Contents

• Sound Synthesis and Physical Modeling
  • Abstract Digital Sound Synthesis
    • Additive Synthesis
    • Subtractive Synthesis
    • Wavetable Synthesis
    • AM and FM Synthesis
    • Other Methods
  • Physical Modeling
    • Lumped Mass-Spring Networks
    • Modal Synthesis
    • Digital Waveguides
    • Hybrid Methods
    • Direct Numerical Simulation
  • Physical Modeling: A Larger View
    • Physical Models as Descended from Abstract Synthesis
    • Connections: Direct Simulation and Other Methods
    • Complexity of Musical Systems
    • Why?
  
  
• Time Series and Difference Operators
  • Finite Differences
    • Time Series
    • Difference and Averaging Operators
    • Frequency Domain Interpretation
  • Problems
  • Programming Exercises
  
  
• The Oscillator
  • The Simple Harmonic Oscillator
    • Sinusoidal Solution
    • Energy
    • As First-Order System
    • Coupled Systems of Oscillators
  • A Simple Scheme
    • As Recursion
    • Initialization
    • Frequency Domain Analysis
    • Accuracy
    • Energy Analysis
    • Computational Considerations
  • Other Schemes
    • Using Time-averaging Operators
    • A Second-order Family of Schemes
    • Hughes-Taylor and Newmark Methods
    • An Exact Solution
    • Further Methods
  • Lumped Mass-Spring Networks
  • Loss
    • Finite Difference Scheme
    • Numerical Decay Time
    • An Exact Solution
  • Problems
  • Programming Exercises
  
  
• The Oscillator in Musical Acoustics
  • Nonlinear Oscillators of Series Type
    • Difference Schemes
  • The Hammer
  • The Reed
  • The Glottis
  • Lip Models
  • The Bow
  • General Collisions
  • Problems
  • Programming Exercises
  
  
• Grid Functions and Finite Difference Operators
  • Grid Functions and Difference Operators in One Dimension
    • Grid Functions
    • Time Difference and Averaging Operators
    • Spatial Difference Operators
    • Mixed Spatial-Temporal Difference Operators
    • Interpolation Operators
    • Frequency Domain Interpretation
    • Matrix Interpretation of Difference Operators
    • Inner Products and Identities
    • Energetic Identities, Manipulations and Inequalities
  • Grid Functions and Difference Operators in Two Dimensions: Cartesian Coordinates
    • Stencil Width
    • Frequency Domain Analysis
    • Anisotropic Behaviour
    • An Inner Product and Energetic Manipulations
  • Grid Functions and Difference Operators in Two Dimensions: Radial Coordinates
  • Problems
  • Programming Exercises
  
  
• The 1D Wave Equation
  • Definition
    • Non-dimensionalized Form
    • Initial Conditions
    • Dispersion Relation
    • Phase and Group Velocity
    • Energy Analysis
    • Boundary Conditions
    • Bounds on Solution Size
    • Modes
    • Characteristics and Travelling Wave Solutions
  • A Simple Finite Difference Scheme
    • Initialization
    • von Neumann Analysis
    • Numerical Dispersion
    • Energy Analysis
    • Numerical Boundary Conditions
    • Bounds on Solution Size and Numerical Stability
    • Matrix Form
    • Output and Interpolation
    • Digital Waveguide Interpretation
  • Other Schemes
    • A Stencil-width Five Scheme
    • A Compact Implicit Scheme
  • A Comparative Study: Physical Modeling Sound Synthesis Methods
    • Lumped Network Models
    • Modal Synthesis
    • Digital Waveguides
  • Problems
  • Programming Exercises
  
  
• Bars and Linear Strings
  • The Ideal Bar
    • Initial and Boundary Conditions
    • Frequency Domain Analysis
    • Energy Analysis
    • Finite Difference Schemes
  • Stiff Strings
    • Frequency Domain Analysis
    • Energy Analysis
    • Finite Difference Schemes
    • Matrix Interpretation
  • Hammer Models and Strings
  • Multiple Strings
  • Prepared Strings
  • Bars of Variable Cross-section
  • Problems
  • Programming Exercises
  
  
• Nonlinear Strings
  • The Kirchhoff-Carrier String Model
    • Pitch Glides
    • Energy Analysis
    • Finite Difference Schemes
    • A Quasi-modal Form
    • A Digital Waveguide Form
  • General Planar Nonlinear String Motion
    • Coupling Between Transverse and Longitudinal Modes and Phantom Partials
    • Finite Difference Schemes
  • Nonplanar String Motion
    • Finite Difference Schemes
    • Instability of Planar Motion and Whirling Behaviour
  • Problems
  • Programming Exercises
  
  
• Acoustic Tubes
  • Cylindrical Tubes
  • Conical Tubes
  • Tonehole Models
  • Coupling with Reed Models
  • Webster's Equation and the Vocal Tract
    • The Kelly-Lochbaum Speech Synthesis Model
    • Coupling with a Glottis Model
  • Problems
  • Programming Exercises
  
  
• Linear Membranes and Plates
  • The 2D Wave Equation
    • Finite Difference Schemes
    • Digital Waveguide Meshes
    • Lumped Networks
    • Modal Synthesis
  • The Kirchhoff Thin Plate Model
    • Finite Difference Schemes
    • Modal Synthesis
  • Realistic Plate Models
    • Finite Difference Schemes
  • Plate Reverberation
  • Soundboards
  • Problems
  • Programming Exercises
  
  
• Nonlinear Plates
  • The Berger Plate Model
    • Definition
    • Energy Analysis
    • A Finite Difference Scheme
    • Pitch Glides
  • The von Karman Plate Model
    • The Operator
    • Energy Analysis
    • A Finite Difference Scheme
  • Cymbals and Gongs
  • Problems
  • Programming Exercises
  
  
• Finite Element Methods
  • Variational Forms
  • Shape Functions
  • Grid Generation
  • The Stiff String Revisited
  • The Linear Plate Revisited
  • Musical Instrument Modeling
  • Problems
  • Programming Exercises
  
  
• Spectral Methods
  • Basis Functions
    • Fourier Series
    • Polynomials
  • Collocation Methods
  • Galerkin Methods
  • Spectral Derivatives
    • As Limiting Case of Finite Difference Approximations
    • Chebyshev Grids
    • Fast Fourier Transforms
  • High Order Time Stepping Methods
    • Forward and Backward Euler
    • Adams-Bashforth Methods
    • Adams-Moulton Methods
    • Runge-Kutta Methods
  • The Stiff String Again Revisited
  • The Linear Plate Again Revisited
  • Problems
  • Programming Exercises
  
  
• Concluding Remarks
  • Physical Modeling Sound Synthesis Methods: Strengths and Weaknesses
    • Modal Synthesis
    • Digital Waveguides
    • Lumped Mass Spring Networks
    • Direct Simulation
  • The Future
  
  
• MATLAB Code Examples
  • The Simple Harmonic Oscillator
  • The 1D Wave Equation: Finite Difference Scheme
  • The 1D Wave Equation: Modal Synthesis
  • The 1D Wave Equation: Digital Waveguide Synthesis
  • The Stiff String
  • A Tension-modulated String
  • Vocal Tract Synthesis
  • The 2D Wave Equation
  • Plate Reverberation
  • Gong
  
  
• Bibliography
• Index

Preface

(introductory remarks to be completed)