Numerical Sound Synthesis: Finite Difference Schemes and Simulation in Musical Acoustics

1 Sound synthesis and physical modeling.

1.1 Abstract digital sound synthesis.

1.2 Physical modeling.

1.3 Physical modeling: a larger view.

2 Time series and difference operators.

2.1 Time series.

2.2 Shift, difference, and averaging operators.

2.3 Frequency domain analysis.

2.4 Energetic manipulations and identities.

2.5 Problems.

3 The oscillator.

3.1 The simple harmonic oscillator.

3.2 A finite difference scheme.

3.3 Other schemes.

3.4 Lumped mass–spring networks.

3.5 Loss.

3.6 Sources.

3.7 Problems.

3.8 Programming exercises.

4 The oscillator in musical acoustics.

4.1 Nonlinear oscillators.

4.2 Lossless oscillators.

4.3 Lossy oscillators.

4.4 Problems.

4.5 Programming exercises.

5 Grid functions and finite difference operators in 1D.

5.1 Partial differential operators and PDEs.

5.2 Grid functions and difference operators.

5.3 Coordinate changes.

5.4 Problems.

5.5 Programming exercises.

6 The 1D wave equation.

6.1 Definition and properties.

6.2 A simple finite difference scheme.

6.3 Other schemes.

6.4 Modal synthesis.

6.5 Loss.

6.6 Comparative study I.

6.7 Problems.

6.8 Programming exercises.

7 Linear bar and string vibration.

7.1 The ideal uniform bar.

7.2 Stiff strings.

7.3 Frequency-dependent loss.

7.4 Coupling with bow models.

7.5 Coupling with hammer and mallet models.

7.6 Multiple strings.

7.7 Prepared strings.

7.8 Coupled bars.

7.9 Helical springs.

7.10 Spatial variation and stretched coordinates.

7.11 Problems.

7.12 Programming exercises.

8 Nonlinear string vibration.

8.1 The Kirchhoff–Carrier string model.

8.2 General planar nonlinear string motion.

8.3 Non-planar string motion.

8.4 Problems.

8.5 Programming exercises.

9 Acoustic tubes.

9.1 Webster’s equation.

9.2 The vocal tract and speech synthesis.

9.3 Reed wind instruments.

9.4 Other wind instruments.

9.5 Problems.

9.6 Programming exercises.

10 Grid functions and finite difference operators in 2D.

10.1 Partial differential operators and PDEs in two space variables.

10.2 Grid functions and difference operators: Cartesian coordinates.

10.3 Grid functions and difference operators: radial coordinates.

10.4 Problems.

10.5 Programming exercises.

11 The 2D wave equation.

11.1 Definition and properties.

11.2 A simple finite difference scheme.

11.3 Other finite difference schemes.

11.4 Digital waveguide meshes.

11.5 Lumped mass–spring networks.

11.6 Modal synthesis.

11.7 Finite difference schemes in radial coordinates.

11.8 Comparative study II.

11.9 Problems.

11.10 Programming exercises.

12 Linear plate vibration.

12.1 The Kirchhoff thin plate model.

12.2 Loss and tension.

12.3 Plate excitation.

12.4 Plate–string connections.

12.5 Anisotropic plates.

12.6 The thin plate in radial coordinates.

12.7 Problems.

12.8 Programming exercises.

13 Nonlinear plate vibration.

13.1 The Berger plate model.

13.2 The von Kármán plate model.

13.3 Spherical shell vibration.

13.4 Problems.

13.5 Programming exercises.

14 Conclusion and perspectives.

14.1 A family of musical systems.

14.2 Comparative study III.

14.3 Beyond finite difference methods.

A Matlab code examples.

A.1 The simple harmonic oscillator.

A.2 Hammer collision with mass–spring system.

A.3 Bowed mass–spring system.

A.4 The 1D wave equation: finite difference scheme.

A.5 The 1D wave equation: digital waveguide synthesis.

A.6 The 1D wave equation: modal synthesis.

A.7 The ideal bar.

A.8 The stiff string.

A.9 The Kirchhoff–Carrier equation.

A.10 Vocal synthesis.

A.11 The 2D wave equation.

A.12 Thin plate.

B List of symbols.

Bibliography.

Index.