Geometric Localization of Waves on Thin Elastic Structures

Abstract

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 studies on related problems have broadly focused on the localization of flexural waves on such structures. Here, using the semiclassical WKB approximation for multicomponent waves, 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. We also see excellent agreement between our numerical experiments and the semiclassical results, which hinges on the vanishing of two extra phases that arise in the semiclassical quantization rule. Our 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.

arXiv comment21 pages, 14 figures

References in corpus (11)

• Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods
• Topological Origin of Equatorial Waves
• Topological waves in fluids with odd viscosity
• Schrödinger potentials solvable in terms of the general Heun functions
• Localised flexural waves in wedges of power-law profile and their relationship with acoustic black holes
• Omnidirectional broadband insulating device for flexural waves in thin plates
• Deviations from Poisson statistics in the spectra of free rectangular thin plates
• Geometric control of topological dynamics in a singing saw
• The effects of curvature on the propagation of undulatory waves in lower dimensional elastic materials
• Rayleigh waves and cyclotron surface modes of gyroscopic metamaterials
• Dynamics of certain Euler-Bernoulli rods and rings from a minimal coupling quantum isomorphism