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Physical Audio Signal Processing
    Digital Waveguide Theory
       Loaded Waveguide Junctions

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Loaded Waveguide Junctions

In this section, scattering relations will be derived for the general case of N waveguides meeting at a load. When a load is present, the scattering is no longer lossless, unless the load itself is lossless. (i.e., its impedance has a zero real part). For $ N>2$, $ v^{+}_i$ will denote a velocity wave traveling into the junction, and will be called an ``incoming'' velocity wave as opposed to ``right-going.''H.7

Figure H.26: Four ideal strings intersecting at a point to which a lumped impedance is attached. This is a series junction for transverse waves.
\includegraphics[width=\twidth]{eps/fNstrings}

Consider first the series junction of $ N$ waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of $ N$ ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance $ R_J(s)$, as depicted in Fig.H.26 for $ N=4$. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., $ V(s) = {\cal L}\{v\}$ for velocity waves and $ F(s) = {\cal L}\{f\}$ for force waves, where $ {\cal L}\{\cdot\}$ denotes the Laplace transform. In the discrete-time case, we use the $ z$ transform instead, but otherwise the story is identical. The physical constraints at the junction are

$\displaystyle V_1(s) = V_2(s) = \cdots = V_N(s)$ $\displaystyle \isdef$ $\displaystyle V_J(s)$ (H.75)
$\displaystyle F_1(s) + F_2(s) + \cdots + F_N(s)$ $\displaystyle =$ $\displaystyle V_J(s) R_J(s) \isdef F_J(s)$ (H.76)

where the reference direction for the load force $ F_J$ is taken to be opposite that for the $ F_i$. (It can be considered the ``equal and opposite reaction'' force at the junction.) For a wave traveling into the junction, force is positive pulling up, acting toward the junction. When the load impedance $ R_J(s)$ is zero, giving a free intersection point, the junction reduces to the unloaded case, and signal scattering will be energy preserving. In general, the loaded junction is lossless if and only if re$ \left\{R_J(j\omega)\right\}\equiv0$, and it is memoryless if and only if im$ \left\{R_J(j\omega)\right\}\equiv0$.

The parallel junction is characterized by

$\displaystyle F_1(s) = F_2(s) = \cdots = F_N(s)$