Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section

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Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section. / Nazarov, S. A.

In: St. Petersburg Mathematical Journal, Vol. 31, No. 5, 10.2020, p. 865-885.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{b05451ecf9744dae894e027db8ce0145,

title = "Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section",

abstract = "Cylindrical acoustic waveguides with constant cross-section ω are considered, specifically, a straight waveguide Ω = R × ω ⊂ Rd and a locally curved waveguide Ωε that depends on a parameter ε ∈ (0, 1]. For d > 2, in two different settings (ε = 1 and ε ≪ 1), the task is to find an eigenvalue λε that is embedded in the continuous spectrum [0,+∞) of the waveguide Ωε and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator Δ + λε arises that vanishes at infinity and implies an eigenfunction in the Sobolev space H1(Ωε). In the first case, it is assumed that the cross-section ω has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide Ωε. In the second case, under an assumption on the shape of an asymmetric cross-section ω, the eigenvalue λε is formed by scrupulous fitting of the curvature O(ε) for small ε > 0.",

keywords = "Asymptotics, Continuous and point spectra, Curved cylinder, Eigenvalue, Expanded scattering matrix, Neumann problem for the laplace operator, expanded scattering matrix, CONTINUOUS-SPECTRUM, Neumann problem for the Laplace operator, BOUND-STATES, curved cylinder, asymptotics, eigenvalue",

author = "Nazarov, {S. A.}",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary 35P15. Key words and phrases. Continuous and point spectra, eigenvalue, Neumann problem for the Laplace operator, curved cylinder, asymptotics, expanded scattering matrix. This work was supported by the Russian Science Foundation. (Project 17-11-01003). Publisher Copyright: {\textcopyright} 2020 American Mathematical Society. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",

year = "2020",

month = oct,

doi = "10.1090/spmj/1626",

language = "English",

volume = "31",

pages = "865--885",

journal = "St. Petersburg Mathematical Journal",

issn = "1061-0022",

publisher = "American Mathematical Society",

number = "5",

}

RIS

TY - JOUR

T1 - Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section

AU - Nazarov, S. A.

N1 - Funding Information: 2010 Mathematics Subject Classification. Primary 35P15. Key words and phrases. Continuous and point spectra, eigenvalue, Neumann problem for the Laplace operator, curved cylinder, asymptotics, expanded scattering matrix. This work was supported by the Russian Science Foundation. (Project 17-11-01003). Publisher Copyright: © 2020 American Mathematical Society. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/10

Y1 - 2020/10

N2 - Cylindrical acoustic waveguides with constant cross-section ω are considered, specifically, a straight waveguide Ω = R × ω ⊂ Rd and a locally curved waveguide Ωε that depends on a parameter ε ∈ (0, 1]. For d > 2, in two different settings (ε = 1 and ε ≪ 1), the task is to find an eigenvalue λε that is embedded in the continuous spectrum [0,+∞) of the waveguide Ωε and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator Δ + λε arises that vanishes at infinity and implies an eigenfunction in the Sobolev space H1(Ωε). In the first case, it is assumed that the cross-section ω has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide Ωε. In the second case, under an assumption on the shape of an asymmetric cross-section ω, the eigenvalue λε is formed by scrupulous fitting of the curvature O(ε) for small ε > 0.

AB - Cylindrical acoustic waveguides with constant cross-section ω are considered, specifically, a straight waveguide Ω = R × ω ⊂ Rd and a locally curved waveguide Ωε that depends on a parameter ε ∈ (0, 1]. For d > 2, in two different settings (ε = 1 and ε ≪ 1), the task is to find an eigenvalue λε that is embedded in the continuous spectrum [0,+∞) of the waveguide Ωε and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator Δ + λε arises that vanishes at infinity and implies an eigenfunction in the Sobolev space H1(Ωε). In the first case, it is assumed that the cross-section ω has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide Ωε. In the second case, under an assumption on the shape of an asymmetric cross-section ω, the eigenvalue λε is formed by scrupulous fitting of the curvature O(ε) for small ε > 0.

KW - Asymptotics

KW - Continuous and point spectra

KW - Curved cylinder

KW - Eigenvalue

KW - Expanded scattering matrix

KW - Neumann problem for the laplace operator

KW - expanded scattering matrix

KW - CONTINUOUS-SPECTRUM

KW - Neumann problem for the Laplace operator

KW - BOUND-STATES

KW - curved cylinder

KW - asymptotics

KW - eigenvalue

UR - http://www.scopus.com/inward/record.url?scp=85091672157&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/3d3f2e74-a797-3162-9cc6-6015c863876b/

U2 - 10.1090/spmj/1626

DO - 10.1090/spmj/1626

M3 - Article

AN - SCOPUS:85091672157

VL - 31

SP - 865

EP - 885

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 5

ER -

ID: 71562148