)

where x is in meters and t is in seconds. Determine thewavelength, frequency, and speed of the interfering waves.

14. Two waves in a long string have wave functions given by

and

where y1 , y2 , and x are in meters and t is in seconds. (a) De-termine the positions of the nodes of the resulting standingwave. (b) What is the maximum transverse position of anelement of the string at the position x " 0.400 m?

Two speakers are driven in phase by a common oscil-lator at 800 Hz and face each other at a distance of 1.25 m.Locate the points along a line joining the two speakerswhere relative minima of sound pressure amplitude wouldbe expected. (Use v " 343 m/s.)

16. Verify by direct substitution that the wave function for astanding wave given in Equation 18.3,

y " 2A sin kx cos $t

is a solution of the general linear wave equation, Equation16.27:

Two sinusoidal waves combining in a medium are de-scribed by the wave functions

y1 " (3.0 cm) sin &(x ! 0.60t)

and

y2 " (3.0 cm) sin &(x # 0.60t)

where x is in centimeters and t is in seconds. Determinethe maximum transverse position of an element ofthe medium at (a) x " 0.250 cm, (b) x " 0.500 cm, and(c) x " 1.50 cm. (d) Find the three smallest values of xcorresponding to antinodes.

18. Two waves that set up a standing wave in a long string aregiven by the wave functions

y1 " A sin(kx # $t ! %) and y2 " A sin(kx ! $t)

Show (a) that the addition of the arbitrary phase constant% changes only the position of the nodes and, in particu-lar, (b) that the distance between nodes is still one half thewavelength.

17.

.2y

.x 2 "

1v

2 .2y.t

2

15.

y2 " (0.015 0 m) cos ! x2

! 40t"

y1 " (0.015 0 m) cos ! x2

# 40t"

Section 18.3 Standing Waves in a String Fixed atBoth Ends

Find the fundamental frequency and the next three frequen-cies that could cause standing-wave patterns on a string thatis 30.0 m long, has a mass per length of 9.00 - 10#3 kg/m,and is stretched to a tension of 20.0 N.

20. A string with a mass of 8.00 g and a length of 5.00 m hasone end attached to a wall; the other end is draped overa pulley and attached to a hanging object with a mass of4.00 kg. If the string is plucked, what is the fundamentalfrequency of vibration?

21. In the arrangement shown in Figure P18.21, an object canbe hung from a string (with linear mass density + "0.002 00 kg/m) that passes over a light pulley. The string isconnected to a vibrator (of constant frequency f ), and thelength of the string between point P and the pulley is L "2.00 m. When the mass m of the object is either 16.0 kg or25.0 kg, standing waves are observed; however, no standingwaves are observed with any mass between these values.(a) What is the frequency of the vibrator? (Note: Thegreater the tension in the string, the smaller the numberof nodes in the standing wave.) (b) What is the largestobject mass for which standing waves could be observed?

19.

µ

L

P

Vibrator

m

Figure P18.21 Problems 21 and 22.

22. A vibrator, pulley, and hanging object are arranged as inFigure P18.21, with a compound string, consisting of twostrings of different masses and lengths fastened togetherend-to-end. The first string, which has a mass of 1.56 g anda length of 65.8 cm, runs from the vibrator to the junctionof the two strings. The second string runs from the junc-tion over the pulley to the suspended 6.93-kg object. Themass and length of the string from the junction to the pul-ley are, respectively, 6.75 g and 95.0 cm. (a) Find the low-est frequency for which standing waves are observed inboth strings, with a node at the junction. The standingwave patterns in the two strings may have different num-bers of nodes. (b) What is the total number of nodes ob-served along the compound string at this frequency,excluding the nodes at the vibrator and the pulley?

23. Example 18.4 tells you that the adjacent notes E, F, and F-sharp can be assigned frequencies of 330 Hz, 350 Hz, and370 Hz. You might not guess how the pattern continues.The next notes, G, G-sharp, and A, have frequencies of392 Hz, 416 Hz, and 440 Hz. On the equally tempered orchromatic scale used in Western music, the frequency ofeach higher note is obtained by multiplying the previousfrequency by . A standard guitar has strings 64.0 cmlong and nineteen frets. In Example 18.4, we found the

12√2

Problems 573

spacings of the first two frets. Calculate the distance be-tween the last two frets.

24. The top string of a guitar has a fundamental frequency of330 Hz when it is allowed to vibrate as a whole, along all ofits 64.0-cm length from the neck to the bridge. A fret isprovided for limiting vibration to just the lower two-thirdsof the string. (a) If the string is pressed down at this fretand plucked, what is the new fundamental frequency?(b) What If? The guitarist can play a “natural harmonic”by gently touching the string at the location of this fretand plucking the string at about one sixth of the way alongits length from the bridge. What frequency will be heardthen?

25. A string of length L, mass per unit length +, and tension Tis vibrating at its fundamental frequency. What effect willthe following have on the fundamental frequency? (a) Thelength of the string is doubled, with all other factors heldconstant. (b) The mass per unit length is doubled, with allother factors held constant. (c) The tension is doubled,with all other factors held constant.

26. A 60.000-cm guitar string under a tension of 50.000 N hasa mass per unit length of 0.100 00 g/cm. What is the high-est resonant frequency that can be heard by a person capa-ble of hearing frequencies up to 20 000 Hz?

A cello A-string vibrates in its first normal mode with a fre-quency of 220 Hz. The vibrating segment is 70.0 cm longand has a mass of 1.20 g. (a) Find the tension in the string.(b) Determine the frequency of vibration when the stringvibrates in three segments.

28. A violin string has a length of 0.350 m and is tuned to con-cert G, with fG " 392 Hz. Where must the violinist placeher finger to play concert A, with fA " 440 Hz? If this posi-tion is to remain correct to half the width of a finger (thatis, to within 0.600 cm), what is the maximum allowablepercentage change in the string tension?

29. Review problem. A sphere of mass M is supported by astring that passes over a light horizontal rod of length L(Fig. P18.29). Given that the angle is / and that f repre-sents the fundamental frequency of standing waves in theportion of the string above the rod, determine the mass ofthis portion of the string.

27.

30. Review problem. A copper cylinder hangs at the bottom ofa steel wire of negligible mass. The top end of the wire isfixed. When the wire is struck, it emits sound with a funda-mental frequency of 300 Hz. If the copper cylinder is thensubmerged in water so that half its volume is below thewater line, determine the new fundamental frequency.

31. A standing-wave pattern is observed in a thin wire with alength of 3.00 m. The equation of the wave is

y " (0.002 m) sin(&x) cos(100&t)

where x is in meters and t is in seconds. (a) How manyloops does this pattern exhibit? (b) What is the fundamen-tal frequency of vibration of the wire? (c) What If? If theoriginal frequency is held constant and the tension in thewire is increased by a factor of 9, how many loops are pres-ent in the new pattern?

Section 18.4 Resonance32. The chains suspending a child’s swing are 2.00 m long. At

what frequency should a big brother push to make thechild swing with largest amplitude?

33. An earthquake can produce a seiche in a lake, in which thewater sloshes back and forth from end to end with remark-ably large amplitude and long period. Consider a seicheproduced in a rectangular farm pond, as in the cross-sec-tional view of Figure P18.33. (The figure is not drawn toscale.) Suppose that the pond is 9.15 m long and of uni-form width and depth. You measure that a pulse producedat one end reaches the other end in 2.50 s. (a) What is thewave speed? (b) To produce the seiche, several peoplestand on the bank at one end and paddle together withsnow shovels, moving them in simple harmonic motion.What should be the frequency of this motion?

L

M

θ

Figure P18.29

Figure P18.33

34. The Bay of Fundy, Nova Scotia, has the highest tides in theworld, as suggested in the photographs on page 452. As-sume that in mid-ocean and at the mouth of the bay, theMoon’s gravity gradient and the Earth’s rotation make thewater surface oscillate with an amplitude of a few centime-ters and a period of 12 h 24 min. At the head of the bay,the amplitude is several meters. Argue for or against the

574 C H A P T E R 18 • Superposition and Standing Waves

proposition that the tide is amplified by standing-wave reso-nance. Assume the bay has a length of 210 km and a uni-form depth of 36.1 m. The speed of long-wavelength waterwaves is given by , where d is the water’s depth.

35. Standing-wave vibrations are set up in a crystal goblet withfour nodes and four antinodes equally spaced around the20.0-cm circumference of its rim. If transverse waves movearound the glass at 900 m/s, an opera singer would haveto produce a high harmonic with what frequency to shat-ter the glass with a resonant vibration?

Section 18.5 Standing Waves in Air Columns

√gd

cylinder is r, and at the open top of the cylinder a tuningfork is vibrating with a frequency f. As the water rises, howmuch time elapses between successive resonances?

Note : Unless otherwise specified, assume that the speed ofsound in air is 343 m/s at 20°C, and is described by

at any Celsius temperature TC.

v " (331 m/s) √1 !TC

2730

36. The overall length of a piccolo is 32.0 cm. The resonatingair column vibrates as in a pipe open at both ends.(a) Find the frequency of the lowest note that a piccolocan play, assuming that the speed of sound in air is340 m/s. (b) Opening holes in the side effectively shortensthe length of the resonant column. If the highest note apiccolo can sound is 4 000 Hz, find the distance betweenadjacent antinodes for this mode of vibration.

Calculate the length of a pipe that has a fundamental fre-quency of 240 Hz if the pipe is (a) closed at one end and(b) open at both ends.

38. The fundamental frequency of an open organ pipe corre-sponds to middle C (261.6 Hz on the chromatic musicalscale). The third resonance of a closed organ pipe has thesame frequency. What are the lengths of the two pipes?

39. The windpipe of one typical whooping crane is 5.00 ftlong. What is the fundamental resonant frequency of thebird’s trachea, modeled as a narrow pipe closed at oneend? Assume a temperature of 37°C.

40. Do not stick anything into your ear! Estimate the length ofyour ear canal, from its opening at the external ear to theeardrum. If you regard the canal as a narrow tube that isopen at one end and closed at the other, at approximatelywhat fundamental frequency would you expect your hear-ing to be most sensitive? Explain why you can hear espe-cially soft sounds just around this frequency.

A shower stall measures 86.0 cm - 86.0 cm - 210 cm.If you were singing in this shower, which frequencies wouldsound the richest (because of resonance)? Assume that thestall acts as a pipe closed at both ends, with nodes at oppo-site sides. Assume that the voices of various singers rangefrom 130 Hz to 2 000 Hz. Let the speed of sound in the hotshower stall be 355 m/s.

42. As shown in Figure P18.42, water is pumped into a tall ver-tical cylinder at a volume flow rate R. The radius of the

41.

37.

R

f

Figure P18.42

If two adjacent natural frequencies of an organ pipeare determined to be 550 Hz and 650 Hz, calculate thefundamental frequency and length of this pipe. (Use v "340 m/s.)

44. A glass tube (open at both ends) of length L is positionednear an audio speaker of frequency f " 680 Hz. For whatvalues of L will the tube resonate with the speaker?

An air column in a glass tube is open at one end andclosed at the other by a movable piston. The air in thetube is warmed above room temperature, and a 384-Hztuning fork is held at the open end. Resonance is heardwhen the piston is 22.8 cm from the open end and againwhen it is 68.3 cm from the open end. (a) What speed ofsound is implied by these data? (b) How far from theopen end will the piston be when the next resonance isheard?

46. A tuning fork with a frequency of 512 Hz is placed nearthe top of the pipe shown in Figure 18.19a. The water levelis lowered so that the length L slowly increases from an ini-tial value of 20.0 cm. Determine the next two values of Lthat correspond to resonant modes.

47. When an open metal pipe is cut into two pieces, the lowestresonance frequency for the air column in one piece is256 Hz and that for the other is 440 Hz. (a) What resonantfrequency would have been produced by the originallength of pipe? (b) How long was the original pipe?

48. With a particular fingering, a flute plays a note with fre-quency 880 Hz at 20.0°C. The flute is open at both ends.(a) Find the air column length. (b) Find the frequency itproduces at the beginning of the half-time performance ata late-season American football game, when the ambienttemperature is # 5.00°C and the musician has not had achance to warm up the flute.

45.

43.

Problems 575

Section 18.6 Standing Waves in Rodsand Membranes

An aluminum rod 1.60 m long is held at its center. It isstroked with a rosin-coated cloth to set up a longitudinalvibration. The speed of sound in a thin rod of aluminum is5 100 m/s. (a) What is the fundamental frequency of thewaves established in the rod? (b) What harmonics are setup in the rod held in this manner? (c) What If? Whatwould be the fundamental frequency if the rod were madeof copper, in which the speed of sound is 3 560 m/s?

50. An aluminum rod is clamped one quarter of the way alongits length and set into longitudinal vibration by a variable-frequency driving source. The lowest frequency that pro-duces resonance is 4 400 Hz. The speed of sound in an alu-minum rod is 5 100 m/s. Find the length of the rod.

Section 18.7 Beats: Interference in TimeIn certain ranges of a piano keyboard, more than one

string is tuned to the same note to provide extra loudness.For example, the note at 110 Hz has two strings at this fre-quency. If one string slips from its normal tension of 600 Nto 540 N, what beat frequency is heard when the hammerstrikes the two strings simultaneously?

52. While attempting to tune the note C at 523 Hz, a pianotuner hears 2 beats/s between a reference oscillator andthe string. (a) What are the possible frequencies of thestring? (b) When she tightens the string slightly, she hears3 beats/s. What is the frequency of the string now? (c) Bywhat percentage should the piano tuner now change thetension in the string to bring it into tune?

A student holds a tuning fork oscillating at 256 Hz. Hewalks toward a wall at a constant speed of 1.33 m/s. (a) What beat frequency does he observe between the tun-ing fork and its echo? (b) How fast must he walk awayfrom the wall to observe a beat frequency of 5.00 Hz?

54. When beats occur at a rate higher than about 20 per sec-ond, they are not heard individually but rather as a steadyhum, called a combination tone. The player of a typical pipeorgan can press a single key and make the organ producesound with different fundamental frequencies. She can se-lect and pull out different stops to make the same key forthe note C produce sound at the following frequencies:65.4 Hz from a so-called eight-foot pipe; 2 - 65.4 "131 Hz from a four-foot pipe; 3 - 65.4 " 196 Hz from atwo-and-two-thirds-foot pipe; 4 - 65.4 " 262 Hz from atwo-foot pipe; or any combination of these. With notes atlow frequencies, she obtains sound with the richest qualityby pulling out all the stops. When an air leak develops inone of the pipes, that pipe cannot be used. If a leak occursin an eight-foot pipe, playing a combination of other pipescan create the sensation of sound at the frequency that theeight-foot pipe would produce. Which sets of stops, amongthose listed, could be pulled out to do this?

Section 18.8 Nonsinusoidal Wave Patterns55. An A-major chord consists of the notes called A, C #, and E. It

can be played on a piano by simultaneously striking stringswith fundamental frequencies of 440.00 Hz, 554.37 Hz, and

53.

51.

49.

659.26 Hz. The rich consonance of the chord is associatedwith near equality of the frequencies of some of the higherharmonics of the three tones. Consider the first five harmon-ics of each string and determine which harmonics show nearequality.

56. Suppose that a flutist plays a 523-Hz C note with firstharmonic displacement amplitude A1 " 100 nm. FromFigure 18.24b read, by proportion, the displacement am-plitudes of harmonics 2 through 7. Take these as the valuesA2 through A7 in the Fourier analysis of the sound, and as-sume that B1 " B2 " . . . " B7 " 0. Construct a graph ofthe waveform of the sound. Your waveform will not lookexactly like the flute waveform in Figure 18.23b becauseyou simplify by ignoring cosine terms; nevertheless, it pro-duces the same sensation to human hearing.

Additional Problems

57. On a marimba (Fig. P18.57), the wooden bar that sounds atone when struck vibrates in a transverse standing wavehaving three antinodes and two nodes. The lowest fre-quency note is 87.0 Hz, produced by a bar 40.0 cm long.(a) Find the speed of transverse waves on the bar. (b) Aresonant pipe suspended vertically below the center of thebar enhances the loudness of the emitted sound. If thepipe is open at the top end only and the speed of sound inair is 340 m/s, what is the length of the pipe required toresonate with the bar in part (a)?

Figure P18.57 Marimba players in Mexico City.

58. A loudspeaker at the front of a room and an identical loud-speaker at the rear of the room are being driven by thesame oscillator at 456 Hz. A student walks at a uniform rateof 1.50 m/s along the length of the room. She hears a sin-gle tone, repeatedly becoming louder and softer. (a) Modelthese variations as beats between the Doppler-shiftedsounds the student receives. Calculate the number of beatsthe student hears each second. (b) What If? Model the twospeakers as producing a standing wave in the room and thestudent as walking between antinodes. Calculate the num-ber of intensity maxima the student hears each second.

Two train whistles have identical frequencies of 180 Hz.When one train is at rest in the station and the other is

59.

Mur

ray G

reen

berg

576 C H A P T E R 18 • Superposition and Standing Waves

moving nearby, a commuter standing on the station plat-form hears beats with a frequency of 2.00 beats/s when thewhistles sound at the same time. What are the two possiblespeeds and directions that the moving train can have?

60. A string fixed at both ends and having a mass of 4.80 g, alength of 2.00 m, and a tension of 48.0 N vibrates in its sec-ond (n " 2) normal mode. What is the wavelength in airof the sound emitted by this vibrating string?

61. A student uses an audio oscillator of adjustable frequencyto measure the depth of a water well. The student hearstwo successive resonances at 51.5 Hz and 60.0 Hz. Howdeep is the well?

62. A string has a mass per unit length of 9.00 - 10#3 kg/mand a length of 0.400 m. What must be the tension in thestring if its second harmonic has the same frequency as thesecond resonance mode of a 1.75-m-long pipe open at oneend?

Two wires are welded together end to end. The wires aremade of the same material, but the diameter of one is twicethat of the other. They are subjected to a tension of 4.60 N.The thin wire has a length of 40.0 cm and a linear mass den-sity of 2.00 g/m. The combination is fixed at both ends andvibrated in such a way that two antinodes are present, withthe node between them being right at the weld. (a) What isthe frequency of vibration? (b) How long is the thick wire?

64. Review problem. For the arrangement shown in FigureP18.64, / " 30.0°, the inclined plane and the small pulleyare frictionless, the string supports the object of mass M atthe bottom of the plane, and the string has mass m that issmall compared to M. The system is in equilibrium and thevertical part of the string has a length h. Standing wavesare set up in the vertical section of the string. (a) Find thetension in the string. (b) Model the shape of the string asone leg and the hypotenuse of a right triangle. Find thewhole length of the string. (c) Find the mass per unitlength of the string. (d) Find the speed of waves on thestring. (e) Find the lowest frequency for a standing wave.(f ) Find the period of the standing wave having threenodes. (g) Find the wavelength of the standing wave hav-ing three nodes. (h) Find the frequency of the beats result-ing from the interference of the sound wave of lowest fre-quency generated by the string with another sound wavehaving a frequency that is 2.00% greater.

63.

the string are fixed. When the vibrator has a frequency f,in a string of length L and under tension T, n antinodesare set up in the string. (a) If the length of the string isdoubled, by what factor should the frequency be changedso that the same number of antinodes is produced? (b) Ifthe frequency and length are held constant, what tensionwill produce n ! 1 antinodes? (c) If the frequency istripled and the length of the string is halved, by what fac-tor should the tension be changed so that twice as manyantinodes are produced?

66. A 0.010 0-kg wire, 2.00 m long, is fixed at both ends and vi-brates in its simplest mode under a tension of 200 N.When a vibrating tuning fork is placed near the wire, abeat frequency of 5.00 Hz is heard. (a) What could be thefrequency of the tuning fork? (b) What should the tensionin the wire be if the beats are to disappear?

67. Two waves are described by the wave functions

y1(x, t) " 5.0 sin(2.0x # 10t)

and

y2(x, t) " 10 cos(2.0x # 10t)

where y1, y2, and x are in meters and t is in seconds. Showthat the wave resulting from their superposition is alsosinusoidal. Determine the amplitude and phase of thissinusoidal wave.

68. The wave function for a standing wave is given in Equation18.3 as y " 2A sin kx cos $t. (a) Rewrite this wave functionin terms of the wavelength ' and the wave speed v of thewave. (b) Write the wave function of the simplest standing-wave vibration of a stretched string of length L. (c) Writethe wave function for the second harmonic. (d) General-ize these results and write the wave function for the nthresonance vibration.

69. Review problem. A 12.0-kg object hangs in equilibriumfrom a string with a total length of L " 5.00 m and a lin-ear mass density of + " 0.001 00 kg/m. The string iswrapped around two light, frictionless pulleys that are sep-arated by a distance of d " 2.00 m (Fig. P18.69a). (a) De-termine the tension in the string. (b) At what frequencymust the string between the pulleys vibrate in order toform the standing wave pattern shown in Figure P18.69b?

Mθ

h

Figure P18.64 m

d

(b)

m

d

(a)

g

A standing wave is set up in a string of variable length andtension by a vibrator of variable frequency. Both ends of

65.Figure P18.69

Answers to Quick Quizzes 577

70. A quartz watch contains a crystal oscillator in the form ofa block of quartz that vibrates by contracting and expand-ing. Two opposite faces of the block, 7.05 mm apart, areantinodes, moving alternately toward each other andaway from each other. The plane halfway between thesetwo faces is a node of the vibration. The speed of soundin quartz is 3.70 km/s. Find the frequency of the vibra-tion. An oscillating electric voltage accompanies the me-chanical oscillation—the quartz is described as piezoelec-tric. An electric circuit feeds in energy to maintain theoscillation and also counts the voltage pulses to keeptime.

Answers to Quick Quizzes18.1 The shape of the string at t " 0.6 s is shown below.

18.5 (d). Choice (a) is incorrect because the number of nodesis one greater than the number of antinodes. Choice (b)is only true for half of the modes; it is not true for anyodd-numbered mode. Choice (c) would be correct if wereplace the word nodes with antinodes.

18.6 For each natural frequency of the glass, the standing wavemust “fit” exactly around the rim. In Figure 18.17a we seethree antinodes on the near side of the glass, and thusthere must be another three on the far side. This corre-sponds to three complete waves. In a top view, the wavepattern looks like this (although we have greatly exagger-ated the amplitude):

1 cm

18.2 (c). The pulses completely cancel each other in terms ofdisplacement of elements of the string from equilibrium,but the string is still moving. A short time later, the stringwill be displaced again and the pulses will have passedeach other.

18.3 (a). The pattern shown at the bottom of Figure 18.9a cor-responds to the extreme position of the string. All ele-ments of the string have momentarily come to rest.

18.4 (d). Near a nodal point, elements on one side of thepoint are moving upward at this instant and elements onthe other side are moving downward.

18.7 (b). With both ends open, the pipe has a fundamental fre-quency given by Equation 18.11: fopen " v/2L. With oneend closed, the pipe has a fundamental frequency givenby Equation 18.12:

18.8 (c). The increase in temperature causes the speed ofsound to go up. According to Equation 18.11, this will re-sult in an increase in the fundamental frequency of agiven organ pipe.

18.9 (b). Tightening the string has caused the frequencies to befarther apart, based on the increase in the beat frequency.

fclosed "v

4L" 1

2 v

2L" 1

2 fopen