A Digital Waveguide Scattering Approach

Figure 9: A bowed string system block diagram.

• The various components of the model are combined in the general system block diagram of Fig. 9.

• The bowing mechanism effectively divides the string into two parts and is implemented as a nonlinear two-port junction (in contrast to wind instrument reed mechanisms which are one-port junctions).

• As mentioned above, the applied bow force $f_b$ must at all times balance with the reactive force of the string ( $f_s = R_s [v_s^+ - v_s^-]$).

• The bow friction curve relates the bow force and the differential velocity in terms of a friction coefficient, $f_b(v_{\Delta}) = R_b(v_{\Delta}) \cdot v_{\Delta}$.

• Smith (1986) recasts the relationship in terms of a differential velocity of known, incoming junction velocities ($v_{\Delta}^+$) to allow an expression of the form:
  $$R_b(v_{\Delta}) \cdot v_{\Delta} = R_s \left[ v_{\Delta}^{+} - v_{\Delta} \right]$$
  where $v_{\Delta}^{+} = v_b - [v_{s,l}^{+} + v_{s,r}^+]$.

• Traveling-wave components entering the bow-string junction from either side have $+$ superscripts.

• Ignoring possible non-zero phase in the bow-hair dynamics, this relationship can be solved simultaneously with the friction curve and represented in terms of a memoryless reflection coefficient as:
  $$\begin{eqnarray*} v_{s,r}^- &=& v_{s,l}^+ + \hat{\rho}( v_{\Delta}^{+} ) \cdot v... ...=& v_{s,r}^+ + \hat{\rho}( v_{\Delta}^{+} ) \cdot v_{\Delta}^{+} \end{eqnarray*}$$
  where
  $$\hat{\rho}( v_{\Delta}^{+} ) = \frac{r(v_{\Delta}(v_{\Delta}^{+}))}{1 + r(v_{\Delta}(v_{\Delta}^{+}))}$$
  and $r(v_{\Delta}) = 0.25 R_b(v_{\Delta}) / R_s$.

• An example piece-wise linear reflection coefficient table is shown in Fig. 10.

  Figure 10: A bow-string reflection coefficient table.

  

• When the bow and string are stuck together, the velocity reflection coefficient is 1 and the $v_{s,r}^{-}$ and $v_{s,l}^{-}$ are computed so that the physical string velocity is equal to $v_{b}$.

• A velocity reflection coefficient of 0 corresponds to the absence of the bow discontinuity altogether.

• A complete digital waveguide implementation for the bowed string system, using a reflection coefficient table, is diagrammed in Fig. 11.

  Figure 11: A complete bowed string implementation using a reflection coefficient table.