A diagram of two ideal strings coupled at a load
is shown in
Fig. C.32. This situation is a special case of the
loaded waveguide junction, Eq.(C.97), 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
![]() |
(C.132) |
Given the filter output
, the outgoing traveling velocity waves are
given by
![]() |
![]() |
![]() |
(C.133) |
![]() |
![]() |
![]() |
(C.134) |
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.97), we have
where
Thus,
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
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.33) 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.