### Finite Difference Time Domain (FDTD) Scheme

As discussed in §C.2, we may use centered finite difference approximations (FDA) for the second-order partial derivatives in the wave equation to obtain a finite difference scheme for numerically integrating the ideal wave equation [481,311]:

 (E.1) (E.2)

where is the time sampling interval, and is a spatial sampling interval.

Substituting the FDA into the wave equation, choosing , where is sound speed (normalized to below), and sampling at times and positions , we obtain the following explicit finite difference scheme for the string displacement:

 (E.3)

where the sampling intervals and have been normalized to 1. To initialize the recursion at time , past values are needed for all (all points along the string) at time instants and . Then the string position may be computed for all by Eq.(E.3) for . This has been called the FDTD or leapfrog finite difference scheme [127].

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Digital Waveguide (DW) Scheme
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Convergence in Audio Applications
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