## 2 Model Description
### The Isolated String
#### 2.4.2 Bow and Finger  
At a physical level, in the vertical polarisation, the bow and finger are essentially modelled the same way--that is, a lumped, flexible body pushing down on the string. The distinction lies in the values of the various parameters which define the collision force. For instance, the bow hair may have different stiffness properties than that of the fingertip, and the latter definitely exhibits higher damping than the former.  
Adding the finger and bow forces to the string model described in Equation (1a) gives:  
\[\mathcal{L}w=-J_{\mathrm{F}}f_{\mathrm{F}}-J_{\mathrm{B}}f_{\mathrm{B}} \tag{5}\]  
where the downward forces exerted by the finger and the bow onto the string are respectively denoted by \(f_{F}(t)\) and \(f_{\mathrm{B}}(t)\). Their action on the string is localised as defined by the continuous distributions \(J_{\mathrm{F}}(x,t)\) and \(J_{\mathrm{B}}(x,t)\), possibly time-varying (one can use, e.g., a Dirac delta function to model a pointwise interaction). \(f_{F}(t)\) and \(f_{\mathrm{B}}(t)\) can be written using the [PERSON] and Crossley collision model mentioned in Section 2.4.1:  
\[f_{\mathrm{F}}(\Delta_{F}) =\frac{\Phi_{F}}{\Delta_{F}}+\dot{\Delta}_{F}\Psi_{F} f_{\mathrm{B}}(\Delta_{B}) =\frac{\dot{\Phi}_{B}}{\dot{\Delta}_{B}}+\dot{\Delta}_{B}\Psi_{B} \tag{6b}\]  
where the dot notation is used for total time differentiation (\(\frac{d}{dt}\)).  
These contact forces are nonlinear functions of the penetration \(\Delta_{F}(t)\) and \(\Delta_{B}(t)\), corresponding to the distance by which the colliding object (here, the fingertip and the bow hair, respectively) would deform from its resting shape. Figures 2a,b provide a visual interpretation of the finger and the bow penetrations. \(\Delta_{F}(t)\) and \(\Delta_{B}(t)\) are defined as:  
\[\Delta_{F}(t) =\int_{\mathcal{D}}J_{F}(x,t)w(x,t)dx-w_{F}(t) \Delta_{B}(t) =\int_{\mathcal{D}}J_{B}(x,t)w(x,t)dx-w_{B}(t) \tag{7b}\]  
where \(w_{F}(t)\) and \(w_{B}(t)\) are respectively the vertical positions of the finger and bow at time \(t\) (see Figure 2).  
Figure 2: Visualisation of the penetration variables \(\Delta_{F}\) and \(\Delta_{B}\). **(a)**\(\Delta_{F}(t)\) represents the distance by which the finger deforms under the surface of the string; **(b)**\(\Delta_{B}(t)\) can be interpreted as the deformation of the flexible bow hair against the string.  
\(\Phi_{F}(\Delta_{F})\) and \(\Phi_{B}(\Delta_{B})\) are nonlinear potential functions, related to the stored collision energy. \(\Psi_{F}(\Delta_{F})\) and \(\Psi_{B}(\Delta_{B})\) are nonlinear damping coefficients. They can be written as:  
\[\Phi_{F} =\frac{K_{F}}{\alpha_{F}+1}\left[\Delta_{F}\right]_{+}^{\alpha_{F}+1} \tag{8a}\] \[\Psi_{F} =K_{F}\beta_{F}\left[\Delta_{F}\right]_{+}^{\alpha_{F}} \tag{8b}\]  
where \(\left[\cdot\right]_{+}\) means \(\max(\cdot,0)\).  
The parameters \(K_{F}\) and \(K_{B}\), both strictly positive, define the respective stiffnesses of the finger and the bow. \(\alpha_{F}\) and \(\alpha_{B}\) are power law exponents, both larger than 1. \(\beta_{F}\) and \(\beta_{B}\) are positive damping factors.  
\(w_{F}(t)\) and \(w_{B}(t)\) are respectively the vertical positions of the finger and bow at time \(t\) (see Figure 2). Their behaviour is governed by:  
\[M_{F}\tilde{w}_{F} =f_{F}+f_{\text{ext}\,w,F} M_{B}\tilde{w}_{B} =f_{B}+f_{\text{ext}\,w,B} \tag{10b}\]  
where \(M_{F}\), \(M_{B}\) are the finger and bow masses, respectively (in kg), and \(f_{\text{ext}\,w,F}(t)\), \(f_{\text{ext}\,w,B}(t)\) are the external forces applied vertically on the finger and bow, respectively (in N).