Fractal Drums Research Articles

Fractal drums and the Weyl-Berry-Lapidus conjecture

We investigate the vibrational characteristics of drums with fractal perimeters, especially focusing our attention to the validity of the Weyl-Berry-Lapidus conjecture for the eigenfrequency distribution. The integrated densities of states of large-scale drums with fractal perimeters have been computed. We have confirmed that the Weyl-Berry-Lapidus conjecture should hold under the condition λ/ l ≪ 1, where λ is the wavelength of mode and l the length of the fractal segment, in addition to the condition of the high-frequency limit.

Counterexamples to the modified Weyl–Berry conjecture on fractal drums

Let Ω be a non-empty open set in ℝn with finite ‘volume’ (n-dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H1(Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let

Generalized Minkowski Content and the Vibrations of Fractal Drums and Strings

Generalized Minkowski Content and the Vibrations of Fractal Drums and Strings

FRACTALS AND VIBRATIONS: CAN YOU HEAR THE SHAPE OF A FRACTAL DRUM?

We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).

The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Motivated in part by the first author's work [23] on the Weyl-Berry conjecture for the vibrations of ‘fractal drums’ (that is, ‘drums with fractal boundary’), M. L. Lapidus and C. Pomerance [31] have studied a direct spectral problem for the vibrations of ‘fractal strings’ (that is, one-dimensional ‘fractal drums’) and established in the process some unexpected connections with the Riemann zeta-function ζ = ζ (s) in the ‘critical interval’ 0 < s < 1. In this paper we show, in particular, that the converse of their theorem (suitably interpreted as a natural inverse spectral problem for fractal strings, with boundary of Minkowski fractal dimension D ∈ (0,1)) is not true in the ‘midfractal’ case when D = 1 2 , but that it is true for all other D in the critical interval (0,1) if and only if the Riemann hypothesis is true. We thus obtain a new characterization of the Riemann hypothesis by means of an inverse spectral problem. (Actually, we prove the following stronger result: for a given D ∈ (0,1), the above inverse spectral problem is equivalent to the ‘partial Riemann hypothesis’ for D, according to which ζ = ζ (s) does not have any zero on the vertical line Re s = D.) Therefore, in some very precise sense, our work shows that the question (à la Marc Kac) “Can one hear the shape of a fractal string?” – now interpreted as a suitable converse (namely, the above inverse problem) – is intimately connected with the existence of zeros of ζ = ζ(s) in the critical strip 0 < Res < 1, and hence to the Riemann hypothesis.

Spectral distribution of drums with fractal perimeters: The Weyl-Berry-Lapidus conjecture.

The Weyl-Berry-Lapidus conjecture on vibrational spectra of drums with fractal perimeters (fractal drums) is examined in terms of large-scale numerical simulations. The integrated densities of states of fractal drums with the perimeter of the Koch curve are computed by employing a powerful numerical method. It is confirmed that the Weyl-Berry-Lapidus conjecture holds in the frequency regime higher than a crossover frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ related to the length scale characterizing the fractal boundary.

Probabilistic computation of Poiseuille flow velocity fields

Velocity fields for Poiseuille flow through tubes having general cross section are calculated using a path integral method involving the first-passage times of random walks in the interior of the cross sectional domain 𝒟 of the pipe. This method is applied to a number of examples where exact results are available and to more complicated geometries of practical interest. These examples include a tube with ‘‘fractal’’ cross section and open channel flows. The calculations demonstrate the feasibility of the probabilistic method for pipe flow and other applications having an equivalent mathematical description (e.g., torsional rigidity of rods, membrane deflection). The example of flow through a fractal pipe shows an extended region of diminished flow velocity near the rough boundary which is similar to the suppressed vibration observed near the boundaries of fractal drums.

Fractal drums and then-dimensional modified Weyl-Berry conjecture

In this paper, we study the spectrum of the Dirichlet Laplacian in a bounded (or, more generally, of finite volume) open set Ω∈R n (n≧1) with fractal boundary ∂Ω of interior Minkowski dimension δ∈(n−1,n]. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the “counting function”N(λ) (i.e. the number of positive eigenvalues less than λ) as λ→+∞, which is of the form λδ/2 times a negative, bounded and left-continuous function of λ. This explains the reason why the modified Weyl-Berry conjecture does not hold generally forn≧2. In addition, we also obtain explicit upper and lower bounds on the second term ofN(λ).

Beating a Fractal Drum

Beating a Fractal Drum

Vibrations of strongly irregular or fractal resonators.

It is shown on a specific example that fractal boundary conditions drastically alter the properties of wave excitations in space. The low-frequency part of the vibration spectrum of a finite-range fractal drum is computed using an analogy between the Helmoltz equation and the diffusion equation. The irregularity of the frontier is found to influence strongly the density of states at low frequency. The fractal perimeter generates a specific screening effect. Very near the frontier, the decrease of the wave form is related directly to the behavior of the harmonic measure. The possibility of localization of the vibrations is qualitatively discussed and we show that localized modes may exist at low frequencies if the geometrical structures possess narrow paths. Possible application of these results to the interpretation of thermal properties of binary glasses is briefly discussed.

An Example of a Two-Term Asymptotics for the "Counting Function" of a Fractal Drum

An Example of a Two-Term Asymptotics for the "Counting Function" of a Fractal Drum

The 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 conjecture'. We discover, in the process, some unexpected and intriguing connections between spectral geometry, fractal geometry and the Riemann zeta-function. We therefore show that one can 'hear' (that is, recover from the spectrum) not only the Minkowski fractal dimension of the boundary—as was established previously by the first author—but also, under the stronger assumptions of the conjecture, its Minkowski content (a 'fractal' analogue of its 'length'). We also prove (still in dimension one) a related conjecture of the first author, as well as its converse, which characterizes the situation when the error estimates of the aforementioned paper are sharp.

An example of a two-term asymptotics for the ‘‘counting function” of a fractal drum

In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain $\Omega \subset {\mathbb {R}^n}$ with fractal boundary $\partial \Omega$. We construct an open set $\mathcal {Q}$ for which we can effectively compute the second term of the asymptotics of the "counting function" $N(\lambda ,\mathcal {Q})$, the number of eigenvalues less than $\lambda$. In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In $\lambda$, not to a constant. We also establish some properties of the $\zeta$-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set $\mathcal {O}$ derived from $\mathcal {Q}$. Analogous periodic functions still appear in our inequalities. These results have been announced in $[{\text {FV}}]$.

Fractal drums and phonons in binary glasses

The low frequency vibration spectrum of a surface fractal with a rigid fractal boundary presents a smaller density of states than an ordinary resonator. Such vibrations, called fractinos, should be present in partially phase separated glasses if one component is soft. In addition the interfaces between the phase separated regions are fractal diffusion fronts which are mass fractals close to percolation cluster hulls. It is proposed that the fractons which have claimed to exist in these non-fractal materials are the vibrations of these internal interfaces.

Beating a Fractal Drum

Beating a Fractal Drum

Beating a Fractal Drum

Beating a Fractal Drum

Vibrations of fractal drums.

Fractal boundary conditions drastically alter wave excitations. The low-frequency vibrations of a membrane bounded by a rigid fractal contour are observed and localized modes are found. The first lower eigenmodes are computed using an analogy between the wave and the diffusion equations. The fractal frontier induces a strong confinement of the wave analogous to superlocalization. The wave forms exhibit singular derivatives near the boundary.

Fractal Drum, Inverse Spectral Problems for Elliptic Operators and a Partial Resolution of the Weyl-Berry Conjecture

Let $\Omega$ be a bounded open set of ${\mathbb {R}^n}\;(n \geq 1)$ with "fractal" boundary $\Gamma$. We extend Hermann Weyl’s classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order $2m\;(m \geq 1)$ on $\Omega$. We consider both Dirichlet and Neumann boundary conditions. Our estimate—which is expressed in terms of the Minkowski rather than the Hausdorff dimension of $\Gamma$—specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry’s conjecture—which extends to "fractals" Weyl’s conjecture—is closely related to Kac’s question "Can one hear the shape of a drum?"; further, it has significant physical applications, for example to the scattering of waves by "fractal" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated "partition function" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible "fractal" (i.e., Minkowski) dimension. The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and—to a lesser extent—geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and "fractal" geometry.

Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture

Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture

Experimental observation of local modes in fractal drums

It is shown that fractal boundary conditions drastically alter the properties of wave excitations in space. The low-frequency vibration modes of soap bubbles deposited on a fractal contour are observed. There exist localized modes. The vibrations have a singular behavior along the boundary. Consequently fractal resonators are necessarily non-linear resonators and have special damping properties.