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 [450]. Inspection of Fig.E.2 makes it clear that terminating the left endpoint may be accomplished by setting

and the right termination corresponds to

By allowing an additional two samples of round-trip delay in each endpoint reflectance (one sample in the chosen subgrid), we can implement these reflections within the state-transition matrix:

(E.38) |

The simplest choice of state transformation matrix is obtained by cropping it to size :

An advantage of this choice is that its inverse is similarly a simple cropping of the case. However, the corresponding FDTD system is not so elegant:

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

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