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Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       Introduction
          Finite Difference Time Domain (FDTD) Scheme

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Finite Difference Time Domain (FDTD) Scheme

As discussed in §H.2, we may use centered finite difference approximations (FDA) for the second-order partial derivatives in the to obtain a finite difference scheme for the ideal wave equation [495,319]:

$\displaystyle {\ddot y}(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2}$ (P.1)
$\displaystyle y''(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2} \protect$ (P.2)

where $ T$ is the time sampling interval, and $ X$ is a spatial sampling interval.

Substituting the FDA into the wave equation, choosing $ X=cT$, where $ c \isdeftext \sqrt{K/\epsilon }$ is sound speed (normalized to $ c=1$ below), and sampling at times $ t=nT$ and positions $ x=mX$, we obtain the following explicit finite difference scheme for the string displacement: