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:
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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More General Velocity Excitations
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Localized Displacement Excitations
