Quasibound states of analytic black-hole configurations in three and four dimensions

In this work we analyze the sound perturbation of Unruh's acoustic effective geometry in both (2 + 1) , (3 + 1) spacetime dimensions and present an exact analytical expression for the quasibound states of these idealized black-hole configurations by using a new approach recently developed, which uses the polynomial conditions of the hypergeometric functions. Our main goal is to discuss the effects of having an event horizon in such effective metrics. We also discuss the stability of the systems and present the radial eigenfunctions related to these quasibound state frequencies. These metrics assume just the form it has for a Schwarzschild black hole near the event horizon , therefore may, in principle, shed some light into the underlying classical and quantum physics of astrophysical black holes through analog acoustic probes.