Trapped Modes in Piezoelectric and Elastic Waveguides

Abstract

We derive a sufficient condition for the existence of trapped modes in a cylindrical piezoelectric waveguide $\varOmega$ with a compact void $\varTheta$ . An infinite part $\varGamma_{D}$ of the exterior boundary $\partial\varOmega$ is clamped along an electric conductor and the remaining part $\varGamma_{N}=(\partial\varOmega\setminus \varGamma_{D})\cup\partial\varTheta$ is traction-free and is in contact with an insulator, e.g., a vacuum. The condition permits the limit passage either to an elastic or a compound waveguide but it crucially differs from the pure elastic case due to the involved electric enthalpy instead of the energy functional. Examples of concrete waveguides and voids supporting trapped modes are given, and open questions are formulated. In particular, in contrast to a pure elastic waveguide where “almost” any crack traps a wave, no example of a crack trapping a wave in a piezoelectric waveguide is known yet.