A Bose condensate, subject to periodic modulation of the two-body interactions, was recently observed to emit matter-wave jets resembling fireworks [Nature (London) 551, 356 (2017)NATUAS0028-083610.1038/nature24272]. In this Letter, combining experiment with numerical simulation, we demonstrate that these "Bose fireworks" represent a late stage in a complex time evolution of the driven condensate. We identify a "density wave" stage which precedes jet emission and results from the interference of matter waves. The density waves self-organize and self-amplify without breaking long range translational symmetry. This density wave structure deterministically establishes the template for the subsequent patterns of the emitted jets. Moreover, our simulations, in good agreement with experiment, address an apparent asymmetry in the jet pattern, and show that it is fully consistent with momentum conservation.