Edge resonance: exact results and fresh insights

Abstract

The eigensystem underpinning the classical edge resonance phenomenon in linear elasticity is studied. Several exact results are presented, the most significant being an identically zero sum demonstrating the linear dependence of the stresses. An exact condition for edge resonance is derived. This is cast into a form that is independent of the Lamb modes, robust and highly convergent, enabling the system to be explored by varying Poisson’s ratio (PR) or frequency. An improved estimate of the value of PR for real resonance is determined, as is the non-Lamé frequency corresponding to resonance when PR is zero. Quasi-resonances are explored. It is demonstrated that, for fixed PR, these occur at more than one frequency, and that they occur for negative PR. It is shown that quasi-resonances are associated with one of two distinct families of complex resonances: real PR and complex frequency or real frequency and complex PR. Higher Lamé frequencies are considered. It is demonstrated that a real pure shear resonance exists at the second Lamé frequency when PR is zero. The corresponding edge displacement is simple in form, and it is anticipated that such resonances exist at every Lamé frequency. Finally, point-wise convergence for Lamb-mode eigenfunction expansions is established.