Harmonicsofeach at Atime.Have Thestudentsnote the Notesame Onvarious Instruments One Pitch

Home , Node (physics), Piano wire, Standing wave

Plan andPrepare Demonstration 418 oscilloscope screen. instrument asthey are displayed onthe characteristic harmonicsofeach at atime.Have thestudentsnote the notesame onvarious instruments one pitch. fork, orastudent’s voice at asingle oscillator, sinewave generator, tuning sound by thesound from inputting an characteristic ofasinglefrequency screen oftheoscilloscope. Show the the studentsnote thepatterns onthe speak orsinginto themicrophone. Have human voice by having several students scope. Display thesound pattern ofthe amplifier to theinputofoscillo- the outputfrom themicrophone and Procedure proper use oftheoscilloscope. user’s manualfor onthe instructions Caution sound sources small amplifier(ifneeded),assorted Materials a variety ofsources. Purpose Seeing Sounds frequency ofastandingwave). lowest the is frequency fundamental as just structure, a of lowestpoint frequency (i fundamental of definition the to root “foundation.” this meaning Apply comes from theLatin fundamentum, Latin roots. Fundamentalinstance, for , their from meaning their take physics in Latin W Preview V SECTION Teach Then have several studentsplay the Chapter 12 ord Origins Consult theoscilloscope’s Observe sound waves from oscilloscope, microphone, . Connect themicrophone or e ., the foundation is the the is foundation the ., ocabulary 3 Often terms used used terms Often Differentiated Instruction Untitled-99 418 the second nodeasthecenter. them to visualize folding the string in half with length anddoublethefrequency. Itmay help addition ofonenodeequalshalfthewave- Students may beconfused asto whythe Below L evel wave frequency ofvibrationastanding fundamental frequency 418 FIGURE 3.1 depend on the string lengths. standing waves whose frequencies vibrating strings of a violin produce Stringed Instruments second. number of beats heard per between two waves to the Relate the frequency difference Relate and harmonics timbre. closed pipes. vibrating string and of open and Calculate the of harmonics a closed pipes. series harmonic of open and Differentiate between the SECTION 3 Chapter 12 Objectives thelowest The Standing Waves on a Vibrating String harmonic series fundamental frequency Key Terms Harmonics is equal to the speed ofthe wave to thespeed is equal by length. divided thestring twice that see ofthisvibrationequation we thefrequency forfrequency, By substituting thevalue forwavelength found above into this frequency timesthewavelength,frequency which can rearranged be asshown. equal λ equal to thenextnode isalways half awavelength, length thestring (L)must displacement, itisanantinode andso (A).Because thedistance from one of thefundamental frequency. half theprevious wavelength, ofthiswave thefrequency that istwice lengthstring to onewavelength. isequal Because thiswavelength is a single pitch, actually consist ofmultiple frequencies. sounds you hear from instrument, astringed that even those soundlike time, each ofwhich has adifferent wavelength So, andfrequency. the Figure 3.1, The vibrations ofamusical onthestring instrument, such astheviolinin waves can isfixed whenastring at occur into endsandset both vibration. inthechapter andWaves,”As discussed “Vibrations ofstanding avariety of The next possible standing wave row isshowninthesecond forastring Harmonics are integral multiples of the fundamental frequency. wave onthisstring. the fundamental isthelowest ofastanding frequency possible frequency length are andbecause we thegreatest considering possible wavelength, vibrating string. isinversely Because frequency proportional to wave- row of (N nodes be idealized string. The endsofthestring, which cannot vibrate, must always Figure 3.2. This ofvibration frequency iscalled the Figure 3.2, As described inthechapter onwaves,As described ofawave thespeed the equals Figure 3.2. 1 /2. Thus, thewavelength length thestring (λ istwice usually consist ofmany standing waves at together thesame In thiscase, there are three instead nodes oftwo, the so ). The simplest vibration that can isshowninthefirst occur onthenext page, shows several possible vibrations onan fundamental frequency In thiscase, the most thecenter experiences ofthestring beat timbre v = f f f 2 so so λ,

2 = f f 1 = = _ λ v f 1

fundamental frequency

= = _ λ v 1

= _ 2L v

1 =2L). ofthe 5/18/2011 6:42:48AM

©RubberBall/Getty Images ©RubberBall/Getty Images Untitled-99 419 To test theirunderstanding offrequency’s changed by varying eitherofthese factors. string, so thefundamentalfrequency canbe unit lengthaffect thespeedofwaves onthe frequency. Thestring’s tension andmass per not necessarily have fundamental thesame series. Thus,two lengthwill strings ofthe same as shown by theequation for theharmonic depends onbothstringlengthandwave speed, Point outto advanced studentsthat frequency Pre-AP N N N N N N N N N N N N N N N N Figure 3.2 in thethird row of ized to include theentire harmonicseries. Thus, f theequationfrequency, forthefundamental can general frequency be - harmonic,on. andso to thesecond corresponds to thefirst harmonic, thenext (f frequency form whatform iscalled a integral multiples ofthefundamental Thesefrequencies frequency. More generally, thefrequencies ofthestanding wave patterns are all of these wavesof these that be by divided will speed thefrequency. wavesthese of soundwaves thespeed be will inair, andthewavelength surrounding have airwill However, thesame frequency. of thespeed vibrates at frequencies, oneofthese thesoundwaves produced inthe oftheresultant andnotthespeed string soundwaves inair. Ifthestring harmonic. The general oftheequation asfollows: iswritten form fundamental (f frequency PH99PE-C13-003-002-A PH99PE-C13-003-002-A PH99PE-C13-003-002-A PH99PE-C13-003-002-A × =harmonicnumber frequency A A A HRW •HoltPhysics A HRW •HoltPhysics HRW •HoltPhysics This pattern continues, ofthestanding andthefrequency wave shown Because each harmonicisanintegral multiple ofthefundamental Note that ofwaves vinthisequation isthespeed onthevibrating on aVibratingString Harmonic SeriesofStandingWaves HRW •HoltPhysics A A A A N N N A A A N A N N N N A A A A N N N N A A N A A N A A N N A A A A A A N N N N N N A N A A N A A A A A A A A A Figure 3.2 N N N N N N N N N N N N N N N N The harmonicSerieS harmonic series. f n =n λ λ λ λ 4 3 2 1

1 = = = 2 = = isthree timesthefundamental frequency. _ 2L L

__ __ 2 1 3 2 v __ 2L

v L L

)andf n =1,2,3,. f f f f 4 3 2 1

4 = 3 = 2 = The fundamental (f frequency n ___ is the frequency ofthenth isthefrequency (speed of waves on the string) ofwaves onthe (speed (2)(length ofvibrating string) (2)(length f f f 1 1 1 fourth harmonic third harmonic second harmonic first harmonic fundamental frequency, or n =nf 1 2 , where f ) corresponds series equation aswell assolve for unknowns. them to substitute values into theharmonic students with multiple scenarios that encourage dependence onstringlengthandspeed,supply 1 isthe 1 )

fundamental frequency frequency andintegralmultiplesofthe cies thatincludesthefundamental harmonic series portion of the string that is vibrating. equation on this page, of fundamental frequencies. In the string can be used to create a variety the string vibrates. As a result, a single becomes a node and only a portion of a guitar string at any point, that point When a guitar player presses down on Did Know? YOUKnow?

a seriesoffrequen- Sound L refers to the 5/18/2011 6:42:49AM 419 string mustequal harmonic (one loop), thelengthLof each end must benodes.For thefirst equations. Bothendsofastringfixed at to helpthemrecall theappropriate Have studentsvisualizetheharmonics Teaching Tip Ask wavelength. that each loopcorresponds to halfa FIGURE 3.2 appropriate multiplesof techniquecan use this same withthe λ =2L.For otherharmonics,students frequency (f Answer: λ in theharmonicseries. TEACH FROM VISUALS Find thewavelength (λ 5 = 5 Be sure students understand ) for thenextpossible case _ 2 5

L, f _ 2 1

5 λ. Thus, =5f _ 2 1 1

λ. 5 _ 2 1 ) and

Sound λ =Land 419 FIGURE 3.3 Standing Waves in an Air Column Waves in a Pipe The harmonic Standing waves can also be set up in a tube of air, such as the inside of a Teach continued series present in each of these organ trumpet, the column of a saxophone, or the pipes of an organ like those pipes depends on whether the end of shown in Figure 3.3. While some waves travel down the tube, others are the pipe is open or closed. reflected back upward. These waves traveling in opposite directions Teaching Tip combine to produce standing waves. Many brass instruments and The memory device used for a vibrating woodwinds produce sound by means of these vibrating air columns. string can be used for open and closed pipes as well. Ask students to visualize If both ends of a pipe are open, all harmonics are present. The harmonic series present in an organ pipe depends on whether the the standing wave, remembering that reflecting end of the pipe is open or closed. When the reflecting end of the closed ends must be nodes and open pipe is open, as is illustrated in Figure 3.4, the air molecules have complete ends must be antinodes. Sketching freedom of motion, so an antinode (of displacement) exists at this end. diagrams for each case will show that If a pipe is open at both ends, each end is an antinode. This situation is the exact opposite of a string fixed at both ends, where both ends are nodes. pipes closed at one end must have half 1 Because the distance from one node to the next (__ λ) equals the __1 2

a loop ( 4 λ) more than pipes open at distance from one antinode to the next, the pattern of standing waves that both ends. can occur in a pipe open at both ends is the same as that of a vibrating string. Thus, the entire harmonic series is present in this case, as shown in Figure 3.4, and our earlier equation for the harmonic series of a vibrating string can be used.

Harmonic Series of a Pipe Open at Both Ends f = n _ v n = 1, 2, 3, . . . Did YOU Know? n 2L A flute is similar to a pipe open at (speed of sound in the pipe) both ends. When all keys of a flute frequency = harmonic number × ____ (2)(length of vibrating air column) are closed, the length of the vibrating air column is approximately equal to the length of the flute. As the keys are In this equation, L represents the length of the vibrating air column. opened one by one, the length of the Just as the fundamental frequency of a string instrument can be varied by vibrating air column decreases, and the changing the string length, the fundamental frequency of many wood- fundamental frequency increases. wind and brass instruments can be varied by changing the length of the vibrating air column.

FIGURE 3.4

Harmonics in an Open-Ended A A A Pipe In a pipe open at both ends, each N N end is an antinode of displacement, and all A

harmonics are present. L N A =L N 2 = 2L 2 =-L 1 3 3 v A v f = - = f f = N 2 2 1 3v 1 L f = = 3f 2L N 3 2L 1 A A A

(a) (b) (c)

(a) First harmonic (b) Second harmonic (c) Third harmonic (tl) ©Joseph Barnell/SuperStock Problem420 Chapter Solving 12 Take It Further Untitled-99Allow 420 students to take a concrete look at 5/18/2011 6:42:50 AM how speed affects frequency by comparing the frequency of an open pipe filled with water to one filled with air. The speed of sound is 346 m/s in air and 1490 m/s in water. The frequency in water is 4.3 times the frequency in air.

420 Chapter 12 FIGURE 3.5

Harmonics in a Pipe Closed at One End In a pipe closed at one end, the closed end is a node of displacement and the open end is an anti- MATERIALS QuickLab node of displacement. In this case, only the odd harmonics are present. • straw N N N • scissors Teacher’s Notes A SAFETY This lab demonstrates the effect of pipe A Always use caution when N 4 L = 4L = -4 = -L 1 3 3L 5 5 working with scissors. length on pitch. The experiment works v A 5v f = N f = 3v = f = = 5 f 1 4L 3 3 f 5 4L 1 4L 1 N best with narrow paper straws. A PIPE CLOSED AT A A A ONE END Students should find that the shorter Snip off the corners of one end the straw is, the higher the fundamental (a) (b) (c) of the straw so that the end frequency is. One length of straw can tapers to a point, as shown (a) First harmonic (b) Second harmonic (c) Third harmonic below. Chew on this end to produce more than one possible tone flatten it, and you create a because there are several possible If one end of a pipe is closed, only odd harmonics are present. double-reed instrument! Put wavelengths of standing waves for a your lips around the tapered When one end of an organ pipe is closed, as is illustrated in Figure 3.5, the end of the straw, press them given length of straw. movement of air molecules is restricted at this end, making this end a together tightly, and blow The sound frequencies can be node. In this case, one end of the pipe is a node and the other is an through the straw. When you antinode. As a result, a different set of standing waves can occur. hear a steady tone, slowly snip demonstrated visually by blowing into a As shown in Figure 3.5(a), the simplest possible standing wave that can off pieces of the straw at the microphone connected to an oscillo- exist in this pipe is one for which the length of the pipe is equal to one- other end. Be careful to keep scope and watching the results on the fourth of a wavelength. Hence, the wavelength of this standing wave about the same amount of equals four times the length of the pipe. Thus, in this case, the fundamen- pressure with your lips. How oscilloscope screen. tal frequency equals the velocity divided by four times the pipe length. does the pitch change as Homework Options This QuickLab can the straw becomes shorter? f = _ v = _v How can you account for this easily be performed outside of the 1 λ 4L 1 change in pitch? You may be physics lab room. For the case shown in Figure 3.5(b), the length of the pipe is equal to able to produce more than one three-fourths of a wavelength, so the wavelength is four-thirds the length tone for any given length of the 4 __ straw. How is this possible? of the pipe (λ3 = 3 L). Substituting this value into the equation for frequency gives the frequency of this harmonic.

f = _ v = _ v = _ 3v = 3f 3 4 1 λ __ 4L 3 3 L The frequency of this harmonic is three times the fundamental frequency. Repeating this calculation for the case shown in Figure 3.5(c) gives a frequency equal to five times the fundamental frequency. Thus, PHYSICS only the odd-numbered harmonics vibrate in a pipe closed at one end. Spec. Number PH 99 PE C13-QKL-011-A Boston Graphics, Inc. We can generalize the equation for the harmonic series of a pipe closed 617.523.1333 at one end as follows:

Harmonic Series of a Pipe Closed at One End f = n _ v n = 1, 3, 5, . . . n 4L (speed of sound in the pipe) frequency = harmonic number × ____ (4)(length of vibrating air column) (tl) ©Joseph Barnell/SuperStock Differentiated Instruction Sound 421 Below Level Untitled-99 421 Review with students the difference between 5/18/2011 6:42:51 AM pitch and loudness. Explain that pitch is how frequency is perceived by the human ear. Pitch is described on a continuum of low and high, because pitch is determined by frequency. Loudness, on the other hand, is determined by amplitude, which is the maximum displacement of a periodic wave.

Sound 421

422 Answers: b. a. 34.5 cmlong. One stringonatoy guitaris Harmonics Classroom Practice Teach continued three harmonics? What are thefrequencies ofthefirst of waves onthestringis410m/s. The stringisplucked, andthespeed harmonic? What isthewavelength ofitsfirst b. a. Chapter 12 590 Hz,12001800Hz 69.0 cm Problem Solving Untitled-99 422 4(2.45m) = 9.80= 4(2.45m) m. be antinodes;thusL= For theopenpipe,bothendsmust f = now befound withtheequation m)=4.902(2.45 m.Thefirstharmoniccan one is an antinode; hence, For theclosed pipe,oneendisanodeand Al 422 ternative Appr _ λ v

, asfollows: Tips andTricks pres ent, son=1, 3, 5, etc. end, are onlyoddharmonics etc. Forapipeclosedatone at bothends, n=1, 2, 3, situation. Forapipeopen numbersforeach harmonic Be sure tousethecorrect f 1 = Chapter 12 Assume that the speed of sound in air is 345 of m/s. sound Assume that the speed is one end of closed? harmonics of when the pipe this pipe 2.45 m that ends? Whatlong at pipe is both areopen the first three Sample Problem B Harmonics _ λ v 1

= SOLVE PLAN ANALYZE 345 m/s _ 4.90 m _ 2 1

λ L

1 , orλ

o = 70.4 Hz = aches What are the first harmonics in three a __ 4 1

1 λ =2L 1 , or the harmonicnumbers by thefundamental frequency. In cases, both harmonicscan found two be by thesecond multiplying at thefollowing isclosed oneend,use equation: thepipe When Unknown: The next possible harmonicsare thethird andthefifth: For at closed oneend: apipe The nextharmonicsare two andthethird: thesecond For at open apipe ends: both solve: and into equations the values the Substitute foundbe by using theequation fortheentire harmonicseries: at isopen thepipe When ends, both thefundamental can frequency orsituation: anequation Choose Given: λ 1

= 4

L f f 24 L =2.45m f f at closed oneend: Pipe at open Pipe ends: both f f f f n n 5 3 1 3 2 1 =5f =3f =n =3f =2f =n =n =n Thus, thefirstharmonicisasfollows: Problem B. the funda multiplying theharmonicnumberby The otherharmonicscanbefound by _ _ _ _ 4L 2L 4L 2L 1 1 1 1 v v v v =(5)(35.2Hz) =176Hz =(3)(35.2Hz) =106Hz =(3)(70.4Hz) =211Hz =(2)(70.4Hz) =141Hz

=(1) =(1)

, n=1,3,5,. , n=1,2,3,. f 1 mental frequency, asinSample = ( ( __ __ _ (2)(2.45m) (2)(2.45m) λ v 345m/s 345m/s v =345m/s 1

= f f 1 1

345 m/s _ PREMIUM CONTENT 9.80 m

) ) HMDScience.com Interactive Demo

= 35.2Hz = 70.4Hz f f 3 2

= 35.2Hz

Continued f f 5 3 5/18/2011 6:42:52AM

one end.For example, shownin although thetrumpet different instruments. different by produced be can that sounds of variety the for responsible part in are deviations These open. is phone saxo- the of end one only though even ends both at open pipe cylindrical a in that to similar is saxophone a in series harmonic the that such is saxophone a of shape The intensities. small relatively at clarinet’stone a in harmonics even some are there but cylindrical, ily primar- is clarinet a Forexample, harmonics. the affect instruments many in holes open the that is reason Another instrument. an of series harmonic the affects pipe a of shape cylindrical the from deviation any that is apply not does equation the reason One instruments. such to apply directly not does pipes of series harmonic the for equation our end, one at closed pipe a and ment. In asaxophone oraclarinet, thereed oneend. closes ends,open theplayer’s mouth oneendoftheinstru- effectively closes affect theirfinalanswer. than anincorrect value for wavelength will tip to checktheirequation writing.Emphasize at bothends, itis4L.Studentsshoulduse this possible wavelength is2L.Whenapipeopen When apipeisopenat oneend,thefirst Dec Harmonics 4. 3. 2. 1. Trumpets, saxophones, are andclarinets at similar closed to apipe Despite the similarity between these instruments instruments these between similarity the Despite a. the speed the of speed the waves on this string? A violin string that is 50.0 cm long has a fundamental frequency of 440 Hz. What is the string is 115 m/s and the effective string lengths are as follows? What is the fundamental frequency of a guitar string when the of speed waves on the flute? The of speed sound in the flute is 340 m/s. are makingclosed, the vibrating air column approximately equal to the length of approximately 66.0 cm. What are the first three harmonics of a flute when all keys A flute is essentially a pipe open at both ends. The length of a flute is one end, when the of speed sound in the pipe is 352 m/s? What is the fundamental frequency of a 0.20 m long organ pipe that is atclosed CHECKYOUR WORK 00c 70.0 cm onstructing Pr (continued) 00c 50.0 cm b. the closed pipe,the closed that is, 70.4=(2)(35.2). thatfundamental should pipe twice oftheopen be frequency of andwavelengthBecause frequency are inversely proportional, the at closed oneend,thefirstin apipe possible wavelength is4L. In at open apipe ends, both thefirst possible wavelength is2L; 40.0 cm c. oblems Figure 3.6 FIGURE 3.6 each instrument a different harmonic series. Shape and Harmonic Series has two

Variations in shape give Sound 5/18/2011 6:42:53AM 423 Practice B Answers 4. 2. 3. *Challenging Problem L v f Solving for: ProblemPB =Sample SetII(online) P SE =StudentEditionTextbook Use thisguide to assign problems. PR 1. n Sample ProblemW =Sample SetI(online) 440 m/s c. b. a. 260 Hz,520 Hz,780 Hz 440 Hz OBLEM guide 144 Hz 115 Hz 82.1 Hz PB P SE PB P SE PB P SE W W W Sample, 1–4 Sample, 4–6 Ch. Rvw. 36a,41* 8–10 1–3 Sample, 4 5–7 7–9 Ch. Rvw. 34–35, 36b, 40 1–3; Sample, Sound 423

TEACH FROM VISUALS 424 ( are for themusicalnote A-natural Ask instrument. harmonics produced by each ofwavespectrum amplitudesof column inFigure 3.7 represents the Figure 3.7 considered to benoise donot. repeating patterns, whilethose that are musical typicallyhave waveforms with Sounds that are considered to be the resultant waveform isnonrepeating. individual waveforms are notperiodic, patterns. Ontheotherhand,when complex, iscomposed ofrepeating the resultant waveform, although When individualwaveforms are regular, between musicalsounds andnoise. sometimesExplain thedistinction made Teaching Tip 1540 Hz,8801100Hz Answer: 660Hz,220440 of intensity, from greatest to least. produced by theclarinetinorder Teach continued f 1 =220Hz).Listthefrequencies The harmonicsshown inFigure 3.7 Chapter 12 Point outthat thesecond Differentiated Instruction Untitled-99 424 different from that of a shrill whistle. This does patience.” Thesound qualityofaflute is as thisone:“Joe possesses thequalityof ing attribute.” Use theword inasentence such It isbeingused to mean “trait” or“distinguish- imply whetherornotsound isgood orbad. the word qualityisnotbeingused here to quality ofthisfabric ispoor.”) Emphasizethat only initseveryday use (for example, “The Students may have heard theword quality BELOW LEVEL intensities harmonics presentatdifferent resulting fromthecombinationof timbre 424 Viola Clarinet Tuning fork Figure 3.7

the musicalqualityofatone Chapter 12 Harmonics ofaTuningfork,clarineT and aViola its owncharacteristic mixture intensities. ofharmonicsat varying viola wheneach soundsthemusical note A-natural. Each instrument has timbre. Figure 3.7 Harmonics account for sound quality, or timbre. the perspective ofthelistener,the perspective results thisspectrum or insound quality, sound ofaninstrument isreferred thesound. of From to asthespectrum frequency. plex than asine wave because each individualwaveform has adifferent harmonic waveform isasine wave, but theresultant wave ismore com - consist ofmany harmonics, each at different intensities. Each individual waveforms oftheotherinstruments are more complex because they vibratealso at higher frequencies are whenthey struckhard enough.) The mental itswaveform frequency, issimply asine wave. tuning (Some forks shown inthethird column.Since atuning fork vibrates at only itsfunda - according ofsuperpositionto give theresultant to theprinciple waveform factors. With aviolin,forexample, theintensity ofeach harmonic amplitudedepending onfrequency, ofvibration, ofother and avariety fuller soundthan that ofatuning fork. same volume. harmonicsofmost Theinstruments provide rich amuch timbre, eveninstruments whenboth are sounding thesame note at the The harmonics shown in the second columnof The harmonicsshowninthesecond In music, themixture ofharmonicsthat produces thecharacteristic The instrument, intensity aparticular ofeach within harmonicvaries A clarinet soundsdifferent Aclarinet from aviola because ofdifferences in shows theharmonicspresent inatuning fork, acla T THesamePicH

one another. butes andthetwo canbedistinguishedfrom other; itmeans that each hasits own attri- not mean orworse oneisbetter thanthe Relative intensity Relative intensity Relative intensity 1 1 1 2 2 2 3 3 3 Harmonics Harmonics Harmonics 4 4 Relative intensity 4 Relative intensity Relative intensity 5 5 5 1 1 1 PH99PE-C13-003-005b,c-A 6 6 6 PH99PE-C13-003-003b,c-A PH99PE-C13-003-004b,c-A 2 2 2 7 7 7 HRW •HoltPhysics HRW •HoltPhysics HRW •HoltPhysics 3 3 3 Harmonics 8 8 8 Harmonics Harmonics 4 4 4 9 9 9 5 5 5 10 10 10 PH99PE-C13-003-005b,c-A 6 6 PH99PE-C13-003-003b,c-A 6 PH99PE-C13-003-004b,c-A 7 7 7 HRW •HoltPhysics HRW •HoltPhysics HRW •HoltPhysics 8 8 8 Resultant waveform 9 Resultant waveform 9 9 Resultant waveform 10 10 10 , Resultant waveform Resultant waveform Resultant waveform Figure 3.7 add together addtogether rinet, anda 5/18/2011 6:42:54AM

©Fred Maroon/Photo Researchers, Inc. frequency. pitch and anotherwithhalfordoubleits to signify theinterval between onemusical being used to imply anumber;itisbeingused than withtwelve. Clarifythat theword isnot they may associate octave witheightrather the religious orliterary meaning oftheword, Latinate roots, orifthey have beenexposed to octave. Iftheirnative language contains many English learners may beconfused by theterm Reverberation English Le depends on where the string is bowed, the speed of the bow on the string, fundamental frequency. patterns occur because each frequency is an integral multiple of the patterns. Such waveforms are said to be plex than those of a tuning fork, note that each consists of repeating factors involved, most instruments can produce oftones. variety awide and theforce thebow onthestring. exerts Because there are many so octave above that note. har of thesecond monic ofanote corresponds ofthe to thefrequency instruments, wind instruments andopen-ended stringed thefrequency that ofthefirst note, the13notes constitute andtogether anoctave. note isexactly The ofthethirteenth twice frequency. frequency teristic (half-step) musical scale, there are 12notes, each ofwhich has acharac- Other harmonicsare referred sometimes to asovertones. In thechromatic the fundamental ofavibration frequency pitch. typically determines itspitch.The ofasounddetermines frequency In musical instruments, Fundamental frequency determines pitch. A the typeofmusicusuallyplayedthere. For example, rock echo ofeachwordcouldbecomeconfusingtolisteners. short. timeisrelatively the reverberation A repeated sound’s intensitytodecreaseby60dB. timeistheamountofittakesfora reverberation This repetitiveechoiscalledreverberation. The theceiling,forth against walls, floor, andothersurfaces. a speakerormusicalinstrumentbouncebackand damped andmuffled. youtohearaspeakerwellbutmakebandsound allow a lecturehall. Your school’s auditorium, forinstance, may for rockconcerts, whileanotherisconstructedforuseas Music hallsmaydifferinconstructiondependingon For speech, theauditoriumshouldbedesignedsothat soundsmadeby Rooms areoftenconstructedsothat Even though the waveforms of a clarinet and a viola are more com- functions inmind. Oneauditoriummaybemade and musicroomsaredesignedwithspecific uditoriums, churches, concerthalls, libraries, arners periodic. These repeating desired fororchestralandchoralmusic. reverberation, issometimes butmorereverberation music isgenerallylesspleasingwithalargeamountof accommodate theaudi accommodate functionofaroom. tory these differentfactorsareconsideredandcombinedto arrangedtoabsorbsound.can alsobestrategically All of tomufflesounds.material Padded furnishingsandplants other quietplacesareoftenmadeofsoftortextured andhard. areflat reverberation Ceilingsinlibrariesand different rooms. Ceilingsdesignedforalotof way ceilings, walls, andfurnishingsaredesignedin For these reasons, youmaynoticeadifferenceinthe For Sound 5/18/2011 6:42:56AM 425 interested inacoustics. career choice for studentswhoare acoustical engineeringas apossible canbeused toThis opportunity discuss ation whenbuildings are beingdesigned. reverberation consider isanimportant - Students may besurprised to learn that Extension and withoneanother. interfere withtheoriginalsound waves interference that occurs whenechoes ment. Thisfeature discusses the the sound waves from asingleinstru- section considers theinterference of The discussion ofharmonicsinthis Reverberation Why ItMatters Sound 425 FIGURE 3.8 Teach continued (a) Superposition and Beats Beats are formed by the interference of TEACH FROM VISUALS two waves of slightly different frequencies traveling in the same direction. In this (b) Figure 3.8 Be sure students under- case, constructive interference is greatest at t2, when the two waves are in phase. stand that this figure depicts two waves t1 t2 t3 at a particular point in space as time passes (rather than an expanse of space Destructive Constructive Destructive interference interference interference at an instant of time, as in previous wave representations).

Ask At what time(s) are the two waves Beats HRW • Holt Physics PH99PE-C13-003-009-A exactly out of phase? At what time(s) are So far, we have considered the superposition of waves in a harmonic the two waves exactly in phase? series, where each frequency is an integral multiple of the fundamental Answer: t and t ; t frequency. When two waves of slightly different frequencies interfere, 1 3 2 the interference pattern varies in such a way that a listener hears an alternation between loudness and softness. The variation from soft to beat the periodic variation in the loud and back to soft is called a beat. amplitude of a wave that is the superposition of two waves of Answers slightly different frequencies Sound waves at slightly different frequencies produce beats. Conceptual Challenge Figure 3.8 shows how beats occur. In Figure 3.8(a), the waves produced by 1. The fundamental frequencies are two tuning forks of different frequencies start exactly opposite one another. These waves combine according to the superposition principle, getting closer together because if as shown in Figure 3.8(b). When the two waves are exactly opposite one the number of beats heard each another, they are said to be out of phase, and complete destructive interference occurs. For this reason, no sound is heard at t . second is decreasing, the two waves 1 are closer to being completely in Because these waves have different frequencies, after a few more cycles, the crest of the blue wave matches up with the crest of the phase at all points. red wave, as at t2. At this point, the waves are said to be in phase. 2. When the two flutists play the same note, the number of beats heard each second is the frequency Conceptual Challenge

difference between the two flutes. Concert Violins Before a performance, musicians tune Sounds from a Guitar Will the speed of waves on a If one of the flutes is adjusted until their instruments to match their fundamental frequencies. vibrating guitar string be the same as the speed of the no beats are heard, the two flutes If a conductor hears the number of beats decreasing as sound waves in the air that are generated by this vibra- two violin players are tuning, are the fundamental frequen- tion? How will the frequency and wavelength of the waves will be in tune with each other. cies of these violins becoming closer together or farther on the string compare with the frequency and wavelength 3. The speed of the sound waves in air apart? Explain. of the sound waves in the air? will not be the same as the speed of Tuning Flutes How could two flute players use beats to ensure that their instruments are in tune with each other? waves on the string. There is no frequency change because the (br) ©Photodisc/Getty Images vibrations are still occurring at the same rate, so the wavelength must Differentiated426 Chapter 12 Instruction change with the wave speed (because v = λf ). Below Level Untitled-99Students 426 can remember that destructive 5/18/2011 6:42:57 AM interference means waves are out of phase by remembering that the Latin prefix de- typically indicates “reversal,” “removal,” or “lack.” There is a lack of sound when there is destructive interference.

426 Chapter 12 Now constructive interference occurs, and the sound is louder. Because the blue wave has a higher frequency than the red wave, the waves are out of phase again at t3, and no sound is heard. Teaching Tip As time passes, the waves continue to be in and out of phase, the Point out that the number of beats is interference constantly shifts between constructive interference and destructive interference, and the listener hears the sound getting softer always the absolute value of the and louder and then softer again. You may have noticed a similar phe- difference between two frequencies. nomenon on a playground swing set. If two people are swinging next to Measuring the number of beats does one another at different frequencies, the two swings may alternate between being in phase and being out of phase. not tell you which of two instruments has a higher frequency. If two waves are The number of beats per second corresponds to the difference exactly in phase, there are no beats and between frequencies. the sounds are exactly in tune.

In our previous example, there is one beat, which occurs at t2. One beat corresponds to the blue wave gaining one entire cycle on the red wave. This is because to go from one destructive interference to the next, the red wave must lag one entire cycle behind the blue wave. If the time that lapses Assess and Reteach from t1 to t3 is one second, then the blue wave completes one more cycle per second than the red wave. In other words, its frequency is greater by 1 Hz. Assess Use the Formative Assessment By generalizing this, you can see that the frequency difference between two sounds can be found by the number of beats heard per second. on this page to evaluate student mastery of the section. Reteach For students who need additional instruction, download the SECTION 3 FORMATIVE ASSESSMENT Section Study Guide. Response to Intervention To reassess Reviewing Main Ideas students’ mastery, use the Section Quiz, 1. On a piano, the note middle C has a fundamental frequency of 262 Hz. available to print or to take directly What is the second harmonic of this note? online at HMDScience.com. 2. If the piano wire in item 1 is 66.0 cm long, what is the speed of waves on this wire? 3. A piano tuner using a 392 Hz tuning fork to tune the wire for G-natural hears four beats per second. What are the two possible frequencies of vibration of this piano wire? 4. In a clarinet, the reed end of the instrument acts as a node and the first open hole acts as an antinode. Because the shape of the clarinet is nearly cylindrical, its harmonic series approximately follows that of a pipe closed at one end. What harmonic series is predominant in a clarinet?

Critical Thinking 5. Which of the following are different for a trumpet and a banjo when both play notes at the same fundamental frequency? a. wavelength in air of the first harmonic b. which harmonics are present c. intensity of each harmonic d. speed of sound in air (br) ©Photodisc/Getty Images Answers to Section Assessment Sound 427

1. 524 Hz

Untitled-99 2. 427 348 m/s 5/18/2011 6:42:58 AM 3. 388 Hz and 396 Hz 4. the odd harmonics 5. b, c

Sound 427