As Recursion

By expanding out the behaviour of the operator $\delta _{tt}$, and reintroducing the time index $n$, one loses compactness of representation, but has a clearer picture of how to program such a scheme on a computer. One has

$$\frac{1}{k^2}\left(u^{n+1}-2u^{n}+u^{n-1}\right) = -\omega_{0}^2 u^n$$ (3.19)

which involves values of the time series at three levels $n+1$, $n$, and $n-1$. Rewriting (3.19) so that $u^{n+1}$ is isolated gives

$$u^{n+1} = \left(2-\omega_{0}^2 k^2\right) u^n-u^{n-1}$$ (3.20)

which is a two-step recursion in the time series $u^{n}$. This difference equation is identical in form to a two-pole digital filter under zero-input conditions.