Isospectral Cavities

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.

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