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### Losses, Sources, and Spatially-varying Coefficients

We can deal with spatially-varying 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

 (4.72) (4.73)

for , half-integer such that is odd, and

for , integer. For the sources, we have used

where

Again, we have applied a semi-implicit approximation to the constant-proportional 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 self-loop 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

 (4.74)

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

Next: Type I: Voltage-centered Mesh Up: The Waveguide Mesh Previous: The Waveguide Mesh
Stefan Bilbao 2002-01-22