Digital filtering structures have recently been applied toward the numerical simulation of distributed physical systems. In particular, they have been used to numerically integrate systems of partial differential equations (PDEs), which are time-dependent, and of hyperbolic type (implying wave-like solutions, with a finite propagation velocity). Two such methods, the *multidimensional wave digital filtering* and *digital waveguide network* approaches both rely heavily on the classical theory of electrical networks, and make use of *wave variables*, which are reflected and transmitted throughout a grid of *scattering junctions* as a means of simulating the behavior of a given model system. These methods possess many good numerical properties which are carried over from digital filter design; in particular, they are numerically robust in the sense that stability may be maintained even in finite arithmetic. As such, these methods are potentially useful candidates for implementation in special purpose hardware.

In this thesis, the subtext is that such scattering-based methods can and should be treated as finite difference schemes, for purposes of analysis and comparison with standard differencing forms. In many cases, these methods can be shown to be equivalent to well-known differencing approaches--we pay close attention to the relationship between digital waveguide networks and *finite difference time domain* (FDTD) methods. For this reason, it is probably most useful to think of scattering forms as alternative realizations of these schemes with good numerical properties, in direct analogy with ladder, lattice and orthogonal digital filter realizations of direct form filters. We make use of this correspondence in order to import (from the finite difference setting) two techniques for approaching problems with irregular boundaries, namely coordinate changes, and a means of designing interfaces between grids of different densities and/or geometries. We also make use of the finite difference formulation in order to examine initial and boundary conditions, parasitic modes, and take an extended look at the numerical properties of all the commonly encountered forms of the waveguide network in two and three spatial dimensions.

Another question is of the relationship between wave digital and waveguide network schemes. Although they are quite similar from the standpoint of the programmer, in that the main operation, scattering, is the same in either case, conceptually they are very different. A multidimensional wave digital network is derived from a compact circuit representation of model system of PDEs. The numerical routine is itself a discrete time and space image of the original network. Waveguide meshes, however, are usually formulated as collection of lumped scattering junctions which span the problem domain, connected by bidirectional delay lines. Lacking a multidimensional representation, then, it is not straightforward to design a mesh which numerically solves a given problem. A useful result is that waveguide meshes can be obtained directly from a system by almost exactly the same means as a wave digital network. This unification of the two methods opens the door to a larger class of methods which are of neither type, and yet which consist of the same numerically robust basic building blocks.

On the applied side, special attention is paid to problems in beam, plate and shell dynamics; though these systems are in general much more complex than the transmission line and parallel-plate problems which have been discussed extensively in the literature, they can be dealt with using both wave digital filters and waveguide networks, though several new techniques must be introduced. Several simulations are presented.

Preface

This thesis is, if anything, long; there have been a few reasons for this. At the beginning stages of research, the focus was on the digital synthesis of musical sound through the use of physical modeling techniques. Since all physical models of vibration in acoustic instruments can be framed in terms of coupled sets of partial differential equations, the problem, then, is one of the numerical integration of these equations, subject to initial and boundary conditions and external excitations. There are, of course, many ways of designing such simulation algorithms. We began by looking at digital waveguide networks, which have been used successfully for this purpose for some time, but soon turned to multidimensional wave digital filtering methods, which are based on some similar ideas, yet within a powerful framework for attacking a much more general (and not necessarily musical) class of problems. Wave digital filters, even for filtering applications, are hardly as well-known here in the U.S. as they are in Europe, so it would not have been particularly helpful to anyone (or wise) to present a few results with only passing nods to the literature. Some rather extensive background information was thus compiled, in the form of a summary of most of the work that has gone on in this field to date (to this author's knowledge)^{}. Because of their fundamental similarities to these wave digital filtering simulation methods, waveguide networks were always slated for a (presumed cursory) second look; upon this reexamination, however, they seemed deserving of an in-depth parallel development all their own, requiring yet more background material.

Traditional approaches to numerical integration usually involve the direct discretization of a given set of equations by a variety of techniques, such as finite difference, finite element and spectral or collocation methods. The methods we will discuss, however, have their roots elsewhere, in electrical network theory, digital filtering and scattering theory. The most general goal of this author has been to provide a unified treatment of wave digital filtering and digital waveguide network simulation techniques, and also to answer, or at least pose some questions about how they fit into the larger picture of numerical integration methods as a whole. As might be expected, this thesis suffers in certain respects (notation among them) from the mismatch between the points of view of the electrical engineer and the specialist in numerical methods. Needless to say, there is much insight to be gained in the attempt to resolve some of the many outstanding distinctions.

Looking back, one of the few regrets of this author has been the erosion of the emphasis on musical sound synthesis applications. Although we will spend a good deal of time later on looking at ways of extending these techniques to simulate the vibration of stiff systems such as beams, plates, and shells, which are the sound-producing mechanisms (resonators) in many musical instruments, we have not done the hard work of optimizing the algorithms for the audio frequency range--the computer program one writes in order to listen to a struck xylophone bar will assuredly be very different from one designed to check the modal frequencies of an I beam under stress. We have tried to lay down the basic principles, however, and nothing would be more rewarding than listening to a real-time waveguide or wave digital chime based on a cylindrical shell model.

Acknowledgements

Stanford Electrical Engineering advising is notoriously overbooked. I thus feel very lucky to have had two advisers with whom I had frequent contact. My principal adviser, Professor Julius Smith, allowed me complete freedom in my choice of topic for this project, and through the CCRMA affiliates program, provided welcome financial support from the outset. Julius is a master of the trade, and I learned a lot by watching him at work. I thank my associate adviser, Ivan Linscott, of STAR Lab, for arranging funding during the latter part of my program, and for the numerous coffee breaks during which all sorts of wayward subjects, technical or not, were fair game. Both were always very cheerful, brimming with ideas, helpful in getting me motivated, and fair-minded. I also thank Professor Robert Gray of ISL, who served on my reading committee, and Professor Simon Wong of CIS, my orals chair. Finally, I thank the third member of my orals and reading committee, Professor Perry Cook of Princeton, who made a long trip out to the West Coast for my defense only to be greeted by foul October weather.

Professor Alfred Fettweis was very obliging and helped me to get my initial footing in this field, as was Gunnar Nitsche, who, during the course of a five-hour dinner (on him) in an empty Chinese restaurant in Hildesheim, Germany, took the trouble to explain his dissertation to me in English, page by page. This thesis is an outgrowth of their work, more than anyone else's, and I am greatly indebted to them. I also thank Dirk Dahlhaus, of the Swiss Federal Institute of Technology and Xiaomin Wang, formerly of the University of Notre Dame, for setting me straight on a few technical matters at the heart of wave digital filtering. Thanks also to the several members of the Stanford Mechanical Engineering department who took the time to talk with me: Professors Charles Steele and Thomas Hughes were especially helpful, as was Kian-Meng Lim. I'd also like to thank Jim Donohoe at Ames Research, as well as the members of the Radioscience Group at STAR lab. Thanks also to professor Eli Turkel of Tel Aviv University for getting me untangled with regard to some specifics of perfectly matched layers.

Thanks to the CCRMA DSP group members of my generation, all of whom I consulted frequently, namely Tim Stilson, Bill Putnam, Scott Levine, Gary Scavone and especially Scott Van Duyne (who got me interested in this topic at the outset), Tony Verma and Dave Berners, one of the few people I know who understands how things work. Thanks also to the many other friends I've made at CCRMA over the past few years, including Bob Sturm, Fabien Gouyon, Tamara Smyth, Hendrik Purwins, Patty Huang, Peer Landa, Stefania Serafina and Sile O'Modhrain. CCRMA is an exceptionally smoothly run lab, with a nearly invisible bureaucracy, and for this I thank Julius, Chris Chafe, John Chowning, Max Mathews, Heidi Kugler, Gary Scavone, and Vibeke Cleaver. A special thanks to our system administrator Fernando Lopez-Lezcano for not only keeping the network afloat, but for taking the time to get my own computer working (and I thank Dave Berners in this regard as well). Thanks also to Jay Kadis for allowing me to inhabit the CCRMA studios at a below-market rate. I wish them all well.

During my time as a student intern at IRCAM in Paris, Miller Puckette (now at UCSD) and David Zicarelli (now at Cycling'74) were instrumental in helping me get my bearings in computer music. I also send my best wishes to the late, great Professor Ivan Tcherepnin at Harvard.

Thanks to the many friends I made at Stanford over the last few years. I look forward to joining my fellow EE parolees Tony Verma and Rob Batchko on the outside.

My biggest thanks go to my family. I'd like to take the opportunity to dedicate this thesis to my sister, Maya. Catch her TV show ``Structures'' weeknights on Rogers Cable 10, in Toronto.