## Abstract

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.

Original language | English |
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Number of pages | 13 |

Journal | Physical Review E |

Volume | 67 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 2003 |