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Losses, Sources, and Spatiallyvarying Coefficients
We can deal with spatiallyvarying material parameters as well as losses and sources in a manner similar to the (1+1)D case. The full (2+1)D transmission line equations, as originally presented in §3.8, are

(4.70a) 
where we have
and
, and , and are driving functions of , and .
The centered difference approximation to (4.58) is
where
for , halfinteger such that is odd, and
for , integer. For the sources, we have used
where
Again, we have applied a semiimplicit approximation to the constantproportional terms of (4.58).
The waveguide network shown in Figure 4.21 is a direct generalization to (2+1)D of Figure 4.15. To the structure of Figure 4.20 we have added an extra port to each scattering junction, series or parallel, which is connected to a selfloop of impedance and doubled delay length, as well as a port with impedance to introduce losses and sources. All immittances are indexed by the coordinates of their associated junctions. As before, we set the admittance = for any impedance in the network. In Figure 4.21, the linking admittances of the bidirectional delay lines are indicated only at the parallel junction, since we must have
The junction admittances and impedances are thus
Beginning from series and parallel junctions, and proceeding through derivations similar to (4.32) yields the difference scheme (4.59) in the junction variables , and , provided we set
Figure:
(2+1)D waveguide mesh for the varyingcoefficient system (4.58), with losses and sources.

We can again identify three useful ways of setting the immittances:
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Stefan Bilbao
20020122