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### 2D Mesh and the Wave Equation

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.33). Then from Eq.(C.105) 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.34).

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 :

 (C.113)

where, for instance, is the outgoing wave to the north'' from node . Similarly, the outgoing waves leaving become the incoming traveling-wave components of its neighbors at time :

 (C.114)

This may be shown in detail by writing

so that

Adding Equations (C.113-C.113), 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.,

with

 (C.115)

Discussion regarding solving the 2D wave equation subject to boundary conditions appears in §B.8.3. Interpreting this value for the wave propagation speed , we see that every two time steps of seconds corresponds to a spatial step of meters. This is the distance from one diagonal to the next in the square-hole mesh. We will show later that diagonal directions on the mesh support exact propagation (of plane waves traveling at 45-degree angles with respect to the or axes). In the and directions, propagation is highly dispersive, meaning that different frequencies travel at different speeds. The exactness of 45-degree angles can be appreciated by considering Huygens' principle on the mesh.

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