Edge resonance in an elastic semi-infinite cylinder |
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Authors: | Anders Holst Dmitri Vassiliev |
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Affiliation: | 1. Centre for Mathematical Sciences , Lund University , Box 118, Lund, SE, 22100, Sweden E-mail: ah@maths.lth.se;2. School of Mathematical Sciences , University of Bath , Bath BA2 7AY, Claverton Down, UK E-mail: D.Vassiliev@bath.ac.uk |
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Abstract: | We study the three-dimensional elasticity operator in a semi-infinite circular cylinder subject to free boundary conditions, in the case of zero Poisson ratio. We prove, adapting the method from [15], i.e., by first finding an invariant subspace for the elasticity operator such that the essential spectrum has a strictly positive lower bound and then finding a test function in this space for which the variational quotient takes a value below the bottom of the essential spectrum, that there is an eigenvalue embedded in the continuous spectrum. Physically, an eigenvalue corresponds to a "trapped mode", that is, a harmonic oscillation localized near the edge. This effect, known in mechanics as the "edge resonance" has been extensively studied numerically and experimentally. Our paper extends the mathematical justification of such phenomena provided by [15] to a three-dimensional setting |
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Keywords: | Edge resonance elastic waves spectral theory trapped modes |
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