Sonic black holes (SBHs) are waveguides intended to slow down the wave propagation speed and focus the energy towards the end of the device. However, the extent to which these effects occur, as well as the degree of wave dispersion introduced, has not been systematically quantified. This article investigates these aspects through transient finite-element computations, analyzing the properties of a novel, numerically optimized SBH with enhanced wave-focusing capabilities. The investigation utilizes the lossless acoustic wave equation as well as a linearized compressible flow formulation to account for viscothermal losses. We analyze the wave focusing and filtering properties of the SBH by monitoring the pressure amplitude and the transmission and reflection coefficients. Moreover, we examine the effective wave propagation speed along the centerline of SBH and assess the similarity of pressure wave packets using cross-correlations. Our results reveal that the optimized SBH not only enhances wave focusing but also on average effectively slows down wave propagation, demonstrating the device's potential as a true sonic black hole. By investigating two crucial aspects - wave-slowing effect and signal dispersion - that were not previously explored, we provide a deeper understanding of the device's functionality and operational mechanisms, including how its design influences wave-focusing performance and local wave speed.