Stability

It is easy to see that the MDKC of Figure 3.6(a) will be MD-passive if the inductances $L_{1}$ and $L_{2}$, and consequently the port resistances $R_{1}$ and $R_{2}$ of the MDWDF in Figure 3.6(b) are non-negative. From (3.54), this gives a constraint on $v_{0}$, the space step/time step ratio, namely that we must have

$$v_{0} = \frac{\Delta}{T} \geq \vert\alpha\vert$$

Any such value of $v_{0}$ yields a passive, and thus stable algorithm.

It is important to mention, however, that the instances of the MDWDF, sampled at every grid point as in Figure 3.7 are not connected port-wise, as must be true for a traditional lumped WD-network. The output wave at the bottom port at spatial location $x=i\Delta$ is sign-inverted and then sent as input to the same port, at location $x=(i-1)\Delta$ at the next time step. Thus the realization of Figure 3.7 can not be analyzed directly as a chain of lumped elements; passivity follows from the multidimensional representations shown in Figure 3.6.