Effects of curvature on the propagation of undulatory waves in lower dimensional elastic materials

The mechanics of lower dimensional elastic structures depends strongly on the geometry their stress-free state. Elastic deformations separate into in-plane stretching and energy out-of-plane bending deformations. For with a curved state, these two modes are coupled within linear elasticity. We investigate effect that curvature-induced coupling wave propagation in structures, focusing simplest example -- rod. find dispersion relation waves becomes gapped presence finite curvature; absent below frequency proportional to curvature By studying scattering undulatory off regions uniform curvature, we frequencies gap associated region tunnel through via conversion compression waves. These results should be directly applicable spectrum spatial distribution phonon number rod-like solids, including carbon nanotubes biopolymer filaments.