Homolya Steven, Osborne Charles F, Svalbe Imants D
School of Physics and Materials Engineering, Monash University, Wellington Road, Clayton, Victoria 3800, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Feb;67(2 Pt 2):026211. doi: 10.1103/PhysRevE.67.026211. Epub 2003 Feb 19.
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.
研究了具有分形边界的膜（分形鼓）的振动。给出了分形维数(D(f)=3/2)的科赫鼓在预分形世代1 - 3时的数值结果，以及科赫雪花鼓（(D(f)=\ln 4 / \ln 3)）在世代3和4时的数值结果。结果表明，鼓的低频态密度积分（IDOS）可以很好地由修正的魏尔 - 贝里（MWB）猜想给出的形式的两项渐近式近似，该猜想预测对主导阶魏尔项有一个与(\Delta N(\Omega))成比例、形式为(\Omega^{D(f)})的修正。在高频区域，当半波长小于预分形周长的最小特征尺寸时，两项魏尔渐近式适用，其中(\Delta N(\Omega))近似为(\Omega)。结果还表明，(\Delta N(\Omega))中的振荡是由于波幅在预分形周长附近的局域化引起的。有人认为，对于自相似分形边界，振荡幅度渐近地与(\Omega^{D(f)})成比例，这意味着MWB猜想给出的渐近IDOS的误差项是(O(\Omega^{D(f)}))，而不是推测的(o(\Omega^{D(f)}))。
