Irregular and fractal resonators with Neumann boundary conditions: Density of states and localization

Vibrations of two-dimensional systems with free irregular or fractal boundaries are studied on specific examples. The eigenmodes are calculated numerically using an analogy between Helmholtz and diffusion equations. We discuss the influence of the fractal boundary on the low-frequency part of the spectrum and on wave forms. The density of states is increased by the irregularity and exhibits oscillations at special frequencies which depend on the geometry. Surprisingly, many states are found to be confined at the fractal boundary. Increasing the perimeter fractality induces increased confinement.