The mainstream literature in digital signal processing does not routinely include methodology for using digital filters as physical modeling elements. On the other hand, literature on general numerical simulation techniques for physical dynamic systems does not generally cover relevant good practice in digital signal processing. Recent applications [91,56,64,122,134] have shown that many benefits can be derived from taking a physical point of view with respect to digital filtering computations. For example, robust, rapidly time-varying, nearly lossless digital filters (such as needed, e.g., for a guitar string simulation) can be developed more easily using a physical approach, and conditions for the absence of unnatural artifacts become more clear. Moreover, nonlinear extensions are much more straightforward when there is a physical interpretation for each of the processing elements. While some benefits of the physical modeling perspective have been realized in the form of general-purpose digital filter structures with good numerical properties, such as ladder and lattice filters,1 more general use of signal processing elements and analysis/optimization methods in physical modeling applications appears to be rare.
The original Kelly-Lochbaum (KL) speech model  employed a ladder filter with delay elements in physically meaningful locations, allowing it to be interpreted as a discrete-time, traveling-wave model of the vocal tract (see the allpass portion of Fig. 5 for a similar diagram). Assuming a reflecting termination at the lips, the KL model can be transformed via elementary manipulations to the ladder/lattice filters used in linear predictive coding (LPC) of speech . The early work of Kelly and Lochbaum appears to have been followed by two main lines of development: (1) ``articulatory'' speech synthesis which utilizes increasingly sophisticated physical simulations [50,22], and (2) linear predictive coding of speech . The all-pole synthesis filter used in LPC, when implemented as a ladder filter, can be loosely interpreted as a transformation of the KL model . There have been ongoing efforts to develop low bit-rate speech coders based on simplified articulatory models [89,29]. The main barrier to obtaining practical speech coding methods has been the difficulty of estimating vocal-tract shape given only the speech waveform.
There have been a few developments toward higher quality speech models retaining the simplicity of a source-filter model such as LPC while building on a true physical model interpretation: An extended derivative of the Kelly-Lochbaum (KL) model, adding a nasal tract, neck radiation, and internal damping, has been used to synthesize high-quality female singing voice . Sparse acoustic tubes, in which many reflection coefficients are constrained to be zero, have been proposed [54,32]. Conical (rather than the usual cylindrical) tube segments and sparsely distributed interpolating scattering junctions have been proposed as further refinements [117,115].
In musical sound synthesis and delay-effects applications, digital waveguide models have been used for distributed media, such as vibrating strings, bores, horns, plates, solids, acoustic spaces, and the like [93,94,97,14,15,16,18,17,,37,103,44,46,47,8,,98,123,119,,116,,115,118,120,130,6,,49,82,121,41,,68,100,45,77,,87,107,,74,101,102,30]. A digital waveguide is typically defined as a bidirectional delay line containing sampled traveling waves. This paper is concerned with general properties of digital waveguide models.
Digital waveguide models often include lumped elements such as masses and springs. Lumped element modeling is the main focus of wave digital filters (WDF) as developed principally by Fettweis [26,27]. Wave digital filters derive from a scattering-theoretic formulation  and bilinear transformation  of lumped RLC elements. The formal traveling-wave signals in the scattering-theoretic description of lumped circuit elements are known as wave variables. Interfacing to digital waveguide models is simple since both use scattering-theoretic formulations. In acoustic modeling applications, WDFs can provide digital filters which serve as explicit physical models of mass, spring, and dashpot elements, as well as more exotic elements from classical network theory such as transformers, gyrators, and circulators. An example of using a WDF to model a nonlinear mass-spring system is the ``wave digital piano hammer'' . For realizability of lumped models with feedback, wave digital filters also incorporate short waveguide sections called ``unit elements,'' but these are ancillary to the main development. The digital waveguide formulation is actually more closely related to ``unit element filters'' which were developed much earlier than WDFs in microwave engineering .
Vaidyanathan and Mitra  developed a class of digital filters containing both the WDFs and the Gray-Markel normalized ladder structures , and proved low passband sensitivity to coefficient quantization and the absence of limit cycles, even under time-varying conditions. The generalized filter structure in  can itself be interpreted as a DWN consisting of a single normalized scattering junction (see Section 4) joining an input/output port and waveguides which are each sample long and reflectively terminated at their other end.
Vaidyanathan and Mitra also defined a general family of multivariable digital ladder filters  and derived a synthesis procedure for -input -output transfer-function matrices based on recursive extraction of generalized scattering junctions from an allpass transfer-function matrix in which the desired transfer function is embedded. Each ``vector'' scattering junction is an elegant generalization of that in the Gray-Markel normalized ladder filter . The extraction approach starts from the specification of a matrix fraction description  of the desired allpass matrix and proceeds directly in the domain.
Digital waveguide models, in contrast to WDFs and generalized Gray-Markel ladder filters, are typically derived directly from the geometry and physical properties of a desired acoustic system. They are often used to simulate nearly lossless distributed vibrating structures such as strings, tubes, rods, membranes, plates, and so on, but only up to a certain bandwidth (the limit of human hearing in audio applications). In many cases we do not want simply to implement a transfer function; we need instead a complete model with which we can interact in a physically consistent way, even when parameters are time-varying and nonlinearities are introduced. In such cases, digital waveguide networks can provide a solid foundation.