How the singing saw gets its voice

The ability to sustain notes or vibrations underlies the design of most acoustic devices, ranging from musical instruments to nanomechanical resonators. Inspired by the singing saw that acquires its musical quality from its blade being unusually bent, we ask how geometry can be used to trap and insulate acoustic modes from dissipative decay in a continuum elastic medium. By using experiments, theoretical and numerical analysis, we demonstrate that spatially varying curvature in a thin shell can localize topologically protected modes at inflection lines, akin to exotic edge states in topological insulators. A key feature is the ability to geometrically control both spatial localization and the dynamics of oscillations in thin shells. Our work uncovers a novel mechanism for designing robust, yet reconfigurable, high quality resonators across scales, simply through geometry.