Elements of numerical analysis with application to sound synthesis

Antoine Chaigne $<$achaigne at ccrma$>$ (CCRMA Visiting Scholar, Professor, ENSTA and Ecole polytechnique, France)

This presentation is essentially a tutorial on basic concepts related to the use and properties of numerical analysis methods in the context of sound synthesis. The main results are illustrated by simple examples, such as piano strings, xylophone bars, and so on.

The physical behavior of musical instruments is described by continuous equations. These equations include derivatives and integrals vs. space and time. In order to solve these equations it is necessary to approximate the model by discrete formulations. Several strategies are possible: finite differences, finite elements, modal truncation, and others. In each case, fundamental questions arise: which spatial/time step should be selected? What are the consequences of these choices? How to ensure the stability of the model? Is it possible to predict the accuracy of a given method? The lecture will focus on the basic problems of stability and numerical dispersion, in the case of simple finite difference modeling of string and bar equations. Analogies and differences with respect to other methods will be discussed. It will be shown, in particular, that the well-known Courant-Friedrich-Levy (or CFL) condition for a second-order explicit finite difference scheme is equivalent to the Shannon (or Nyquist) sampling theorem in signal processing. Fruitful discussion on the comparison between waveguide modeling and finite difference modeling will certainly be of great interest.