Two Ideal Strings Coupled at an Impedance

A diagram of two ideal strings coupled at a load is shown in Fig. C.30. This situation is a special case of the loaded waveguide junction, Eq. (C.98), with the number of waveguides being , and the junction load being the transverse driving-point impedance . If the bridge is passive, then its impedance is a positive real function (see §C.11.2). For a direct derivation, we need only observe that (1) the string velocities of each string endpoint must each be equal to the velocity of the bridge, or , and (2) the sum of forces of both strings equals the force applied to the bridge: . The bridge impedance relates the force and velocity of the bridge via . Expanding into traveling wave components in the Laplace domain, we have

or

where is the wave impedance of string , and

(C.108) |

Thus, in the time domain, the incoming velocity waves are scaled by their respective wave impedances, summed together, and filtered according to the transfer function to obtain the velocity of the bridge .

Given the filter output
, the outgoing traveling velocity waves are
given by

(C.109) | |||

(C.110) |

Thus, the incoming waves are subtracted from the bridge velocity to get the outgoing waves.

Since
when
, and vice versa exchanging strings
and
,
may be
interpreted as the *transmission admittance filter* associated with
the bridge coupling. It can also be interpreted as the bridge admittance
transfer function from every string, since its output is the bridge
velocity resulting from the sum of incident traveling force waves.

A general coupling matrix contains a filter transfer function in each
entry of the matrix. For
strings, each conveying a single type of
wave (*e.g.*, horizontally polarized), the general linear coupling
matrix would have
transfer-function entries. In the present
formulation, only one transmission filter is needed, and it is shared
by all the strings meeting at the bridge. It is easy to show that the
shared transmission filter for two coupled strings generalizes to
strings coupled at a common bridge impedance: From
(C.98), we have

where

Thus, is the shared portion of the bridge filtering, leaving only a scaling according to relative impedance to be done in each branch.

The above sequence of operations is formally similar to the *one
multiply scattering junction* frequently used in digital lattice
filters [299]. In this context, it would be better termed the
``one-filter scattering termination.''

When the two strings are identical (as would be appropriate in a model for coupled piano strings), the computation of bridge velocity simplifies to

where is the

Note that a yielding bridge introduces losses into all attached
strings. Therefore, in a maximally simplified implementation, all
string loop filters (labeled
LPF
and
LPF
in
Fig.C.31) may be *eliminated*, resulting in only one
filter--the transmission filter--serving to provide all losses in a
coupled-string simulation. If that transmission filter has no
multiplies, then neither does the entire multi-string simulation.

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