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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 L)
[458,292]. For example
|
(G.2) |
and
|
(G.3) |
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:
|
(G.4) |
|
(G.5) |
The odd-order derivative approximations suffer a half-sample delay error
while all even order cases can be compensated as above.
Subsections
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