Bow Response to String Force
Robert T. Schumacher
Department of Physics
Carnegie Mellon University, Pittsburgh PA 15213
e-mail: rts@andrew.cmu.edu
Abstract
The force that a bowed string exerts on the bow can be reconstructed from the forces on
the string's termination. By using one or more accelerometers attached to the bow, the
response of the bow to that reconstructed force can be recorded. I will show bow
responses to a bowed violin open E string and A string using a bowing machine. At some
harmonic frequencies of the string the signal from the longitudinal standing wave excited
on the bow hair allows one to deduce the velocity of propagation of that wave. The
consequence of the variable excitation of various harmonics of the standing wave as the
bow stroke progresses is a rapidly changing excitation of the bow’s various normal
modes. Three data channels, requiring at least a 128 kHz sampling rate for E-strings, are
required for bow force reconstruction and motion detection of the bow. I explore
practical validity of substituting the force of the string on the bridge instead of the
reconstructed bow force. That allows a reduction of the minimum number of data
channels from three to two, a reduction in the necessary sampling rate, and use of a bow
stroke by a player instead of a bowing machine.
Introduction
The bow may be the most elusive component of the violin family of music-making apparatus. The violin
(used here generically for all members of the family) has been studied using computer modeling, normal
mode measurements, acoustic measurements in anechoic chambers and in highly reverberant
surroundings. Radiation efficiency of its important normal modes, and of groups of modes at higher
frequencies using statistical energy analysis are two of the recent tools applied to the instrument. But
engineering and physics studies of the violin family are nearly two centuries old. Yet, almost no similar
level of activity has been devoted to the physical behavior of "good" and "bad" bows, in spite of the
strongly held belief of even moderately skilled amateur players that the quality of sound from an
instrument as perceived by the player is affected by the bow being used.
Part of the problem may be that collecting a body of knowledge of the linear physical properties of a
large selection of bows, and correlating them according to some standard of quality does not seem
interesting or challenging, particularly since it is not so obvious that the result would be of value. In this
paper I provide a "demonstration of principle" of a method of measuring the response of the bow to the
forces exerted on it by the string as it is being bowed. Although the bow is mounted in this study on a
bowing machine, what is proposed can be done with a bow played by a player and a relative minimum
of signal digitizing hardware and software.
One might reasonably wonder why an adaptation of the method that works with the violin itself could
not be used. In the most efficient version of that method the bridge is given a sharp impulse in the
bowing plane, at one side of the bridge or at a string notch, and a variety of responses is measured:
radiation (as a function of direction), or corpus vibrations ‘everywhere’ on and inside the instrument.
This is famously tedious, and would be more so if each measurement were made by exciting the bridge
with a sine wave, and making separate measurements at a variety of interesting frequencies. The bow
presents surprisingly equivalent challenges. Unlike the violin, the excitation point on the bow varies as
the bow is drawn across the string. It is possible to excite the bow on the hair with a sinusoidal
oscillation, sweep the frequency, move the excitation point on the hair a little, repeat, etc., etc. That
method, but applied only at one point on the bow, was used in the first attempt to investigate bow
dynamics, although variation with position was not knowingly attempted [1]. As we will see later, when
results are displayed, the distance that one must move to produce a different result is surprisingly small
in some cases, although the degree of significance of that observation is at this point undetermined.
The bow of course exerts a force on the string to set it and keep it in oscillation. By Newton's third law
of dynamics, the string exerts an equal and opposite force on the string. Woodhouse, Schumacher, and
Garoff [2] showed that the frictional force of the bow on the string can be reconstructed from the forces
on the terminations of the string, as measured by force transducers mounted there. In principle the
reconstructed force should be correlated with the bow response to that force, which is regarded as the
force of the string driving the bow at the ever-changing position of the string on the bow. The response
is, in this demonstration, recorded from a one gram accelerometer mounted on the bow at the inside of
the tip, with its sensitivity axis parallel to the bow hair. Clearly many other mounting positions are
possible, but they are not explored here.
By “correlate’, above, is meant, in technical terms, find the transfer function. In practice that is done in
the frequency domain. So, in simple terms, the frequency spectrum for each signal is obtained by
Fourier transform (FFT) of each, and then the ratio of the amplitude of the response at a selected
frequency to the amplitude of the excitation is calculated.
The procedure in the ideal case described above would be as follows: record a bow stroke by digitizing
simultaneously three or more channels, two from the string termination force transducers and the rest
from the accelerometer(s). However, as explained in [2], to this date the reconstruction works with
assured accuracy only using the violin E string (preferably not overwound). The only dispersive
mechanism should be from string stiffness - the solution of the inverse problem required has not been
worked out for lossy terminations or strings. Consequently a digitizing rate must be used that is
sensitive to the effects of bending stiffness. For an E string, 50 kHz bandwidth is required, so the
digitizing rate should be at least 128 kHz. The subsequent calculation of the transfer function is then up
to the experimenter, who can employ available or user-created software.
In fact, our laboratory apparatus has only two data channels, so a subterfuge is required. The details of
the subterfuge are given in Appendix 1. Suffice it to say that even with three channels available
considerable rather specialized analysis is required to extract the force on the bow from the data. So in
addition to presenting some limited results using the reconstructed bow force – done entirely for
validation purposes – the major thrust of this paper is to explore the use of the force of the string on the
bridge in place of the bow force. Thus a minimum of two signal channels is required, one for the bridge
force and one for the bow accelerometer.
The purpose of this investigation has been to illustrate a method that it is hoped will be useful to the bow
maker in the shop. To that end, the minimum equipment required is a violin (family) instrument with a
transducer on the bridge that measures the transverse force the string exerts on the bridge, an
accelerometer light enough not to seriously disturb the bow when mounted on it, and a two channel
digitizer with a path into a computer. The commercial digitizing rate of 44.1 kHz should be sufficient
for almost all purposes.
Experimental Apparatus.
The apparatus used was adapted easily from a bowing machine designed for friction studies with a glass
bow [3]. A viola bow was held in place by the stick with small lumps of modeling clay. The bow hair
was constricted in width so that about 5 mm was in contact with the string. This width contrasts
strongly with the typical width of 0.1 mm characteristic of the glass bow. That introduces another
uncertainty in the reconstruction, which assumes bowing at a point, but the reconstruction software gave
results that showed no clue that the assumed conditions were not being met. The apparatus limits the
length of the bow stroke accurately to 20 cm, which was taken in the middle third of the length of the
bow. The steady state velocity of the bow was 20 cm/s, with the speeds ramped up and down at roughly
constant acceleration in 25 milliseconds, resulting in a data file that represents slightly more than one
second. Figures 1(a) and 1(b) show a few periods of the waveforms of the bridge force and the
accelerometer signal. As explained in Appendix 1, an E string was used, and the digitizing rate was 128
kHz.
Bridge signal (blue) and bow accelerometer signal (red)
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Figure 1a: Typical bridge and bow accelerometer signals. A frequency of 650 Hz corresponds to about
197 samples at 128000 samples per second digitizing rate.
Bow force signal (blue) and accelerometer signal (red)
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Figure 1b: Bow force and accelerometer signals corresponding to the same times as figure 1.
The accelerometer signal in 1b is the same as in 1a, but the vertical scale differs in the two figures.
Analysis Methods
One difficulty facing the experimenter to know what to look for, particularly if the experimenter is not a
bow maker and has no experience-based ideas. A quick survey of the data to reveal clues is needed. The
survey was done by taking in sequence FFTs of the data (bow force or bridge force plus bow
accelerometer) of length 2048 samples, at intervals of several thousand samples from data files with
typically 150,000 samples. These spectra, in 20 to 50 frames, can then be examined for behavior that
may merit a more time-consuming examination.
Examples of potentially interesting behavior were revealed in the initial survey by the following
characteristics: 1) large amplitudes of the bow signal at or around the 9th, 18th and 27th harmonics of the
string signal; 2) many bow signal harmonics vary with time (and hence bowing point position on the
bow) compared to the excitation amplitude – the bow or bridge force; 3) bow signals that are at
frequencies not at harmonics of the driving signal must be used with care because in the sonogram
method used for detailed information beyond the initial survey, any windowing method used –and using
none is not an option, since that would be the worst possible case, a rectangular window – produces
spurious low amplitudes or zeros in the FFT signal that produce misleading results.
The sonogram uses a 2048 sample, hanning-windowed [4] FFT that advances in steps of one string
period, for 600 to 800 periods in a run. A string period for the E string used (at typically 645 to 655 Hz
in various runs) is about 197 samples. Based on the preliminary survey, a position of a peak of the
string signal is identified, and a bow signal frequency - preferably the same harmonic as the string signal
- is also selected. The FFT is interpolated to four times as many points to improve accuracy and insure
that peak positions are not misidentified, and written to an array and to a file if desired. (MATLAB is
used throughout). The amplitude of the peaks is plotted and if desired the ratio of the bow peak to the
string peak is also plotted. In none of the graphs shown below is there any significance to the relative
size of the peaks or the scale of the ordinate, since there has been no effort to adjust the relative gains of
the acquisition channels to provide physically meaningful numbers to the ordinate of the graphs. The
scales were selected for visual clarity only. When ratios are plotted, they are ratios of the selected peak
of the bow signal to the selected peak of the string signal, and it is the change of that ratio as the bow
position changes that is of interest.
Some Results
The accelerometer was oriented to be sensitive to accelerations in the direction of the bow hair. The
force on the string terminations and the reconstructions of the force on the bow are of forces in the
bowing plane (plane defined by bow and string) only, so the force on the bow hairs to which the
accelerometer data is sensitive is in the longitudinal direction, parallel to the bow hairs. That is why the
accelerometer is mounted to respond to motions in that direction. The time dependent force on the hair
produces a velocity wave that propagates with a finite velocity on the hair in both directions away from
the point of contact. The boundary conditions for the bow hair are probably somewhat ill-defined, but
very likely correspond to attachment of each hair (assumed to be independent of all of its neighbors) to a
large mass compared to the hair mass, so movement at the ends is constrained. It is reasonable to
consider the ends to be nodes. That means there can be a standing wave (with longitudinal
displacements) on the bow hair at integral multiples of frequencies given by c/2L, the same frequency
formula that applies to the fundamental of a string’s oscillation. Here c is the velocity of propagation of
the longitudinal excitation of the bow hair, and L is its length, just as, in the case of the string, the
velocity of propagation of a transverse wave and the string’s length, respectively. Appendix 2 explores
some consequences of these remarks.
It is important to recognize that the principal force acting on the hair is longitudinal, and the principal
communication of the string with the corpus of the bow is via that force, as long as the vertical force of
the bow on the string is kept constant. Thus the response of the bow to quick changes in vertical bow
force, caused by an almost infinite variety of bowing gestures, is not part of this discussion. However, it
is quite possible that tone color of a legato bow stroke during which the vertical force of the bow is
reasonably constant is influenced by the mechanism investigated here.
In summary of the Appendix 2 discussion, it is important to understand that the excitation of standing
longitudinal waves on the bow hair must occur in the vicinity of nodes of harmonics of the fundamental,
whose wavelength is 2L, and the exciting frequency st be at or near the frequency of that longitudinal
wave frequency. That limitation is not particularly great, since the harmonic content of the bow force,
and the force on the bridge (which has the same harmonic content except for relative amplitudes) is very
rich, and over even a small region of travel along the bow the coincidence of position near a longitudinal
wave node of some harmonic of the bow hair fundamental and relevant string harmonic frequency is by
no means a rare occasion.
Bridge amplitude (red), accelerometer amplitude (green), Harmonic 9
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Figure 2(a): The blue line is the amplitude of the peak of the 9th harmonic of the bridge transducer
signal, the red (not green!) line is the amplitude of the bow accelerometer signal. Notice that the
acceleration and deceleration intervals are shown in the data. The ratio of the accelerometer amplitude to
the bridge force amplitude at this frequency is the green data plotted in figure 2(b).
Ratios acc/brg signals: Har 9 (green), Har 18 (blue), Har27 (red)
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Figure 2(b) above, 2(c) below. See text.
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Ratios acc/bowforce signals: Har 9 (green), Har 18 (blue), Har27 (red)
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Figure 2(b) is a plot of the results of three sonogram runs in which the successive FFTs are taken one
string period apart – in this case, the period is the reciprocal of 653 Hz, the frequency the E string was
tuned to. The ratio is of the bow accelerometer signal to the bridge force signal. In the digitized signal
the period of the string was about 196 samples, and the FFT used was 2048 samples. Thus there is
considerable averaging in the sense that neighboring FFTs contain contributions from several neighbors
of the “central” period, but ever diminishing contributions since the hanning window goes smoothly to
zero at each end of the 2048 sample. The plot is of the ratios of the amplitudes of the peaks of the 9th,
18th and 27th harmonics of the bow to bow force signal. Figure 2(c) shows the same results using the
bow force signal. The similarity between 2(b) and 2(c), remembering that the ordinate scale is arbitrary,
illustrates the point that the substitution of bridge force for the reconstructed bowing point force gives
the same information [5].
The interpretation of the data is as follows. The frequencies of 5810 Hz (the ninth harmonic of 653 Hz),
and twice and three times that, excite standing longitudinal waves of the bow hair. Remembering that
the abscissa of the graphs really represents position along the bow (0.306 mm per period at 200 mm/s
bow speed), the positions between the recurring minima represent distances of half a wavelength of the
standing wave. The velocity of propagation of that wave was calculated from the data to be (2630 +/50) m/s, in agreement with a previous rough (time of flight) measurement of 2500 m/s on a single bow
hair by the author [1]. The determination of the velocity from the data in figure 2 is explained in
appendix 2.
It is natural to investigate frequencies near the natural modal frequencies of the bow. Figure 3a is an
FFT showing the spectral response of the bow to a tap on the bow’s end parallel to the bow hair. The
horizontal axis is frequency. The arrow points to the fifth harmonic of the string’s fundamental at 3265
Hz.
Viola Bow response to parallel tap.
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Frequency in Hz
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Figure 3a
Ratios: accelerometer to bow force (blue) and bridge force (red)
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Fifth harmonic of E string (3250 Hz),
at a normal mode frequency of bow.
Response normalized to last peak.
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Figure 3b
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Figure 3b compares the ratios of the accelerometer response to the reconstructed bow force (blue) and
the bridge force (red). The data has been renormalized to the value of the last peak. This plot shows
more convincingly than figures 2a and 2b the approximate interchangeability of the easily obtainable
data from the bridge transducer and the reconstructed bow force. Many details are reproduced with
remarkable accuracy – the comparison from about 620 periods to the end is remarkable – yet with
differences, such as the relative amplitudes of the last three peaks. The interchangeability is more
convincing because it could be argued that figures 2 probe only properties of the bow hair, which will be
independent of the bow, whereas figure 3b explores the vicinity of a peak in the impulse response
spectrum of the bow stick itself.
Figure 4a is the equivalent for that bow of figure 3a for the viola bow. The spectra shown are with the
hair in contact with the string at various positions within the range of bow travel on the bowing machine,
and also free of the string. This juxtaposition shows that string contact has minimal effect on the
positions of the normal modes of the stick. The amplitudes are normalized to the lowest frequency free
bow data.
Glass cello bow spectra
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Figure 4a. Black: free bow; red: hair string contact 20 mm from start of bowing; green: hair-string
contact 155 mm from start of bowing; blue: 300 mm from start of bowing.
Figure 4b is similar to figure 3b. The string frequency is 422 Hz, near the frequency of the first peak in
4a (433 Hz), achieved with a slightly down-tuned gut A string. In this example the string’s fundamental
is close to the first and largest peak of the impulse response. In addition, the accelerometer signal was
integrated, so that the response of the bow was a velocity response, and the ratio is technically a transfer
admittance, rather than a transfer accelerance. It makes absolutely no difference which is used, of
course, because only the relative amplitudes of the frequency spectrum altered by the integration. The
data points of the complicated ratio response (red dots) are connected by straight lines to aid the eye.
Glaser Cello Bow: bridge force (green, accelerometer (blue), ratio (red)
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Figure 4b
Conclusion
I have demonstrated that the bow’s response to the force exerted on it by the vibrating string during a
bow stroke can be probed by digitizing signals from two sensors, one of which senses the transverse
force the string exerts on the bridge during bowing, while the other simultaneously senses the response
of the bow. The data can be accumulated by a player doing a normal bow stroke. Although in this
demonstration a bowing machine was used, that is not necessary, and not even desirable. Several
distinct results have been obtained:
1. The substitution of the bridge force for the bow force produces the same results, at least qualitatively,
in the transfer function. That conclusion has of course not been universally verified, but is a reasonable
one given the role of the bridge force in reconstructing the bow force, as shown in reference 2.
2. The bow responds to the longitudinal force exerted on the tip and (in an obvious extension) on the
frog of the bow by the hair. The excitation of that force occurs most prominently at the anti-nodes of the
longitudinal standing waves set up in the bow hair when the bow force (or bridge force) spectrum
coincides with the bow hair standing wave frequencies, and when the position of the string on the bow is
at or near a node of those longitudinal waves. That means the excitation varies with time during the bow
stroke. As a consequence, the propagation velocity of longitudinal disturbances on bow hair was
measured to be about 2600 m/s, a reasonable propagation velocity for solid materials, and in reasonable
agreement with the time of flight result quoted in reference 1 of 2500 m/s.
3. Particularly interesting responses as a function of position of the string on the bow as the bow stroke
is proceeding occur when a string harmonic and a bow’s normal mode frequency coincide. That
requirement is not in any way a limitation of the method. Of course, a player can select any desired
frequency by playing fingered notes in order to probe any frequency that may look interesting based on
the bow’s normal mode spectrum.
It is possible, but not of course proven, that the mechanisms demonstrated here are the origin of the
rather subtly perceived influence of the bow on the tone of a violin. It is always important to realize in
thinking about the violin, either in the abstract or as it is played, that it obeys the rules of what is
technically known as an autonomous oscillator. That means in this context that one must beware of
regarding the violin string as an driver of the bow, or the bow as a passive driver of the violin. They
collaborate in the tone production process, and each influences the other cooperatively and
simultaneously in producing the final result. The virtue of this method of investigating the bow response
is that, except for a small disturbance of the bow’s mass distribution by the accelerometer, there is no
interruption or intervention in the closed circuit of the violin, the bow, and the player, whose perception
of the tone during its production allows her to modify that tone as the occasion requires. I recommend
that the reader not interested in the technical details of Appendix 2 at least read its last paragraph, which
elaborates on the ideas generated by this report.
Appendix 1: The Subterfuge
The primary observation that allows at least a plausible force of the string on the bow to be used in
conjunction with the accelerometer signal is the following: Reference 2 shows that the bow force and
string velocity reconstruction from the string termination signals are done in two ways. The force
reconstruction F(t) requires the bridge transducer signal at time (t-tbrg) and the nut transducer at (t+tnut),
or, independently, the bridge transducer signal at (t+tbrg) and the nut transducer at (t-tnut). Clearly both
transducers are required. However, the velocity reconstruction requires either bridge data at (t-tbrg) and
at (t+tbrg), or, independently, nut data at (t-tnut) and (t+tnut). Here, tbrg and tnut are the propagation times
between bowing point and bridge and nut respectively. In the reconstruction process the two velocity
reconstructions are used to tweak the parameters relevant to the note played: the distance from the
bridge, the gains of the transducers, the frequency of the note’s fundamental, the string’s wave
impedance, and the bending stiffness of the string. The initial approximation of all these parameters is
established by a preceding calibration pluck at the bowing position. Small changes in all of these
parameters except bending stiffness are then made to minimize the difference between the two velocities
over a 25 period interval in the data, chosen arbitrarily.
The critical observation is that the string’s velocity at the bowing point can be established with the
bridge transducer alone, although of course without the small corrections that are made by comparing
the two velocity reconstructions. The subterfuge (kluge, or scam) is to make two runs. The first, using
just the string transducers, produces both the force of the bow on the string (and its negative, the force of
the string on the bow), and the string velocity. Then the second run uses the bridge transducer and the
accelerometer. The bow velocity is reconstructed from the second run, and compared with the bow
velocity reconstructed from the first run. Since no two transients are quite alike, it is necessary to align
the two velocity signals (in time) at the first sign that the noise part of the initial transient is over, and
then compare the alignment over longer times. That allows the appropriate readjustment of the time of
onset of the reconstructed bow force signal to be aligned with the accelerometer signal.
The use of a bow instead of a coated glass rod, for which the apparatus is normally used, results in a less
noisy and more reproducible signal than occurs when the glass rod is used as a bow. That gives some
confidence that when the magnitude of the transfer function is calculated between the reconstructed bow
force of the first run and the accelerometer signal of the second run, the results can be reasonably
assumed to be the same as if three transducers were available for a single run. The confidence is
increased when one gets the same (qualitative) result when calculating the transfer function between
either of the bridge forces and the accelerometer response, and between the reconstructed bow force of
the first run and the accelerometer response of the second run. By “qualitative” above is meant that the
graphs look the same if scales are adjusted to give the same relative magnitude. The accelerometer was
not calibrated, and no attempt was made to adjust for the difference between amplitudes of the bow and
bridge forces.
The subterfuge tends toward the “scam” when one realizes that the reconstruction assumes that the
bowing point is physically just that - a point. With the glass rod, the width of the rod in contact with the
string is about 0.12 mm, whereas in the real bow being used here, it is between 5 and 10 mm. The
reconstruction also implicitly assumes that the “bow” is non-compliant, contrary to the spirit of this
investigation. The reconstruction program of course does not know this, so it just tweaks the parameters
to do the best it can. The difference between the two velocity reconstructions from the first run, which
is a measure of the quality of the reconstruction, turned out to be similar quantitatively to some worse
than average runs using the glass rod.
An E string was almost exclusively used in this demonstration because its properties are particularly
well-suited to the reconstruction method of reference 2. In particular, it has a well-defined, easily
measurable bending stiffness, crucial to the reconstruction. It has the smallest wave impedance of any
string in the traditional violin family, thus increasing the impedance mismatch between string and
terminations, which is assumed infinite (non-compliant terminations), and constructed to be so as much
as possible in the bowing machine used here. It also produces negligible rotational effects when bowed
at about a tenth the string length from the bridge. But if the bridge force is to be used in place of the bow
force, reconstruction is unnecessary, and the above considerations are irrelevant, except possibly the
concern about rotational impedance. However, conversion of rotational motion to transverse motion at
the terminations is expected to produce some signature of rotational motion in the bridge force.
Appendix 2. Longitudinal standing wave modes
The velocity c of longitudinal waves on the bow hairs has been deduced from the data to be about 2600
m/s. The fundamental for a bow hair with length L (0.661 m for the viola bow, and 0.61 m for the glass
fiber cello bow) is c/2L, just as for the fundamental of the transverse wave of a violin string (with
different numbers for c and L, of course). If the terminations of the ends of the hairs are fixed – that is, if
the ends are nodes – then there is no node of the fundamental along the length of the bow, and the nodes
are at (1/2)L for the first harmonic, (1/3)L and (2/3)L for the second harmonic, etc., with the harmonic
frequencies being integral multiples of the fundamental frequency. The nodes of the nth harmonic are
separated in space by n/2, half a wavelength of the nth mode. However, for these arguments to be valid
it is not necessary for the terminations to be nodes, only that they be identical. Thus, if the boundary
conditions are “free-free”, rather than “fixed-fixed”, the arguments are the same except that the
fundamental will have a node in the center. Since that fundamental frequency of about 2 kHz did not
coincide with any harmonic of the string, the nature of the boundary conditions was not determined. If
the boundary conditions approximated “fixed-free”, that would lead to a bow hair fundamental of about
1 kHz, and the harmonics would be at odd multiples of that frequency, and the distribution of nodal
positions would be different.
The question arises, then, how many nodes are there in a typical, randomly selected region of the bow,
near the center, of length 0.2 m, or about L/3, and what are their frequencies, and how many of those
frequencies match multiples of the string’s fundamental? That question is unnecessarily restrictive,
however, because a node of a longer wavelength than will fit into the 0.2 m bow stroke of the bowing
machine might well appear within the region being bowed, and might also have a frequency that
matches closely enough a harmonic of the string. Even more latitude is allowed if the hair modes are
somewhat lossy, so that they can be excited over a broad enough frequency range to encompass more
than one string harmonic. In addition, the bow stroke will not be confined, as it is here for practical
reasons, to the roughly middle third of the bow, but will probably encompass nine tenths of the bow.
For the viola bow, the fundamental is, using the above numbers, 1967 Hz, and for the fiberglass cello
bow, 2131 Hz, on the order of the third harmonic of the string’s fundamental. Given that the bowing
region is on the order of a third of the bow hair length, the lowest non-serendipitous mode to fit into the
bowing region – those having two nodes within that region – has frequency on the order of 6 kHz, about
nine times the string’s fundamental. So one expects that strong excitation would occur at least at
multiples of the 9th string fundamental. That indeed is what happens. The frequencies of some of the
shorter wavelength waves, those with half wavelengths in the low single digit centimeters, might be at
such a high frequency as to be uninteresting in practice. That may well be the case, but it is important to
note that even if events occur in string dynamics that are characterized by time scales corresponding to
frequencies well above the audible range, in such a highly non-linear system the modes can couple
cooperatively, not independently, so that ignoring high frequencies in the dynamics can in principle, at
least, be misleading.
This discussion cannot be regarded as definitively restrictive. As was shown in the main text, the
serendipitous frequency coincidence of a stable normal mode peak of the string and of the bow has led
to features in the transfer function that seem to fall outside the bounds of simple considerations. The
lesson to be learned is that one can expect that a bow stroke will produce numerous excitations many of
the bow’s modes that wax and wane rapidly as the string moves on the bow moves from one end of it to
the other. In that property the bow’s dynamics, even in playing a “steady” note, differs from the
excitation of the violin’s corpus, for which without any type of pitch change, including vibrato, the
modes would be invariant during the note. That is not true for the normal modes of a bow, even if bow
velocity and normal force are rigorously controlled. That may be an important reason that, to an extent
that is quantitatively different from notes produced by most of the standard orchestral wind instruments,
there is no such thing as a periodic note from a violin.
References
1.There are some articles grouped under the subject “The Bow”, in the Hutchins/Benade collection:
Research Papers in Violin Acoustics, 1975-1993, published by the Acoustical Society of America.. The
only ones that seem relevant to the present effort are by Bissinger (paper 30, from 1993) and my own
early contribution (paper 33, from 1975). The references in the text apply to that 1975 note, originally
published in the Catgut Acoustical Society Newsletter 24, 5-8, November, 1975.
2. J. Woodhouse, R. T. Schumacher, and S. Garoff, “Reconstruction of bowing point friction force in a
bowed string”, J. Acoust. Soc. Am. 108 (1) [ July 2000], pgs. 357-368.
3. For a reasonably up-to-date summary of that work see “Bowing with a Glass Bow: An Update”, R. T.
Schumacher, CAS Journal V. 4, No.4 (Series II) Nov. 2001, pg. 7. Notice the last section, “Bow
Properties”, for an early forecast of the present paper.
4. A hanning window (perhaps a somewhat jokey name derived from the closely related Hamming
window, which is actually named after someone – hence the capitalization) is just one period of a cosine
function raised so its minima are at zero. Its period corresponds to the length of the section of the data
that is being analyzed. Thus the data at the ends are given zero weight, and maximum weight is given to
the data in the center of the selected section, where the amplitude of the window is unity.
5. In order to verify that the sinusoidal-like behavior of figures 2 is characteristic of the bow hair, and
not the bow used, I made similar measurements on a very inexpensive fiber-glass cello bow (Glaser,
purchased circa 1970), and the results were identical, including the propagation speed, within
experimental error.