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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 [441]. 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 $ K$ to obtain force waves, and finally dividing by the wave impedance $ R=\sqrt{K\epsilon }$ to obtain velocity waves:

$\displaystyle v^{+}$ $\displaystyle =$ $\displaystyle -cy'^{+}= \frac{f^{{+}}}{R}$  
$\displaystyle v^{-}$ $\displaystyle =$ $\displaystyle \;cy'^{-}= -\frac{f^{{-}}}{R},
\protect$ (E.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.$ \,$ (E.4) with respect to $ x$ 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 $ y_{n,m}$ and $ y_{n,m\pm1}$ . For example, the initialization

\begin{eqnarray*}
y_{n-1,m+1} &=& +1\\
y_{n-1,m} &=& -1
\end{eqnarray*}

corresponds to an impulse velocity excitation at position $ m+1/2$ . In this case, both interleaved grids are excited.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA