Boundary Conditions

The two most important types of boundary condition are

Both are lossless, and have the form of (3.8). The first of these conditions is easy to deal with by short-circuiting the left-hand port. In this case, we may also remove the self-loop, as well as the combined loss/source port from the junction at . is forced to zero by the short-circuit condition; this boundary termination is shown in Figure 4.17(a). We have assumed that , so that the termination degenerates to a simple sign inversion of the wave incoming from the right-hand side.

Beginning from system (4.43), where we assume no source (to avoid conflicting conditions at the boundary), we can apply centered differences in time, and use a *one-sided* [176] approximation for the spatial derivative,

where we have set in accordance with the boundary condition. This yields the difference approximation to (4.43b) given by

The difference formula obtained from the network termination of Figure 4.17(b) is

where we now have

It is easy to see that if we set

then the waveguide network is indeed performing a calculation equivalent to (4.46). and may be set according to the type of network we are using (i.e., I, II or III in §4.3.6) as long as (4.47) and (4.48) are satisfied; the stability bounds are unchanged. It is important to note that this realization of the boundary condition will be first-order accurate in the grid spacing , due to our use of a one-sided difference approximation. It is recommended, then, to align the waveguide network such that a series junction lies at the left-most grid point, where it can be simply terminated by an open circuit. Such a termination will be second-order accurate.

Because *any* bounded reflectance may be used to terminate the waveguide network, it has been proposed [96] (in the context of the Karplus-Strong algorithm [103], which can be shown to be equivalent to a particular digital waveguide configuration [168]) that one can model wave propagation in a dispersive medium (like a beam [77]) by using a non-dispersive waveguide for the interior with an all-pass terminating reflectance. Such an all-pass filter will introduce a frequency-dependent phase delay which can be chosen so as to match the dispersion of the medium itself, without dissipating energy. It should be kept in mind, however, that one means of synthesizing such an all-pass filter would be by adding an additional chain of junctions on the opposing side of the boundary junction, and terminating it with an open- or short-circuit. Synthesis (for a given dispersion profile) presumably proceeds along the lines of methods used in related filter-design areas [215].