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
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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 [177], the bow-string interaction can be simulated using a memoryless table lookup or segmented polynomial in a manner similar to single-reed woodwinds [435].
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.
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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.