The Bow-String Interaction

Figure 4: (left) Friction force exerted on string by the bow; (right) Modeled friction coefficient from measured data (from Woodhouse (2014)).

• An approximation to the friction force exerted by the bow on the string is shown in Fig. 4 (Friedlander, 1953; Keller, 1953). The curve is antisymmetric because bowing can happen in two opposite directions. This force is dependent on $v_{\Delta} = v_{b} - v_{s}$, the difference between the bow and string velocities.

• The bow and string are stuck together for $v_{\Delta} = 0$ (the point of infinite slope in the figure). In this case, the friction force is based primarily on static friction.

• For $\vert v_{\Delta} \vert > 0$, the string is “slipping” and the friction force is based roughly on kinetic friction, which is significantly less than the static friction (especially when rosin is applied to the bow).

• The maximum friction force is roughly proportional to the normal force between the bow and string.

• At all times, the force applied by the bow on the string must balance the reactive force of the string.

• The reactive force can be expressed in terms of the string wave impedance and traveling-wave components of velocity as $f_{s} = R_s [v_{s}^{+} - v_{s}^{-}] = R_s [v_{s} - 2v_{s}^{-}]$, where $R_{s}$ is the string wave impedance.

• A graphical solution can be found by plotting this expression together with the friction force expression to determine a resulting outgoing traveling-wave component, as shown in Fig. 5.

  Figure 5: Friedlander graphical approach for solving the friction force and string reactive force at a given moment. Case I: sliding; II: sticking; III: case with ambiguity, resolved by a hysteresis loop (from Woodhouse (2014)).