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The Physics of the Bowed Siring

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The Physics of the Bowed Siring String The Physics of the Bowed What actually happens when a violin string is bowed? Modern circuit concepts and an electrom,agnetic method of observing string motion have stimulated new interest in the question by John C. Schelleng he heart of the violin or any of its simplicity its action under the bow pre­ bowed. Helmholtz' procedure is a good relatives, the center from which sents many unanswered questions. The example of how a well-conceived experi­ Tflows the acoustic pulse that is the elementary physics of its behavior can ment combined with simple mathematics very life of the music, is the bowed nonetheless be of considerable impor­ can illuminate a problem that could not string. The string-its action under the tance to the player. at the time be solved with either ap­ fingers and the bow, its gratifying re­ Of the many papers published by Her­ proach alone. Today we would call his sponsiveness and even the problems it mann von Helmholtz, ranging through apparatus an oscilloscope; to him it was forces the player to solve-plays a major physiology, anatomy, physics and the a "vibration microscope," an invention role in establishing the musical identity fine arts, there is one titled "On the Ac­ he credited to the French physicist Jules of this family of instruments. Conceptu­ tion of the Strings of a Violin" that ap­ Antoine Lissajous. Through this instru­ ally the string is the simplest of com­ peared in the proceedings of the Glas­ ment he looked at a grain of starch fas­ ponents, although its manufacture calls gow Philosophical Society in 1860. Up tened to a black string, which he set in for meticulous care: it must be flexible, to that time little was understood about vibration by bowing. The objective lens uniform and strong. In spite of this what actually happens when a string is of the microscope was mounted on a MOVABLE NUT TAILPIECE MAGNET _.-(i�<:::::;::ElI=��:====n PREAMPLIFIER AMPLIFIER OSCILLOSCOPE MONOCHORD, a simple experimental arrangement used by the put signal that can be amplified and displayed on an oscilloscope author to study the motion of a bowed string, consists of a single screen (see circllit diagram at bottom). With the two switches in the electrically conducting string mounted between two massive up position the system displays string velocity; with the two bridges on a firm base. The movement of the string through the switches down the system displays string displacement. The string magnetic field set up by a small movable magnet generates an out· can be bowed by hand, by a pendulum·driven bow or rotary bow. 87 © 1973 SCIENTIFIC AMERICAN, INC large tuning fork so as to vibrate slowly was broken were always in the same ra­ to view the string with the aid of a parallel to the length of the string. When tio as the two lengths into which the stroboscopic lamp, the boundary would both string and fork were set inmotion point of observation divided the string. have disappeared and he would have at suitable rates, Helmholtz saw a "Lis­ Something can be learned merely by seen the string as a sharply bent straight sajous figure," a form of oscillogram that looking at the lowest string of an in­ line. When the bow changes from "up displayed the position of the starch par­ strument while it is being vigorously bow" to "down bow," the motion of the ticle as it varied within the period of bowed [see illustration on opposite bend around the edge changes from vibration of the fork. By similarly ex­ page]. In appearance it widens into a counterclockwise to clockwise. amining the motion at other points he ribbon bounded end to end by two The sideways velocity of the string experimentally acquired the basis for a smooth curves. (Actually the position at any point has two alternating values, mathematical description of the motion around which the string vibrates is unequal in magnitude and opposite in of the string as a whole. moved a little to one side by the average sign. As a result a typical zigzag dis­ Helmholtz wrote that "during the force exerted by the bow in its direction placement curve has a corresponding ve­ greater part of each vibration the string of motion. ) Helmholtz found mathemati­ locity curve that is rectangular in shape. is carried on by the bow. Then it sud­ cally that the boundaries are parabolas; The ratio of the two alternating veloc­ denly detaches itself and rebounds, because of their shallowness they are in­ ities is the same as the ratio of the whereupon it is seized by other parts of distinguishable from arcs of a circle. It lengths into which the point of observa­ the bow and again carried forward." would be a mistake, however, to suppose tion divides the string. Plotting the position of the bit of starch that the string itself has this shape at Two simple physical facts underlie as a function of time, he found that ev­ any time. The string, Helmholtz found, the action of the bowed string. The first ery aspect of the picture as he found it has at any instant the shape it would is that "sliding" friction is less than except one could be represented by have if it were pulled aside by the finger "static" friction and that change from straight lines. During one period of vi­ to some point on the arc: it is a straight one to the other is almost discontinuous­ bration, almost regardless of where on line sharply bent at one point. The bend ly abrupt. The second is that thG flexible the string he looked or where he bowed, races around this edge once in every vi­ string in tension has a succession of nat­ the curve was a zigzag of two straight bration; for the open A string of a violin, ural modes of vibration whose frequen­ lines [see illustration below]. The two for example, it goes around 440 times in cies are almost exact whole-number periods of time into which the vibration one second. If Helmholtz had been able multiples of the lowest frequency; as a result the duration of a single vibration in the first, or lowest, mode precisely equals the duration of two vibrations in the second mode, three in the third and so on. Without outside compulsion the string is therefore by its very nature giv­ en to supporting a "periodiC" wave, that is, a repetitive series of similar vibrations with a wave form dictated by the "stick­ slip" process. The string allows the co­ existence of a multitude of harmonics; the peculiarities in friction require it. Helmholtz' shuttling discontinuity is the timekeeper that precisely triggers the capture and release of the string at the bow. There is a perennial explanation that views the string as a spring peri­ odically pulled sideways to the breaking TIM E ------'7 point of static friction. The spring re­ d coils and is again captured. This view cannot explain the constancy inrepeti­ ·--· -- ------ tion rate over the wide range of bow t -----+-f---------f-+-------- forces, or "pressures," applied by the " � hand. The correct explanation must be a: I­ CJ) given in dynamic terms; such an ex­ LL planation is suggested by the constancy o � in time needed in the flexible string for o the bend to travel twice its length. The --'o w timing is vividly illustrated by striking > a long, taut clothesline near one end TIM E ------'7 with a stick. A discontinuity is clearly seen hurtling to the far end, where it is DISPLACEMENT OF BOWED STRING from its average position is plotted as a function of time in the first three curves in this illustration. The characteristically zigzag curves were reflected. On its return one feels through obtained by bowing near one end of the string and observing at the middle (a), near the the stick (still resting on the line) an im­ bridge end (b) and near the nut end (c). In each case the two periods of time into which pulse much like the momentary frictional the vibration is broken are in the same ratio as the two lengths into which the point of ob· force on the string, which fails to hold servation divided the string. Rectangular string-velocity curve (d) corresponds to curve c. at escape but succeeds at recapture. 88 © 1973 SCIENTIFIC AMERICAN, INC A simple experiment partially con­ firms this picture of the action of the bowed string [see illustration on next page]. An instrument is mounted with the strings horizontal. A light bow, sus­ pended at its heavy end by a long thread, rests on one of the strings at a point near the bridge. A second bow sets the 2 string in vigorous vibration. Before the hanging bow can begin to move slipping occurs at all times except at moments when the string reverses. Since friction in slipping is nearly independent of �--� speed of slipping, the forces in the two directions of vibration are the same but the impulses imparted to the bow are 3 proportional to the duration. The di­ rection of the acceleration of the hang­ ing bow will therefore indicate the di­ rection of string motion during the <=----� Ion ger dura tion. The experiment shows, however, that the direction in which the hanging bow 4 moves is the same as the direction in which the driving bow is moving. There­ fore it is in the longer interval that the string moves with the driving bow; rela­ tive motion between driver and string is accordingly less than in the shorter in­ terval and sticking is presumably occur­ ring.
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