Abstract
A circular elastic ring or a segment of such a ring may vibrate linearly in a dilatationless natural mode, such that the particles oscillate normal to the cross sections and the deformations of all cross sections are identical. Circumferential modes of a solid disk or a sector of such a disk constitute a limiting case. This type of motion was discovered by L. Pochhammer, who observed that nontorsional circumferential modes of vibration can exist in a solid circular cylindrical bar. He also studied dilatationless longitudinal modes of vibration of a circular cylindrical bar, which may be regarded as circumferential modes for a torus of infinite radius. Pockhammer did not note that the modes that he discussed can occur in rings and prismatic bars. Each shape of cross section presents a boundary-value problem, since the lateral surface must be free of stress.
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