There are several possible ways to extend the bowed-string simulation model to allow for a finite-width bow. The model used here is based on one developed by Roland Pitteroff [1]. We assume some number of separate, discrete “bow-hairs”, spread over a chosen total width as sketched in Fig. 1. These “hairs” are not assumed to be rigid: instead, measured behaviour of individual strands of horsehair is used to deduce an approximate model consisting of a parallel spring-dashpot combination. This model is expressed in terms of values per unit width across the bow by assuming the total number of “active” hairs in potential contact with the string to be 50, representing a single layer of hairs out of the total of 200 or so for a typical violin bow. The properties of each “hair” in the discrete model are then deduced by subdividing the total bow width into the chosen number. The details of all this can be found in reference [2].

The model then uses a finite-difference approach to the section of string lying under the bow, combined with the digital waveguide method to represent the two sections of string outside the width of the bow. At each time step, outgoing waves from the two edges of the bow are converted into incoming waves by convolution with reflection functions as before. Both transverse and torsional waves are included.

The short section of string under the bow is treated in the simplest possible way: transverse and torsional waves are each assumed to obey the simple wave equation, with the appropriate wave speeds. Local effects of damping and bending stiffness within this short length are ignored: but for the string as a whole they are allowed for via the reflection functions. Each wave equation can be represented approximately by a central-difference form for the spatial second derivative, involving the displacements of the “hairs” on either side of the one being considered, and by a backward-difference form for the time derivative. The spring-dashpot model for the “hairs” is also expressed in a finite-difference form. Putting all this together, the new value of displacement of string and bow-hair at each discrete “hair” can be calculated, from a knowledge of the displacement of it and its neighbours at previous time steps together with the friction force acting on that particular “hair”. Details of all this can be found in reference [1].

The calculation of the friction force at each “hair” involves nothing new: each “hair” is treated exactly like the single-point “bow” we have studied in earlier sections. There is only one minor twist. The particular way the Pitteroff method was implemented assumes that the friction curve is a hyperbolic function. This allows an analytic solution for the force and velocity at the next time step, rather than relying on a numerical approximation. In order to produce results that are reasonably comparable with the earlier ones, it is therefore useful to make a hyperbolic fit to the measured friction coefficient that we have seen before. The result is shown in Fig. 2. The explicit functional form is

$$\mu(v) =\mu_d + \dfrac{(\mu_s – \mu_d) v_0}{v + v_0}$$

with $\mu_s = 1.2$, $\mu_d =0.35$ and $v_0=0.02$. So the maximum sticking coefficient of friction and the asymptotic coefficient of sliding friction at high speed are exactly the same as before, and the only difference lies in the exact shape of the curve at intermediate speeds. (Note that, for consistency with earlier plots, the curve is shown here as a function of $-v$.)

We can illustrate some typical results of this model. These simulations, and the ones shown in section 9.7, use parameter values suitable for a violin G string, rather than the cello D string that was used for most of the earlier examples. The reason is a technical one, arising from the simulation method. An open cello string has a length around 680 mm, and the bow width is about 13 mm. An open violin string has a length around 330 mm, and the bow width is about 10 mm: a much bigger fraction of the string length than for the cello.

Now, the resolution for the spatial variation within the bow width obviously depends on the number “bow-hairs” used. We don’t want this number to be too small: the results here have 11 “hairs”. But having made that choice, there is a lower limit on the chosen sampling frequency (or equivalently, an upper limit on the time step length) in order to achieve numerical stability with the finite-difference method. It turns out that with cello-like parameter values, the required sampling rate becomes embarrassingly high, and the process of simulation becomes slow, unwieldy and inaccurate. Everything is much easier with violin-like parameter values.

Some individual simulation results were shown in section 9.7, but it is useful to show some more systematic results here, to illustrate the effect of options within the model. For a first set of results, we show some columns from a simulated Schelleng diagram. Four particular values of bow position $\beta$ (referred to the centre point of the finite bow) have been selected, and for each $\beta$ value simulations have been run with 10 logarithmically-spaced values of bow force. For each simulation the model was initialled with ideal Helmholtz motion, then it was run for 50 nominal period-lengths to give the motion an opportunity to settle down into a periodic state (or not, of course). Finally, we plot an extract of the bridge force waveform for the last few period-lengths.

Figure 3 shows the results, for simulations using the friction-curve model with the hyperbolic shape plotted in Fig. 2. The format of the plot is similar to Fig. 5 of section 9.3, where we looked at a measured Schelleng diagram (for a cello D string on the Galluzzo rig). Each column shows the characteristic sawtooth waveform in the lower part, giving way to some kind of irregular motion towards the top. This transition, a version of the Schelleng maximum bow force, marks out a downward-slanting line across the set of plots, much as we expect from the earlier discussion. As the maximum bow force is approached, irregular “spikes” associated with differential slipping across the finite width of the bow become more evident: we will see more detail of this shortly.

More surprisingly, we do not see any clear example of a transition to double slipping motion at low values of bow force. Instead, the sawtooth waveform gets more and more rounded, and in the lower left of this figure they seem to be decaying slowly away. This absence of a clear example of Schelleng’s minimum bow force is probably not a consequence of the finite-width bow model, though: similar behaviour has been seen in some simplified single-point bowing models [3].

Most likely, it is a result of other simplifications that have been made in the development of this particular model. There are two possible candidates: there is no representation of coupling to an instrument body, and the intrinsic damping of transverse string vibration has been approximated by a simplified model representing a constant Q-factor for all modes. Both these omissions were for purposes of numerical efficiency: in particular the digital filter method of representing string damping more accurately, developed by Hossein Mansour [4], proved hard to adapt to the higher sampling rate used here.

More interesting for the present purpose, we can produce corresponding results using the alternative friction models developed in section 9.6. By running the heat-flow calculation for each of the “bow-hairs” separately, we can estimate the contact temperature across the width of the bow. We don’t really know the details of contact size between individual bow-hairs and the string. For the purposes of these results, I have simply assumed the same total area of contact as in the earlier single-point models, divided equally between the discrete “bow-hairs”. We can use the computed temperature either with the original thermal model, or with the modified thermal model. Results corresponding to Fig. 3 for these two cases are shown in Figs. 4 and 5, respectively.

The immediate impression from these plots is that all three friction models give quite similar-looking results. However, if we look in a little more detail we can see significant differences between the three models. We will choose a particular case to illustrate: the third waveform from the top in the second column of Figs. 3, 4 and 5. This corresponds to $\beta=0.0449$ and bow force $1.23$ N. The following sequence of plots shows this bridge force waveform from the three different friction models, accompanied by maps to show what is going on under the bow. All cases have the stick-slip map, and the two thermal models also have a temperature map.

Finally, we can show some transient results using the three finite-width models. Each is compared to the corresponding point-bow model — these point-bow results are not quite the same as the ones showed in section 9.5, because they are using the parameters for a violin G string rather than a cello D string for reasons explained above. Figure 14 shows this set of 6 Guettler diagrams. All cases use the bow position $\beta = 0.0899$ as in earlier figures. These are the same diagrams that were shown in section 9.7, but now each Guettler diagram is annotated with a $3 \times 3$ grid of green circles to mark the pixels corresponding to acceleration values 5, 11 and 17 (counting from the left-hand side), and force values 5, 10 and 15 (counting from the bottom). The waveforms for these 9 cases are shown in Fig. 15, for all 6 models. The 9 panels of that figure are laid out in the same arrangement as the $3 \times 3$ grid of green circles. In each panel, results for the 6 different models are identified by plot colour.

These waveforms show what lies behind the automatic classification in the Guettler diagrams. The black curves, for the finite-width friction-curve model, show that this model finds it very hard to settle into any kind of periodic motion. Individual sticks and slips at the separate “bow hairs” are all subject to the jumps arising from the frictional hysteresis mechanism, and probably this injects too much “sensitive dependence” into the nonlinear system, leading, often, to chaotic response. The corresponding point-bow model (blue curves) only has one place for such jumps to occur, and the behaviour is a little less wild.

All the thermal models show some evidence of periodic motion occurring by the end of the time segment plotted here — not in every case, and not always Helmholtz motion, but often enough to suggest more benign behaviour from the perspective of a violinist. Comparing each waveform with the corresponding pixel in the Guettler diagram gives reassuring evidence that the automatic classification routine is behaving quite reliably. The only apparent anomaly is associated with the point-bow thermal model (magenta curves). In the middle row of Fig. 15, two of the waveforms look as if they lead to Helmholtz motion, and yet the Guettler plot shows black pixels. This is a manifestation of the “rounded bottoms” phenomenon highlighted in section 9.6: the effect is not visible when plotted at this resolution, but the waveforms are sufficiently different from a sawtooth form that they fail the “Helmholtz test” used by the classification routine.

[1] R. Pitteroff and J. Woodhouse, “Mechanics of the contact area between a violin bow and a string. Part II: simulating the bowed string”; *Acta Acustica united with Acustica*, **84**, 744—757 (1998).

[2] R. Pitteroff and J. Woodhouse, “Mechanics of the contact area between a violin bow and a string. Part I: reflection and transmission behaviour”; *Acta Acustica united with Acustica*, **84**, 543—562 (1998).

[3] J. Woodhouse; “On the playability of violins, Part II minimum bow force and transients”; *Acustica* **78**, 137–153 (1993).

[4] Hossein Mansour, Jim Woodhouse and Gary P. Scavone, “Enhanced wave-based modelling of musical strings, Part 1 Plucked strings”; *Acta Acustica united with Acustica*, **102**, 1082–1093 (2016).

[5] R. Pitteroff and J. Woodhouse, “Mechanics of the contact area between a violin bow and a string. Part III: parameter dependence”; *Acta Acustica united with Acustica*, **84**, 929—946 (1998).