A film clip would be helpful here. A few days ago I noticed a swarm of newly hatched flying bugs about 20 feet away from me with the sun behind them so the shape of swarming (such as it was) was easy to make out. Several times, after creating various spiral shapes or chaotic masses they returned to a sort of 'holding pattern' which I either never noticed or never saw before. To a pretty good approximation the bugs arranged themselves in what seemed like an "oscillating drum head" pattern that is well approximated by a rotated sinc function. The peak at zero is a little exaggerated here but the bugs were more or less dispersed on a shape like:
subject to a light breeze and continuous adjustment. I know that people have studied swarming behavior and made models but are there any that account for this one? It has a lot of significance in ODEs and engineering but I never associated it with swarming behavior.