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Figure G.27:
Zoom-in about node at time in a
rectilinear waveguide mesh, showing traveling-wave components entering
and leaving the node. (All variables are at time ,)
|
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. G.26).
Then from
Eq. (G.89) 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. G.27).
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 :
|
(G.100) |
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 :
|
(G.101) |
This may be shown in detail by writing
so that
Adding Equations (G.100-G.100), replacing
terms such as
with
, yields a finite difference
equation in terms of physical node velocities:
Thus, the rectangular waveguide mesh satisfies this difference 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 and approach zero, maintaining their original
ratio, 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
|
(G.102) |
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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