This paper concerns axially symmetric solutions for waves propagating along a cylinder in an infinite elastic solid. Solutions are presented describing unattenuated propagation along the axis at phase velocities higher than shear and compressional speeds in the solid, in contradiction to earlier publications. Special attention is given to the limiting case of phase velocity equal to compressional speed in the solid, which at low frequencies very closely approximates the coupling of a fluid-filled borehole to a plane compressional wave in the surrounding solid. Comparison with some experiments in a uniform section of Pierre Shale [Cretaceous] show excellent agreement at low frequencies. In the low-frequency limit, these solutions reduce to an approximate expression for borehole coupling published earlier by the author.