Strange Behavior of Natural Oscillations of an Elastic Body with a Blunted Peak

Abstract

The point of a peak on the surface of an elastic body Ω generates a continuous spectrum inducing wave processes in a finite volume (“black holes” for elastic waves). The spectrum of a body Ω^h with a blunted peak is discrete, but the normal eigenvalues take on “strange behavior” as the length h of the broken tip tends to zero. In different situations, eigenvalues are revealed that do not leave the small neighborhood of the fixed point or, conversely, fall off along the real axis with high velocity, but smoothly decrease to the lower limit of the continuous spectrum of the body Ω. The chaotic wandering of eigenvalues above the second limit may occur. A new way of forming the continuous spectrum of the body Ω with a peak from the family of discrete spectra of the bodies Ω^h with a blunted peak, h > 0, has been discovered.