Digital waveguides are but one example of a scattering-based numerical method , for which the underlying variables propagated are of wave type, which are reflected and transmitted throughout a network by power-conserving scattering junctions (which can be viewed, under some conditions, as orthogonal matrix transformations). Such methods have appeared in various guises across a wide range of (often non-musical) disciplines. Perhaps the best known is the transmission-line matrix method [50,108], which is popular in the field of electromagnetic field simulation, and dates back to the early 1970s , but multidimensional extensions of wave digital filters [81,80] intended for numerical simulation have also been proposed [83,22]. Most such methods are designed based on electrical circuit network models, and make use of scattering concepts borrowed from microwave filter design ; their earliest roots are in the work of Kron in the 19402 .
Scattering-based methods also appear in standard areas of signal processing, such as inverse estimation , fast factorization and inversion of structured matrices , and linear prediction  for speech signals (leading directly to the Kelly-Lochbaum speech synthesis model, which is a direct antecedent to digital waveguide synthesis).
In the musical sound synthesis community, scattering methods, employing wave (sometimes called ``W") variables are sometimes viewed  in opposition to methods which employ physical (correspondingly called ``K," for Kirchhoff) variables, such as lumped networks, and, as will be mentioned shortly, direct simulation techniques, which are employed in the vast majority of simulation applications in the mainstream world.
In recent years, moves have been made towards modularizing physical modeling ; instead of simulating the behaviour of a single musical object, such as a string or tube, the idea is to allow the user to interconnect various predefined objects in any way imaginable. In many respects, this is the same point of view as that of those working on lumped network models--this is reflected by the use of hybrid or ``mixed" K-W methods, i.e., methods employing both scattering methods, such as wave digital filters and digital waveguides, and finite difference modules (typically lumped) [119,118]. See Figure . In some situations, particularly those involving the interconnection of physical ``modules," representing various separate portions of a whole instrument, the wave formulation may be preferable, in that there is a clear means of dealing with the problem of non-computability, or delay-free loops--the concept of the reflection-free wave port, introduced by Fettweis long ago in the context of digital filter design , can be fruitfully employed in this case. The automatic generation of recursible structures, built around the use of wave digital filters, is a key component of such methods , and can be problematic when multiple nonlinearities are present, requiring specialized design procedures . One result of this work has been a completely modular software system for physical modeling sound synthesis, incorporating elements of both types, called BlockCompiler . More recently the scope of such methods has been hybridized even further through the incorporation of functional transformation (modal) methods into the same framework .