As we mentioned above, all the numerical methods to be discussed in this thesis have their origin in digital filter design, even though they are intended, ultimately, for use in simulation, and not filtering. Though these two goals may seem to be at cross purposes, there is a very early instance of an engineering problem which straddles both worlds.
Kelly and Lochbaum [104] developed a digital speech synthesis model by treating the vocal tract as a slowly time-varying circular one-dimensional acoustic tube of variable cross-sectional area, excited at one end (periodically by the glottis, or by turbulent noise), and radiating a speech waveform at the other--see Figure 1.1(a). At any given time 
, the shape of the tube as a function of the spatial coordinate 
determines the system resonances, or formants [145], which serve as important perceptual cues for the listener in distinguishing among various voiced and unvoiced vocal sounds. The problem, then, is to develop a numerical method, suitable for computer implementation, which somehow simulates the time-evolution of the acoustic ``state'' of the vocal tract, i.e., the pressure and velocity distributions in the interior. We follow the standard exposition of the Kelly-Lochbaum model here, as per [30,145].