There is hope to discover dark matter subhalos free of stars (predicted by the current theory of structure formation) by observing gaps they produce in tidal streams. In fact, this is the most promising technique for dark substructure detection and characterization as such gaps grow with time, magnifying small perturbations into clear signatures observable by ongoing and planned Galaxy surveys. To facilitate such future inference, we develop a comprehensive framework for studies of the growth of the stream density perturbations. Starting with simple assumptions and restricting to streams on circular orbits, we derive analytic formulae that describe the evolution of all gap properties (size, density contrast etc) at all times. We uncover complex, previously unnoticed behavior, with the stream initially forming a density enhancement near the subhalo impact point. Shortly after, a gap forms due to the relative change in period induced by the subhalo's passage. There is an intermediate regime where the gap grows linearly in time. At late times, the particles in the stream overtake each other, forming caustics, and the gap grows like $t$. In addition to the secular growth, we find that the gap oscillates as it grows due to epicyclic motion. We compare this analytic model to N-body simulations and find an impressive level of agreement. Importantly, when analyzing the observation of a single gap we find a large degeneracy between the subhalo mass, the impact geometry and kinematics, the host potential and the time since flyby.