A more detailed diagram of the digital waveguide implementation of the
bowed-string instrument model is shown in Fig.9.52.
The right delay-line pair carries left-going and right-going velocity waves
samples
and
, respectively, which sample the traveling-wave
components within the string to the right of the bow, and similarly for the
section of string to the left of the bow. The `
' superscript refers to
waves traveling into the bow.
String velocity at any point is obtained by adding a left-going
velocity sample to the right-going velocity sample immediately
opposite in the other delay line, as indicated in
Fig.9.52 at the bowing point. The reflection
filter at the right implements the losses at the bridge, bow, nut or
finger-terminations (when stopped), and the round-trip
attenuation/dispersion from traveling back and forth on the string. To
a very good degree of approximation, the nut reflects incoming
velocity waves (with a sign inversion) at all audio wavelengths. The
bridge behaves similarly to a first order, but there are additional
(complex) losses due to the finite bridge driving-point impedance
(necessary for transducing sound from the string into the resonating
body). According to [95, page 27], the bridge of a
violin can be modeled up to about
kHz, for purposes of computing
string loss, as a single spring in parallel with a
frequency-independent resistance (``dashpot''). Bridge-filter
modeling is discussed further in §9.2.1.
Figure 9.52 is drawn for the case of the lowest
note. For higher notes the delay lines between the bow and nut are
shortened according to the distance between the bow and the finger
termination. The bow-string interface is controlled by differential
velocity
which is defined as the bow velocity minus the total
incoming string velocity. Other controls include bow force and angle which are changed by modifying the reflection-coefficient
. Bow position is changed by taking samples from one delay-line
pair and appending them to the other delay-line pair. Delay-line
interpolation can be used to provide continuous change of bow position
[269].