
Figure C.34:
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.
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

:
![$\displaystyle v_{lm}(n+1) = \frac{1}{2}\left[ v_{l,m+1}^{-\textsc{s}}(n) + v_{l...
...}}(n) + v_{l,m-1}^{-\textsc{n}}(n) + v_{l-1,m}^{-\textsc{e}}(n)\right] \protect$](http://www.dsprelated.com/josimages_new/pasp/img4024.png) |
(C.116) |
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

:
![$\displaystyle v_{lm}(n-1) = \frac{1}{2}\left[ v_{l,m+1}^{+\textsc{s}}(n) + v_{l...
...}}(n) + v_{l,m-1}^{+\textsc{n}}(n) + v_{l-1,m}^{+\textsc{e}}(n)\right] \protect$](http://www.dsprelated.com/josimages_new/pasp/img4026.png) |
(C.117) |
This may be shown in detail by writing
so that
Adding Equations (
C.116-
C.116), replacing
terms such as

with

, 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.118) |
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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