Numerical Sound Synthesis: Finite Difference Schemes and Simulation in Musical Acoustics
Digital sound synthesis has long been approached using standard digital filtering techniques. Newer synthesis strategies, however, make use of physical descriptions of musical instruments, and allow for much more realistic and complex sound production and thereby synthesis becomes a problem of simulation. This book has a special focus on time domain finite difference methods presented within an audio framework. It covers time series and difference operators, and basic tools for the construction and analysis of finite difference schemes, including frequency-domain and energy-based methods, with special attention paid to problems inherent to sound synthesis. Various basic lumped systems and excitation mechanisms are covered, followed by a look at the 1D wave equation, linear bar and string vibration, acoustic tube modelling, and linear membrane and plate vibration. Various advanced topics, such as the nonlinear vibration of strings and plates, are given an elaborate treatment. Key features:* Includes a historical overview of digital sound synthesis techniques, highlighting the links between the various physical modelling methodologies.* A pedagogical presentation containing over 150 problems and programming exercises, and numerous figures and diagrams, and code fragments in the MATLAB(r) programming language helps the reader with limited experience of numerical methods reach an understanding of this subject.* Offers a complete treatment of all of the major families of musical instruments, including certain audio effects. Numerical Sound Synthesis is suitable for audio and software engineers, and researchers in digital audio, sound synthesis and more general musical acoustics. Graduate students in electrical engineering, mechanical engineering or computer science, working on the more technical side of digital audio and sound synthesis, will also find this book of interest.
Why Read This Book
You will learn how to move beyond filter-based sound synthesis to physically informed, time-domain simulation methods that produce highly realistic musical sounds. The book gives a rigorous but application-minded treatment of finite-difference time-domain (FDTD) schemes, stability and energy analysis, and practical implementation details tailored to audio and musical acoustics.
Who Will Benefit
Engineers, researchers, and advanced hobbyists with a background in DSP or acoustics who want to build accurate, physically based sound synthesis systems or prototype instrument simulations.
Level: Advanced — Prerequisites: Undergraduate calculus and differential equations, linear algebra, basic digital signal processing (z-transform, FFT), and some programming experience (MATLAB/C/Python recommended).
Key Takeaways
- Implement finite-difference time-domain (FDTD) schemes for 1D and 2D wave equations relevant to strings, bars, and plates.
- Analyze and ensure numerical stability, consistency, dispersion, and energy conservation of time-domain schemes.
- Design and simulate realistic excitation and boundary conditions for musical instruments and couple exciter–resonator interactions.
- Apply frequency-domain and energy-based methods to predict and control modal behaviour and spectral content.
- Optimize and discretize damping, nonlinearities, and loss mechanisms for perceptually convincing audio synthesis.
- Integrate numerical simulations into real-time audio workflows and validate results with spectral analysis and listening tests.
Topics Covered
- Introduction: Physical Modeling and Time-Domain Simulation
- Time Series, Difference Operators, and Discretization Basics
- Stability, Consistency, and Dispersion Analysis
- Energy Methods and Numerical Conservation
- Lumped Systems and Basic Excitation Models
- The 1D Wave Equation: Strings and Wave Propagation
- Bars and Beams: Higher-Order Spatial Operators
- Plates and 2D Wave Systems
- Boundary Conditions, Coupling, and Connected Systems
- Damping, Losses, and Nonlinear Effects
- Spectral Analysis, FFTs, and Validation of Simulations
- Implementation Considerations and Real-Time Issues
- Examples, Case Studies, and Perceptual Considerations
- Appendices: Numerical Tools and Reference Material
Languages, Platforms & Tools
How It Compares
Compared with Julius O. Smith's Physical Audio Signal Processing, Bilbao focuses more on finite-difference PDE methods and energy-based numerical analysis, while Smith emphasizes digital waveguides and filter-based physical modeling; The Computer Music Tutorial is broader and less mathematically rigorous.
