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Vibration d'une membrane smectique : role de la forme du contour
Vibration d'une membrane smectique : role de la forme du contour
C. Even et P. Pieranski ,
"On hearing the shape of drums : an experimental study using vibrating smectic films",
Europhysics Letters, 47 (1999) 531
Marc Kac
The spectra of eigenmodes of drums in two dimensions depend on their shapes. Can one deduce univoquely the shape of a drum from its spectrum ? Or, in other words, as it has been asked by Mark Kac :
Can one hear the shape of a drum ?
The answer to this question is NO and has been given by Wolpert, Gordon and Web who invented a pair of drums of different shapes but having the same spectra of eigenmodes. This first pair of isospectral drums (called "cocotte" and "flèche") is shown in the figure below. We used Smectic Films vibrating in vacuum for an experimental demonstration of the isospectrality of these drums .
Carolyn Gordon and David Webb
Experimental set-up used in this study. The detection of vibrations is optical. The spectra of smectic drums recorded with this set-up are shown below.
As expected, the frequencies of eigenmodes are identical.
Spectrum of eigenmodes of the "cocotte" drum.
Fractal drums
C. Even, S. Russ, V. Repain, P. Pieranski, and B. Sapoval.
"Localizations in fractal drums: an experimental study."
Physical Review Letters 83, no. 4 (1999): 726.
Principle of construction of a fractal drum by iterations.
In experiments, the frame obtained by three iteration (on the right) was used
Above: Spectrum of the smectic fractal drum vibrating in vacuum.
Right: Comparison between computed (left) and experimental (right) amplitude distribution for modes (a) n =1, (b) n =8, and (c) n=38. Two frames are drawn on experimental plots: The larger corresponds to the real frame and the smaller to the zone scanned by the electrode. Clearly the n=38th mode is localised in a creek of the fractal drum.
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