Hearing the Shape of Drums
Can one hear the shape of the drum? -M. Kac (1966).
No! -Gordon, Webb and Wolpert (1992).
Experiments on `Not' Hearing the Shape of Drums
While the theorem of isospectral domains was proved on abstract mathematical
grounds, the actual resonance frequencies and wave functions of the geometries
were not known. The microwave experiments were able to make an unique contribution
by exploiting the the fact that below a certain frequency, the equation
obeyed in thin microwave cavities is the same as the Helmholtz wave equation.
The resonances were obtained by actually constructing the shapes using
copper. The wavefunctions are obtained using a
cavity perturbation technique. Therefore these experiments were not only
able to physically realize drums which sounded alike, but also were able
to show that there were no degeneracies in the first 54 resonances.
Wavefunctions of the Isospectral Geometries
The 1st (top), 3rd (middle) and 6th (bottom) pairs of wavefunctions
of the isospectral cavities. Even though the spatial details of the wavefunctions
are quite different, the eigenvalues are identical. The wavefunctions however
are related via a non-isometric transformation, which forms the basis of
the original mathematical proof.
Recently there have also been numerical
investigations by Tody Driscoll, and by Peter Knipp into the properties
of these isopectral domains.
Click here to see wavefunctions of Chaotic
and Disordered cavities.
Further details can be found in the publications.