FDS for the Ideal String

Consider first the ideal (lossless, dispersionless, unforced) string wave equation:

$$\frac{\partial^{2}y}{\partial t^{2}} = c^{2}\frac{\partial^{2}y}{\partial x^{2}}$$

Replacing the second-derivatives by their finite-difference approximations gives

$$\begin{eqnarray*} y_{m}^{n-1}-2y_{m}^{n}+y_{m}^{n+1} = \frac{c^{2}T^{2}}{X^{2}} (y_{m-1}^{n}-2y_{m}^{n}+y_{m+1}^{n}) \end{eqnarray*}$$

If we choose $X = cT$ (which is most natural physically), the equation reduces further to

$$\begin{eqnarray*} y_{m}^{n+1} = y_{m-1}^{n}+y_{m+1}^{n}-y_{m}^{n-1} \end{eqnarray*}$$

Let's examine this recursion on the time-space grid, assuming for the moment no boundary conditions: