Introduction

The theory of lossless scattering of traveling waves has had an enormous influence on theory and practice in science and engineering. For example, classical scattering theory has found applications in transmission-line and microwave engineering [3], geoscience and speech modeling [39,35], numerically robust filter structures [27,35,111,113,112,114,110], and estimation theory [9]. Moreover, signal processing models of acoustic systems based on a sampled traveling-wave physical description have recently led to extremely efficient structures for musical sound synthesis based on physical models [93,97,118,115,134,76,100] and for delay effects such as artificial reverberation [91,106,79,122,84,125].

The solution of the wave equation in terms of traveling waves began with d'Alembert's first publication of it in 1747 [20]. While it may seem likely that scattering theory would have developed with transmission-line analysis in the early part of this century, the concept of the scattering matrix was introduced from general physics to the microwave engineering literature in 1948 (see, e.g., [3, p. 851]). The scattering formalism was developed independently and simultaneously for lumped networks by Belevitch [3, p. 851]. Since then, many fruitful lines of development have grown from the traveling-wave formalism, including results in insertion-loss filter theory, distributed amplifier design, the design of $n$-ports in telephone applications, wave digital filters, the applications mentioned in the previous paragraph, and the many results discussed in the references.