The curved acoustic black hole (ABH), a thin structure with a curved baseline and a thickness that tapers according to a power law, has been proposed and investigated numerically and experimentally as a candidate for an effective and compact absorber of flexural waves. To understand physically the wave motions within a curved ABH and to utilize a spiral ABH in practical applications, systematic investigations of the effects of geometrical parameters such as the curvature or the damping-layer thickness are needed, as well as a theoretical approach to facilitate fast and accurate calculations. In this paper, we propose a wave-based theoretical method and investigate the “cut-on frequency” of a curved ABH, the frequency from which waves start to propagate within the ABH, using this method. The governing equation of the wave motion is derived using Hamilton's principle, and the solution of the governing equation is obtained numerically by recasting the original equation into impedance-equation form. Using the proposed method, we calculate the reflection coefficients of arc ABHs and Archimedean spiral ABHs with various geometrical parameters and frequencies to investigate their cut-on frequencies. The cut-on frequency increases almost linearly in proportion to the curvature in arc ABHs, and decreases with increasing gap distance in spiral ABHs.