### 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
[442]. The FDA approach to numerical simulation
was used by Pierre Ruiz in his work on vibrating strings [392],
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

(C.7) |

while the fully general linear, time-invariant 2D case is

(C.8) |

A nonlinear example is

(C.9) |

and a time-varying example can be given by

(C.10) |

**Next Section:**

Traveling-Wave Partial Derivatives

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3D Sound