(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 (Keller, 1953; Friedlander, 1953). The curve is antisymmetric because bowing can happen in two opposite directions. This force is dependent on
, the difference between the bow and string velocities.
The bow and string are stuck together for
(the point of infinite slope in the figure). In this case, the friction force is based primarily on static friction.
, 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
fs = Rs [vs+ - vs-] = Rs [vs - 2vs-], where Rs 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.
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)).