![]() |
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
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
:
so that
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
with