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