Edge resonance in an elastic semi-strip

We study the elasticity operator in a semi-strip subject to free boundary conditions. In the case of zero Poisson ratio we prove the existence of a positive eigenvalue embedded in the essential spectrum. Physically, the eigenvalue corresponds to a 'trapped mode', that is, a harmonic oscillation of the semi-strip localized near the edge. This effect, known in mechanics as the 'edge resonance', has been extensively studied numerically and experimentally. Our result provides a mathematical justification.