Unloaded Junctions and Alpha Parameters

In the unloaded case, we have $R_J(s)=0$ , so there are no dynamics at the junction, and Eq.(C.99) for the series junction reduces to

$$\alpha_i = \frac{2R_i}{R_1 + \cdots + R_N}$$ (C.126)

These we call the alpha parameters, and they are analogous to those used to characterize ``adaptors'' in wave digital filters (§F.2.2). For unloaded junctions, the alpha parameters obey

$$0\leq\alpha_i \leq 2$$ (C.127)

and

$$\sum_{i=1}^N\alpha_i = 2$$ (C.128)

In the unloaded case, the series junction scattering relations are given (in the time domain) by

$$v_J(t) = \sum_{i=1}^N \alpha_i v^+_i(t) \protect$$ (C.129)

$$v^-_i(t) = v_J(t) - v^+_i(t)$$ (C.130)

The alpha parameters provide an interesting and useful parametrization of waveguide junctions. They are explicitly the coefficients of the incoming traveling waves needed to compute junction velocity for a series junction (or junction force or pressure at a parallel junction), and losslessness is assured provided only that the alpha parameters be nonnegative and sum to $2$ . Having them sum to something less than $2$ simulates a ``resistive load'' at the junction.

Note that in the lossless, equal-impedance case, in which all waveguide impedances have the same value $R_i=R$ , (C.126) reduces to

$$\alpha_i = \frac{2}{N}$$ (C.131)

When, furthermore, $N$ is a power of two, we have that there are no multiplies in the scattering relations (C.129). This fact has been used to build multiply-free reverberators and other structures using digital waveguide meshes [434,522,399,524].

An elaborated discussion of $N=2$ strings intersection at a load is given in in §9.2.1. Further discussion of the digital waveguide mesh appears in §C.14.