Miscellaneous types of localization of natural oscillations of a gasket between parallel flanges

Both bases of a thin cylindrical gasket are fixed but the lateral side stays traction-free. For middle cross-sections with a smooth boundary as well as for a square one, we describe the asymptotics of frequencies of natural oscillations. Based on an investigation of the spectrum of the mixed boundary-value problem in the theory of elasticity in a quarter of the unit layer, we discover a new effect of localization of modes of natural oscillations near vertices of the square section and of the exponential decay at a distance from them. In the case of the smooth boundary of the cross-section eigenmodes concentrate either near the whole edge (a circular plate), or at the vicinity of points of maximum of the contour curvature.