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]:
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:
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 .
Digital Waveguide (DW) Scheme
Convergence in Audio Applications