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Harmonics of <I>S</I> Motion on Bowed Strings

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Harmonics of <I>S</I> Motion on Bowed Strings 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 bowing point. 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</•< 1. the monochord could be monitored by means of strain • is the dimensionlessobservation distance, i.e., gauges.In order to bring out the different,modesof vibra- the ratio of the lengthLo betweenthe bridgeand tion,we useda thin stringof "rocketwire "7 with 0.16-mm the point of observation and the length L; diameter and 0.2-g/m mass.It had a length of 33 cm and 0<•5< 1. frequencyof 440 Hz. The stringpassed through the vertical y is rr//3. field of a permanentmagnet and the inducedsignal was fed 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 this in 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<p< 1), (4) bar of the solid line except near the poles.The arrows (labeledby their/• •rpn values)show the positionsof somepoles. the velocitywaveform at the bow becomesrectangular, i.e., it hasa constantvelocity ( = Vo}during the stickingtime and anotherconstant velocity during the slippingtime. The infi- nite set of parametersF(n} is therebyfixed and replacedby tion.4 For small forcesthe sinusoidalampEtude increases one arbitrary parameterp (butp < 1}, definedby p/3 = slip- with increasingbow force until a plateauis reached.(The ping time at the bow point, i.e., the durationof the square forceat the onsetof the plateauwas used • to define,opera- wave at the bowing point.
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