Two-Port Scattering Junctions

Force and velocity must be continuous at every point

$$\begin{eqnarray*} f_{i-1}(t,cT) &=& f_i(t,0) \\ v_{i-1}(t,cT) &=& v_i(t,0) \nonumber \end{eqnarray*}$$

Velocity is positive to the right.
Stress (compression pressure or force density) is scalar.

Scattering relations:

$$\begin{eqnarray*} f^{{+}}_i(t) &=& \left[1+k_i\right]f^{{+}}_{i-1}(t-T) - k_if^{... ...(t+T) &=& k_if^{{+}}_{i-1}(t-T) + \left[1-k_i\right]f^{{-}}_i(t) \end{eqnarray*}$$

where

$$\zbox{k_i\mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{ R_i-R_{i-1}}{R_i+R_{i-1}}}$$

is the $i$th reflection coefficient (we'll derive this later as a special case of $N$-port scattering)

The Kelly-Lochbaum scattering junction