Modal Structure of Plate Boundaries and Klein Bottle Reverberation
Physical modeling sound synthesis has achieved remarkable success in terms of its fidelity to reality. In many cases, since modeling of the physical system is performed on the sounding objects that already exist in the real world, observation precedes the model itself. Departing from this convention, this paper aims to physically model the acoustic characteristics of objects that do not necessarily exist in reality. Specifically, we study wave propagation on compact two-dimensional (2D) manifolds that are non-orientable surfaces, such as the Klein bottle that cannot be embedded in three-dimensional Euclidean space without self-intersection. We derive closed-form expressions for the eigenfrequencies and mode shapes of non-orientable 2D topologies and study their acoustic characteristics. The modal structures are verified through comparison with finite-difference time-domain simulations. The results demonstrate how the topological character formed by the boundaries influences the acoustic resonances, and how the quotient-space framework provides a practical route to reverb synthesis on geometries with no physical counterpart.