### Localized Velocity Excitations

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 [437]. 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 to obtain velocity waves:
 (E.13)

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.
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Localized Displacement Excitations