Density of states for vibrations of fractal drums

Vibrations of membranes with fractal boundaries (fractal drums) are investigated. Numerical results are presented for Koch drums of fractal dimension [Formula presented] at prefractal generations 1–3, and for Koch snowflake drums [Formula presented] at generations 3 and 4. The results show that the low-frequency integrated densities of states (IDOS’s) of the drums are well approximated by a two-term asymptotic of the form given by the modified Weyl-Berry (MWB) conjecture, which predicts a correction of [Formula presented] to the leading-order Weyl term. In the high-frequency regime, where the half wavelength is smaller than the smallest features of the prefractal perimeter, the two-term Weyl asymptotic is applicable, with [Formula presented] The results also indicate that oscillations in [Formula presented] arise due to localization of the wave amplitude near the prefractal perimeter. It is argued that for a self-similar fractal boundary, the amplitude of the oscillations is asymptotically proportional to [Formula presented] which implies an [Formula presented] rather than the conjectured [Formula presented] error term for the asymptotic IDOS given by the MWB conjecture.