Harmonicsof S motionon bowedstrings Bo Lawergren Hunter Collegeof the City Universityof New York, 695 Park Avenue,New York, New York 10021 (Received3 February 1981;accepted for publication10 March 1983) Losslessbowed strings have usually been thought to possessa motion discovered by Helmholtzin 1863.However, it wasshown [Acustica 44, 194-206 {1980)]by the authorthat a more complicatedstanding wave motion, the $ motion,exists on suchstrings provided both the bowing distanceand bowingforce are abovecertain minimum values. This paperexplores S-motion harmonicswhich givearise to waveformsof considerablecomplexity on very thin strings. Equationsare found which describethe experimentallydetermined waveforms as a functionof bow position,bow velocity,and observationpoint. In the specialcase of squarevelocity waves at the bow point, the equationsgive quantized values for the bow/stringsticking duration. That resultagrees with Raman's[Proc. Ind. Assoc.Adv. Sci. 15, 1-158 {1918)]prediction. In general, however, the waveforms have rounded corners.
PACS numbers: 43.75.De
INTRODUCTION r is the dimensionlesstime variable,i.e., the ratio Earlier•'2 we havedescribed studies of violin strings of the time and the duration of the fundamental bowedon a monochord.The generaltransverse motion of a periodof the oscillatingstring; 0 < r < 1. losslessstring was found to be differentfrom that described Intervalais - •5/2 < r < + •5/2 and interval b is + •5/2 by Helmholtz.3 We derivedan expressionwhich accounted < r < {1 -- •5/2). {Intervalsa and b denoteregions in the well for the motionprovided the bowingpoint lay more than space-timediagram for travelingwaves on the stringas illus- 5 cm from the bridgeon a normal violin string{33 cm long) trated in Fig. 15a, Ref. 1.) and the downward bowing force on the string exceededa As mentioned above, these expressionsapply when certain minimum value (the critical force). In essence,the /3 > 0.18 and the bowingforce exceeds the critical force.The motionexpression consisted of a prominentsinusoidal com- criticalforce varies inversely 4with/3 froma lowvalue of 0.05 ponent{S motion)superposed on the Helmholtz• motion. N at/3 = 0.36 to 0.4 N at/3 = 0.18. The periodof the sinusoidis/3 (/3 is definedbelow). Perma- As with other standing-wavephenomena, the $ motion nent nodepoints existed at the two endsof the stringand a mustpossess upper partials of its fundamentalmode. Such temporary, intermittent, node occurred at the bow. The modesof vibrationshould be characterizedby {n -- 1) nodes expressionwas chosen to givecontinuity at the bowingpoint betweenthe bridge and the bowingpoint. This paper deals {in positionand spatialderivative). This choiceled to an in- with the form of these,modes and their occurrence on real verse relationshipbetween bowing positionand S-motion stringsas well as on ideal strings.The latter topic connects velocity amplitude.The fact that $ motion is not observed harmonic$ motionto Raman'stheory of bowedstring mo- for small bowingdistances can then be understood,at least tion. 5 qualitatively' the bow is unable to sustainthe large kinetic energyresulting from the S-motionamplitudes. I. APPARATUS During the time interval a the transversevelocity was found to be Expression(1) was derived from velocitywaveforms re- cordedat severalpositions along the string.The recordshad (la) V= vo (COS(2yr) ]•sin sin[(1- y •5)y] 1--•5) ]• to be measuredat constantbow velocity,bow position,and and during time interval b, bow force. For this purposea bowing machinewas devel- oped.A violinbow ran on horizontaltracks pulled by a servo (lb) controlled,printed armature, motor 6 whichautomatically v= vo(• cos[(r--0.5)2y]/3sin y sin(6y})' • alternatedup- and down-bow strokesand kept a constant Vois the velocityof the bow (in absoluteunits). bow velocityof 9 cm/s. The bow hairswere squeezed to a 4- /3 is the dimensionlessbow position,i.e., the ratio mm-widebundle but, otherwise,the bowingconditions were of the lengthLo betweenthe bridgeand the bow similar to those in a normal violin. The monochord rested on point and the total length L of the string; a platform with adjustableheight. The downwardforce on 0
2174 J. Acoust.Soc. Am. 73 (6), June 1983 0001-4966/83/062174-06500.80 ¸ 1983 AcousticalSociety of America 2174
Downloaded 19 Mar 2013 to 146.95.253.17. Redistribution subject to ASA license or copyright; see http://asadl.org/terms to a waveformrecorder sfor off-lineplotting and analysis in 0.093 0.314 0.647 0.868 the time domain. [ [ [ [ r--I [ ::i.
II. HARMONICS OF THE S MOTION For normal violin stringsharmonics of the $ motion are,essentially, damped out andEq. {1)adequately describes the motion("normal $ motion").However, our experimen- tal data showthat Eqs. (2) and (3) mustbe usedto describe 0.159 0.713 0.907 the motionof thinnerstrings. For transversedeflection .•.
d= • d{n), (2) • :•'":;: F
ß
d{n}=(•) F{n}( sin(2•/•-n) 2•rsin[{1 sin(•/n)-- 6)•'n ] {2a}
0.192 0.5s• o.?'zs 0.055 d{n) =(•) F{n}( ,6 {•' --0'5} (2b) _ sin[2•/n{•-2•r--sin0/n) 0.5)] sin(?'6n)) ' andEq. {3)for transversevelocity 9
:. _ v= • v(n), (3)
v(n)=voF(n)(COS(2•'•'•sin[(1--6)•'n]sin0/n) 1--•.)/3 ' (3a) FIG. 1. Velocitywaveforms of a bowedrocket wire with a bowingposition of/3 = 0.418 obtained at the fractional observation distance 6 indicated v(n)= Vo F(n) (6]3 cos[2•/n(•'--O.13sin0/n) 5)]sin(•'6n!) ' (3b)above each waveform. The verticalvelocity scale is arbitrarybut the same Equations(la), (2a),and (3a)apply in the sameinterval as do for all waveforms.The horizontalscale gives fractional time r whereone Eqs.(lb), (2b),and 3(b),respectively. F(n)is the amplitudeof unit (indicatedby thebracket) equals one period of the fundamentalstring the nth mode andf is the frequencyof the fundamental vibration.The dottedcurves are measurements while the solid lines are giv- en by Eq. (3). stringvibration. The F factorsmust be normalizedto make the infinitesum • F(n) = 1. As required,the harmonicsare characterizedby additionaltemporary nodes with spacings equalto thedistance between the bow and the bridge divided into n equal segments.These nodesexist only during the stickingperiod. String shapes and waveformscreated by the additionof S-motion harmonics as in Eqs.(2) and (3) have the samecontinuity properties as in Eq. (1) discussedprevious- ly.1 Harmonic$ motion can lead to vastlymore complex waveformsthan normal$ motion. Figure 1 showsa set of thin-stringmeasurements at a bowingposition of/3 = 0.418. Five termsin Eq. (3) [ F(1) = 1.7;F(2) = -- 1.3;F(3) = 1.0; F(4) = -- 0.7;F(5) = 0.3] wereneeded to givethe fit in Fig. 1. Thesewaveforms are moredifficult to matchwith Eq. (3) thanprevious measurements 1because the presentdata are more highly dependenton the bowingposition and the ob- servationpoints (the nodesare more closelyspaced) and somewhatdependent on the bow force(which must exceed the criticalvalues for eachrelevant harmonic). There are no adjustableparameters in normal$ motionbut in harmonic$ motionthe amplitudesF (n)are arbitrary.With thisin mind, the data of Fig. 1 lendreasonable support to the predictions of Eq. (3). 0.1 0.2 0.3 0.4 0.5 Often(but not always,as can be seen in Fig. 6) harmonic Bow Force (N) motion requireshigher critical bow forcethan normal $ mo- FIG. 2. Velocity amplitude (arbitrary units) of the sinusoidalpart of the tion. The dependenceof $ motion amplitude on the bow waveformsin $ motion of a rocket wire at/3 = 0.36. The two curveslabeled I force, qualitatively,resembles the behaviorof normal mo- and II showthe n = 1 and 2 components,respectively.
2175 J.Aceust. Sec. Am., Vol. 73, No. 6, June1983 BeLawergren: S motion on bowed strings 2175
Downloaded 19 Mar 2013 to 146.95.253.17. Redistribution subject to ASA license or copyright; see http://asadl.org/terms III. HARMONICS ON INFINITELY FLEXIBLE STRING On losslessstrings without stiffnessall harmonicsare undamped.One might guessthat the $ motionis impossible sinceany value of/5' can be thought of as lying closeto a rationalfraction M/N and, thus,close to polesfor the M th • Fund. harmonic, indeed close to all the (MK)th harmonics
- - - 3/8 (K = 1,2,3,...). Nevertheless,$ motion may occur because,
-- -- 2/5 analytically,the amplitudesF(MK) can be made zero and,
****3/7 physically,this can be justifiedfor all higher harmonicsby
oooo 4/9 the finite dampingof any real string. Let us first consider certain, analytically simple,waveforms. Two caseswith an eee$ 5/11 infinite number of harmonicsin Eq. (2) or (3) are of special interest.
A. Square waveform at the bowing point
. With • =/5' in Eq. (3) the waveformat the bowingpoint FIG. 3. Velocity amplitudes(arbitrary units) of $ motion (measuredand calculated)for variousbowing positions near/• = 2/5. Each measurement is obtained.If an infinitenumber of termsare includedin Eq. was taken at a string position where the amplitude was optimal, i.e., 6 (3) with =/•/2. The solidlines are givenby Eq. (3). For the sakeof visualclarity the measurementshave not been drawn. However, the data fell within one error F(n,square)= (-- 1) n+• 2 sin(•rpn). (0 2176 J. Acoust.Soc. Am., Vol. 73, No. 6, June 1983 Bo Lawergren:S motionon bowed strings 2176 Downloaded 19 Mar 2013 to 146.95.253.17. Redistribution subject to ASA license or copyright; see http://asadl.org/terms I Period case)can be expressedby a finite continuedfraction' a(1)-I- 1 1 + 1 a(3)+ SINEN•$ • 1 • (5) wherea(1),a(2) .... ,a(k ) arepositive integers called partial quo- tients.If onlya( 1),a(2),...,a(l) areincluded, one obtains the l th convergent.The convergentsare, in fact, the desiredap- SOUARE N• $ proximations( ----n/Q) to ]5'.The lowestorder convergents are SQUAREN•I0 ]5'(1)= l/a(1), ]5'(2) = a(2)/[a(1)a(2)+ 1], ]5'(3)= [a(2)a(3)+ 1]/[a(1)a(2)a(3)+a(1) + a(3)]. FIG. 4. Waveforms(arbitrary velocity units along the verticaldirection and timealong the horizontal) calculated according to Eq. (3)at bowingposition ]• -- 4/9 and measuringposition •5 = 4/9. The summationsin Eq. (3) have This implies that terms with n = 1, n =a(2), been terminated at the n values shown. n = [a(2)a(3)+ 1] are large. [The first convergentgives a poor approximation,consequently, the denominatorof the firstterm [v( 1 ) in Eq.(3)] is generally not close to zero.} If the irreduciblefraction ]• = M/N, where M < N/2 < about 7, ofp. In fact, the sum in Eq. (3) fails to converge. Eq. (5) becomesexactly equal to ]5'with only a few partial The large terms occur for n values that make the de- quotients.In that caseall convergentsare poorapproxima- nominatorsin Eqs. (2)and (3) small, i.e., sin(ny)•0. This tionsexcept for thelast which corresponds to thelowest pole conditionarises when ny = Qrr q- e (Q = 1,2,3,...)or, equiv- [which,of course, is removed ifp ischosen to bea multipleof alently,n/•Q = ]5'__+ 1' (eandl' aresmall numbers). One finds p(M)]. In other words,for such]5' values as 3/7, 3/8, 3/10, thesen valuesby searchingfor rational fractionsn/Q that 3/13; 4/9, 4/11, 4/13; 5/11, 5/13; 6/13 the waveformat the approximatethe valueof]5' and yet haven and Q lessthan M bowingpoint is a perfectsquare wave while the waveformsat and N. The algorithm for sucha searchcan be found in the otherlocations consist of line segmentsjoined with sharp theory of continuedfractions. corners.On the other hand, if M and N are large, partial Accordingto thattheory TM a rational fraction ( ]5'in this quotientsarise which make the higherorder convergents III. 1/4 1/16 (2+e)18 - • (a-e)/11 - •• 4/13 1/8 . 5/18 (a+e)/7 - • 8/7 1/15 I I I I FIG. 5.Transverse deflection (arbitrary units in thevertical direction) waveforms. The horizontal direction shows time scaled to givean oscillation period of thelength shown by brackets under each column. Column I: Raman's(Ref. 5, Fig. 9)waveforms; column II: waveformsgiven by Eqs. (2) and (6) withp = 2/M whereN andM are integerssuch that ]• -- N/M; columnIII: waveformsmeasured on rocketwire. 2177 J. Aceust.Sec. Am., Vol. 73, No.6, June1983 Be Lawergren:$ motionon bowedstrings 2177 Downloaded 19 Mar 2013 to 146.95.253.17. Redistribution subject to ASA license or copyright; see http://asadl.org/terms closeapproximations to/•. The numeratorsof theseconver- Our Eq. (2) may be usedto calculatethe 64 waveforms. gentsgive the n valuesof the very largeterms in Eq. (3}. For definitenesswe take sinusoidal waveforms and p The valuesof a(n}can be obtainedfrom • by useof the - 1/M. At Raman's"irrational" points,i.e.,/• = (r _+e), p continuedfraction algorithm. TM is chosennear the valuesthat would be usedat/• = r. These It followsthat the string cannotcarry squarewaves for waveformsare in goodagreement with themeasurements 15 every value of •. To enable comparisonwith Raman's re- anda comparisonwith a typicalset of Raman'swaveforms is sults, it is desirable to find a waveform (with selectable shown17 in Fig. 5. Theonly exception to theagreement isthe widths)that existseven when/• = r -+-e on a perfectlyflexi- third waveformfrom the top which will be discussedbelow. ble string.Sine waves satisfy this requirement. V. HARMONIC $ MOTION IN GENERAL B. Sinusoidal,waveform at the bowing point In reality, as the data in Fig. 1 shows,the waveformat Here we have thebowing point {or, equivalently, at 6 = 1 --/• ) is neithera squarewave nor a sinewave. The data of Krigar-Menzel and F(n,sine}= (-- 1)n 2sin(•rpn} . (0 In this manner we can find sinewave expansionsof Eq. •E 2 ./.•] ...... II. (3} when• = (r q- e} sincethe termsthat werelarge in the caseof squarewave expansion (i.e., terms with M>>5} now are ß _. I 1 negligiblysmall. Furthermore, the slippingduration p • is 0.1 0.2 0.3 not quantizedin that case. Bow Force (N) IV. COMPARISON OF RAMAN'S AND OUR WAVEFORMS I. In 1891Krigar-Menzel and Raps •s measured 64 trans- ß 3 versedeflection waveforms and subsequently Raman 5 pre- senteda theory of thesewaveforms based on geometrical = 2 constructions.The measurementsgave a few rather sharp- E edgedwaveforms but the majoritywere rounded. However, Raman'smethod assumed a squarewave at the bowingpoint and all his resultsare sharpedged. Apart from this, thereis good agreementbetween experiment and theory with re- •0.24 0.26 0.28 gardsto relativesizes and widthsof the featuresin the wave- FIG. 6. Variousmeasurements (dots and boxes)and calculations (solid lines) forms.Raman also noticed 16 the quantization ofp (calledw pertinentto the secondwaveform (from the top)of columnIII in Fig. 5. In in his work} at bowingpoints with/3 = M/N if M < 5. At all casesthe verticalaxes show transverse velocities proportional to the val- thesepoints Raman assumedp = 1/M in the waveforms. uesgiven by Eq. (3) whenrelevent F valuesare unity. 2178 J. Acoust.Soc. Am., Vol. 73, No. 6, June 1983 Bo Lawergren:S motionon bowedstrings 2178 Downloaded 19 Mar 2013 to 146.95.253.17. Redistribution subject to ASA license or copyright; see http://asadl.org/terms Occasionallyour measurementsgive rather peculiar morelikely to haverounded (e.g., sinusoidal) waveforms be- waveformsat the bowingpoint. The secondwaveform from causethe harmonic content drops off fast [as n ¾3,see Eq. (6), the top in Fig. 5 (columnIII) is measuredat a/3 valueslightly providedp is not too small]. Both squareand sinusoidal abovethe/3 = 1/4 pole; each cycle shows 11 small peaks waveformsmay give $ motion even when a very flexible superposedon a ramp. The waveformcan be generatedby stringis bowed at a "rationalpoint" ( 13= M/N) if the dura- Eq. (2)ifone takesF(1) = F(n > 2) = 0 andF(2) = 1, i.e., the tion of the slippingtime is quantizedto valuesthat aremulti- fundamentalis suppressed.Figure 6 showsthe correspond- plesof/3/N. ing velocitywaveforms at •5= 1/15 (11 peaks/cycle)and at the bowingpoint (a W shapeduring the slippingtime). These resultsoccur at a moderatebowing force (about0.075 N). With larger force (about0.11 N) the fundamentalof the $ lB.Lawergren, "On the motion of bowed violin strings," Acustica 44, 194- motiondominates as shown in the third waveform(from the 206 (1980). top) in Fig. 6. The reasonthis happenscan be learnedfrom 2B.Lawergren, "A newand a rediscoveredtype of motionof thebowed violinstring," Catgut Acoust. Soc. Newslett. 30, 8-13 (1978). the two lowercurves: at this point (/3 = 0.26) the harmonic 3H.L. F. Helmholtz,On the Sensations of Tone as a PhysiologicalBasis for (n = 2) of the $ motion can be elicitedat a lower bow force the Theoryof Music, translatedby A. $. Ellis from the German editionof than the fundamental(n = 1). 1877(Longmans, Green and Co., London,1895; reprinted by Dover),Ap- pendix VI. VI. CONCLUSIONS 4SeeRef. 1, Figs.4, 9, and 10. 5C.V. Raman,"On the mechanical theory of bowed strings and of musical Equations (2)and (3) account for all observedwave- instrumentsof the violin family, with verificationof results,"Proc. Indian formsproduced by sufficientlylarge bow force (0.02 to 0.4 iV Assoc.Advance. of Sci. 15, 1-158 (1918),Figs. 9-12. 6ServodiscTMmotor made by PMI Motors,5 AerialWay, Syosset, NY. dependingon the valueof/3' ) andsufficiently large/3 ( > 0.18). 7Suppliedby SuperSensitive String Co., R.R. 4, Box30-V, Sarasota, FL Most violin stringsare thick enoughto preventthe forma- 33577. tionof harmonicsin whichcase Eq. (3)reduces to Eq. (1).For 8Model805, Biomation, Cupertino, CA 95014. moreflexible strings the harmonicscome into play.If a high- 9Theequation for harmonic S motionsuggested earlier [Ref. 1, Eq. (5)] is incorrectsince it doesnot accountfor the observedpoles although it may ly flexiblestring is bowedat/3 = M/iv, the (KM)th term in givesome waveforms with fair degreeof accuracy,see Fig. 19,Ref. 1. De- Eq. (2)or (3)(i.e., at the pole)becomes very large and the tailed experimentalstudy of/3 dependencenear the secondharmonic at expansionin the equationsfail to converge.$ motion may /3 = 2/5 showsthat Eqs.(2) and (3) are correct. •øHowever,waveforms without S motioncan be obtained at/3 = 1/m if the stillbe possible at suchpoints if F (KM) = 0 in whichcase the bowingforce is verysmall. This resultsin Helmholtzmotion [i.e., the non- pole is cancelled.(For real stringsof some thicknessthis sinusoidalpart of Eq. (1)] with a missing(total string) harmonic, namely situationprobably implies that all harmonicswith n > M are the onewith a nodeat the bowingpoint, i.e., the mth harmonic. dampedout as well.) Very flexiblestrings permit high har- llSeeRef. 1, Fig. 5. •2SeeRef. 1, Figs.12 and 13. monics and they are, thus, difficult to bow at "rational" 13Weuse Raman's (Ref. 5) notation of e insteadof theusual e. points(where/3 = M/iv with low M values). •4G.H. Hardyand E. M. Wright,An Introduction tothe Theory of Numbers Another way to cancelthe polesis to assumethat the (OxfordU. P., London, 1938),Chap. 10. waveformis square(as Raman 5 did)but the F valuesmust •50. Krigar-Menzeland A. Raps,"Ueber Saitenschwingungen," Berl. Klin. Preuss.Akad. Wiss., Sitzungsber.44, 613-629(1981). thenbe arrangedin specificways [i.e., accordingto Eq. (4)] •6SeeRef. 5, Figs.13-15. that are unlikely to occurexperimentally. Real stringsare •7SeeRef. 2, Fig.9 whichincludes the waveforms of Raman,Ref. 5, Fig.9. 2179 ' J. Aceust.Sec. Am., Vol. 73, No.6, June1983 Be Lawergren:$ motionon bowedstrings 2179 Downloaded 19 Mar 2013 to 146.95.253.17. 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