Abstract
High frequency localized vibration modes for linear isotropic elastic plates and rods of slowly varying thickness are shown to occur within the vicinity of maximal or minimal cross-sections. Long wave asymptotic integration close to the eigenmodes of such cross-sections is provided. The cases of a one integer parameter of thickness resonances (for a plate in plane stress) and two integer parameters (for a rod of circular cross-section) are fully investigated. One-dimensional Hermite-type equations for the eigenmodes are derived and associated frequency shifts from those of the thickness resonances of the maximal or minimal cross-section obtained. It is established that the existence of this type of localization phenomena is strongly dependent on the sign of the group velocity of the associated uniform thickness structure this being the same as that of the local group velocity calculated at the maximal or minimal cross-section. It is confirmed that positive group velocity allows localization for elevated structures with negative group velocity allowing localization only for ligaments. In each case intervals of the Poisson's ratio providing localization are given for the first few modes.
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