Reflection Coefficient, Series Case

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

$$\rho^v_i \isdef \left. \frac{v^{-}_i(n)}{v^{+}_i(n)} \right\vert _{v^{+}_j(n)=0, \forall j\neq i} \protect$$ (F.38)

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

$$v^{-}_i(n) = \rho^v_i v^{+}_i + \sum_{j\neq i} \tau^v_{ji} v^{+}_j \protect$$ (F.39)

where $\tau^v_{ji}$ denotes the velocity transmission coefficient from port $j$ to port $i$ . Substituting Eq.(F.36) into Eq.(F.37) yields

$$\begin{eqnarray*} v^{-}_i(n) &=& v_J(n) - v^{+}_i(n)\\ &=& \left(\sum_{j=1}^N \beta_j v^{+}_j(n)\right) - v^{+}_i(n)\\ &=& (\beta_i - 1)v^{+}_i(n) + \sum_{j\neq i} \beta_j v^{+}_j(n) \end{eqnarray*}$$

Equating like terms with Eq.(F.39) gives

$$\rho^v_i = \beta_i - 1 \protect$$ (F.40)

$$\tau^v_{ji} = \beta_j, \quad (i\neq j)$$ (F.41)

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.19)). They are exactly the dual of Equations (F.19-F.20) for force waves at a parallel junction.