In discrete-time modeling of acoustic systems, it is often useful to
attach waveguide junctions to external dynamic systems which act as
a *load*. We speak in this case of a *loaded junction*. The
load is expressed in general by its complex admittance and can be
considered a lumped circuit attached to the distributed waveguide
network.

Examples of loaded junctions can be found in certain representations of finger holes in wind-instrument modeling [93,80], or in bridge terminations in stringed-instrument modeling [98] (see the example in the next section). They are also a natural way to load a waveguide mesh to simulate air above a membrane [31]. Sometimes the load can be time-varying and even nonlinear, as in the case of a piano hammer hitting a string [127,7]; in this case, the hammer is modeled as a mass-spring system in which the spring stiffness nonlinearly depends on its compression, and the impedance ``seen'' by the string changes continuously during contact. Finally, the loaded junction equations can be used to interface a DWN with other types of physical modeling simulations (e.g., a WDF): In such a case, the attached load is simply the driving-point impedance of the external simulation.

To derive the scattering matrix for the *loaded* parallel junction of
lossless acoustic tubes, the Kirchhoff's node equation is
reformulated so that the sum of velocities meeting at the junction
equals the exit velocity (instead of zero). For the series
junction of transversely vibrating strings, the sum of forces exerted
by the strings on the junction is set equal to the force acting on the
load (instead of zero).

The load admittance is regarded as a *lumped
driving-point admittance* [128], and the equation

(48) |

In the impulse-invariant case, to avoid aliasing, we require
for frequencies near and above half the sampling
rate. Since the driving-point admittances of acoustic systems are not
generally bandlimited (consider the driving-point impedance of an
elementary mass
or spring
),
anti-aliasing filtering should be applied to . The
bilinear transform is inherently free of aliasing since the entire
axis in the plane is mapped only once to the unit circle
in the plane, but as a result of this mapping, the frequency axis
is *warped*, and the behavior at continuous-time frequencies can
be mapped to the same discrete-time frequencies at only three points
(dc, half the sampling rate, and one more frequency which is
arbitrary). For this reason, the impulse-invariant method is normally
preferred when the precise tuning of more than one spectral feature is
important.

In case of a time-varying or nonlinear load, as in the piano-hammer model, to avoid aliasing, the load impedance should vary sufficiently slowly and/or be sufficiently weakly nonlinear so that the low-pass filtering in the DWN will attenuate modulation and/or distortion products to insignificance before aliasing occurs in the recursive network.

The scattering matrix for a *loaded* parallel junction of lossless,
scalar waveguides with real wave impedances can be rewritten as [93]

where

(49) |

For the more general case of -variable physical waveguides,
the expression of the scattering matrix is that of (48), with

(50) |

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