The Oscillations of a Bowed String
by A.H. BENADE
Bowed string instruments such as the violin, viola, cello, and bass viol are like the wind instruments in their ability to produce steady tones. The wind player uses a control device to convert the steady air supply from his lungs into the longitudinal oscillations of his instrument's air column; in place of the wind player's air supply, the violinist uses a bow that he pulls steadily across a string. The periodically varying frictional force between the string and the bow maintains the transverse oscillations of the string. The nature of the interaction between bow hair and string can be compared to that between reed and air column, since the end result of both systems is the setting up of regimes of oscillation. Cooperation among the various string resonances is mediated in familiar fashion by the production of heterodyne components that transfer oscillatory energy generated at the frequencies of some pair of sinusoidal components to oscillations taking place at other frequencies in the total vibration recipe.
23.1. The Excitation Mechanism of a Bowed String
Let us begin our investigation of the bowed-string excitation mechanism by looking, as is our usual custom, at the behavior of a simple mechanical device. The top part of figure 23.1 shows a mass M mounted between a pair of springs S to form an oscillatory system having its own characteristic natural frequency. The presence of some cotton stuffed into the springs provides the frictional damping D that is an inevitable part of any real oscillatory system. This damped springand-mass system represents any one of the characteristic vibrational modes of the violin string in which we are interested (compare this with figure 22.10, which uses a bottle within which air oscillates to represent any mode of a flute). The oscillating mass in figure 23.1 rests on a moving motor-driven belt B; the belt runs steadily toward the right with velocity V, which represents the speed at which a musician might drive his bow across the strings of his instrument. The oscillating block M represents the string, and the belt B represents the bow hair; the friction between them is the focus of our attention at this time.
If the block shown in figure 23. 1 is oscillating horizontally with a varying velocity v while the belt is moving steadily, the velocity of the block relative to the belt will be less during the time that the block is itself moving in the direction of belt movement (to the right in the figure) than it is during the half of the oscillation in which the block moves in the opposite direction (leftward in the figure). Some friction always exists between the belt and the block; in the system shown in figure 23.1 this friction is at all times directed toward the right. During the block's rightward travel (when the sliding velocity is smaller than V), the frictional force is exerted in a direction that is helpful for the maintenance of oscillation, whereas the frictional force tends to kill off the oscillation during the return half of the cycle (when the sliding velocity is large). Sustained oscillation is of course only possible if the helpful frictional force has an overall contribution that is larger than the negative effect of the force during the return.
The lower part of figure 23.1 shows curves that relate the block's frictional sliding force to the belt speed V, assuming that enough cotton is packed into the springs to prevent any oscillation (v = 0). The upper member of the beaded curves (labeled "bow pressed hard") shows the behavior of the sliding force for a block pressing firmly against the moving belt, which is analogous to a player exerting considerable pressure on the string with his bow. The other beaded curve (marked "bow pressed lightly") shows the very similar variation of sliding friction produced when the rosined surfaces are not pressed so firmly together. These curves slope downward from left to right, which tells us that the frictional force is largest when the sliding velocity (V - v) is smallest; this is exactly the condition required for the maintenance of oscillation. Notice that the interpretation of the bow-friction curves in figure 23.1 is exactly analogous to the interpretation of the flow-control curves for woodwind reeds shown in figure 21.4.
The lightly dotted curve chat rises from left to right in figure 23.1 shows how the sliding friction would vary if the belt or violin bow were to be treated with wax or grease instead of with rosin. Such a treatment will not permit the maintenance of oscillation, since under these conditions the frictional force is largest rather than smallest when the sliding velocity is high. The slight leveling-off of the dotted curve at the right-hand side of the diagram calls our attention to the fact that when the sliding speed is very large the wax begins to melt, which reduces the friction.
The important features of the bowed string excitation mechanism are outlined below in numbered statements, many of which are closely analogous to those given for reeds in section 21.2.
1. The sliding-friction behavior of a bow acting on a string can only sustain oscillations
when the surface treatment (i.e., rosin) is such as to give a downward slope to the force versus-bowing-speed curve. (See the analogous statement for woodwind reeds in sec. 21.2.)
2. The steeply sloping portions of this curve correspond to operating conditions in which the excitatory force is sensitively controlled by oscillatory variations in the string velocity v at the bowing point.
3. The shapes of the excitatory friction curves are such that the player can move the operating point for the oscillation toward a region of greater steepness either by pressing harder or by bowing more slowly.
4. The fact chat the excitatory-friction characteristic curve is not straight (i.e., the slope varies from point to point along it) is an indication that heterodyne effects can occur, giving rise to regimes in which oscillation is maintained by excitations taking place at several frequencies simultaneously. The bowing conditions which increase the steepness of the curve also increase its curvature.
23.2 The Resonance Curves and Regimes of Oscillation of a Bowed String
In the case of wind instruments, we found it convenient to study the response of an air column to an. excitation produced by pumping a constant-amplitude sinusoidal flow of air in and out of the mouthpiece, measuring the resulting pressure variations inside the mouthpiece by means of a tiny microphone. A response curve measured in this way has peaks at certain frequencies, and these peaks represent frequencies at which the air column can exert maximum influence on the reed, thereby setting up a regime of oscillation. In figure 20.3 we saw one of the ways to measure the response curve for a reed-instrument air column.
To help our understanding of violin family instruments, we can similarly imagine measuring the response of a string to an excitatory force applied at the spot where a bow will eventually be placed, making use of a (slightly impractical) machine of the sort sketched in figure 23.2. Here a long, thin spring connects the rotating crank to the driving point on the string. This spring plays a role analogous to that of the capillary tube shown in figure 20.3, transmitting excitation from the crank to the string while at the same time leaving the string able to respond without any direct constraint coming from the position of the crank pin. The oscillatory response of the string to its sinusoidal force excitation can be measured by means of some sort of motion detector placed near the string at the driving point. A pickup similar to those used on electric guitars is particularly suitable for this purpose, since it responds to the velocity v with which the string vibrates (see also experiment 5 in sec. 7.4). A response curve plotted with the held of such a machine contains information on the frequencies at which an applied sinusoidal force gives the maximum oscillatory velocity; the peaks on the response curve tell which frequencies best communicate with the bow friction in setting up regimes of oscillation.
We already know that the string will respond strongly to the driving force at the frequency of each one of its characteristic modes. The frequencies of these modes are of course in very nearly harmonic relationship. We also have met and repeatedly applied the idea that an excitation applied near the middle of a vibratory hump will produce much more of a response than will a driving force applied near a node (see secs. 7.3 and 10.7). All these ideas are apparent in figure 23.3. The uppermost resonance curve shown here is what one calculates for a hypothetical string in which the damping of all the modes is the same. The length of the string is L, and it is driven (and measured) at a distance B=L/16 from one end. Notice that the resonance peaks corresponding to modes 1 through 8 are progressively taller as the excitation point finds itself lying ever closer to the middle of a hump of the corresponding standing are more and more weakly excited, until at mode 16 there is essentially no response, because the driving point lies at a node for that oscillation.
The middle diagram of figure 23.3 shows how our hypothetical string would respond if it were excited at a point one eighth of the way from one end, so that the distance B = L/8. Here we see that modes 4 and 12 have particularly tall peaks, since the driver acts on them at mid-hump, whereas modes 8, 16, etc., are hardly excited at all, because the driving force is applied at a node for each of them. The bottom diagram of figure 23.3 shows in exactly similar fashion the driving-point resonance curve expected when B = L/4.
If the bowed string behaves like reed instruments (which, as the dynamic level increases, progress from using a single resonance to using many), then we would expect from these curves that a lightly pressed bow, moving quickly over the string at a distance B=L/16 from the end, would preferentially excite mode 8 (since this mode has the tallest peak in the top curve of figure 23.3). Increasing the bow pressure might be expected to set up cooperative regimes involving two resonance peaks (e.g., 6 and 12, or 5 and 10), and then three peaks (e.g., 4, 8, and 12), etc., leading ultimately to a fully developed oscillation in which all the peaks collaborate to give a strongly controlled regime whose fundamental frequency is equal to that of the mode1 resonance, in exact analogy to the low-register regime of a woodwind instrument.
Actual experimentation on a violin shows that none of these theoretically based expectations are borne out in practice, nor are the analogous expectations based on bowing points at B = L/8 or L/4. It turns out that the resolution of our difficulty lies in correcting the assumption that the damping of all the string modes is the same. It is extremely difficult to make a direct measurement of the string's resonance curve, because the narrowness of the resonance peaks and the associated long duration of any transient behavior would require the complete measurement to be spread over an hour or so, during which time tiny temperature and humidity changes could easily destroy the validity of the experiment. It proves possible, however, to calculate the needed resonance curve on the basis of measurements of the characteristic frequencies and ringing times of the various string modes when the string is plucked or struck. Painstaking measurements made in 1967 by Walter Reinicke at Lothar Cremer’s laboratory at the Technical University in Berlin, West Germany, provide us with an example of this sort of information. His measurements show that the half-amplitude time for the decay of mode 1 of the A-string on a particular violin is about 0.5 seconds, about five times longer than the decay times for modes 2 and 3, and about fourteen times longer than the decay times of modes 4 through 10. The damping rises very rapidly for the higher modes beyond mode 10. Since the heights of the resonance curve peaks are closely related to the decay times of the corresponding string modes, the heights of the response peaks for any modes that are heavily damped will be reduced.
Figure 23.4 shows the response curve calculated (using Reinicke's data) for a real violin string driven at a point located at B=L/8. Notice that (in contrast to the corresponding curve in figure 23.3) resonance peak 1 is the tallest. The next two or three peaks are also quite tall, so that light bowing can be expected to give a tone whose fundamental component has a frequency of 440 Hz, in agreement with the mode-1 natural frequency. Pressing harder on the bow simply adds cooperative contributions from the other string modes, and the system plays in a regime of oscillation dominated by half a dozen peaks. This behavior is in accord with experiments one can carry out on the open (i.e., full-length) A-string of an actual violin. When the string is shortened by pressing it against the fingerboard with the tip of a finger, the string-mode frequencies will of course be raised because of the shortened string, but at the same time the damping of the modes will he increased by frictional effects at the fingertip. The small open circles drawn part way up each resonance peak in figure 23.4 show the height of each peak when a typical amount of finger damping is added to everything else. Notice that when one bows at B = L/8 using finger damping, peak 3 (rather than peak 1) is the tallest, which explains why the lightest possible bowing now produces a sound whose fundamental frequency is 3 X 440 = 1320 Hz (at a pitch of E6, a twelfth above A4). Heavier bowing causes the pitch to drop down to the normal A4 as the main, low-register regime takes over.
The above example will be recognized as a type of what violinists call a harmonic. Harmonics are almost exact counterparts of second- and third-register tones on a woodwind, in that the normal, low-register regime of oscillation involving all of the modes is somehow disrupted so as to favor regimes based on peaks 2, 4, 6, etc. (an octave higher), or on peaks 3, 6, 9, etc. (giving a tone whose pitch is a twelfth higher). When a violinist wishes to play an octave harmonic, he fingers the string very lightly at its midpoint, which has the selective effect of damping the odd-numbered modes; this lowers the corresponding resonance peaks without altering the heights of the even-numbered ones. As long as the how is lightly wielded, it is only necessary to lower the tallnesses of peaks 1, 3, 5, etc., below those of peaks 2, 4, 6, etc., in order to permit the second-register tone to come forth. With heavier bowing, the pitch drops back to the normal position. Notice that there is a complete parallelism between this behavior and what we met in woodwinds in connection with the pianissimo-type (resistive) register hole. The violin player ordinarily lacks a cognate to the fortissimo-type (reactive) register hole, which functions by displacing the frequencies of certain modes rather than by increasing their damping.
23.3 The Effect of Inharmonicity and Damping on the Setting-Up of Regimes
In the course of our earlier studies of wind instruments, we learned of the advantages that come with the proper alignment of air-column resonances into a harmonic relationship. We also came to recognize that a given resonance can participate to some extent in a regime of oscillation even when it is not perfectly aligned, provided that some harmonic of the generated tone lies reasonably well up on the resonance peak. Let us formulate this remark with some care and outline its implications in a set of three numbered statements:
1. In any multi-resonance oscillating system, a given resonance peak can take part in the regime only if its own natural frequency differs from that of the nearest harmonic of the tone by an amount that is less than the half-amplitude bandwidth W1/2 of the peak (see sec. 10.3).
2. Increasing the damping of a given mode of oscillation has two effects on the nature of the resonance curve: (a) the height of the peak is reduced, and (b) the width is increased by the same factor. These in turn have two opposing effects on the ability of the resonance to participate in a regime of oscillation: (1) a reduction in the height of the peak means that the influence of this resonance is reduced, and (2) for a given small amount of detuning, an increase of the width means that the peak is given additional influence over the regime.
3. In wind instruments it has been unambiguously verified that for reasonably small misalignments the benefits of increased resonance width usually offset the disadvantages of reduced peak height. This means that if a peak cannot be aligned quite perfectly, it is worthwhile to make sure that there is enough damping to give reasonable overlap of the peak with the closest sound component. A similar behavior appears to manifest itself among the bowed strings.
Let us see what implications these statements have for the violin family of instruments. John Schelleng, a retired Bell Laboratories engineer whose skillful experimentation and imaginative use of mathematics have made him a recognized leader in violin physics research, has measured the coefficients for stiffnessproduced inharmonicity for many kinds of violin and cello strings (see sec. 16.5).2 Using his data for a typical unfingered violin A-string, we can work out the amount by which the frequencies of successive string modes are raised by stiffness effects (this assumes the string to be mounted on a solid metal frame rather than on an actual violin, where resonances of the front plate and of the bridge can alter the inharmonicity; see sec. 16.5 once again). These upward shifts of frequency away from harmonicity are tabulated below for modes 1, 4, 8, and 12, along with the resonance widths calculated from dara obtained by Reinicke for such a string:
mode no. 1 4 8 12
freq. shift 0 + 1.5 + 13.0 +44.0 Hz
res. width 0.3 1.0 2.6 6.2 Hz
We can see at once that the upper resonances of a rigidly mounted violin string are not at all well aligned: above mode 4 the various harmonics of a 440-Hz tone lie considerably more than the half-amplitude bandwidth W1/2 away from the resonances which might contribute to their support. In other words, we are led to expect that only the first few resonances participate directly in the regime of oscillation, and any higher partials that may be present in the tone arise only as the result of heterodyne action via the lower components. That is, the upper partials are produced in very much the same way as are those partials of a woodwind tone that lie above the tone-hole lattice cutoff frequency. The essential correctness of these deductions relating string inharmonicity to the nature of the spectrum has been verified by Schelleng. I should point out that, from the point of view of the player, a rigidly mounted violin string does not always seem to respond well to the bow. Some of the reasons for this are to be found in the remarks concerning figure 23.3 and some of them have to do with the consequences of inharmonicity.
Let us now turn our attention to the behavior of strings mounted on a violin. We already know that the measurements made by Reinicke show that the string is more heavily damped when it is in its normal surroundings than when clamped on a rigid frame. The numbered statements earlier in this section should then lead us to expect the string mounted on a violin to be more forgiving of any inharmonicities that may be present. We also recall that the inharmonicity itself will be altered when the string is mounted on a violin. In the following tabulation you will find the frequency shifts that I have measured for an A-string on a violin of good quality; also listed are the resonance widths appropriate for a string so mounted (once again calculated from Reinicke's data):
This tabulation shows that the resonance frequencies of a violin string in its normal environment are considerably closer to being harmonic than they are when the string is mounted on a rigid frame. We also notice that the resonance widths are sufficiently broad (even for the open string) that the peaks all find it easy to join in a regime of oscillation according to the requirements outlined in the numbered statements given at the beginning: of this section.[ 3]
On the violin that I measured, mezzo-forte bowed A4 sounds at a pitch that is about 5 cents higher than A-440 ( when the string is tuned in such a way as to place its plucked first-mode frequency at 440 Hz. Examination of the line drawn during such a screech shows that it is made up of a series of fine dots or clashes. If the piece of, halk is long and it is held lightly at one end while the other end hops along the blackboard, one can easily observe the chalk alternately sticking to the board (making a mark)' and then leaping forward to where it recatches during the return trip of its more or less sinusoidal oscillation. This sort of oscillation can arise whenever the frictional force between two bodies is less when they are in relative motion than it is when they are stationary.
The bowing of a violin string works in very similar fashion to the screeching of chalk. When the bow is placed on the string and drawn to one side, the string sticks to the bow, which pulls it aside until the elastic restoring force produced by the string tension becomes large enough to break the string loose from the bow. It now swings back in much the same way it would after slipping off the plectrum of a harpsichord jack; there is, however, a small amount of damping produced by the rapid (and therefore lowfriction) sliding of the string against the steadily moving bow hair. At the end of its backward swing the string will come to rest and then recommence its motion in the direction of the bow velocity. At this time it is once again caught by the large sticking friction of the bow and carried forward to begin a new cycle of the oscillation, just as the chalk alternately caught on the board and broke free of it.
Helmholtz studied the motion of the bowed string at the bowing point and at other points along it by observing an illuminated speck of starch attached to an otherwise blackened string, using what he called a vibration microscope (this device is an optical cousin of today's oscilloscope). The top part of figure 23.5 shows the sort of vibratory pattern that one normally sees at the bowing point of a string. The longer, more gently sloping part of the oscilloscope trace shows the steady upward motion of the string as it is carried along by the bow. The duration of this part of the cycle is known as the sticking time. When the string reaches the upper limit of its travel, it breaks away from the bow and runs downward quickly to the opposite extreme of its motion, where it is recaught by the bow for a steady upward trip. The time during which the string is sliding quickly back against the motion of the bow is called the flyback time. Helmholtz was able to show that the theory of undamped vibrating strings agrees quite well with experiment in predicting that the ratio of the flyback time to the total repetition time will be equal to the ratio of the bowing point distance B to the total string length L. For example, in figure 23.5 the diagram is drawn to show a flyback time that lasts one-quarter of the time for a complete cycle of oscillation, which means that we are dealing with a string that is bowed one-quarter of the way along the string from the bridge.
On the assumption of zero damping of the string, Helmholtz was able to show that the vibration recipe observed at the bowing point (corresponding to the motion we have been discussing) is the same as the recipe for the amplitudes of the modes of a plucked string (which we met in section 7.2). He also pointed out that the expected effects of large bow-hair width on a stiff string would be similar to those of a broad plectrum exciting it (see secs. 8.1, 8.4, and 8.5). In particular, he noted that any frequency component having a node at the bowing point is expected to be missing unless the bow has appreciable width. If B = I/4, as in our present example, we are led to expect that harmonic partials 4, 8, 12, etc., will be very nearly missing from the vibration recipe. Helmholtz also described observations of the unsteady oscillations produced by bowing a badly made violin: the steady sawtooth motion is replaced by a spluttery one in which extra kinks appear randomly from time to time.
We must not forget that what we hear is not the vibration recipe at the bowing point but rather the excitation transmitted to the violin and thence to the room by means of forces exerted by the end of the string where it passes over the bridge. As a first step in working out the driving forces at the bridge, we should turn our attention to the lower part of figure 23.5 this shows the motion of the string driven as before at B = L/4, but now ob served with the vibration microscope focused on a point near the end of the string (either bridge or nut end will do) Here we still see the basically sawtooth waveform, but superposed on it are smal steplike wiggles (crumples is the name given to them in the English translation of Helmholtz's book). It turns out that crumples are visible on a steadily maintained waveform if one bows at L/4, 7 crumples if one bows at L/7, and so or You can perhaps deduce from the diagram that these crumples are themselvs made up of precisely those harmonic partials that were missing from the recipe c too weak to detect easily at the bowing point. Helmholtz recognized the existence of a problem here. The essential invisibility of these extra components at the bowing point is not in itself surprising since one does not expect to see evidence of their presence at a place where they all have nodes. However, how the bow has managed to excite them by means of forces exerted at the bowing point is not instantly apparent. Helmholtz expressed a suspicion that the phenomenon had something to do with damping of the string modes; as we shall sec, his suspicion proved correct.
During the period from 1909 through 1921, the Indian physicist C. V. Raman published a series of papers on the properties of bowed strings, along with the first half of a book on the same subject. Raman's scientific reputation today rests chiefly on his later work in optics (which earned him the Nobel Prize in 1930), but his careful experiments and thorough analysis of the properties of bowed strings underlie or anticipate most of the more recent work in the field. In 1969 Raman sent me a copy of his out-of-print book, On the Mechanical Theory of the Vibrations of Bowed Strings (published in 1918).7 The book includes (among many other things) a large number of photographs showing the motions of a bowed string, excited and observed at many different points along it. These photographs and Raman's analysis of the string motions confirm and greatly extend Helmholtz's work, taking into account the presence of string damping. Raman assumed, however, that all modes are equally damped, as we did for the sake of simplicity in our discussion of the hypothetical string shown in figure 23.3. He also assumed a pure form of stick-slip friction which ignores the way the frictional force varies with the sliding velocity, as sketched in figure 23.1. Nevertheless, Raman was able to account quite well not only for oscillations of the type which we might describe as involving all of the string modes equally in the oscillatory regime, but also for those in which only a selected set of these participate.
Before we continue our examination of the consequences of the stick-slip bowing-point motion discussed by Helmholtz and Raman, we should summarize its salient features as we have met them so far:
1. In normal operation the string of a violin-type instrument remains "stuck" to the bow hair and travels along with it for a considerable fraction of each vibratory cycle, after which the string flies back abruptly to begin the next cycle, which takes place at very nearly the first-mode frequency of the string. (The physics of this sort of oscillation is very reminiscent of that of a reed instrument in which the aperture snaps open for only a short fraction of each cycle.)
2. The string motion at the bowing point has a simple appearance: the ratio of flyback time to repetition time is, to a good approximation, equal to the ratio of bowing-point distance to string length.
3. As a consequence of statement 2, the vibration recipe observed at the bowing point shows very little oscillation at the frequencies of modes that have nodes at or near the bowing point.
4. The components which are apparently missing at the bowing point are easily detected when the motion of the string is studied at points other than the bowing point. The origin of these components will be discussed later in this section.
Let us turn our attention now to some of the practical consequences of the Helmholtz - Raman approach to bowed strings and its later developments. If one wishes to play more loudly, for instance, it is clear that any given point on the string must make a wider excursion to each side of center in the course of each oscillation. Since the number of these back-and-forth trips per second is fixed by the playing frequency, we are led to conclude that the point must move with greater velocity to cover a larger roundtrip distance in the time of each oscillation. Because the bow and the string are moving together during one part of each cycle and because the string cannot move faster than the bow, it is obvious that loud playing demands fast bow motion. These observations lead to two additions to our numbered statements:
5. Since the amplitude of oscillation of a bowed string is directly determined by the velocity of the bow, loud playing calls for a faster bow velocity V. The firmness with which the bow is pressed on the string does not affect the amplitude of the oscillation as long as the bow pressure is within a certain range of suitability.
6. The amplitude of motion at the bowing point itself is smaller than the amplitude of the motion measured at the string's midpoint. As a result we realize that the bow velocity required to produce a given oscillation amplitude is less when the bowing point B is near the bridge than it is when B is a larger fraction of the total string length.
The next question that concerns us is what limitations on bow pressure arise from the necessity for the bow to control the string vibrations adequately. Besides elucidating the general behavior of bowed strings, Helmholtz observed the effect of applying what a musician would call low bow pressure. Raman extended these studies and clarified the manner in which the velocity and position of the bow affect the minimum pressure that is required for proper tone generation. He also observed that the required bow pressure is alter when the played note contains frequencies that match some of the resonances of r violin itself. In 1937, the physicist Frederick Saunders of Harvard University published an account of further work , the relationship of minimum bow pr( sure to the body resonances.' He is al credited with the first recognition of upper limit to the usable force, thou, one can find a brief section devoted this subject in the middle of Raman's book.
In 1973 John Schelleng published article, "The bowed string and the player." The account he gives in it of I own work and that of others on the relationship of the bow to its strings provides the background for my discussion here the bowing-pressure requirements. Let us begin our examination of this question by noticing that the minimum pressure that which is just sufficient to carry t string along with the bow. In other words, the bow must be able to synchronize all the string modes into a motion the desired sawtooth type (i.e., to set a fully developed regime of oscillation] Obviously, if some of the string modes are somewhat inharmonic or if the damping is high, more bow pressure a be required. On the other hand, the bow pressure must be small enough to allow the string to break loose cleanly at the end of its sawtooth swinging motion order to make a good flyback.
We have already learned that a large bowing velocity is needed to produce large-amplitude oscillation of the string. The bow supplies the frictional force necessary to produce these large deflection and this force is proportional to the downward pressure exerted by the bow against the strings (compare the two force curves in fig.23.1.
We conclude, therefore, that the minimum required bow pressure that the player must exert increases and decreases in proportion to the speed with which he propels the bow. The complete relationship of the minimum bowing pressure to all of the properties of the string. to the bowing point, and to the nature of the frictional force can be summarized with the help of the formula:
Here Kstick and Kslip are the coefficients that determine the size of the frictional forces produced by the bowing pressure P under sticking and slipping conditions, and V is the velocity of the bow itself. Since increasing the damping makes the resonance peaks less tall, such a change leads to a higher value for Pmin. Detuning a peak so that the string harmonic does not lie directly on top of it will also raise Pmin.
Because mode 1 has the largest amplitude at the bowing point, the aggregate tallness of the resonance peaks is effectively dominated by the behavior of peak 1 (even though it is not generally the tallest peak). If we take the simplified view that only peak 1 is to be taken into account and if we assume the bowing point to be fairly near one end of the string, it is not hard to verify with the help of figure 23.3 that for every doubling of the bowing-point distance B, there is a fourfold decrease in the minimum required bow pressure. That is, in going from B = L/16 to B = L/8, we notice that peak 1 has risen fourfold in height. A comparison of the peak-1 heights for B = L/8 and B = L/4 shows slightly less than a fourfold increase, the reason being that the bowing point at U4 is getting rather close to the point of maximum excitation for the string. Even when the influence of additional peaks is taken into account, the above conclusions remain valid and can be stated briefly as follows:.
7. The minimum bowing pressure required to maintain oscillation of the normal (i.e., simplest) Helmholtz-Raman type is proportional to the velocity V with which the bow is moved across the strings.
8. The minimum bowing pressure required to maintain normal oscillation on a string is large when one bows near the bridge, and falls to a quarter of its value for every doubling of the distance B from bridge to bowing point.
The final item in our investigation of the bowing properties of strings is the limitation placed on the maximum bow pressure. If the pressure is too high between the moving bow and its string, the string simply pulls to one side, scraping and stuttering against the bow hair without ever going into oscillation. The following more or less describes what goes on when heavy bowing pressure is employed. When the string sticks to the bow and is carried forward with it, an impulse is sent along the string toward its fixed end. This impulse is reflected at the fixed end and comes back in inverted form to the bowing point. 1f the bow pressure is not excessive, the impulse succeeds in breaking the string free in a manner that is quite reminiscent of the way in which the reflected pulse from a piano hammer blow returns to throw the hammer off the string (see sec. 17.4). Schelleng shows that the maximum bow pressure that permits the string to break loose properly is proportional to the bow velocity and inversely proportional to the distance B between bridge and bowing point. This is interesting in itself, but its practical implications are better displayed if we consider the way in which the ratio Pmax /Pmin depends on the bowing conditions:
This tells us, for example, that the nearer the bow is to the bridge, the narrower is the range within which the player must maintain the pressure it exerts, a fact well known to string players. Notice also that an instrument having heavily damped string resonances (so that the peaks are less tall) is one that is less forgiving of chance variations in the bowing pressure. One cannot, however, leap from this observation to a statement that a musician would automatically prefer to play on lightly damped strings-there must always be enough damping to permit proper cooperation among the string modes, as outlined in section 23.3.
23.5. The Bridge Driving Force Spectrum
In our study of wind instruments we found it necessary to distinguish between the sound spectrum produced inside the mouthpiece (as a result of the cooperation between reed and air column) and the spectrum of sounds transmitted out into the room by way of the bell and/or the tone holes. The problem for stringed instruments is very similar, though it is somewhat more complicated: we must go from the bowing-point spectrum to the vibration recipe of the forces exerted by the string on the bridge before we can usefully consider how these forces drive the wooden parts of the instrument to make them act as a sound source in the concert hall.
In chapter 7 we learned how to estimate the amplitudes of the string modes themselves when they are excited by plucking or striking. In section 23.4 of l this chapter we learned that these same rules apply very nearly unchanged to the( recipe produced by the bowed excitations of a string when it it observed at the bowing) point. We must now learn how to trans late the recipe for the amplitudes of the N various modes of a string into the driving! force recipe which these modes give rise to at the bridge. The sideways force exerted by a string on its anchorage at any instant during its vibration depends not) only on the tension under which the string is kept, but also on the angle t( which the string end is momentarily deflected.
Figure 23.6 shows the vibrations, shapes of the first four modes of a uniform string, drawn in such a way that at the left-hand end all of these modes cause the string to be tilted to the same angle. It other words, the amplitudes of these par ticular vibrations have been chosen ii such a way as to make them all exert the same amount of driving force on the left hand string anchorage. You can vent from the diagram that mode 2, because its shorter and more abruptly rounded humps, can run at half the amplitude ( mode 1 and still exert the same driving force on the anchorage. Similarly, mode 3 and 4 are three and four dines as efficacious as mode 1 in driving the anchorage, which accounts for their proportionately smaller amplitudes in the diagram. Let us distill the content of these observations into a single statement:
1. In order to estimate the magnitude of the driving force Fn exerted on the bridge by the nth vibrational mode of a string, one must multiply the amplitude A„ of the mode by its serial number n and by the tension T of the string, according to the formula:
Fn = nTAB x a (a numerical constant)
The numerical constant in all cases turns out to be (ð/L), where L is the vibrating length of the string.
Look now ac the vertical line at the point B along the string in figure 23.6. Notice that the amplitude of motion observed at B associated with each mode is considerably less than the amplitude of the mode itself. This calls to our attention the fact that one must convert the bowing-point amplitudes a1, a2, etc., into the corresponding mode amplitudes A1, A2, etc., before making use of the formula given in statement 1. When we do this as a strictly mathematical problem, considering the strings to be ideally flexible and completely undamped (in accordance with Helmholtz's simplified description of the motion), we obtain the following rather simple result:
Fn =A1 x a , (a numerical constant)
In other words, this simplified calculation implies that all the driving force components are equal in magnitude. This almost-true formula applies to all modes except those that happen to have nodes precisely ac the bowing point, and the formula is entirely independent of the bowing position! Clearly we have oversimplified something in the mathematics leading to the formula, since every string player knows that his tone can be altered by changes in bowing point and bowing pressure. We have looked at the behavior of enough oscillatory regimes to know that the bowing-point oscillation is itself altered when the bowing point is changed, and these alterations cannot be compensated by changes in the bridge driving behavior of the sort we have been discussing. If we look again at the two parts of figure 23.5, we will find that hints are available as to the source of the trouble: the upper figure shows slightly rounded corners which already signal the departure of the real bowing-point motion from a simplistic pattern made up of straight line segments joined together.
John Schelleng, Lothar Cremer, and Cremer's co-worker Hans Lazarus have studied the way in which the roundingoff of these corners (which represents incipient slipping at the point of release and the beginnings of sticking at the end of the flyback time) depends on bow pressure, damping, inharmonicity, etc. They also have studied the small wiggling motions that arise because the string undergoes a sort of twisting and rolling oscillation about its own axis under the influence of the bow. All of these give rise to departures from the straight line segment motion used to calculate the formula given above. To summarize, we can say that ac the minimumforce end of the useful range of bowing pressures, the higher partials in the bridge force recipe are weaker than those given in our formula, whereas heavy bowing makes these partials stronger.
Helmholtz's own observation of the crumples shown in the lower part of figure 23.5 and Raman's later study of them show that these early workers were well aware of the very thing chat most clearly shows the limitations of their pioneering efforts to describe the behavior of bowed strings. Because of the central position of the crumples in our recognition of the shortcomings of the simplest version of the Helmholtz-Raman theory, let us give some attention to the way in which the unexpected vibration components these crumples represent may be excited.
Those of you who have assimilated the ideas presented earlier in this book about the production of heterodyne components by the action of a nonlinear system will not find it impossible to imagine that the strongly generated pairs of components in the motion of a bowed string, such as the second and sixth, the first and fifth, or the third and seventh components, might each act with the nonlinearity of the bow friction to generate a contribution to the fourth harmonic vibration of a string bowed at L/4. Similarly, we might expect a double-frequency heterodyne component arising from component 2, and so on. We have already met examples of this sort of behavior among the wind instruments. For instance, the pedal tone of a crumpet contains a fundamental component char is almost totally derived by heterodyne action from the upper partials of the tone, there being no resonance peak at the fundamental frequency. The even-numbered partials making up the internal spectrum of the clarinet tone are similarly strengthened by heterodyne ac tion far above the levels one would expect from direct cooperation of the reed wit[ the resonance-curve minima chat are found at these frequencies. The question still remains how the bow can communicate this extra vibrational excitation to string modes that nominally display n( motion at the driving point.
It is at this point that we come to understand the shrewdness of Helmholtz's suspicion that string damping must haw something to do with the phenomenon o the excitation of vibration component that have a node at the bowing point. It section 20.3 we learned that the reso nance peaks and dips measured at one end of an air column could be understood in terms of the relationship between a wave sent down the air column by the excita tion mechanism and the reflected wave that comes back from the far end. We learned there that the reason the resonance dip does not fall all the way to zero is that the reflected wave is reduced in amplitude because of the damping it has Buttered, and so cannot totally cancel the initiating wave on its return. It is perfectly correct to look at what goes on at the driving point on a string in exactly the same way (although account must be taken of the waves on both sides by adding their impedances). In other words, the so-called node is not a point of true rest in a driven system that suffers damping. The resonance curves shown in figures 23.3 and 23.4 do not, as a result, fall all the way to zero at the positions of the dips, any more than do the resonance curves of wind instruments. There is a small motion even at what we might call a nodal driving point, so that a bow or other driving mechanism can in fact provide some excitation to the corresponding modes.
Figure 23.6 shows us that even a very small excursion produced at or near a node (e.g., at or near B =L/4) can be associated with a reasonably large vibrational amplitude for the mode in question, which results in a considerable contribution at this frequency to the bridge driving force spectrum. The physics of this behavior is exactly the same as that underlying the unusually large transmission of the even-numbered harmonic components out of a clarinet (see sec.22.4)
Recall that these components are produced at frequencies for which the air column resonance curve is not tall, showing that the pressure standing waves corresponding to these even numbered modes have an approximate node within the mouthpiece.
23.6. Examples, Experiuients, and Questions
1. A vivid way to demonstrate the importance of cooperative effects among the various modes of a bowed string is to disrupt the harmonic relationships of their frequencies. A small strip of masking tape can be rolled tightly around a violin A-string at a point about threeeighths of the way from either end of the string. The added load will lower the frequencies of the various modes in an amount determined by the position of the paper relative to the nodes. If the load has a mass of two percent of the string mass, the calculated frequency shifts are as follows:
mode number frequency shift
1 -3.8 Hz
2 -2.2 Hz
3 -0.6 Hz
4 -4.4 Hz
5 -0.6 Hz
6 -2.2 Hz
The negative signs here indicate a lowering of the mode frequency. When these shifts are added to those tabulated in section 23.3, we find that the resulting inharmonicity is enough to prevent the playing of a normal tone, although various raucous screeches are sometimes possible. You might find it worthwhile to experiment with different loadings and points of application. It will be possible for you to figure out the regimes of oscillation that sustain some of the resulting sounds.
2. When a string is working well with its bow, the two are caught together over a considerable fraction of each cycle of the oscillation. In figure 23.5 the stickingtime is shown to be three-quarters of the repetition time for a bowing point B =L/4. You should estimate the sticking times associated with more normal points of bow application. See if you can put together the information contained in the friction curve shown in figure 23.1 with what you have learned about the sticking time to deduce the reason why a bow seems to skate across the strings when oscillation conditions are unfavorable, and why it seems to require a considerable push to move it along when it takes hold to generate a proper tone. You can study the phenomenon very easily as follows. First, bow good vigorous strokes on an open string of a cello or a violin to establish the feel of the bow in your hand. Next, while continuing to bow vigorously, close your eyes and have a friend periodically press a large rubber sponge or wadded up sweater against the string and fingerboard, so as to provide enough damping to kill the oscillation. Whenever the damping is applied the bow will appear to slip abruptly ahead in its travel. String players generally recognize that putting a heavy mute onto the bridge of an instrument not only changes the sound, but also makes quite an alteration in the feel of the bow in the player's hand. Investigate this change by simple playing experiments, keeping in mind that a massive object attached to the bridge will not only alter the inhamonicity of the strings but also reduce their damping.
3. It is possible to learn a great deal about the motion of a violin bridge by attaching various objects to it in various ways, or by touching it. To begin with, verify by plucking the open strings one by one that their vibrational decay times are little influenced by any sort of pinching or sideways pressure on the bridge that you can exert with your fingers (as long as you do not touch the strings themselves). Notice at the same time that the tone color of the various twanging sounds from the plucked strings is also very little changed by such finger pressure. These preliminary experiments tell us that the wave impedance of the bridge and violin body is high enough that the additional effect of the finger pressure is fairly small.
Now take a steel or brass rod some 10 cm long and 1 cm in diameter (the size of a fountain pen) and press its end firmly against the bridge in various directions. For example, press straight down on the bridge at right angles to the violin's top surface at the E-string end of the bridge, at the G-string end, and in the middle. The tone color of both plucked and bowed sounds will be altered in various ways depending on the point of contact. You should also try pressing on either end of the bridge, the rod being held parallel to the top surface of the instrument and at right angles to the strings. Repeat the experiment with the rod held more or less parallel to the strings, pressing from the tailpiece side for convenience. The relative magnitudes of the resulting tonal changes indicate the amount of bridge motion that occurs in the various directions assumed by the rod. It is possible to estimate the amplitudes of these various components of the vibration by pressing the bar very lightly against the bridge so that it buzzes. If you are scientifically inclined, you may also be able to estimate the acceleration of the vibrating bridge in terms of the mass of the rod and the magnitude of the force with which it must be pressed in order to just barely stop the buzzing.
1. Walter Reinicke, "Ubertragungseigenschaften des Streichinstrumentenstegs," Catgut Acoust. Soc. Newsletter 19 (May 1973): 26-34. (See note 7 below for two other sources for some of the data shown in this article.) The Catgut Acoustical Society was founded in the early 60s by a group of people (centered around Frederick Saunders of Harvard) interested in research on bowed-string acoustics. In recent years its membership has expanded to several hundred, including musicians and scientists all over the world. The Newsletter has come to be the leading source of technical and semi-technical information on the stringed instruments. It is published from the society headquarters at 112 Essex Avenue, Montclair, N.J. 07042.
2. John C. Schelleng, "The bowed string and the player,"J. Acoust. Soc. Ain. 53 (1973): 26--41, see especially part III. See also John C. Schelleng, "The Physics of the Bowed String," Scientific American, January 1974, pp. 87-95.
3. Very recently it has become possible to make useful measurements of the resonance curves of violin strings. The response peaks show all kinds of asymmetry, splitting, and displacement as a result of the various influences that act on the string. See Maurice Hancock, "The Mechanical Impedances of Violin Strings," Catgut Acoust. Soc. Newsletter 23 (May 1975); 17-26. An earlier brief preliminary report appeared in the Catgut Acoust. Soc. Newsletter 22 (November 1974): 25.
4. R. S. Shankland and J. W. Coltman, "The Departure of the Overtones of a Vibrating Wire from a True Harmonic Series,"J. Acoust. Soc. Am. 10 (1939): 161-66.
5. R. T. Schumacher, C. J. Amick, and C. B. Croke, "The bowed string: an integral equation formulation" (abstract),J. Acoust. Soc.. Am. 56 supplement (1974): S26. A more extended account of this work (including some application to wind instruments) has been submitted for publication in the same journal. The title is "Self-sustained musical oscillators: an integral equation approach."
6. Hermann Helmholtz, On the Sensations of Tone, trans. Alexander Ellis from 4th German ed. of 1877, with material added by translator (reprint ed., New York: Dover, 1954), pp. 80-88, 384-87; and A. B. Wood, A Textbook of Sound, 3rd rev. ed. (1955; reprinted., London: Bell, 1960), pp. 101-3. See also Schelleng, "The bowed string and the player," and Schelleng, "The Physics of the Bowed String."
7. C. V. Raman, On the Alechauical Theory of the Vibrations of Bowed Strings, Bulletin no. 15, The Indian Association for the Cultivation of Science (Calcutta, Indian Association for the Cultivation of Science, 1918). Raman also provided me with a listing of his published acoustical papers; this list is reproduced in Catgut Acoust.Soc. Newsletter 13 (May 1970): 6-7.
8. F. A. Saunders, "The Mechanical Action of Violins,"J. Acoust. Soc. Am. 9 (1937): 81-98.
9. Schelleng, "The bowed string and the player," and Schelleng, "The Physics of the Bowed String." See also L. Cremer, "Des Einfluss des 'Bogendrucks' auf die selbsterregten Schwingungen der gestrichenen Saite," Acxstica 30 (1974): 119-36. There is an earlier version of this article in English: "The Influence of 'Bow Pressure' on the Movement of a Bowed String: I and ll," Catgut Acoust. Soc. Newsletter 18 (November 1972): 13-19, and 19 (May 1973): 21-25. Reinicke's string damping data (which form the basis for the example in the present chapter) are summarized in figure 7 of Cremer ’s Acustica article, and in figure 3 of part 1 of Cremes’ s Catgut Acoust. Soc. Newsletter article.
10. The arrangement and tallness of the peaks is strongly influenced by the resonances of the violin bridge and body, so that the minimum bow presure satisfying our requirements is not at all constant as we go from note to note in the scale. See figures 7.6 and 7.9 in Alexander Wood, The Physcs of Music, 6th ed., rev. J. M. Bowsher (New York: Dover, 1961), p. 103.
11. See the references in note 9 above, and also Hans Lazarus, "Dynamical Theory of String Excitation by Bowing" (abstract), J. Acoust. Soc. Am. 48 (1970):