Consider the 2D rectilinear mesh, with nodes at positions and , where and are integers, and and denote the spatial sampling intervals along and , respectively (see Fig.C.35). Then from Eq.(C.129) the junction velocity at time is given by
where is the ``incoming wave from the north'' to node , and similarly for the waves coming from east, west, and south (see Fig.C.36).
These incoming traveling-wave components arrive from the four neighboring nodes after a one-sample propagation delay. For example, , arriving from the north, departed from node at time , as . Furthermore, the outgoing components at time will arrive at the neighboring nodes one sample in the future at time . For example, will become . Using these relations, we can write in terms of the four outgoing waves from its neighbors at time :
Adding Equations (C.137-C.137), replacing with , etc., yields a computation in terms of physical node velocities:
Thus, the rectangular waveguide mesh satisfies this equation giving a formula for the velocity at node , in terms of the velocity at its neighboring nodes one sample earlier, and itself two samples earlier. Subtracting from both sides yields
Dividing by the respective sampling intervals, and assuming (square mesh-holes), we obtain
In the limit, as the sampling intervals approach zero such that remains constant, we recognize these expressions as the definitions of the partial derivatives with respect to , , and , respectively, yielding
This final result is the ideal 2D wave equation , i.e.,