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 must at all times balance with the reactive force of the string (
).
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. 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 and are computed so that the physical string velocity is equal to .
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.