We now consider the scattering matrix for the parallel junction of
-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 -variable branch as

(58) |

where , is the pressure at the junction, and we have used pressure continuity to equate to for any .

Using conservation of velocity we obtain

(60) |

and

where

(62) |

(63) |

where the scattering matrix is deduced from (59):

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

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