Animation of Moving String Termination and Digital Waveguide Models

In the force wave simulation of Fig.6.5b,^{7.4} the termination
motion appears as an additive injection of a constant force
at the far left. At time 0
, this initiates a force step from
0
to
traveling to the right. Since force waves are negated
slope waves multiplied by tension, *i.e.*,
, the slope of
the string behind the traveling force step is
. When the
traveling step-wave reaches the right termination, it reflects with
*no* sign inversion, thus sending back a doubling-wave to the left
which elevates the string force from
to
. Behind this
wave, the slope is then
. This answers the question of
the previous paragraph: The string is in fact piecewise linear during
the first return reflection, consisting of two line segments with slope
on the left, and twice that on the right. When the return
step-wave reaches the left termination, it is reflected again and
added to the externally injected dc force signal, sending an amplitude
positive step-wave to the right (overwriting the amplitude
signal in the upper rail). This can be added to the amplitude
samples in the lower rail to produce a net traveling force step
in the string of amplitude
traveling to the right. The slope
of the string behind this wave is
, and the slope in
front of this wave is still
. The force applied to the
string by the termination has risen to
in order to keep the
velocity steady at
. (We may interpret the
input as the
*additional* force needed each period to keep the termination moving
at speed
--see the next paragraph below.)
This process repeats forever, resulting in
traveling wave components which grow without bound, and whose sum
(which is proportional to minus the physical string slope) also grows
without bound.^{7.5}The string is always piecewise linear, consisting of
at most two linear segments having negative slopes which differ by
. A sequence of string displacement snapshots is shown in
Fig.6.6.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University