It's pretty easy to write up what blurring means as a partial differential equation:

du / dt = k * d

or in the terse form preferred by PDE folks:

u

This says that to blur any function, the peaks (negative second derivative) go down, and the valleys (positive second derivative) come up.

It's also pretty clear that lots of interesting stuff, like waves, aren't blurring. But we can get from blurring to waves by modifying the blurring PDE to second-order in time:

u

You can derive this in several ways, including directly from Hooke's spring law.

There are quite a few different ways to impliment the wave equation--the most obvious is to just compute the second derivative, and set it equal to the acceleration of each point (the second derivative of position).