Figure C.34:
Zoomin about node at time in a
rectilinear waveguide mesh, showing travelingwave 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
travelingwave components arrive from the four
neighboring nodes after a onesample
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.116) 
where, for instance,
is the ``outgoing wave to the
north'' from node
. Similarly, the outgoing waves leaving
become the incoming travelingwave components of its
neighbors at time
:

(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 meshholes), 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 squarehole mesh. We will show later that diagonal
directions on the mesh support
exact propagation (of
plane
waves traveling at 45degree 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 45degree angles can be
appreciated by considering Huygens' principle on the mesh.
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