Electromagnetics and Acoustics on a bounded domain are governed by the Helmholtz's equation; when such a domain is a [pre-]fractal described by means of a 'just-touching' Iterated Function System (IFS) spectral decomposition of the Helmholtz's operator is self-similar as well. Renormalization of the Green's function proves this feature and isolates a subclass of eigenmodes, called diaperiodic, whose waveforms and eigenvalues can be recursively computed applying the IFS to the initiator's eigenspaces. The definition of spectral dimension is given and proven to depend on diaperiodic modes only for a wide class of IFSS. Finally, asymptotic equivalence between box-counting and spectral dimensions in the fractal limit is proven. As the "self-similar" spectrum of the fractal is enough to compute box-counting dimension, positive answer is given to title question.

Can you hear the fractal dimension of a drum?

Arrighetti, W;Gerosa, G
2005

Abstract

Electromagnetics and Acoustics on a bounded domain are governed by the Helmholtz's equation; when such a domain is a [pre-]fractal described by means of a 'just-touching' Iterated Function System (IFS) spectral decomposition of the Helmholtz's operator is self-similar as well. Renormalization of the Green's function proves this feature and isolates a subclass of eigenmodes, called diaperiodic, whose waveforms and eigenvalues can be recursively computed applying the IFS to the initiator's eigenspaces. The definition of spectral dimension is given and proven to depend on diaperiodic modes only for a wide class of IFSS. Finally, asymptotic equivalence between box-counting and spectral dimensions in the fractal limit is proven. As the "self-similar" spectrum of the fractal is enough to compute box-counting dimension, positive answer is given to title question.
978-981-256-368-2
978-981-270-181-7
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1656407
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