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Chapters

Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       Excitation Examples
          More General Velocity Excitations

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More General Velocity Excitations

From Eq.$ \,$(P.11), it is clear that initializing any single K variable $ y_{n,m}$ corresponds to the initialization of an infinite number of W variables $ y^{+}_{n,m}$ and $ y^{-}_{n,m}$. That is, a single K variable $ y_{n,m}$ corresponds to only a single column of $ \mathbf{T}^{-1}$ for only one of the interleaved grids. For example, referring to Eq.$ \,$(P.11), initializing the K variable $ y_{n-1,m}$ to -1 at time $ n$ (with all other $ y_{n,m}$ intialized to 0) corresponds to the W-variable initialization

\begin{eqnarray*} y^{+}_{n,m-(2\mu+1)}&=&+1, \quad \mu =0,1,2,\cdots\\ y^{-}_{n,m-(2\mu+1)}&=&-1, \quad \mu =0,1,2,\cdots \end{eqnarray*}

with all other W variables being initialized to zero. In view of earlier remarks, this corresponds to an impulsive velocity excitation on only one of the two subgrids. A schematic depiction from $ \mu = m-5$ to $ m+5$ of the W variables at time $ n$ is as follows:

\begin{displaymath}\begin{array}{crrrrr\vert rrrrrrc} \cdots & 1 & 0 & 1 & 0 & 1... ... \cdots\\ & & & & & & m & & & & & \mu & \rightarrow \end{array}\end{displaymath} (P.14)

Below the solid line is the sum of the left- and right-going traveling-wave components, i.e., the corresponding K variables at time $ n$. The vertical lines divide positions $ \mu=m-1$ and $ \mu=m$. The initial displacement is zero everywhere at time $ n$, consistent with an initial velocity excitation. At times $ \nu=n+1,n+2,n+3,n+4$, the configuration evolves as follows:

\begin{displaymath}\begin{array}{crrrrr\vert rrrrrrc} \cdots & 0 & 1 & 0 & 1 & 0... ... 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots \end{array}\end{displaymath} (P.15)