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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 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.
Spectra of the two isospectral drums called "cocotte" (red) and flêche" (blue). As expected, the frequencies of eigenmodes are identical.
Four spectra of eigenmodes of the "cocotte" drum. They were recorded with different settings of positions (xe,ye) of the exciting sharp electrode and (xl,yl) of the laser beam.
Four spectra of eigenmodes of the "cocotte" drum. They were recorded with different settings of positions (xe,ye) of the exciting sharp electrode and (xl,yl) of the laser beam.
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