Lossless Scattering

**Figure 2:** A four-port scattering junction.
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A four-port scattering junction is shown in Fig. 2.

When viewed as a mechanical system, each junction is bounded by the following conditions:

The port velocities at a given junction must be equal because the medium is continuous at that point: $v_1 = v_2 = \ldots = v_N$
The forces exerted by all the connections must balance or sum to zero at the junction: $f_1 + f_2 + \ldots + f_N = 0$

The traveling-wave components at the junction, $v_i = v^{+}_{i} + v^{-}_{i}$ and $f_i = f^{+}_{i} + f^{-}_{i}$ , are related by their respective port wave impedances: $f_i^{+} = R_{i}v^{+}_{i}$ and $f_i^{-} = -R_{i}v^{-}_{i}$ .

When these relationships are combined, the following lossless scattering equations for a generalized $N$

-port junction can be derived:

$\displaystyle v_{J} = \frac{2 \sum_{i} R_{i}v_{i}^{+}}{\sum_{i} R_{i}} \hspace{0.2in} \mbox{and} \hspace{0.2in} v_{i}^{-} = v_{J} - v_{i}^{+}$

For homogeneous, isotropic media, the various wave impedances are equal, allowing the junction velocity calculation above to be simplified to:

$\displaystyle v_{J} = \frac{2}{N} \sum_{i} v_{i}^{+}$

For the case of a four-port ( $N=4$

) junction, the scale factor $2/N$

reduces to $1/2$

, which can be implemented in fixed-point arithmetic with a bit shift.

The junctions are separated from each other by unit delays, so an input wave variable of one port of one junction is equal to the output at the adjacent junction port from the previous time step:

$\displaystyle v^{+}_{x,y}(n) = v^{-}_{x-1,y}(n-1)$

The various junction equations can be manipulated to show that the 2D digital waveguide mesh structure implements the standard second-order finite difference scheme applied to the partial differential wave equation of the ideal membrane (see this link for details),

$\displaystyle u_{tt}(t,x,y) = c^{2} [ u_{xx}(t,x,y) + u_{yy}(t,x,y) ],$

with wave propagation speed $c = \sqrt{2}/2$ and the time and spatial sampling intervals ( $X=\Delta x$ , $Y=\Delta y$ , $T=\Delta t$ ) all equal to one another.

The 2D rectilinear mesh implements a wave propagation speed equivalent to one-half unit diagonal distance per time sample. Intuitively, this can be understood as occuring because of a doubling of waveguide mass density.