  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 $\sqrt{t}$. 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.
