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
:
![\begin{eqnarray*}
\mbox{Applied Force} &=& \mbox{Bow-String Friction Curve} \tim...
... &=& \mbox{String Wave Impedance}
\times \mbox{Velocity Change}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2563.png)
or, equating these equal and opposite forces, we obtain
![$\displaystyle R_b(v_{\Delta})\times v_{\Delta}= R_s\left[v_{\Delta}^{+}- v_{\Delta}\right]
$](http://www.dsprelated.com/josimages_new/pasp/img2564.png)
![$ v_{\Delta}=v_b-v_s$](http://www.dsprelated.com/josimages_new/pasp/img2565.png)
![$ v_s=v_{sl}^{+}+v_{sl}^{-}=v_{sr}^{+}+v_{sr}^{-}$](http://www.dsprelated.com/josimages_new/pasp/img2566.png)
![$ R_s$](http://www.dsprelated.com/josimages_new/pasp/img2400.png)
![$ R_b(v_{\Delta})$](http://www.dsprelated.com/josimages_new/pasp/img2567.png)
![$ F_b(v_{\Delta}) =
R_b(v_{\Delta}) \cdot v_{\Delta}$](http://www.dsprelated.com/josimages_new/pasp/img2568.png)
![$ v_{sr}$](http://www.dsprelated.com/josimages_new/pasp/img2569.png)
![$ v_{sl}$](http://www.dsprelated.com/josimages_new/pasp/img1252.png)
![]() |
In a bowed string simulation as in Fig.9.51, a velocity input (which is injected equally in the left- and right-going directions) must be found such that the transverse force of the bow against the string is balanced by the reaction force of the moving string. If bow-hair dynamics are neglected [176], the bow-string interaction can be simulated using a memoryless table lookup or segmented polynomial in a manner similar to single-reed woodwinds [431].
A derivation analogous to that for the single reed is possible for the
simulation of the bow-string interaction. The final result is as follows.
![]() |
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|
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where
![$ v_{\Delta}^{+}\isdef v_b-(v_{sr}^{+}+v_{sl}^{+})$](http://www.dsprelated.com/josimages_new/pasp/img2575.png)
![$ v_b$](http://www.dsprelated.com/josimages_new/pasp/img2576.png)
![$\displaystyle \hat\rho (v_{\Delta}^{+})=\frac{r(v_{\Delta}(v_{\Delta}^{+}))}{1 + r(v_{\Delta}(v_{\Delta}^{+}))}
$](http://www.dsprelated.com/josimages_new/pasp/img2577.png)
![$ r(v_{\Delta})=0.25R_b(v_{\Delta})/R_s$](http://www.dsprelated.com/josimages_new/pasp/img2578.png)
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