It is interesting, but it's a standard way of demonstrating normal modes, or the applications of eigenvectors. I would think most physics and engineering undergraduates have studied this. That's not to say you shouldn't watch it, but just to put it in perspective - this is a topic studied in the 1st/2nd year of university, not a poorly-understood topic for research. https://en.wikipedia.org/wiki/Normal_mode
Forgive my ignorance, but I don't have a physics or engineering background. Given it's a well understood phenomenon, is there some way of generating mathematically what the patterns will be for a given plate shape and frequency?
You solve the wave equation, a differential equation for the small-amplitude motion of the surface. The solutions depend strongly on the shape of the boundary, as is usually the case. For some shapes the patterns can be found analytically. For more complicated boundaries it's not too hard to solve this equation numerically - it's linear. Orders of magnitude easier than, say, calculating the flow of water down a pipe.
