Substituting the FDA into the wave equation gives
which can be solved to yield the following recursion for the string displacement:
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