## Appendix  
Consider the simplifiedsystem of an ideal string, coupled to a pointwise bow at constant position \(x_{B}\), with constant velocity \(v_{B}\), pushing down with a constant force \(f_{B}\). Using the same notations as established in Section 2, the transverse displacement of the string \(u(x,t)\) obeys:  
\[\rho\partial_{t}^{2}u=T\partial_{x}^{2}u-J_{B}f_{B}\varphi_{B}(v_{\text{rel},B}) \tag{1}\]  
where \(J_{B}(x)=\delta\left(x-x_{B}\right)\) is a Dirac delta function, and \(v_{\text{rel},B}(t)\) is defined as:  
\[v_{\text{rel},B}(t)=\int_{\mathcal{D}}J_{B}(x,t)\dot{u}(x,t)dx-v_{B}(t)=\dot{u }(x_{B},t)-v_{B}(t) \tag{2}\]  
System Equation (1) can be discretised as:  
\[\rho\delta_{tt}\mathbf{u}^{n}=T\mathbf{D}_{xx}\mathbf{u}^{n}-\mathbf{j}g_{B} \varphi_{B}(v_{\text{rel},B}^{n}) \tag{3a}\] \[v_{\text{rel},B}^{n}=h\mathbf{j}_{B}\Upsilon_{\delta_{t}}\mathbf{u}^{n}-v_{B} \tag{3b}\]  
The stability condition for this scheme can straightforwardly be deduced from that of the full system Equation (67):  
\[h\leqslant ck\ \Leftrightarrow\ \lambda\leqslant 1 \tag{4a}\] \[\lambda=\frac{h}{ck}\] (4b) \[c=\sqrt{\frac{T}{\rho}} \tag{4c}\]  
Equation (3a) can be rearranged to derive an update equation for \(\mathbf{u}^{n+1}\):\[\mathbf{u}^{n+1}=\left(2+\frac{Tk^{2}}{\rho}\mathbf{D}_{xx}\right)\mathbf{u}^{n}- \mathbf{u}^{n-1}-\frac{k^{2}}{\rho}\mathbf{j}_{B}f_{B}\varphi_{B}(v^{n}_{\text{ rel},B}) \tag{100}\]  
Rewriting Equation (100) in terms of \(v^{n}_{\text{rel},B}\) yields the following nonlinear equation:  
\[v^{n}_{\text{rel},B}+\sigma\varphi_{B}(v^{n}_{\text{rel},B})+b=0 \tag{101}\]  
where \(\sigma\) and \(b\) are constants given by:  
\[\sigma =\frac{k}{2\rho}\|\mathbf{j}_{B}\|^{2}f_{B} \tag{102a}\] \[b =v_{B}-\frac{h}{2k}\mathbf{j}_{B}\,^{\mathsf{T}}\left(\left(2+ \frac{Tk^{2}}{\rho}\mathbf{D}_{xx}\right)\mathbf{u}^{n}-2\mathbf{u}^{n-1}\right) \tag{102b}\]  
The Friedlander construction, detailed in Section 3.6.3, can be used again, giving a condition on \(\sigma\) so that the solution of Equation (101) is unique:  
\[-\frac{1}{\sigma}\leqslant\min(\varphi^{\prime}_{B})\quad\Rightarrow\quad k \leqslant-\frac{2\rho}{\min(\varphi^{\prime}_{B})\|\mathbf{j}_{B}\|^{2}f_{B}} \tag{103}\]  
Suppose that \(\mathbf{j}_{B}\) is normalised as usual, so that \(h\|\mathbf{j}_{B}\|^{2}=1\). Bound Equation (103) now becomes:  
\[f_{B}\leqslant\frac{2c\rho\lambda}{-\min(\varphi^{\prime}_{B})} \tag{104}\]  
which is independent of the bow velocity. With the friction curve defined in Equation (17a), \(-\min(\varphi^{\prime}_{B})=44.5\). The minimum bow force beyond which pitch flattening occurs therefore only depends on the string's physical parameters, and the Courant number \(\lambda\).  
Scheme Equation (100), without the bow force term, can be proved to provide the exact solution for the 1D wave equation, when the grid spacing and time step are set at the stability limit, that is \(\lambda=1\) ([18], pp. 133-136). In this limit, adding the bow model yields a bound on \(f_{B}\) that is indeed the same as that found analytically by [PERSON] [56], and such a bound does not seem to vanish, even when the time step and grid spacing are made very small (keeping \(\lambda\) constant).