Black body radiation problem

Question: suppose one starts out with a cavity of size L at temperature T and at first there is no electromagnetic radiation in the cavity. How long will it take for the radiation in the cavity to come into equilibrium with the walls of the cavity? Assume that the walls of the cavity are a black body.

Does the answer depend on the frequency? That is, is it possible that some modes are already described by Planck's formula, while others still have very little energy compared to that predicted by Planck's formula?