For the KellyLochbaum vocal tract model, it is straightforward to arrive at a numerical scattering formulation of the problem; the approximation of a smoothlyvarying tube by a series of concatenated tubes is intuitively satisfying, and leads immediately to a wave variable numerical solution to Webster's equation. The identification of the mesh of onedimensional tubes of Figure 1.8(a) as a numerical solver for the twodimensional wave equation is more difficult, because it is by no means clear that such a mesh behaves like a twodimensional acoustic medium (say). Although as we have seen, it is possible to prove (through a finite difference treatment) that the tube network is indeed solving the right equation, we have not shown a way of deducing such a structure from the original defining PDE system. If one wants to develop a DWN for a more complex system (such as a stiff vibrating plate of variable density and thickness, for example), then guesswork and attempts at invoking Huygens' principle will be of limited use.
The scattering operation we introduced in §1.1.1 and §1.1.2 is at the heart of all the numerical methods we will discuss in this thesis, whether they are based on digital waveguide networks or wave digital filters, which we will shortly introduce. A given system of PDEs is numerically solved by filling the problem domain with scattering nodes, or junctions, such as that shown in Figure 1.7(b), which calculate reflected waves from incident waves according to (1.13) (or its series dual form). The topology of the network of interconnected junctions will be dependent on the particulars of the system we wish to solve. As we have seen, these scattering junctions act as powerconserving signal processing blocks, and in a DWN, they are linked by discretetime acoustic tubes, or transmission lines, which are also powerconserving, and serve to transport energy from one part of the network to another. The key concept here is the losslessness of the network components, which is dependent on the positivity of the various circuit element values (admittances); as we have seen, this positivity condition ensures that some squared norm of the signals in the discretetime network will remain constant as time progresses. In other words, the simulation routine that such a network implies is guaranteed stable by enforcing this condition.
Wave digital filters (WDFs) are also based on the idea of preserving losslessness (and more generally passivity) in a discretetime simulation of a physical system, though the approach is somewhat different from what we have just seen. As they were originally intended to transfer analog electrical filter (RLC) networks to discrete time, it is best to begin by looking briefly at lumped circuit elements. A oneport element, such as that shown at left in Figure 1.9 is characterized by a voltage , and a current , both of which are functions of time . In the time domain, the oneport generally relates and through some combination of differential or integral operators. If the oneport (or more generally, port) is linear and timeinvariant, then there is a simple description of its behavior in the frequency domain, but we will wait until Chapter 2 before entering into the details. An analog filter is simply an interconnected network of such elements; it is operated by applying a voltage at one pair of free terminals, and then reading the filtered output at another pair. In particular, if the network is made up of passive elements such as resistors, capacitors, inductors etc., then it must behave as a stable filter.
Fettweis [46] developed a procedure for mimicking the energetic behavior of an analog filtering network in discrete time. The input and filtered output become digital signals, and the filtering network becomes a recursion, to be realized as a computer program. Most importantly, the digital network has the same topology as the analog network, and can be thought of as its discretetime ``image.'' Oneports (or more generally ports) are first characterized in terms of wave variables,

Consider a parallel connection of two oneport circuit elements, as shown in Figure 1.10(a). The oneports are defined by some relationship between their respective voltages and currents, which we will write as , and , . For such a parallel connection, Kirchoff's Laws dictate that
Kirchoff's Laws for the parallel connection can then be rewritten in terms of the wave variables as