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Edge resonance in an elastic semi-infinite cylinder

Holst, Anders LU and Vassiliev, Dmitri G. (2000) In Applicable Analysis 74(3-4). p.479-495
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 I. Roitberg, D. G. Vasilʹev and T. Weidl [Quart. J. Mech. Appl. Math. 51 (1998), no. 1, 1--13; MR1610688 (98m:73041)], 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... (More)
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 I. Roitberg, D. G. Vasilʹev and T. Weidl [Quart. J. Mech. Appl. Math. 51 (1998), no. 1, 1--13; MR1610688 (98m:73041)], 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 Roitberg et al. [op. cit.] to a three-dimensional setting. (Less)
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Contribution to specialist publication or newspaper
publication status
published
subject
categories
Popular Science
in
Applicable Analysis
volume
74
issue
3-4
pages
479 - 495
publisher
Taylor & Francis
ISSN
0003-6811
language
English
LU publication?
yes
id
4f934eb0-0a22-47a9-9971-3ff4d550a8fb (old id 1220484)
date added to LUP
2008-09-01 11:08:43
date last changed
2017-02-09 12:02:28
@misc{4f934eb0-0a22-47a9-9971-3ff4d550a8fb,
  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 I. Roitberg, D. G. Vasilʹev and T. Weidl [Quart. J. Mech. Appl. Math. 51 (1998), no. 1, 1--13; MR1610688 (98m:73041)], 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 Roitberg et al. [op. cit.] to a three-dimensional setting.},
  author       = {Holst, Anders and Vassiliev, Dmitri G.},
  issn         = {0003-6811},
  language     = {eng},
  number       = {3-4},
  pages        = {479--495},
  publisher    = {Taylor & Francis},
  series       = {Applicable Analysis},
  title        = {Edge resonance in an elastic semi-infinite cylinder},
  volume       = {74},
  year         = {2000},
}