Lossless Scattering

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

$${\bf p}^- = {\bf A}{\bf p}^+$$ (22)

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

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

Considering the parallel junction of $N$ lossless acoustic tubes, each having characteristic admittance $\Gamma_j$, the continuity of pressure and conservation of volume velocity at the junction give us the following scattering matrix for the pressure waves [28]:

$${\bf A} = \left[ \begin{array}{rrrr} \frac{2 \Gamma_{1}}{\Ga... ...ots & \frac{2 \Gamma_{N}}{\Gamma_J} -1\\ \end{array}\right]$$ (23)

where

$$\Gamma_J = \sum_{i=1}^N\Gamma_{i}$$ (24)

(23) 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 (23).