Eigenfrequencies of Fractal Drums Eigenfrequencies of Fractal Drums
نویسنده
چکیده
A method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented. The approach involves first mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method. The numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in [2]. The linear system arising from the spectral discretization is large and dense. To circumvent this problem we devise a fast method for the inversion of such a system. Consequently the eigenvalue problem is solved iteratively. We obtain 8 digits for the first eigenvalue of the Koch snowflake and at least 5 digits for eigenvalues up to the 20th. Numerical results for two more fractals are shown.
منابع مشابه
The Distribution of Eigenfrequencies of Anisotropic Fractal Drums
Motivated by some aspects of boundary value problems for partial differential equations, several authors have recently been concerned with the study of function spaces on and of fractals. We refer mainly to the papers by A. Jonsson and H. Wallin [7–10] and to the book [16] where complete references to this topic are given. Let Ω be a bounded domain in 2# having C ¢ boundary ¦Ω and let 0! d A ! ...
متن کاملIs it possible to tune a drum?
It is well known that the sound produced by string instruments has a well defined pitch. Essentially, this is due to the fact that all the resonancefrequencies of the string have integer ratio with the smallest eigenfrequency.However, it is enough to use Ashbaugh-Benguria bound for the ratio of thesmallest two eigenfrequencies to conclude that it is impossible to build a drumwith a uniform dens...
متن کاملPeriodic orbit theory in fractal drums
The level statistics of pseudointegrable fractal drums is studied numerically using periodic orbit theory. We find that the spectral rigidity ∆3(L), which is a measure for the correlations between the eigenvalues, decreases to quite small values (as compared to systems with only small boundary roughness), thereby approaching the behavior of chaotic systems. The periodic orbit results are in goo...
متن کاملThe Riemann Zeta-function and the One-dimensional Weyl-berry Conjecture for Fractal Drums
Based on his earlier work on the vibrations of 'drums with fractal boundary', the first author has refined M. V. Berry's conjecture that extended from the 'smooth' to the 'fractal' case H. Weyl's conjecture for the asymptotics of the eigenvalues of the Laplacian on a bounded open subset of W (see [16]). We solve here in the one-dimensional case (that is, when n = 1) this 'modified Weyl-Berry co...
متن کاملLocalizations in Fractal Drums: An Experimental Study
The low-frequency eigenmodes of a fractal drum are studied through experimental observation of the acoustical resonances of a fractal-shaped liquid crystal film. The resonance frequencies agree with the numerical predictions for the Laplacian eigenvalues in the fractal domain with Dirichlet boundary conditions. The amplitude distribution of the modes is detected by scanning the excitation posit...
متن کامل