FDA of the Ideal String

Substituting the FDA into the wave equation gives

$$K\frac{y(t,x+X) - 2 y(t,x) + y(t,x-X)}{X^2} = \epsilon \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x)}{T^2}$$

which can be solved to yield the following recursion for the string displacement:

$$y(t+T,x) = \frac{KT^2}{\epsilon X^2} \left[ y(t,x+X) - 2 y(t,x) + y(t,x-X)\right]$$

$$\qquad\qquad\qquad\qquad + 2 y(t,x) - y(t-T,x) \protect$$

In a practical implementation, it is common to set $T=1,\, X=(\sqrt{K/\epsilon })T$ , and evaluate on the integers $t=nT=n$ and $x=mX=m$ to obtain the difference equation

$$y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m). \protect$$ (C.6)

Thus, to update the sampled string displacement, past values are needed for each point along the string at time instants $n$ and $n-1$ . Then the above recursion can be carried out for time $n+1$ by iterating over all $m$ 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 [445]. The FDA approach to numerical simulation was used by Pierre Ruiz in his work on vibrating strings [395], 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 $y(n-l,m-k)$ 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

$$\sum_{k=0}^\infty \alpha_k \frac{\partial^k y(t,x)}{\partial t^k} = \sum_{l=0}^\infty \beta_l \frac{\partial^l y(t,x)}{\partial x^l},$$ (C.7)

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

$$\sum_{k=0}^\infty \sum_{l=0}^\infty \alpha_{k,l} \frac{\partial^k\partial^l y(t,x)}{\partial t^k \partial x^l} = \sum_{m=0}^\infty \sum_{n=0}^\infty \beta_{m,n} \frac{\partial^m\partial^n y(t,x)}{\partial x^m \partial x^n}.$$ (C.8)

A nonlinear example is

$$\frac{\partial y(t,x)}{\partial t} = \left(\frac{\partial y(t,x)}{\partial x}\right)^2,$$ (C.9)

and a time-varying example can be given by

$$\frac{\partial y(t,x)}{\partial t} = t^2\frac{\partial y(t,x)}{\partial x}.$$ (C.10)