Table Of ContentSPECIALFEATURE: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
