### The Bow-String Scattering Junction

The theory of bow-string interaction is described in [95,151,244,307,308]. 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

*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 [176], the bow-string interaction can be simulated using a memoryless table lookup or segmented polynomial in a manner similar to single-reed woodwinds [431].

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.

where , is bow velocity, and

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 [308,95].

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.

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Bowed String Synthesis Extensions

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Digital Waveguide Bowed-String