Spectral characteristics in resonators with fractal boundaries

Vibrations of drums with self-similar (fractal) boundaries are investigated in terms of large-scale simulations, for elucidating the characteristics of their spectral densities of states. It is found that the integrated density of states Delta I(omega) is proportional to omega(Df) (D-f the fractal dimension of the boundary) in the frequency regime higher than a characteristic frequency omega(c) with oscillating but small amplitude. The frequency omega(c) is related to the length scale characterizing the fractal boundary. We show that there exist edge modes localized near the fractal boundary under the stress-free boundary condition (Neumann condition), which vibrate at both ends of the drum with antiphase.