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 (13) and (11) can be rewritten for each $m$-variable branch as

$$\begin{array}{rcccc} {\mbox{\boldmath$u$}}_i^+ &=& & {\mbox{... ...h$\Gamma$}}^{*}_i(1/z^*)& {\mbox{\boldmath$p$}}_i^- \end{array}$$ (43)

and

$$\begin{array}{l} {\mbox{\boldmath$u$}}_i = {\mbox{\boldmath$... ...\mbox{\boldmath$p$}}_i^- = p_J{\mbox{\boldmath$1$}} \end{array}$$ (44)

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

Using conservation of velocity we obtain

$$0 = {\mbox{\boldmath$1$}}^T{\sum_{i=1}^{N}{{\mbox{\boldmath$u$}}_i}}$$

$$= {\mbox{\boldmath$1$}}^T{\displaystyle \sum_{i=1}^{N}}\left\{\left... ...{\mbox{\boldmath$\Gamma$}}^{*}_i(1/z^*)\right]{\mbox{\boldmath$p$}}_i^+ \right.$$

$$\left. \qquad - {\mbox{\boldmath$\Gamma$}}^{*}_i(1/z^*)p_J{\mbox{\boldmath$1$}}\right\}$$

and

$$p_J= S {\mbox{\boldmath$1$}}^T {\sum_{i=1}^{N}{\left[{\mbox{... ...math$\Gamma$}}^{*}_i(1/z^*)\right] {\mbox{\boldmath$p$}}_i^+}}$$ (45)

where

$$S = \left\{{\mbox{\boldmath$1$}}^T\left[{\sum_{i=1}^{N}{{\mb... ...ma$}}^{*}_i(1/z^*)}}\right]{\mbox{\boldmath$1$}}\right\}^{-1}.$$ (46)

Hence, the scattering matrix is given by

$${\mbox{\boldmath$A$}}= S \left[\begin{array}{l}{\mbox{\boldm... ...\end{array} \right] \end{array}\right] - {\mbox{\boldmath$I$}}$$ (47)

It is clear that (38) is a special case of (48) when $m=1$. If the branches do not all have the same dimensionality $m$, we may still use the expression (48) by letting $m$ be the largest dimensionality and embedding each branch in an $m$-variable propagation space.