### Kelly-Lochbaum Scattering Junctions

Conservation of energy and mass dictate that, at the impedance discontinuity, force and velocity variables must be*continuous*

where velocity is defined as positive to the right on both sides of the junction. Force (or stress or pressure) is a scalar while velocity is a vector with both a magnitude and direction (in this case only left or right). Equations (C.57), (C.58), and (C.59) imply the following

*scattering equations*(a derivation is given in the next section for the more general case of waveguides meeting at a junction):

where

is called the th

*reflection coefficient.*Since , we have . It can be shown that if , then either or is negative, and this implies an active (as opposed to passive) medium. Correspondingly, lattice and ladder recursive digital filters are

*stable*if and only if all reflection coefficients are bounded by in magnitude [297]. The scattering equations are illustrated in Figs. C.19b and C.20. In linear predictive coding of speech [482], this structure is called the

*Kelly-Lochbaum scattering junction*, and it is one of several types of scattering junction used to implement

*lattice*and

*ladder*digital filter structures (§C.9.4,[297]).

**Next Section:**

One-Multiply Scattering Junctions

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Longitudinal Waves in Rods