Wave Propagation in Scale Free
Media
Thomas Michelitsch, Gérard Maugin, Franck Nicolleau*,
Andrzej Nowakowski*, Shahram Derogar*
Institut Jean le Rond d'Alembert (Paris 6), CNRS UMR 7190
Paris, France
Email: michel@lmm.jussieu.fr
*University of Sheffield
Sheffield, United Kingdom
Email: F.Nicolleau@sheffield.ac.uk
6th Workshop on Synthetic Turbulence Models
5th-7th July 2010, Lyon
ERCOFTAC/SIG 42
Overview
• Motivation
• Previous works and approaches
• Self-similar functions and linear operators
• Self-similar Laplacian operator and relationship to fractional
derivatives
• Physical model of a linear chain with self-similar harmonic
interactions
• Self-similar (fractal) dispersion relations and oscillator density
• Perspectives
Motivation
Fractals (self-similar objects) in nature
Why nature choses so often self-similarity (scaling invariance) as intrinsic symmetry?
Which (mechanical) dynamic properties are optimized in such
arborescent self-similar structures?
--- The answer requires deeper understanding on:
how the self-similarity (fractality) affects the mechanical
especially dynamic properties, such as wave propagation,
dispersion relation, vibrational density.
--- However there is so far no generally accepted approach, hence,
simple models catching essential features of self-similar and
fractal systems are desirable.
Motivation
Previous works and approaches
• Kigami J., A harmonic calculus on the Sierpinski spaces, Japan J. Appl.
Math. 8 (1989), 259-290.
• Bondarenko A.N., Levin, V.A., Self Similar Spectrum of a Fractal Lattice,
Science and Technology (2005) KORUS Proceedings. The 9th Russian-
Korean Int. Symposium, 33-35, ISBN: 0-7803-8943-3
• Tarasov V.E. , Chains with fractal dispersion law, J Phys A: Math Theor. 41
(2008), doi: 10.1088/1751-8113/41/3/035101
• Ostoja-Starzewski, M., Towards Thermomechanics of Fractal Media,
Journal of Applied Mathematics and Physics (ZAMP), 58:6, 1085-1096,
2007.
• Epstein M., Sniatytcki J., Fractal Mechanics , Physica D,
2006, vol. 220, no1, pp. 54-68
•