From Eq. (E.11), it is clear that initializing any single K variable corresponds to the initialization of an infinite number of W variables and . That is, a single K variable corresponds to only a single column of for only one of the interleaved grids. For example, referring to Eq. (E.11), initializing the K variable to -1 at time (with all other intialized to 0) corresponds to the W-variable initialization
with all other W variables being initialized to zero. In view of earlier remarks, this corresponds to an impulsive velocity excitation on only one of the two subgrids. A schematic depiction from to of the W variables at time is as follows:
Due to the independent interleaved subgrids in the FDTD algorithm, it is nearly always non-physical to excite only one of them, as the above example makes clear. It is analogous to illuminating only every other pixel in a digital image. However, joint excitation of both grids may be accomplished either by exciting adjacent spatial samples at the same time, or the same spatial sample at successive times instants.
In addition to the W components being non-local, they can demand a larger dynamic range than the K variables. For example, if the entire semi-infinite string for is initialized with velocity , the initial displacement traveling-wave components look as follows:
where denotes the set of all integers. While the FDTD excitation is also not local, of course, it is bounded for all .
Since the traveling-wave components of initial velocity excitations are generally non-local in a displacement-based simulation, as illustrated in the preceding examples, it is often preferable to use velocity waves (or force waves) in the first place .
Another reason to prefer force or velocity waves is that displacement inputs are inherently impulsive. To see why this is so, consider that any physically correct driving input must effectively exert some finite force on the string, and this force is free to change arbitrarily over time. The ``equivalent circuit'' of the infinitely long string at the driving point is a ``dashpot'' having real, positive resistance . The applied force can be divided by to obtain the velocity of the string driving point, and this velocity is free to vary arbitrarily over time, proportional to the applied force. However, this velocity must be time-integrated to obtain a displacement . Therefore, there can be no instantaneous displacement response to a finite driving force. In other words, any instantaneous effect of an input driving signal on an output displacement sample is non-physical except in the case of a massless system. Infinite force is required to move the string instantaneously. In sampled displacement simulations, we must interpret displacement changes as resulting from time-integration over a sampling period. As the sampling rate increases, any physically meaningful displacement driving signal must converge to zero.