Consider the grid arrangement of Figure 4.38(a). The interface between region and region was discussed in the previous section; in that case, we assumed the waveguide delays to be identical everywhere in regions and , including the boundary. We can, of course, apply the same idea again in order to introduce a grid of quadrupled point density, by adjoining it to region . In this case, however, we would like to take advantage of the fact that waveguide lengths in region are half those of region ; it is more natural, then, to use a delay of half that of region throughout the interior of region . Waveguides which run along the boundary will still operate at the rate of regions and , as will the self-loop (of admittance , not shown). A scattering junction at a typical point on the boundary between regions and is shown in Figure 4.38(b). Here, we have abbreviated the depiction of a bidirectional delay line to a single double-headed arrow, and have omitted the self-loop (which contains a full unit delay). Note in particular that for such a boundary junction, the delays of the connecting waveguides are now not all identical. As might be expected, the difference scheme relating the junction voltage at such a point to those of its neighbors is no longer a simple two-step difference method, but a four-step scheme, where each time step is now ; a full derivation of this difference scheme is very lengthy but rewarding, in the sense that it becomes clear why it takes four steps for the wave variables to fully ``recombine'' into junction voltages. We will, however, only present the resulting difference equation for a boundary junction at location as in Figure 4.38(b). Here the junction voltage will be , and the junction voltages at the neighboring points are, referring to Figure 4.38(b), . The admittances of the connecting waveguides will be called , the self-loop admittance and the junction admittance at point is defined as
Because we now have a four-step scheme, the determination of is no longer as simple as in the two-step case, but it can be found, nevertheless, to be
The positivity requirement yields the bound
Corners present essentially the same problems as before, and we will not discuss them further, other than to repeat that, given the settings derived above for the waveguide admittances at the / boundary, which determine completely the scattering behavior at the corners (an example of which is point in Figure 4.38(a)), the mesh will not be consistent with the parallel-plate system at these points.