# Finite-Difference Schemes in Musical Acoustics: A Tutorial

• Stefan Bilbao
• Brian Hamilton
• Reginald Harrison
• Alberto Torin
Chapter
Part of the Springer Handbooks book series (SHB)

## Abstract

The functioning of musical instruments is well described by systems of partial differential equations. Whether one's interest is in pure musical acoustics or physical modeling of sound synthesis, numerical simulation is a necessary tool, and may be carried out by a variety of means. One approach is to make use of so-called finite-difference or finite-difference time-domain methods, whereby the numerical solution is computed as a recursion operating over a grid. This chapter is intended as a basic tutorial on the design and implementation of such methods, for a variety of simple systems. The 1-D wave equation and simple difference schemes are covered in Sect. 19.1, accompanied by an analysis of numerical dispersion and stability, as well as implementation details via vector-matrix representations. Similar treatments follow for the case of the ideal stiff bar in Sect. 19.2, the acoustic tube in Sect. 19.3, the 2-D and 3-D wave equations in Sect. 19.4, and finally the stiff plate in Sect. 19.5. Some more general nontechnical comments on more complex extensions to nonlinear systems appear in Sect. 19.6.

1-D

one-dimensional

2-D

two-dimensional

3-D

three-dimensional

FDTD

finite-difference time domain

GKSO

Gustafsson Kreiss Sundstrom Osher

PDE

partial differential equation

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## Authors and Affiliations

• Stefan Bilbao
• 1
Email author
• Brian Hamilton
• 1
• Reginald Harrison
• 1
• Alberto Torin
• 1
1. 1.Acoustics and Audio GroupUniversity of EdinburghEdinburghUK