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 D) [481,311]. For example
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.
FDA of the Ideal String
Substituting the FDA into the wave equation gives
In a practical implementation, it is common to set , and evaluate on the integers and to obtain the difference equation
Thus, to update the sampled string displacement, past values are needed for each point along the string at time instants and . Then the above recursion can be carried out for time by iterating over all along the string.
Perhaps surprisingly, it is shown in Appendix E that the above recursion is exact at the sample points in spite of the apparent crudeness of the finite difference approximation . The FDA approach to numerical simulation was used by Pierre Ruiz in his work on vibrating strings , and it is still in use today [74,75].
When more terms are added to the wave equation, corresponding to complex losses and dispersion characteristics, more terms of the form appear in (C.6). These higher-order terms correspond to frequency-dependent losses and/or dispersion characteristics in the FDA. All linear differential equations with constant coefficients give rise to some linear, time-invariant discrete-time system via the FDA. A general subclass of the linear, time-invariant case giving rise to ``filtered waveguides'' is
while the fully general linear, time-invariant 2D case is
A nonlinear example is
and a time-varying example can be given by
The Ideal Vibrating String