will initialize only , a solitary left-going pulse of amplitude 1 at time , as can be seen from Eq.(E.11) by adding the leftmost columns explicitly written for . Similarly, the initialization
gives rise to an isolated right-going pulse
to the leftmost column of
plus the first column on the left not
explicitly written in Eq.(E.11). The superposition of these two
examples corresponds to a physical impulsive excitation at time 0 and
Thus, the impulse starts out with amplitude 2 at time 0 and position , and afterwards, impulses of amplitude 1 propagate away to the left and right along the string.
In summary, we see that to excite a single sample of displacement traveling in a single-direction, we must excite equally a pair of adjacent colums in . This corresponds to equally weighted excitation of K-variable pairs the form .
Note that these examples involved only one of the two interleaved computational grids. Shifting over an odd number of spatial samples to the left or right would involve the other grid, as would shifting time forward or backward an odd number of samples.
Localized Velocity Excitations
FDTD and DW Equivalence