Rotating Isospectral Drums
Anton Lebedev
TL;DR
This work investigates whether isospectral planar drums retain isospectrality under uniform rotation. It develops a physical-mathematical framework using differential forms and a 3+1 spacetime foliation to derive a rotating-wave equation for TM modes, including Coriolis-like couplings. The problem is recast as a gyroscopic quadratic eigenvalue problem and solved with a self-written FEM, validated against analytic disc/square results and existing literature, and then applied to known isospectral domain pairs. The key finding is that even isospectral domains at rest diverge when rotated, with the spectral differences growing quadratically in $\left(\tfrac{\omega}{c}\right)$; medium effects and the choice of full versus linearized equations modulate the magnitude but not the quadratic scaling. This implies that rotation renders isospectrality fragile and highlights a robust numerical framework for gyroscopic PDEs in complex geometries with potential applications to rotating structures and metamaterial designs.
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 $\propto\left(\fracω{c}\right)^2$ similarly to a square but different from a circle, whose degenerate eigenfrequencies split $\propto \left(\fracω{c}\right)$. 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.