Analytical solution of black hole equation and some consequences

A complete set of fundamental solutions of the bending oscillating rod equation, sharpened by parabolic law, is suggested. Cutoff frequency, from which and above, this parabolically sharpened rod becomes a black hole, capable to absorb vibrational energy, is defined. These solutions are used to calculate the elements of the input matrix impedance of this rod. For an ideal black hole, corresponding to the sharpening rod strictly to zero value, the impedance is determined by use only two fundamental solutions. For a real black hole, corresponding to the cutting of the sharpening at a finite (non-zero) thickness, the impedance is determined by all four fundamental solutions. Spectrum of eigenfrequencies of a real black hole is calculated. A modified WKB approximation solutions of a rod equation with arbitrary smoothly varying cross-section is proposed.