The theory of bow-string interaction is described in [95,151,246,310,311]. The basic operation of the bow is to reconcile the nonlinear bow-string friction curve with the string wave impedance :
or, equating these equal and opposite forces, we obtain
where is the velocity of the bow minus that of the string, is the string velocity in terms of traveling waves, is the wave impedance of the string (equal to the geometric mean of tension and density), and is the friction coefficient for the bow against the string, i.e., bow force . (Force and velocity point in the same direction when they have the same sign.) Here, denotes transverse velocity on the segment of the bowed string to the right of the bow, and denotes velocity waves to the left of the bow. The corresponding normalized functions to be used in the Friedlander-Keller graphical solution technique are depicted in Fig.9.53.
In a bowed string simulation as in Fig.9.51, a velocity input (which is injected equally in the left- and right-going directions) must be found such that the transverse force of the bow against the string is balanced by the reaction force of the moving string. If bow-hair dynamics are neglected , the bow-string interaction can be simulated using a memoryless table lookup or segmented polynomial in a manner similar to single-reed woodwinds .
A derivation analogous to that for the single reed is possible for the
simulation of the bow-string interaction. The final result is as follows.
The impedance ratio is defined as ,
Nominally, is constant (the so-called static coefficient of friction) for , where is both the capture and break-away differential velocity. For , falls quickly to a low dynamic coefficient of friction. It is customary in the bowed-string physics literature to assume that the dynamic coefficient of friction continues to approach zero with increasing [311,95].
Figure 9.54 illustrates a simplified, piecewise linear bow table . The flat center portion corresponds to a fixed reflection coefficient ``seen'' by a traveling wave encountering the bow stuck against the string, and the outer sections of the curve give a smaller reflection coefficient corresponding to the reduced bow-string interaction force while the string is slipping under the bow. The notation at the corner point denotes the capture or break-away differential velocity. Note that hysteresis is neglected.