Experiments on not ‘‘hearing the shape’’ of drums

Abstract

We present an experimental test of the theorem of isospectral domains which states that certain pairs of planar domains have identical spectra for eigenvalues of the Laplacian operator or the Helmholtz wave equation. The experiments employ thin microwave cavities shaped in the form of two different domains known to be isospectral. We verify the equality of at least 54 of the measured low-lying eigenvalues to a few parts in ${10}^{4}$. We also obtain the eigenfunctions experimentally, and confirm that two nonisometric transformations connect isospectral eigenfunction pairs. An analysis in terms of periodic orbits using the trace formula is also carried out, and shown to be an alternative confirmation of isospectrality.