The propagation of high frequency scalar surface waves along the generators of a homogeneous cylinder which has a cross-sectional boundary of nonconstant curvature is investigated. Asymptotic solutions are obtained to the reduced wave equation, subject to an impedance boundary condition at the surface of the cylinder. In the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, it is found that modes exist for which the disturbance is essentially confined to a region in the neighborhood of the point of maximum curvature, as well as to the neighborhood of the surface. The amplitude of the disturbance decays rapidly on either side of the point of maximum curvature, and the higher order modes have nulls. The case of closed boundary curves is also discussed. In a companion joint paper by L. O. Wilson and the author, the asymptotic procedures developed for the analysis of the scalar problem will be applied to the investigation of the propagation of elastic surface waves (Rayleigh waves) along a homogeneous isotropic cylinder with stress-free boundary. This problem arises in connection with guided acoustic surface waves.