The Essence of “Black Holes” for Elastic Waves in Solids with Cuspidal Sharpening

Abstract

The effect of a “black hole” for elastic waves, discovered by M.A. Mironov and examined in detail by followers, is usually associated with propagation of elastic waves along the cuspidal sharpening of a deformable solid, i.e., the cusp absorbs the energy of elastic oscillations and does not return it to a massive part of a body. At the same time, an ideal cusp cannot be made and its tip is blunted in real constructions. Smoothing of the sharpening crucially changes the spectrum structure: the continuous component disappears, but the concentration of eigenfrequencies in the mid-frequency range occurs. This note provides asymptotic formulas for eigenvalues of the Kirchhoff beam with a thinning end and on their basis describes a new and effectual mechanism of action of the “black hole,” namely, a blunted cusp, in which propagation of waves becomes impossible, but trapping of waves occurs at “almost all frequencies” within a wide enough spectral range. Improvement of the cusp’s quality leads to enhancemen of the concentration of eigenvalues and enlarging of its region of occurrence.