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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
(E.14) |

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:

(E.15) |

(E.16) |

(E.17) |

(E.18) |

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:

(E.19) |

and the variables evolve forward in time as follows:

(E.20) |

(E.21) |

(E.22) |

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

*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.

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Additive Inputs

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Localized Velocity Excitations