The Bow-String Interaction: A Scattering Approach

- The various components of the model are combined in the general system block diagram of Fig. 5.
- 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,
.
- Smith (1986) recasts the relationship in terms of a differential velocity of known, incoming junction velocities () to allow an expression of the form:

where . - 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:

where

and . - An example piece-wise linear reflection coefficient table is shown in Fig. 6.
- 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. 7.

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