Following the classical definition of passivity [4,133],
a medium is said to be *passive* if

To derive a definition of passivity in terms of the wave
impedance, consider a perfectly reflecting interruption in the
transmission line, such that
. For a passive medium,
using (16), (17) becomes

We say that wave propagation in the medium is *lossless* if the
wave impedance matrix satisfies

Most applications in waveguide modeling are concerned with nearly lossless propagation in passive media. In this paper, we will state results for in the more general case when applicable, while considering applications only for constant and diagonal impedance matrices . As shown in Section 2.3, coupling in the wave equation (5) implies a non-diagonal impedance matrix, since there is usually a proportionality between the speed of propagation and the impedance through the non-diagonal matrix .

The wave components of equations (11) travel
undisturbed along each axis. This propagation is implemented digitally
using * bidirectional delay lines*, as depicted in
Fig. 1. We call such a collection of delay lines an
*-variable waveguide section*. Waveguide sections are then
joined at their endpoints via *scattering junctions* (discussed further
in following sections) to form a DWN.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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