Abstract <p>The propagation of torsional waves through rods with a variable cross section is considered. When the flattening of a rod is linearly increased, the propagation velocity of a torsional wave also linearly decreases to become zero at a finite rod length. Meanwhile, the time of propagation to the sharpened end is equal to infinity. In contemporary terminology, such a decelerating structure is called a vibrational black hole. Exact solutions are found for the equations of torsional vibrations in a sharpened rod with an inertia moment and a torsion moment in the form of power functions along with corresponding expressions for the input impedance at the initial cross section.</p>