An acoustic black hole (ABH) is a pit of power law profile inserted in a plate with internal damping. Such a pit with varying thickness leads to local variations of wave propagation properties and has been found to be an efficient vibration damper. In this paper, the ABH is seen as a penetrable obstacle and is studied through its scattering properties. A wave based model is developed within the framework of the Kirchhoff theory for a thin plate of locally varying thickness. It is shown that analytical solutions can be found in the case of a pit of quadratic profile and for uniform damping properties (without added layer). A major outcome of the model is the derivation of the dispersion relations giving a detailed analysis of the behavior of the waves within the ABH. Particularly, cut-on frequencies below which no wave propagates within the ABH are derived and thus give interpretation of the well known typical ABH efficiency threshold frequency for damping vibrations. The analysis is led from numerical computations of the dispersion curves, the scattering diagrams and the scattering cross-section, which are compared to the case of a simple hole. The results show that the ABH behaves as a resonant scatterer, which is a key outcome of this study. The so-called trapped modes, which describe free damped oscillations of the ABH, are responsible for variations of the scattering cross-section with frequency. These variations are investigated thanks to a parametric study on the geometrical properties of the ABH.