initial conditions (for time )
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.
Initial velocity excitations are straightforward in the DW paradigm,
but can be less intuitive in the FDTD domain. It is well known that
velocity in a displacement-wave DW simulation is determined by the
difference of the right- and left-going waves
. Specifically, initial velocity waves can
be computed from from initial displacement waves by spatially
differentiating to obtain traveling slope waves
, multiplying by minus the tension to obtain force
waves, and finally dividing by the wave impedance
obtain velocity waves:
where denotes sound speed. The initial string velocity at each point is then . (A more direct derivation can be based on differentiating Eq.(E.4) with respect to and solving for velocity traveling-wave components, considering left- and right-going cases separately at first, and arguing the general case by superposition.)
We can see from Eq.(E.11) that such asymmetry can be caused by unequal weighting of and . For example, the initialization
corresponds to an impulse velocity excitation at position . In this case, both interleaved grids are excited.
More General Velocity Excitations
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:
Below the solid line is the sum of the left- and right-going traveling-wave components, i.e., the corresponding K variables at time . The vertical lines divide positions and . The initial displacement is zero everywhere at time , consistent with an initial velocity excitation. At times , the configuration evolves as follows:
The sequence consists of a dc (zero-frequency) component with amplitude , plus a sampled sinusoid of amplitude oscillating at half the sampling rate . The dc component is physically correct for an initial velocity point-excitation (a spreading square pulse on the string is expected). However, the component at is usually regarded as an artifact of the finite difference scheme. From the DW interpretation of the FDTD scheme, which is an exact, bandlimited physical interpretation, we see that physically the component at comes about from initializing velocity on only one of the two interleaved subgrids of the FDTD scheme. Any asymmetry in the excitation of the two interleaved grids will result in excitation of this frequency component.
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:
and the variables evolve forward in time as follows:
Thus, the left semi-infinite string moves upward at a constant velocity of 2, while a ramp spreads out to the left and right of position at speed , as expected physically. By Eq.(E.9), the corresponding initial FDTD state for this case is
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.
corresponds to physically driving a single sample of string displacement at position . This is the spatially distributed alternative to the temporally distributed solution of feeding an input to a single displacement sample via the filter as discussed in .
As shown above, driving a single displacement sample in the FDTD corresponds to driving a velocity input at position on two alternating subgrids over time. Therefore, the filter acts as the filter on either subgrid alone--a first-order difference. Since displacement is being simulated, velocity inputs must be numerically integrated. The first-order difference can be seen as canceling this integration, thereby converting a velocity input to a displacement input, as in Eq.(E.23).
State Space Formulation