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
[441]. 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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University