Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Parallel Junction of Multivariable Complex Waveguides

We now consider the scattering matrix for the parallel junction of $N$ $m$-variable physical waveguides, and at the same time, we treat the generalized case of matrix transfer-function wave impedances. Equations (21) and (16) can be rewritten for each $m$-variable branch as

{\mbox{\boldmath$u$}}_i^+ &=& {\mbox{\bol...
...\Gamma$}}^{*}_i(1/z^*){\mbox{\boldmath$p$}}_i^- \,,
\end{array}\end{displaymath} (58)

{\mbox{\boldmath$u$}}_i = {\mbox{\boldmath$...
...ldmath$p$}}_i^- = p_{\rm J}{\mbox{\boldmath$1$}}\,,
\end{array}\end{displaymath} (59)

where ${\mbox{\boldmath$\Gamma$}}_i(z)={\mbox{\boldmath$R$}}_i^{-1}(z)$, $p_{\rm J}$ is the pressure at the junction, and we have used pressure continuity to equate ${\mbox{\boldmath$p$}}_i$ to $p_{\rm J}$ for any $i$.

Using conservation of velocity we obtain

$\displaystyle 0$ $\textstyle =$ $\displaystyle {\mbox{\boldmath$1$}}^T{\sum_{i=1}^{N}{{\mbox{\boldmath$u$}}_i}}$  
  $\textstyle =$ $\displaystyle {\mbox{\boldmath$1$}}^T{\displaystyle \sum_{i=1}^{N}}\left\{\left...
...{\mbox{\boldmath$\Gamma$}}^{*}_i(1/z^*)\right]{\mbox{\boldmath$p$}}_i^+ \right.$  
    $\displaystyle \left. \qquad - {\mbox{\boldmath$\Gamma$}}^{*}_i(1/z^*)p_{\rm J}{\mbox{\boldmath$1$}}\right\}$ (60)

p_{\rm J}= S {\mbox{\boldmath$1$}}^T {\sum_{i=1}^{N}{\left[{...
...$\Gamma$}}^{*}_i(1/z^*)\right] {\mbox{\boldmath$p$}}_i^+}} \,,
\end{displaymath} (61)

S = \left\{{\mbox{\boldmath$1$}}^T\left[{\sum_{i=1}^{N}{{\mb...
...}}^{*}_i(1/z^*)}}\right]{\mbox{\boldmath$1$}}\right\}^{-1} \,.
\end{displaymath} (62)

From (57), we have the scattering relation
$\displaystyle {\mbox{\boldmath$p$}}^-$ $\textstyle =$ $\displaystyle \left[ \begin{array}{l}
{\mbox{\boldmath$p$}}_1^- \\
\vdots \\
\end{array} \right] = {\mbox{\boldmath$A$}}{\mbox{\boldmath$p$}}^+$  
  $\textstyle =$ $\displaystyle {\mbox{\boldmath$A$}}\left[ \begin{array}{l}
1 \\
\vdots \\
\end{array} \right] - {\mbox{\boldmath$p$}}^+ \: ,$ (63)

where the scattering matrix is deduced from (59):
{\mbox{\boldmath$A$}}= S \left[\begin{array}{l}{\mbox{\boldm...
...d{array} \right]
\end{array}\right] - {\mbox{\boldmath$I$}}\,.
\end{displaymath} (64)

If the branches do not all have the same dimensionality $m$, we may still use the expression (62) by letting $m$ be the largest dimensionality and embedding each branch in an $m$-variable propagation space.

Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download gdwn.pdf

``Generalized Digital Waveguide Networks'', by Julius O. Smith III and Davide Rocchesso, preprint submitted for publication, Summer 2001.
Copyright © 2008-03-12 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [About the Automatic Links]