We consider the propagation of free surface waves on an elastic half-space that has a localized geometric inhomogeneity perpendicular to the direction of wave propagation (such waves are known as topographyguided surface waves). Our aim is to investigate how such a weak inhomogeneity modifies the surfacewave speed slightly. We first recover previously known results for isotropic materials and then present additional results for a generally anisotropic elastic half-space assuming only one plane of material symmetry. It is shown that a topography-guided surface wave in the present context may or may not propagate depending on a number of factors. In particular, they cannot propagate if the original two-dimensional surface wave on a flat half-space is supersonic with respect to the speed of anti-plane shear waves. For the case when a topographyguided surface wave may exist, the existence and computation of wave speed correction is reduced to the solution of a simple eigenvalue problem whose properties are previously well understood. As a byproduct of our analysis, we deduce that there exists at least one topography-guided surface wave on an isotropic elastic half-space, and that it is unique when the geometric inhomogeneity has sufficiently small amplitude.