## 2 Model Description
### Horizontal Polarisation
#### 2.5.2 Bow  
The most straightforward way to introduce the tangential friction forces is probably to come back to the simple bowed string, where the friction phenomenon is the most appreciable. The bowed string equation in the horizontal polarisation is:  
\[\mathcal{L}u=-J_{B}\left[f_{B}\right]_{+}\varphi_{B} \tag{16}\]  
Figure 3: Diagram summarising the interactions of the string with the finger, bow, and fingerboard, in the vertical polarisation.  
where \(J_{B}(x,t)\) and \(f_{B}(t)\) are the distribution and the normal collision force defined in Section 2.4.2. Note the negative sign in front of the friction term, reflecting the friction force opposing the motion of the string. \(\varphi_{B}(v_{\text{rel},B})\) is a dimensionless friction coefficient, depending on \(v_{\text{rel},B}\), the relative velocity between the bow hair and the string. These are defined as [42]:  
\[\begin{cases}|\varphi_{B}|\leqslant 1.2&\text{if }\left(v_{\text{rel},B} \right)=0\\ \varphi_{B}=\text{sign}\left(v_{\text{rel},B}\right)\left(0.4e^{\frac{-|v_{ \text{rel},B}|}{0.01}}+0.45e^{\frac{-|v_{\text{rel},B}|}{0.1}}+0.35\right)& \text{if }v_{\text{rel},B}\
eq 0\text{ (kinetic)}\end{cases} \tag{17a}\] \[v_{\text{rel},B}(t)=\frac{d}{dt}\left(\int_{\mathcal{D}}J_{B}(x,t)u(x,t)-u_{B}(t) \right)dx \tag{17b}\]  
where \(u_{B}(t)\) is the bow transverse displacement. As the bow is pushed across the string, the equation governing the transverse motion of the bow is:  
\[M_{B}\ddot{u}_{B}=-\lambda_{B}\dot{u}_{B}+\left[f_{B}\right]_{+}\varphi_{B}+f _{\text{ext},B} \tag{18}\]  
\(\lambda_{B}\) is a positive coefficient quantifying the linear energy absorption by the bow in the horizontal direction. The linear damping term is negligible compared to the friction term, when the string is in contact with the bow; indeed, this linear term mainly has a practical purpose in the numerical simulations, that is to avoid the bow drifting away when it is lifted from the string at the end of a note.  
\(f_{\text{ext}\,u,B}(t)\) is the force with which the player pushes the bow tangentially, in order to establish the desired bow velocity. Note the slight difference with the usual control parameter in most bowed string studies; instead of directly imposing a bow velocity \(v_{B}(t)\), the force applied by the player on the bow is used, resulting in a bow velocity \(\dot{u}_{B}\).  
The choice of a friction coefficient depending on relative velocity, \(\varphi_{B}(v_{\text{rel},B})\), while already quite refined and fairly costly to model, is nonetheless somewhat of a trade-off between computational simplification and physical realism. More elaborate models for the bowed string friction interaction, involving viscothermal effects in the rosin layer coating the bow hair, can be used [13; 14]; however, they require significantly more advanced implementations. Satisfying results and synthetic sound are obtained with the simple friction curve, although it has been shown that using a temperature-based friction model improves playability [25]. The friction curve employed here for the bow is deduced from experimental measurements in the steady sliding case (e.g., at constant velocity) [42]; it is illustrated in Figure 4a.  
Figure 4: Friction curves for (**a**) bow, and (**b**) neck and finger. (**a**) Friction characteristic for the bow, from [PERSON] _et al._[42]; (**b**) Coulomb friction characteristic.