Wave propagation in self-similar structures PACS REFERENCE: 47.53n (Fractals) & 43.40Dx (Vibrations of membranes and plates ) Etienne B. du CHAZAUD*, Vincent GIBIAT**, Ana BARJAU*** *Laboratoire Ondes et Acoustique, ESPCI, 10 rue Vauquelin, 75 005 Paris, phone : +33 (0)1 40 79 44 79, fax : +33 (0)1 40 79 44 68 etienne.bertaud@iutc.u-cergy.fr **Laboratoire Acoustique, Mesures et Instrumentation, Université P. Sabatier, Toulouse III, 118 route de Narbonne, 31 000 Toulouse, phone : +33 (0)5 61 55 81 69, fax : +33 (0)5 61 55 81 54 vincent.gibiat@espci.fr ***ETSEIB, Univ. Polytech. de Catalunya, Diagonal 647, 08028 Barcelona, Spain, phone : +34 93 401 6716, fax :+34 34 93 401 5813 ana.barjau@upc.es ABSTRACT : As for Cantor-like structures, structures generated from the Sierpinski carpet present remarkable vibrational behaviours such as localisation phenomena and scale effects associated with their self-similar geometry. The singular characteristics of structures based on the Cantor set have been discussed since 1992 by Petri et al. [1,2] in ultrasonics and Gibiat et al. [3] in audible range. In particular, it has been shown that two types of vibrational modes can exist in such structures : extended modes (phonons) where the energy is distributed all along the structure, and localised modes (fractons) where energy is trapped in just a fraction of the structure. The frequencies corresponding to these type of modes can be predicted recursively from those of lower order structures. In this paper we will present similar behaviours that we have observed in structures whose geometry has been generated following that of the Sierpinski carpet. I –INTRODUCTION In a former work [Gibiat, 2002] a theoretical and experimental study of the acoustical propagation in a waveguide with a Cantor-like structure was presented. One of the main results is that of the existence of trapped modes and scale effects related with the degree of self- similarity, results that are in good agreement with the previous results obtained by Petri et al. in ultrasonics [Petri 1992]. The present works deals with the same kind of study on a mechanical 2D system consisting of a square membrane loaded with masses (positive or negative -i.e. holes-) whose positions are chosen so that the final object shows a geometry close to that of the Sierpinski carpet [Mandelbrot, 1982] . The loaded membrane has been studied through numerical simulations based on an algorithm following the same philosophy that the cellular automaton presented in Barjau et al. [Barjau 2002]. In the first section we will present the iterative construction of the pre-fractal membrane. The second one will be devoted to the basic concepts leading to the numerical algorithm that has been used. The last section will give some results showing that trapped modes are present on this membrane and that the same kind of scale effects are detectable as in the Cantor-like system. II – ITERATIVE CONSTRUCTION OF THE SIERPINSKY MEMBRANE The iterative process to build a Sierpinski membrane is the following: from a square uniform surface (which will be called Sierpinski order 0), the surface is divided into nine equal squares. The corners of the central one are then loaded with positive or negative masses (Sierpinski order 1). This operation (division and loading) is repeated for the eight remaining