#### Reactive Terminations

In typical string models for virtual musical instruments, the nut end'' of the string is rigidly clamped while the bridge end'' is terminated in a passive reflectance . The condition for passivity of the reflectance is simply that its gain be bounded by 1 at all frequencies [447]:

 (E.42)

A very simple case, used, for example, in the Karplus-Strong plucked-string algorithm, is the two-point-average filter:

To impose this lowpass-filtered reflectance on the right in the chosen subgrid, we may form

which results in the FDTD transition matrix

This gives the desired filter in a half-rate, staggered grid case. In the full-rate case, the termination filter is really

which is still passive, since it obeys Eq.(E.42), but it does not have the desired amplitude response: Instead, it has a notch (gain of 0) at one-fourth the sampling rate, and the gain comes back up to 1 at half the sampling rate. In a full-rate scheme, the two-point-average filter must straddle both subgrids.

Another often-used string termination filter in digital waveguide models is specified by [447]

where is an overall gain factor that affects the decay rate of all frequencies equally, while controls the relative decay rate of low-frequencies and high frequencies. An advantage of this termination filter is that the delay is always one sample, for all frequencies and for all parameter settings; as a result, the tuning of the string is invariant with respect to termination filtering. In this case, the perturbation is

and, using Eq.(E.41), the order FDTD state transition matrix is given by

where

The filtered termination examples of this section generalize immediately to arbitrary finite-impulse response (FIR) termination filters . Denote the impulse response of the termination filter by

where the length of the filter does not exceed . Due to the DW-FDTD equivalence, the general stability condition is stated very simply as

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