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Vibrations d'une membrane smectique : rôle de la forme du contour

Abstract : Due to some specific properties (uniformity of thickness, bidimensional density and tension), liquid crystal films in smectic phase constitute ideal membranes which obey the Helmholtz wave equation, with Dirichlet boundary condition. In this PhD thesis, a new experiment has been developed which allows, for a film spanned on a given shape, to measure, not only its spectrum of eigenfrequencies, but also the geometrical shape of the eigenmodes. It has indeed been possible to analyse the role of the membrane shape. First, a "prefractal" shape has been studied. This shape is made of a quadratic Koch curve, which building has been stopped at a finite iteration order. The agreement between the experimental results and some numerical ones (obtained by other authors) is excellent. Two physically different mechanisms of localization for the wavefunctions have been pointed out. A mathematical problem has also been studied, which can be expressed by "Can one hear the shape of a drum?". The mathematically predicted answer is "no", which means that there exist geometrically different shapes, which lead to the same spectrum of eigenfrequencies : these shapes are named "isospectral". The isospectrality of two such shapes has been checked experimentally with a good precision. It has also been shown in a detailed manner that only the symmetry rules for building the shapes, based on the group theory, matter.
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Contributor : Catherine Even <>
Submitted on : Friday, July 30, 2004 - 2:22:51 PM
Last modification on : Wednesday, September 16, 2020 - 4:32:00 PM
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  • HAL Id : tel-00006615, version 1



Catherine Even. Vibrations d'une membrane smectique : rôle de la forme du contour. Acoustique [physics.class-ph]. Université Paris Sud - Paris XI, 1999. Français. ⟨tel-00006615⟩



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