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We consider the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a singly-curved shell as concrete examples. Previous work on related problems have largely focused on the localization of flexural waves on such structures. Here, using the semiclassical/WKB approximation, we show that in addition to flexural waves, extensional and shear waves also form localized, bound states around points where the absolute curvature of the structure has a minimum. These findings open up novel ways to fine-tune the acoustic and vibrational properties of thin elastic structures, and raise the possibility of introducing new phenomena not easily captured by effective models of flexural waves alone., Comment: 26 pages, 12 figures

Twist-grain-boundary phases in smectics are the geometrical analogs of the Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear elasticity is important in evaluating their energetics. We analytically construct the height function of a pi/2 twist-grain-boundary phase in smectic-A liquid crystals, known as Schnerk's first surface. This construction, utilizing elliptic functions, allows us to compute the energy of the structure analytically. By identifying a set of heretofore unknown defects along the pitch axis of the structure, we study the necessary topological structure of grain boundaries at other angles, concluding that there exist a set of privileged angles and that the \pi/2 and \pi/3 grain boundary structures are particularly simple., Comment: 13 pages, 7 figures