Can one hear the shape of a filament? How geometry controls the localization of waves in thin elastic structures.

Several remarkable applications of metamaterials from cloaking to negative refraction subtly rely on wave localization induced by material geometry. Recent work [1] has examined how elastic waves can localize around the inflection point of a thin sheet with spatially varying curvature. Here we consider the vibrations of a curved filament explicitly accounting for both flexural deformations that predominantly bend the filament, and extensional deformations that predominantly stretch it. Using WKB asymptotics for multicomponent wave fields, we show that flexural waves get trapped around the inflection point, whereas extensional waves do not. The analytically computed frequencies of the localized flexural modes also show very good agreement with numerical results. Thus, we expect infinitely long filaments to have discrete, localized flexural modes coexisting with a continuum of extensional waves. These findings have implications on possible excitations of thin elastic structures and raise the possibility of introducing new phenomena not easily captured by effective models of flexural waves alone.

[1] Shankar et al., PNAS 119, e2117241119 (2022).