Let's begin with simple ``resistive'' terminations at the string
endpoints, resulting in the reflection coefficient at each end of
the string, where
corresponds to nonnegative (passive)
termination resistances [29]. Inspection of
Fig. 3 makes it clear that terminating the left endpoint may be
accomplished by setting
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(39) |
where
and
. We see that the left
FDTD termination is non-local for
, while the right
termination is local (to two adjacent spatial samples) for all
.
This can be viewed as a consequence of having ordered the FDTD state
variables as
instead of
. Choosing the other ordering
interchanges the endpoint behavior. Call these orderings Type I and
Type II, respectively. Then
; that is, the similarity
transformation matrix
is transposed when converting from Type I
to Type II or vice versa. By anechoically coupling a Type I FDTD
simulation on the right with a Type II simulation on the left,
general resistive terminations may be obtained on both ends which are
localized to two spatial samples.
In nearly all musical sound synthesis applications, at least one of
the string endpoints is modeled as rigidly clamped at the ``nut''.
Therefore, since the FDTD, as defined here, most naturally provides
a clamped endpoint on the left, with more general localized terminations
possible on the right, we will proceed with this case for simplicity in what
follows. Thus, we set and obtain