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 . The FDA approach to numerical simulation was used by Pierre Ruiz in his work on vibrating strings , 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
while the fully general linear, time-invariant 2D case is
A nonlinear example is
and a time-varying example can be given by
Traveling-Wave Partial Derivatives