Localized Velocity Excitations

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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 [447]. Specifically, initial velocity waves $v^{\pm}$ can be computed from from initial displacement waves $y^\pm$ by spatially differentiating $y^\pm$ to obtain traveling slope waves $y'^\pm$ , multiplying by minus the tension to obtain force waves, and finally dividing by the wave impedance $R=\sqrt{K\epsilon }$ to obtain velocity waves:

$\displaystyle v^{+}$	$\displaystyle =$	$\displaystyle -cy'^{+}= \frac{f^{{+}}}{R}$
$\displaystyle v^{-}$	$\displaystyle =$	$\displaystyle \;cy'^{-}= -\frac{f^{{-}}}{R}, \protect$	(P.13)

where $c=\sqrt{K/\epsilon}$ denotes sound speed. The initial string velocity at each point is then $v(nT,mX)=v^{+}(n-m)+v^{-}(n+m)$ . (A more direct derivation can be based on differentiating Eq. $\,$ (P.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. $\,$ (P.11) that such asymmetry can be caused by unequal weighting of $y_{n,m}$ and $y_{n,m\pm1}$ . For example, the initialization