The theory of bowstring interaction is described in
[
95,
151,
244,
307,
308]. The basic
operation of the bow is to reconcile the
nonlinear bowstring
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 FriedlanderKeller
graphical solution technique are depicted in
Fig.
9.53.
Figure 9.53:
Overlay of normalized bowstring
friction curve
with the string ``load line''
. The ``capture'' and ``breakaway'' differential velocity is
denoted
. Note that increasing the bow force increases
as well as enlarging the maximum force applied (at the peaks of the
curve).

In a bowed string simulation as in Fig.
9.51, a
velocity input (which is injected equally in the left and rightgoing
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 bowhair dynamics are neglected
[
176], the bowstring interaction can be
simulated using a memoryless table lookup or segmented polynomial in a
manner similar to singlereed
woodwinds
[
431].
A derivation analogous to that for the single reed is possible for the
simulation of the bowstring interaction. The final result is as follows.
where
,
is bow velocity, and
The
impedance ratio is defined as
,
Nominally,
is constant (the socalled static coefficient
of friction) for
, where
is both the
capture and breakaway differential velocity. For
,
falls quickly to a low dynamic coefficient of friction. It
is customary in the bowedstring
physics literature to assume that the
dynamic coefficient of friction continues to approach zero with increasing
[
308,
95].
Figure 9.54:
Simple, qualitatively chosen bow table for the
digital waveguide violin.

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
bowstring interaction force while the string is slipping under the
bow. The notation
at the corner point denotes the capture or
breakaway differential velocity. Note that hysteresis is neglected.
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