We define general DW inputs as follows:
block of the input matrix
is then given by
Typically, input signals
are injected equally to the left and right
along the string, in which case
Physically, this corresponds to applied forces
at a single,
non-moving, string position over time. The state update with this
simplification appears as
Note that if there are no inputs driving the adjacent subgrid
), such as in a half-rate staggered grid
scheme, the input reduces to
To show that the directly obtained FDTD and DW state-space models
correspond to the same dynamic system, it remains to verify that
. It is somewhat easier to show that
A straightforward calculation verifies that the above identity holds,
as expected. One can similarly verify
, as expected.
provides a recipe for translating any
choice of input signals for the FDTD model to equivalent inputs for
the DW model, or vice versa.
For example, in the scalar
input case (
), the DW input-weights
become FDTD input-weights
The left- and right-going input-weight superscripts indicate the
origin of each coefficient. Setting
, we obtain the
result familiar from Eq.
, the weighting
appears in the second column, shifted down one row.
in general (for physically stationary displacement
can be seen as the superposition of weight patterns
left column for even
, and the right column for odd
subgrid), where the
is aligned with the driven sample.
This is the general collection of displacement inputs.
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Third-Order Time Derivative is Ill Posed