###

Boundary Conditions

The relations of the previous section do not hold exactly when the
string length is finite. A finite-length string forces consideration
of *boundary conditions*. In this section, we will introduce
boundary conditions as perturbations of the state transition matrix.
In addition, we will use the DW-FDTD equivalence to obtain physically
well behaved boundary conditions for the FDTD method.

Consider an ideal vibrating string with spatial samples. This is a sufficiently large number to make clear most of the repeating patterns in the general case. Introducing boundary conditions is most straightforward in the DW paradigm. We therefore begin with the order 8 DW model, for which the state vector (for the 0th subgrid) will be

#### Resistive Terminations

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

(E.38) |

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

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

#### Boundary Conditions as Perturbations

To study the effect of boundary conditions on the state transition matrices and , it is convenient to write the terminated transition matrix as the sum of of the ``left-clamped'' case (for which ) plus a series of one or more rank-one perturbations. For example, introducing a right termination with reflectance can be written

where is the matrix containing a 1 in its th entry, and zero elsewhere. (Following established convention, rows and columns in matrices are numbered from 1.)

In general, when is odd, adding
to
corresponds to a *connection* from left-going waves to
right-going waves, or vice versa (see Fig.E.2). When is
odd and is even, the connection flows from the right-going to the
left-going signal path, thus providing a termination (or partial
termination) on the right. Left terminations flow from the bottom to
the top rail in Fig.E.2, and in such connections is even
and is odd. The spatial sample numbers involved in the connection
are
and
, where
denotes the greatest integer less than or equal to
.

The rank-one perturbation of the DW transition matrix Eq.(E.39) corresponds to the following rank-one perturbation of the FDTD transition matrix :

In general, we have

Thus, the general rule is that transforms to a matrix which is zero in all but two rows (or all but one row when ). The nonzero rows are numbered and (or just when ), and they are identical, being zero in columns , and containing starting in column .

#### 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]:

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

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

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

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

#### Interior Scattering Junctions

A so-called *Kelly-Lochbaum scattering junction*
[297,447] can be introduced into the string at the fourth
sample by the following perturbation

A single time-varying scattering junction provides a reasonable model for plucking, striking, or bowing a string at a point. Several adjacent scattering junctions can model a distributed interaction, such as a piano hammer, finger, or finite-width bow spanning several string samples.

Note that scattering junctions separated by one spatial sample (as typical in ``digital waveguide filters'' [447]) will couple the formerly independent subgrids. If scattering junctions are confined to one subgrid, they are separated by two samples of delay instead of one, resulting in round-trip transfer functions of the form (as occurs in the digital waveguide mesh). In the context of a half-rate staggered-grid scheme, they can provide general IIR filtering in the form of a ladder digital filter [297,447].

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Lossy Vibration

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DW State Space Model