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 [414]. 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:

$$v^{+} = -cy'^{+}= \frac{f^{{+}}}{R}$$

$$v^{-} = \;cy'^{-}= -\frac{f^{{-}}}{R}, \protect$$ (M.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. (M.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. (M.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.