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SPECIALFEATURE:SOUNDPHYSICS www.iop.org/journals/physed Building a copper pipe ‘xylophone’ DavidRLapp TheWrightCenterforInnovativeScienceEducation,TuftsUniversity,4ColbyStreet, Medford,MA02155,USA E-mail: [email protected] Abstract Musiciscentraltothelifeofmanystudents. Thisarticledescribesusingthe equationforfrequencyofvibrationofatransverselyoscillatingbarorpipe (withbothendsfreetovibrate)tobuildasimpleandinexpensive xylophone-likemusicalinstrumentorsetofchimesfroma3msectionof copperpipe. Theinstrumentproducesafullmajorscaleandcanalsobe usedtoinvestigatevariousmusicalintervals. With even a quick look around most school campuses, it is easy to see that students enjoy music. Ears are sometimes hard to find, being covered or plugged by headphones that are Pipe length L connected to radios or portable CD players. The music emerging from these headphones inspires, Figure1. Abarorpipevibratingtransverselyinits firstmode. entertains and provides emotional release. A closer look reveals that much of the life of a student either revolves around music or is at least strongly influenced by it. The radio is the first thing to go on in the morning and the Pipe length L last to go off at night (if it goes off at all). T-shirts with the logos and tour schedules of Figure2. Abarorpipevibratingtransverselyinits popularbandsaretheartifactsofthemostcoveted secondmode. teenage activity—the concert. The teacher of physicshastheuniqueabilityandopportunityto tovibrate,atransversestandingwaveconditionis bridge the gap between the sometimes-perceived createdwhenitisstruckonitsside,asinthecase irrelevantcurriculumofferedwithintheschooland of the marimba or glockenspiel. The constraint the life of the student outside the school. This for this type of vibration is that both ends of the articledescribesusingtheequationforfrequency barorpipemustbeantinodes. Thesimplestway of vibration of a transversely oscillating bar or for a bar or pipe to vibrate with this constraint pipe (with both ends free to vibrate) to build a is to have one antinode at its centre, in addition simple and inexpensive xylophone-like musical to the antinodes at each end. The nodes occur instrumentorsetofchimesfroma3msectionof at 0.224L and 0.776L [1]. This first mode of copperpipe(thetypeusedbyplumbersforwater vibrationproducesthefundamentalfrequency,f 1 service). (seefigure1). Thesecondmodeofvibration,f , 2 InabarorpipeoflengthLwithbothendsfree producing the next higher frequency, is the one 316 PHYSICSEDUCATION 38(4) 0031-9120/03/040316+04$30.00 ©2003IOPPublishingLtd Buildingacopperpipe‘xylophone’ Table1. Speedofsoundwithinvariousmetals. instrument can be built which will produce recognizable tones. I recently built such an Material Speedofsound (ms−1) instrument from a 3 m length of copper pipe of 15 mm diameter (see figure 3). Desiring to Aluminium 5150 produce the fundamental frequency, I decided to Brass 3500 supportthecutpipesatthenodesforthefirstmode Copper 3700 Steel 5050 of vibration. This desired mode also required Glass 5200 givingmavalueof3.0112. Theinsideandoutside radii of the pipe were measured to be 7 mm and 8mmrespectively,givingK avalueof withatotaloffourantinodes,includingtheonesat (cid:1) theends. Thismodehasanodeatthecentreofthe K = 1 (7mm)2+(8mm)2 =5.32mm. 2 pipe and two other nodes at 0.132L and 0.868L (see figure 2). There are many higher modes of Wanting to be able to cut eight lengths of pipe vibration. The frequency of the nth transverse representing a C major scale (Cx, D, E, F, G, A, mode of vibration for a bar or pipe is given by B, Cx+1), I first calculated the length for a pipe [1]: vibratingtransverselyinthefrequencyofthenote πvK C . The frequency of C in the Scale of Equal f = m2 (1) 5 5 n 8L2 Temperament [2] is 523.25 Hz. Rearranging the where v is the speed of sound in the material of equationforfrequencyofvibrationandcalculating thebarorpipe(seetable1forthespeedofsound thenecessarylengthforthefirstmodetoproduce in common materials); L is the length of the bar thefrequencyoftheC5notegives: or pipe; m = 3.0112 when n = 1, m = 5 when πv K n=2,m=7whenn=3,... (2n+1);K isthe f = L m2 ⇒ radiusofgyrationandisgivenby 1 (cid:2)8L2 πv Km2 thicknessofbar L= L K = 3.46 (cid:2) 8f1 π(3700ms−1)(0.00532m)(3.0112)2 forrectangularbars,or = 8(523.25Hz) (cid:1) K = 1 (innerradius)2+(outerradius)2 =0.366m. 2 forpipes. If all eight pipes were the same length, then the The values for m indicate that, unlike the total length of pipe needed would be 2.93 m. vibration modes of plucked and bowed strings However, if C5 is defined as the lowest note or of air columns within pipes, the transverse produced by the instrument, then all the other vibrationmodesofbarsandpipesareanharmonic pipeswillbesmaller. Thusa3mlengthisquite (f (cid:1)=2f ,f (cid:1)=3f ). Instead sufficient. Sincetheonlyvariableintheequation 2 1 3 1 fordeterminingthelengthforthesamecopperpipe f 52 is the desired frequency of vibration, the other 2 = =2.76 f 3.01122 elements of the equation can be combined into 1 one constant, simplifying the calculation for the and remaininglengths: f 72 3 = =5.40. (cid:2) (cid:2) f 3.01122 1 πv Km2 70 L= L = m. Given equation (1), students can easily 8f f/Hz calculate the lengths required for a bar or pipe toproducevariousfrequenciesofvibrationwithin Table2providesthenotesintheC majorscale, 5 theaudiblerange. Ifseveralofthesefrequencies their frequencies, and the corresponding lengths are chosen to match certain frequencies within for the copper pipe sections that will vibrate anacceptedmusicalscale, thenasimplemusical transverselyatthosefrequencies. July2003 PHYSICSEDUCATION 317 DRLapp Figure3. Thissimplexylophone-likemusicalinstrument,builtfromasingle3mlengthofcopperpipe,produces allthenotesintheC majorscale. 5 Table2. Lengthsfor15mmdiametercopperpipesto commonly available at most home improvement vibrateatthefrequenciesofthenotesintheC5major stores was used. There are many other options scale. for mounting the pipes depending on the quality Notes Frequencies Pipelengths and permanence desired [3]. Simple mallets for (Hz) (m) strikingthepipescanbemadefromleftoverpieces C 523.25 0.366 of the 3 m section of copper pipe. A 150 mm 5 D5 587.33 0.345 piece of copper pipe wrapped at the end with E 659.26 0.326 5 several layers of office tape produces tones that F 698.46 0.317 5 aresatisfactory. Woodenspheresabout25mmin G 783.99 0.299 5 A 880.00 0.282 diameterorhardrubberballsmountedonwooden 5 B 987.77 0.266 dowels or thin metal rods will also produce 5 C6 1046.5 0.259 satisfactory, but different, tones. Reference [3] provides detailed instructions for constructing a varietyoftypesofmallets. At this point the bars, if they were to be Givenonlyonesignificantfigureofaccuracy used as chimes, could be drilled through with a in the measurement of the radius of the pipe, small hole at one of the nodes. After filing off I was prepared for its vibration frequency to any sharp areas around the holes, fishing line or deviatesignificantlyfromthatusedtocalculatethe othersuitablestringcouldbestrungthrougheach length,buttestingtheC pipe’sfrequencyagainst 5 pipe. The pipes could then be hung from the the 523.25 Hz tone from an audio frequency fishinglineandwouldproduceahighquality,first- generatorproducedabeatfrequencyofonly2Hz, mode-dominated tone if struck at their centres. an unusually good level of accuracy. Yet even However, I chose to make a simple xylophone- if the frequency of the cut pipe had deviated like instrument. This was accomplished by considerably from the desired frequency, the attaching 25 mm sections of ‘self-stick rubber simplexylophoneisstillausefulendeavour—the foamweatherseal’atbothnodesofeachpipe. A instrument would still sound as though it plays 3/8(cid:3)(cid:3)wideand7/16(cid:3)(cid:3)thick(9.5×11.1mm)variety the full scale perfectly. As long as the lengths 318 PHYSICSEDUCATION July2003 Buildingacopperpipe‘xylophone’ arecuttothenearestmillimetre,threesignificant produced if two people work together to strike figuresofprecisionarepossibleinthefrequencies threeorfourpipessimultaneously. Forthosewith of the pipes. The result is a set of pipes whose limited or no knowledge of the physics of the vibrations are consistent with each other. The musicalscale,therearegoodintroductorysources firstandlaststilldifferinfrequencybyanoctave ofinformationavailable[4–6]. (a frequency ratio of 2:1) and the pipes, when Buildingthecopperpipexylophoneprovides played from longest to shortest, still go up in thestudentwithanengagingandmeaningfulway frequency with Equal Temperament half-step or to apply the equations describing the transverse whole-step increments in the same pattern as the vibrations of bars and pipes. The end product Cmajorscale. ‘Do–Re–Me–Fa–So–La–T–Do’is brings together the disciplines of physics, music instantly familiar. Critics might counter that the and art in a coherent and authentic way. Finally, whole scale is either too flat or too sharp, but itprovidesameanstofurtherexploretheideasof the frequency of any standard note (Concert A consonanceandmusicaltemperament. = 440 Hz, for example) is arbitrary. And unless thexylophoneweretobeplayedalongwithsome Received28April2003 PII:S0031-9120(03)62697-3 other instrument, it’s only the intervals between the notes that are important. The copper pipe References xylophoneproducestheseverynicely. Amusician can also easily use the xylophone to play good- [1] FletcherNHandRossingTD1998ThePhysics ofMusicalInstruments2ndedn(NewYork: soundingtunesaslongasallnotesofthetuneare Springer) withinthemajorscale(noflatsorsharps). [2] http://www.rit.edu/∼pnveme/raga/Equaltemp.html Forthosewhoarefamiliarwiththephysical [3] HopkinB1996MusicalInstrumentDesign basis for the musical scale and wish to use the (Tucson,AZ:SeeSharpPress) xylophone to explore the physics of the musical [4] http://hyperphysics.phy-astr.gsu.edu/hbase/music/ tempercn.html scale, it is easy to demonstrate the musical [5] RossingTD,MooreFRandWheelerPA2002 importance of the octave by striking the two TheScienceofSound3rdedn(SanFrancisco: ‘C’ pipes simultaneously or one right after the AddisonWesley) other. They are clearly different frequencies, but [6] BergREandStorkDG1982ThePhysicsof at the same time have a similar sound, hence Sound(EnglewoodCliffs,NJ:Prentice-Hall) thefoundationofthemusicalscalebeingaround the phenomenon of the octave. One can also appreciate the particularly consonant intervals of DavidLapphastaughtphysicsatthe the perfect fifth (a frequency ratio of 1.5:1) and highschoolandintroductorycollege levelssince1984.In2003heisworking the perfect fourth (a frequency ratio of 1.33:1) asaResearchFellowattheWright by striking the C5 and G5 or C5 and F5 pipes CenterforInnovativeScienceEducation simultaneously. Therelativeconsonanceofmany atTuftsUniversityinMedford, Massachusetts.Hisresearchisprimarily other musical intervals can also be explored. In intheareaofthephysicsofmusicand addition, chords of three or four notes can be musicalinstruments. July2003 PHYSICSEDUCATION 319