The eigenfrequencies at which smooth convex objects resonate under the incidence of an acoustic wave correspond to the real parts of those complex frequency values at which circumferential waves generated by the incident signal phase‐match after repeated circumnavigations around the object [H. Uberall, L. R. Dragonette, and L. Flax, J. Acoust. Soc. Am. 61, 711 (1977)]. A resonance condition based on this principle is formulated, and applied to the case of elastic prolate spheroids and cylinders with hemispherical endcaps. Using then the known phase velocities of surface waves on elastic spheres, with a radius equal to the local radius of curvature along the surface path, the elastic resonance frequencies of these objects can be predicted. This was done for the Rayleigh wave on a prolate spheroid, where comparison with resonances in the scattering amplitude as obtained by a T‐matrix calculation led to good agreement.