These are classic standing wave patterns. Now, Garble, to answer your question, the reason that the wave patterns are not perfect circles is that the vibrating medium is not a circle, but a square. If the experiment had been done with a circular flat disc, many of the patterns would have resembled bulls-eye targets. These patterns are actually visible representations of solutions to complicated sets of differential equations, which exist for practically any shape.

Standing waves in circular membranes have been studied extensively, because

(a) they are a relatively simple case, and
(b) they have practical application in the design of drums i.e. controlling the pitch and getting "good sound".