Kelly-Lochbaum Scattering JunctionsConservation 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):
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 .
One-Multiply Scattering Junctions
Longitudinal Waves in Rods