Numerical Sound Synthesis Finite Difference Schemes and Simulation in Musical Acoustics

by Bilbao, Stefan

• ISBN13: 9780470510469
• ISBN10: 0470510463
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-12-02
• Publisher: Wiley

Summary

This book provides numerical methods for the modelling of musical instruments and the simulation of sound through synthesisation, potentially enabling more realistic and complex sound production. Bilbao focuses on physical modelling synthesis techniques rather than the more established DSP techniques, setting out useful programming exercises, worked through problems and MATLAB code examples so that readers can easily implement the methods discussed in the book. It starts with a historical overview of digital sound synthesis techniques, highlighting the link between the various physical modelling methodologies. Bilbao then goes on to introduce time series and difference operators, and the simple harmonic oscillator, analyzing the various difference schemes, especially with respect to numerical stability and accuracy. Different numerical methods are examined next, starting with the finite difference method, then followed by a look at the 1D wave equation, bars and linear strings, nonlinear strings, acoustic tubes, and linear membranes and plates. The final chapters cover more complex methods including nonlinear plates, finite element methods and spectral methods. Bilbao's concluding remarks look to the future direction of the technology, musical applications, and the potential strengths and weaknesses of the methods. Original approach to the modelling and synthesis of sound, focusing on the use of numerical methods and algorithms for the simulation of music Presents numerical methods and algorithms for the synthesis of digital sound, enabling more realistic and complex simulated sound reproduction Focuses primarily on the use of established finite difference methods, as well as exploring newer, emerging techniques such as spectral methods and modal synthesis Provides practical worked through problems and programming exercises, plus MATLAB code examples so that readers can easily implement the methods discussed in the book

Author Biography

Stefan Bilbao, School of Arts, Culture and Environment, University of Edinburgh, UK
Dr Stefan Bilbao is currently a Lecturer of Music at the University of Edinburgh, specializing in sound synthesis based on physical models of musical instruments, with a particular focus on the intersection between digital filtering techniques and numerical simulation methods. He has been involved in computer music research since 1992, working firstly at the Institut de Recherche et Coordination Acoustique Musicale (IRCAM) under a joint fellowship with Harvard University, then at the Space, Telecommunications and Radioscience Laboratory (STAR Lab). Dr Bilbao then moved UK to take a post as a Lecturer at the Sonic Arts Research Centre (SARC) at the Queens University, Belfast, and in 2005, at the School of Arts, Culture and Environment at the University of Edinburgh. He is now undertaking a project supported by the Engineering and Physical Sciences Research Council UK for the development of new numerical techniques for sound synthesis. He teaches both undergraduate and postgraduate courses on musical acoustics, DSP, and musical applications of Fourier theory and has written over 10 published journal articles, 30 conference papers and the book Wave and Scattering Methods for Numerical Simulation for Wiley, published in 2004.

Table of Contents

Preface	p. ix
Sound synthesis and physical modeling	p. 1
Abstract digital sound synthesis	p. 2
Physical modeling	p. 8
Physical modeling: a larger view	p. 18
Time series and difference operators	p. 25
Time series	p. 26
Shift, difference, and averaging operators	p. 27
Frequency domain analysis	p. 31
Energetic manipulations and identities	p. 38
Problems	p. 42
The oscillator	p. 45
The simple harmonic oscillator	p. 46
A finite difference scheme	p. 49
Other schemes	p. 55
Lumped mass-spring networks	p. 61
Loss	p. 63
Sources	p. 67
Problems	p. 68
Programming exercises	p. 71
The oscillator in musical acoustics	p. 73
Nonlinear oscillators	p. 74
Lossless oscillators	p. 74
Lossy oscillators	p. 82
Problems	p. 87
Programming exercises	p. 90
Grid functions and finite difference operators in ID	p. 93
Partial differential operators and PDEs	p. 93
Grid functions and difference operators	p. 98
Coordinate changes	p. 112
Problems	p. 113
Programming exercises	p. 115
The ID wave equation	p. 117
Definition and properties	p. 118
A simple finite difference scheme	p. 131
Other schemes	p. 148
Modal synthesis	p. 152
Loss	p. 153
Comparative study I	p. 155
Problems	p. 157
Programming exercises	p. 161
Linear bar and string vibration	p. 163
The ideal uniform bar	p. 163
Stiff strings	p. 174
Frequency-dependent loss	p. 177
Coupling with bow models	p. 180
Coupling with hammer and mallet models	p. 183
Multiple strings	p. 185
Prepared strings	p. 187
Coupled bars	p. 194
Helical springs	p. 199
Spatial variation and stretched coordinates	p. 206
Problems	p. 213
Programming exercises	p. 218
Nonlinear string vibration	p. 221
The Kirchhoff-Carrier string model	p. 221
General planar nonlinear string motion	p. 232
Non-planar string motion	p. 242
Problems	p. 244
Programming exercises	p. 247
Acoustic tubes 249
Webster's equation	p. 249
The vocal tract and speech synthesis	p. 258
Reed wind instruments	p. 265
Other wind instruments	p. 278
Problems	p. 278
Programming exercises	p. 283
Grid functions and finite difference operators in 2D	p. 287
Partial differential operators and PDEs in two space variables	p. 288
Grid functions and difference operators: Cartesian coordinates	p. 291
Grid functions and difference operators: radial coordinates	p. 299
Problems	p. 301
Programming exercises	p. 303
The 2D wave equation	p. 305
Definition and properties	p. 305
A simple finite difference scheme	p. 310
Other finite difference schemes	p. 312
Digital waveguide meshes	p. 315
Lumped mass-spring networks	p. 316
Modal synthesis	p. 317
Finite difference schemes in radial coordinates	p. 318
Comparative study II	p. 321
Problems	p. 322
Programming exercises	p. 327
Linear plate vibration	p. 331
The Kirchhoff thin plate model	p. 331
Loss and tension	p. 341
Plate excitation	p. 341
Plate-string connections	p. 346
Anisotropic plates	p. 349
The thin plate in radial coordinates 352
Problems	p. 355
Programming exercises	p. 357
Nonlinear plate vibration	p. 361
The Berger plate model	p. 361
The von Kármán plate model	p. 363
Spherical shell vibration	p. 371
Problems	p. 376
Programming exercises	p. 378
Conclusion and perspectives	p. 379
A family of musical systems	p. 379
Comparative study III	p. 382
Beyond finite difference methods	p. 386
Matlab code examples	p. 391
The simple harmonic oscillator	p. 391
Hammer collision with mass-spring system	p. 392
Bowed mass-spring system	p. 393
The ID wave equation: finite difference scheme	p. 394
The ID wave equation: digital waveguide synthesis	p. 395
The ID wave equation: modal synthesis	p. 397
The ideal bar	p. 398
The stiff string	p. 399
The Kirchhotf-Carrier equation	p. 401
Vocal synthesis	p. 402
The 2D wave equation	p. 403
Thin plate	p. 405
List of symbols	p. 407
Bibliography	p. 411
Index	p. 427
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