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We investigate adiabatic oscillations trapped near the inner edge of black hole accretion disks. We analyze the normal modes of oscillations (some of which are analogous to the $p$- and $g$-modes of stars) within thin accretion disks which are terminated by an innermost stable orbit. Another set of modes, the corrugation modes ($c$-modes), exhibit a strong dependence on the angular momentum of the hole. We develop a general formalism for investigating the adiabatic oscillations of arbitrary unperturbed disk models in the Kerr geometry. We obtain the lowest eigenfrequencies and eigenfunctions of various modes for various angular momenta of the black hole. These trapped modes do not exist in Newtonian gravity, and thus provide a new signature of black holes. Our predictions are related to past and future observations which could detect modulation in the X-rays from stellar mass black holes and in the optical and UV from supermassive AGN black holes.
