Buckling

A (finite approximation for a) buckling column has difference
equation
$\Delta^2 a_n = -\lambda a_n$
(this is the convexity at $x$, so $a_{n+1}-2 a_n + a_{n-1}$)

This is just the discrete $x''=-\lambda x$
with free end values,
so the solutions are cosine waves
(obvious similar pictures apply if the bottom is considered fixed, or
whatever; similarly for musical instruments and so forth;
this is just an amusing discrete example)

BTW, these yield tridiagonal matrices;
the idea is that tridiagonal matrices reflect a kinda difference
equation on a line.
