The delay-line inputs (outgoing traveling waves) are computed by multiplying the delay-line outputs (incoming traveling waves) by the feedback matrix (scattering matrix) . By defining , , we obtain the more usual DWN notation
The junction of physical waveguides determines the structure of the matrix according to the basic principles of physics.
Considering the parallel junction of lossless acoustic tubes, each having characteristic admittance , the continuity of pressure and conservation of volume velocity at the junction give us the following scattering matrix for the pressure waves :
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.