The Finite Difference Approximation

In the musical acoustics literature, the normal method for creating a computational model from a differential equation is to apply the so-called finite difference approximation (FDA) in which differentiation is replaced by a finite difference (see Appendix N) [495,319]. For example

and

where is the time sampling interval to be used in the simulation, and is a spatial sampling interval. These approximations can be seen as arising directly from the definitions of the partial derivatives with respect to and . The approximations become exact in the limit as and approach zero. To avoid a delay error, the second-order finite-differences are defined with a compensating time shift:

The odd-order derivative approximations suffer a half-sample delay error while all even order cases can be compensated as above.

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Subsections

• FDA of the Ideal String

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.