Tunnelling around bends - Wave scattering in curved shell structures

Abstract A ray dynamics describing wave transport on curved and smooth thin shells can be obtained from the underlying wave equations via the Eikonal approximation. We analyse mid-frequency effects near the ring frequency for curved plates consisting of a cylindrical region smoothly connected to two flat plates. Using classical shell theory, we treat a corresponding ray-tracing limit derived in the short wavelength regime for bending, shear and pressure incident waves. A treatment based on real rays gives either total reflection or total transmission; the solution of the full wave equations shows in contrast a smooth transition and exhibits resonant states localized in and around the region of maximal curvature. We show here for the first time, that both the smooth transition from total reflection to total transmission and the occurrence of resonant states can be described in a ray tracing approximation by extending the treatment to complex rays. In this sense, wave transmission across bends is a tunnelling phenomenon similar to the tunnelling effects known in optics and quantum mechanics. Based on simplified graph models, we can make qualitative predictions for resonance positions in parameter and frequency spaces. We are thus able to approximate the scattering matrix for waves incident on the bend accounting for tunnelling mediated by resonant states and uniformly treating the transition between the limits of totally reflected and totally transmitted waves.