In the more general case of ports being connected in series, we have the physical constraints
The outgoing wave variables are given by
It is customary in the wave digital filter literature to define the beta parameters as
where are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:
However, we normally employ a mixture of parallel and series adaptors,
while keeping a force-wave simulation. Since
, we obtain, after a small amount of algebra, the following
recipe for the series force-wave adaptor:
We see that we have multiplies and additions as in the parallel-adaptor case. However, we again have from Eq.(F.26) that
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.
Physically, the force-wave reflection coefficient seen at port of a series adaptor is due to an impedance step from , that of the port interface, to a new impedance consisting of the series combination of all other port impedances meeting at the junction. Let
denote this series combination. Then we must have, as in Eq.(F.25),
For port to be reflection free in a series adaptor, we require
That is, the port's impedance must equal the series combination of the other port impedances at the junction. This result can be compared with that for the parallel junction in §F.2.2.
The series adaptor has now been derived in a way which emphasizes its duality with respect to the parallel adaptor.
``Piano hammer in flight''
Two-Port Series Adaptor for Force Waves