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Reflection Coefficient, Series Case

The velocity reflection coefficient seen at port $ i$ is defined as

$\displaystyle \rho^v_i \isdef \left. \frac{v^{-}_i(n)}{v^{+}_i(n)} \right\vert _{v^{+}_j(n)=0, \forall j\neq i} \protect$ (F.31)

Representing the outgoing velocity wave $ v^{-}_i(n)$ as the superposition of the reflected wave $ \rho^v_iv^{+}_i(n)$ plus the $ N-1$ transmitted waves from the other ports, we have

$\displaystyle v^{-}_i(n) = \rho^v_i v^{+}_i + \sum_{j\neq i} \tau^v_{ji} v^{+}_j \protect$ (F.32)

where $ \tau^v_{ji}$ denotes the velocity transmission coefficientvelocity!transmission coefficient from port $ j$ to port $ i$. Substituting Eq.$ \,$(F.29) into Eq.$ \,$(F.30) yields
v^{-}_i(n) &=& v_J(n) - v^{+}_i(n)\\
&=& \left(\sum_{j=1}^N ...
... &=& (\beta_i - 1)v^{+}_i(n) + \sum_{j\neq i} \beta_j v^{+}_j(n)
Equating like terms with Eq.$ \,$(F.32) gives
$\displaystyle \rho^v_i$ $\displaystyle =$ $\displaystyle \beta_i - 1
\protect$ (F.33)
$\displaystyle \tau^v_{ji}$ $\displaystyle =$ $\displaystyle \beta_j, \quad (i\neq j)$ (F.34)

Thus, the $ j$th beta parameter is the velocity transmission coefficient from $ j$th port to any other port (besides the $ i$th). To convert the transmission coefficient from the $ i$th port to the reflection coefficient for that port, we simply subtract 1. These relationships are specific to velocity waves at a series junction (cf. Eq.$ \,$(F.22)). They are exactly the dual of Equations (F.22-F.23) for force waves at a parallel junction.
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Physical Derivation of Series Reflection Coefficient
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Beta Parameters