EFFECTS OF FREE-END BOUNDARY CONDITION ON THE FREQUENCY SPECTRUM OF A VICSEK FRACTAL

Abstract

In this paper, we investigate the effects of free-end boundary condition on the transverse vibrations of a Vicsek fractal. The unique feature of the eigenvalue spectrum corresponding to this case is that both non-degenerate and degenerate modes of a given generation are present in all subsequent generations, whereas in the case of fixed-end boundary condition only the degenerate modes persist. In the first case, the nature of the persistent degenerate modes is found to be either edge-localized or localized in the region between two adjacent clusters or adjacent sub-clusters, whereas in the second case the persistent modes are found to be edge-confined superlocalized modes. In both cases one finds that: (i) the frequency spectrum consists of atomic-like levels superimposed on a point-dense, cantor-set-like background; and (ii) there is side-by-side coexistence of the extended non-degenerate modes and superlocalized degenerate persistent modes.