Trapped modes in a waveguide with a long obstacle

Khallaf, N. S. A.; Parnovski, Leonid; Vassiliev, Dmitri

doi:10.1017/s0022112099007028

Cited by 20 publications

(20 citation statements)

References 8 publications

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“…Trapped modes occuring in two-dimensional acoustic waveguides containing long obstacles symmetric about the centreline were considered by Khallaf, Parnovski, and Vassiliev (2000). In their paper the authors use variational arguments to provide estimates for the trapped mode frequencies and also to prove that the number of trapped modes occuring is asymptotically proportional to the obstacle's length.…”

Section: Variational Methodsmentioning

confidence: 99%

Bound states in coupled guides. I. Two dimensions

Linton

Ratcliffe

2004

Journal of Mathematical Physics

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Bound states in coupled guides. I. Two dimensions.C. M. Linton and K. RatcliffeBound states that can occur in coupled quantum wires are investigated. We consider a two-dimensional configuration in which two parallel waveguides (of different widths) are coupled laterally through a finite length window and construct modes which exist local to the window connecting the two guides. We study both modes above and below the first cut-off for energy propagation down the coupled guide. The main tool used in the analysis is the so-called residue calculus technique in which complex variable theory is used to solve a system of equations which is derived from a mode-matching approach. For bound states below the first cut-off a single existence condition is derived, but for modes above this cut-off (but below the second cut-off), two conditions must be satisfied simultaneously. A number of results have been presented which show how the bound-state energies vary with the other parameters in the problem.

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Section: Variational Methodsmentioning

confidence: 99%

Bound states in coupled guides. I. Two dimensions

Linton

Ratcliffe

2004

Journal of Mathematical Physics

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“…The case of a "window" in the Dirichlet boundary appeared earlier in [ELV94] within the context of a Neumann guide with an obstacle. The main physical motivation comes in this case from acoustics; more general results on embedded eigenvalues in straight channels due to symmetric Neumann obstacles, not necessarily of zero measure, can be found in [DP98] or [KPV00]. The spectrum in the mixed-condition situation of Problem 4, which can be regarded as a twodimensional version of "twisted" boundary conditions, was found numerically in [DKř02b] using mode matching; the result suggests that the first critical value is a 1 ≈ 0.26 d, more recent results on such systems can be found in [BC11,BC12].…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

137

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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“…This paper is related to an extensive series of works on trapped modes in perturbed quantum [1]- [5], acoustic [6]- [10], and elastic waveguides [11]- [15].…”

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confidence: 99%

Trapped modes in an elastic plate with a hole

Förster

Weidl

2012

St. Petersburg Math. J.

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Abstract. For an infinite linear elastic plate with stress-free boundary, the trapped modes arising around holes in the plate are investigated. These are L 2 -eigenvalues of the elastostatic operator in the punched plate subject to Neumann type stress-free boundary conditions at the surface of the hole. It is proved that the perturbation gives rise to infinitely many eigenvalues embedded into the essential spectrum. The eigenvalues accumulate to a positive threshold. An estimate of the accumulation rate is given. §1. Introduction We consider a homogeneous and isotropic linear elastic medium in the domainThen the quadratic forminduces the elastostatic operator A for materials with zero Poisson ratio; A corresponds to the differential expressionwith stress-free (Neumann-type) boundary conditions. 1 The operator A has purely absolutely continuous spectrum filling the nonnegative half-line. Now, consider a punched plate Ω c = G \ Ω with a hole Ω = Ω 0 × J, where Ω 0 ⊂ R 2 is a bounded Lipschitz domain. Let A Ω c be the elastostatic operator corresponding to the differential expression (1.2) on the outer domain Ω c subject to stress-free (Neumanntype) boundary conditions. This geometric perturbation does not change the location of the essential spectrum, but it gives rise to a somewhat unexpected trapping effect. As the main result of this paper we prove the existence of infinitely many eigenvalues {ν k } k∈N of A Ω c , embedded into the essential spectrum, which accumulate at a certain threshold Λ > 0, and we compute the accumulation rate of these trapped modes. We establish the formula2010 Mathematics Subject Classification. Primary 74B05. Key words and phrases. Elasticity operator, trapped modes. 1 Here we have chosen a suitable set of units such that Young's modulus E fulfills E = 2. Moreover, we point out that the special choice of zero Poisson ratio is essential for the results of this paper.

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Trapped modes in a waveguide with a long obstacle

Cited by 20 publications

References 8 publications

Bound states in coupled guides. I. Two dimensions

Bound states in coupled guides. I. Two dimensions

Quantum Waveguides

Trapped modes in an elastic plate with a hole

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