  We measure elastomechanical spectra for a family of thin shells. We show that
these spectra can be described by a "semiclassical" trace formula comprising
periodic orbits on geodesics, with the periods of these orbits consistent with
those extracted from experiment. The influence of periodic orbits on spectra in
the case of two-dimensional curved geometries is thereby demonstrated, where
the parameter corresponding to Planck's constant in quantum systems involves
the wave number and the curvature radius. We use these findings to explain the
marked clustering of levels when the shell is hemispherical.
