# Mechanics of the bowed string and simulation methods

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## Mechanics of the bowed string and simulation methods

This chapter presents some basic results of studies on bowed strings and the cou-pling to the instrument. Since Pythagoras and other early works on vibrating strings, the understanding of the dynamics of the bowed string has increased suc-cessively due to pioneering works by Helmholtz, Rayleigh, and Raman, who gave the basis of the modern view on the problem. By first introducing the basic knowl-edge obtained in the historical works, and then adding the most important results of contemporary studies, some landmarks will be given which allow a comparison between simulations, idealized theory and experimental results.
After the presentation of these important landmarks, we will give an overview of the phenomena that should be taken into account for obtaining a complete physical description of the bowed string and violin. The bowed-string model that will be used in the experiments and analyses in the following chapters uses only a limited set of these ingredients. It is therefore important to show why it can be called a “minimal” model, compared to all the elements that could be included in a more extensive description of the bowed string and instrument.
Finally, diﬀerent techniques for simulation of the motion of the bowed string will be presented. Recent developments of computers and improved eﬃciency of algorithms make it possible to run even rather sophisticated simulation models in real time. These simulation methods form the basis of contemporary sound synthesis based on physical modelling.

### Kinematics of the bowed string

From a radical point of view, the description of the violin in physical terms can be reduced to the study of the bowed string. Questions related to the the coupling between the diﬀerent elements of the violin (string, bridge, body), the radiation into the air, and the classical issues related to tone quality and violin making, can all be regarded as auxiliary compared to the main characteristics of the instrument, which is the excitation through friction by a bow drawn across the string. Whereas an analytical description of the string vibration in the case of free oscillations has been available since D’Alembert and Bernoulli (18th century), the motion of the bowed string remained unknown until the second half of the 19th century and Helmholtz’s pioneering work .
By observing the actual motion of the string, he concluded that the string mo-tion consisted in a sharp corner travelling around a parabolic trajectory. From this idealized motion he predicted the influence of the bow velocity and bow-bridge dis-tance on the vibration amplitude of the string. By considering that real strings can-not show a perfectly sharp corner, Cremer and Lazarus  introduced a smoothing of the Helmholtz corner which enabled to describe the influence of the bow force on the vibrations. It is interesting to notice that all these results were obtained without any precise measurements on the string motion, but entirely based on kine-matic considerations and very strong approximations in the dynamics of the bowed string. Actually, before the access to computers, a detailed description of the vibra-tions of the bowed string was almost impossible to approach. Raman  was the first trying to deal with the problem at the very beginning of the 20th century. In order to be able to solve the problem by hand, he had to simplify the problem by considering a flexible string with purely resistive terminations, bowed at an integer fraction of the string length. In addition to Helmholtz motion, he discovered a great variety of possible periodical motions of the string.
The string equation
The dynamical behaviour of the string depends on the boundary conditions at the terminations and a set of mechanical string properties including the tension, mass and length. If the string is represented by a one-dimensional continuum in the x direction, with tension T0 and linear density ρL, the equation describing the displacement y(x, t) of the string can be written as (see for example ) ρL ∂2y(x, t) = T0 ∂2y(x, t) (1.1) ∂t2 ∂x2
This is a classical equation of wave propagation and D’Alembert (1717-1783) gave a general solution consisting in the sum of two waves travelling in opposite directions y(x, t) = y+(x − ct) + y−(x + ct)with c = T0 (1.2) ρL
In this solution, y+ represents a wave propagating in the +x direction with a velocity c while y− propagates in the -x direction. The finite length L can be taken into account by considering the boundary conditions. The simplest conditions are obtained by assuming that the displacement is zero at the bridge and the nut (fixed ends), giving a total reflection of the incoming waves with opposite polarity. The state of the vibration is identical once the travelling waves have made a round trip on the string, and the fundamental frequency of the oscillation is given by
f0 = 1 T0
2L ρL
Another formulation of the solution has been given by Bernoulli, also considering fixed terminations of the string. The solution can then be written as a superposition of particular solutions with separate variables x and t y(x, t) =an sin nπx sin nωt ω = T0 (1.3)
The string equation can be solved analytically for free oscillations produced by struck and plucked excitations, but sustained excitations produced by drawing a bow across the string are substantially more diﬃcult to examine. As mentioned by Helmholtz (1862):
“No complete mechanical theory can yet be given for the motion of strings excited by the violin bow, because the mode in which the bow aﬀects the motion of the string is unknown” (, cited in ).
A better understanding of the bow-string interaction was necessary for approaching an analytical description of sustained oscillations. It turned out, however, that even with simple models such as the ones used by Raman (1918) , or later, Friedlander (1953) , and Keller (1953) , a number of approximations were necessary for obtaining a solution. Direct observations of the motion of the bowed string gave an invaluable starting point for improving the dynamical description of the vibrations.
Helmholtz and the idealized motion of the bowed string
Using a vibration microscope, Helmholtz observed a surprisingly simple motion of the string when played by a bow. At any position the displacement followed a triangular pattern, and the velocity consequently alternated between two values with opposite polarity. This motion is illustrated in Fig. 1.1, right.
When observing the motion at a given point x1, the oscillation is made up of two successive phases whose total duration corresponds to the period of the vibration of the free string T . During a time T+, the string moves in the same direction as the bow, with velocity v+. Then, during the time T−, the string moves in the opposite direction with velocity v−. The duration of the two phases depends on the position x1 where the motion is observed. If the string is observed between the bridge and the midpoint, T+ is greater than T− and vice versa on the other half toward the nut. At the middle, the two phases have exactly the same duration.
Figure 1.1: Illustration of the idealized Helmholtz motion. Left: At any time, the string is composed of two straight segments connected at a sharp corner (the “Helmholtz corner”). When bowing the string, the corner travels around a parabolic trajectory, the capture and release of the string corresponding to the moment when the corner passes under the bow. Right: The resulting string velocity (top) at any point of the string shows an alternation between two phases with opposite sign, giving a sawtooth pattern for the displacement (bottom).
At the bowing position x0, the interpretation of these two phases is straightfor-ward, showing an alternation between two states of the bow-string interaction: slip and stick. During the time T+, the string sticks to the bow hair and consequently moves with the same velocity (v+ = vb), and during the time T−, the string slips under the bow with a velocity whose sign is opposite to vb.
Using these observations and the basic model of the string described in the previous section, it was possible for Helmholtz to quantify the motion. As a first approximation, he used the general solution for the free oscillation (Bernoulli’s solution, Eq. 1.3) and deduced the Fourier coeﬃcients an for the triangular patterns that he observed experimentally. This gave proportional relations between the time intervals T+, T−, T , the length L, and the observation position x1 T+ = 2(L − x1) , T− = 2×1 , T = 2L c c c
The velocity at any point x1 along the string can be written as a function of the bow velocity vb at the bowing position x0 v − = − (L − x1) v , v = x1 v x0b + x0 b
Finally, the maximum displacement of the string at a position x1 can be written as ym(x1) = vbT (L − x1)x1 (1.4) 2Lx0
The displacement envelope is seen to be composed of two parabolas passing through zero at the string terminations. The corresponding overall motion of the string is illustrated in Fig. 1.1, left. At any moment, the string configuration is made up of two straight-line segments whose corner lies on the parabola, the so-called Helmholtz motion (dotted line, Eq. 1.4). When the string is bowed, the corner travels around the parabolic trajectory in one period. As the eye cannot follow this quick motion of the string, the observer sees only the parabolic enve-lope which gives the impression of a uniform vibration, as if the whole string was vibrating back and forth.
The successive phases of the vibration can be followed in Fig. 1.1, left. At time 1, the string is still sticking to the bow and the displacement of the string between the bridge and point 1 is increasing, whereas the displacement is decreasing on the other part of the string. Between time 1 and 2, the corner travels along the trajectory and when it passes under the bow, the string is released and begins to slip in the opposite direction. Until time 3, when the string is slipping, the corner reaches the bridge termination, is reflected, and begins to propagate toward the nut with an opposite displacement. At time 3, the corner passes under the bow and the string is captured again.
The described motion is an idealization of the observed vibrations. In particular, the corner between the two string segments can only be sharp with an ideal, flexible string. With real strings, it is rounded due to the stiﬀness of the string. However, all cases of bowed string motion characterized by an alternation between one sliding phase and one sticking phase during one nominal period of the string vibrations will be referred to as Helmholtz motion, in contrast to other possible vibrations of the string.
Theoretical inferences
Important results can be drawn from the simplified model described above. In the bowed-string instruments, the sound is radiated from the body, which is excited by the vibrations of the string transmitted via the bridge. The force acting on the bridge Fbridge can be deduced from the spatial derivative of Bernoulli’s solution (Eq. 1.3)
∂y(x, t) vb ∞ 2
Fbridge(t) = T0 (1.5)
x=0 = T0ρL sin nωt
∂x x0 n=1 nπ
This expression corresponds to a perfect sawtooth function with linear ramps and steps. The maximal value of the ramp is
vb
Fmax = T0ρL
x0
The amplitude of the vibration at the bridge, and hence the sound level, in-creases with increasing bow velocity and with decreasing bow-bridge distance1 .
Eq. 1.5 also gives an estimation of the spectrum of the sound. The amplitude of the n-th harmonic is
2 vb
Fn = T0ρl
nπ xb
The spectral slope is −6 dB/octave, which approximately corresponds to the measured spectrum of the force on the bridge. It will be seen in the next section that the highest partials are actually lower when playing with a low bow force. The -6 dB/octave slope corresponds to a limiting case corresponding to Helmholtz motion with a sharp corner, towards which the spectrum tends when the bow force increases. It should be noted that if the string is bowed at a nodal point, the corresponding partials would not be present in the spectrum. Due to the finite width of the bow complete cancellation does not occur, but the corresponding partials are strongly suppressed.
Eﬀect of bow force
The previous analysis of the bowed string is an approximation based on free oscil-lations and on idealized representation of the observed motion of the string under the bow. It does not take into account the eﬀect of external forces such as the frictional force applied by the bow. A complete description of the bowed string behaviour must include this eﬀect. As every string player knows, the string cannot be bowed properly if the bow is not pressed hard enough against the string, and when the force is too high, the resulting sound becomes scratchy. As described above, playing closer to the bridge increases the amplitude of the driving force on the bridge and the sound level. However, a decrease in bow-bridge distance needs to be coordinated with an increase in the force with which the bow is pressed against the string (the bow force), in order to maintain the Helmholtz motion. Studies including forced oscillations of the string were first carried out by Raman [65, 66]. He focused on the velocity waves travelling in opposite directions of the string and studied the diﬀerent solutions that could be obtained, using a simple model with purely resistive terminations of the string. For the particular case of Helmholtz motion, he showed that the vibrations could not occur below a given value of bow force.

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Introduction
1 Mechanics of the bowed string and simulation methods
1.1 Kinematics of the bowed string
1.2 Physical modelling of the bowed string
1.3 Techniques for simulating the bowed string motion
1.4 Conclusions
2 Modal formalism and numerical implementation for sound synthesis
2.1 Introduction
2.2 General principle
2.3 Numerical resolution and sound simulation
2.4 Influence of computation parameters
2.5 Concluding discussion
3 Observations on the playability and sound properties of the model
3.1 Preliminary considerations
3.2 Onset of the vibration: The attack
3.3 Maintaining Helmholtz motion: Schelleng diagrams
3.4 Influence of gesture parameters on the sustained part of the vibration
3.5 Conclusions and applications
4 Measuring bowing parameters in violin performance
4.1 Introduction: On physical modelling and the control
4.2 Introduction to Paper I: The bow force sensor – from the laboratory to the stage
4.3 Introduction to Paper II
Paper I: Measuring bow force in bowed string performance: Theory and implementation of a bow force sensor
Paper II: Extraction of bowing parameters from violin performance combining motion capture and sensors
5 Description, modelling and parametrization of some typical bowing patterns
5.1 Introduction
5.2 Bouncing bow strokes
5.3 Fast martelé
5.4 Tremolo and fast détaché
5.5 Conclusion
6 Observations on sustained bowing patterns
6.1 Playing détaché
6.2 Bow direction changes
6.3 Conclusions
Conclusions
Bibliography

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