Representing the outgoing velocity wave as the superposition of the reflected wave plus the transmitted waves from the other ports, we have
where denotes the velocity transmission coefficientvelocity!transmission coefficient from port to port . Substituting Eq.(F.29) into Eq.(F.30) yields
Equating like terms with Eq.(F.32) gives
Thus, the th beta parameter is the velocity transmission coefficient from th port to any other port (besides the th). To convert the transmission coefficient from the 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.
Physical Derivation of Series Reflection Coefficient