Density of states for vibrations of fractal drums.

Abstract

Vibrations of membranes with fractal boundaries (fractal drums) are investigated. Numerical results are presented for Koch drums of fractal dimension D(f)=3/2 at prefractal generations 1-3, and for Koch snowflake drums (D(f)=ln 4/ln 3) 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 DeltaN(Omega) proportional, variant Omega(D(f)) 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 DeltaN(Omega) approximately Omega. The results also indicate that oscillations in DeltaN(Omega) 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 Omega(D(f)), which implies an O(Omega(D(f))), rather than the conjectured o(Omega(D(f))), error term for the asymptotic IDOS given by the MWB conjecture.