Lossless Scattering

The delay-line inputs (outgoing traveling waves) are computed by multiplying the delay-line outputs (incoming traveling waves) by the $N\times N$ feedback matrix (scattering matrix) $\mathbf{A}= [a_{i,j}]$ . By defining $p^+_i= x_i(n-M_i)$ , $p^-_i= x_i(n)$ , we obtain the more usual DWN notation

$$\mathbf{p}^- = \mathbf{A}\mathbf{p}^+$$ (C.140)

where $\mathbf{p}^+$ is the vector of incoming traveling-wave samples arriving at the junction at time $n$ , $\mathbf{p}^-$ is the vector of outgoing traveling-wave samples leaving the junction at time $n$ , and $\mathbf{A}$ is the scattering matrix associated with the waveguide junction.

The junction of $N$ physical waveguides determines the structure of the matrix $\mathbf{A}$ according to the basic principles of physics.

Considering the parallel junction of $N$ lossless acoustic tubes, each having characteristic admittance $\Gamma_j=1/R_j$ , the continuity of pressure and conservation of volume velocity at the junction give us the following scattering matrix for the pressure waves [437]:

$${\bf A} = \left[ \begin{array}{rrrr} \frac{2 \Gamma_{1}}{\Gamma_J} - 1 & \frac{2 \Gamma_{2}}{\Gamma_J} & \dots & \frac{2 \Gamma_{N}}{\Gamma_J} \\ \frac{2 \Gamma_{1}}{\Gamma_J} & \frac{2 \Gamma_{2}}{\Gamma_J}-1 & \dots & \frac{2 \Gamma_{N}}{\Gamma_J} \\ \dots \\ \frac{2 \Gamma_{1}}{\Gamma_J} & \frac{2 \Gamma_{2}}{\Gamma_J} & \dots & \frac{2 \Gamma_{N}}{\Gamma_J} -1\\ \end{array} \right]$$ (C.141)

where

$$\Gamma_J = \sum_{i=1}^N\Gamma_{i}.$$ (C.142)

Equation (C.142) can be derived by first writing the volume velocity at the $j$ -th tube in terms of pressure waves as $v_j = (p_j^+ - p_j^-)\Gamma_j$ . Applying the conservation of velocity we can find the expression

$$p = 2 \sum_{i=1}^{N}\Gamma_{i} p_i^+ / \Gamma_J$$

for the junction pressure. Finally, if we express the junction pressure as the sum of incoming and outgoing pressure waves at any branch, we derive (C.142). See §C.12 for further details.