Rotating Isospectral Drums

Abstract

In this thesis I demonstrate that isospectral domains, that is domains of differing geometric shapes that possess identical spectra, do not remain isospectral when subject to uniform rotation. One thus *can* hear the shape of a rotating drum. It is shown that the spectra diverge as $\propto\left(\frac{\omega}{c}\right)^2$ similarly to a square but different from a circle, whose degenerate eigenfrequencies split $\propto \left(\frac{\omega}{c}\right)$. The latter two cases are studied analytically and used as test cases for the selection of numerical solution methods. I demonstrate that the presence of a simple medium attenuates the effects of rotation on the spectrum differences. Further I show that the common but linearised augmented wave equation yields eigenmodes with sensible structure for non-physical parameters, whereas a full equation destroys the structure as intended. The full equation is obtained from first principles using Maxwell equations formulated as differential forms and a careful application of the 3+1 foliation of space-time. The differences of the result from literature arXiv:physics/0607016 are highlighted and their effects studied. The equation is treated analytically on simple domains to obtain reference data for numerical analysis. Nodal and modal discretisations of the PDE are tested and a self-written FEM implementation is chosen due to higher precision of the results. As part of the validation process I demonstrate that results of arXiv:physics/0607016 are reproducible. The resulting gyroscopic quadratic eigenvalue problem is linearised into a Hamiltonian/skew-Hamiltonian pencil and a suitable shift operator for the Arnoldi iteration is determined. The resulting formulation is solved using ARPACK.