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Existence theorems for trapped modes

  • D. V. Evans (a1), M. Levitin (a2) and D. Vassiliev (a3)
Abstract

A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.

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Callan, M., Linton, C. M. & Evans, D. V. 1991 Trapped modes in two-dimensional waveguides. J. Fluid Mech. 229, 5164.

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Evans, D. V., Linton, C. M. & Ursell, F.1993Trapped mode frequencies embedded in the continuous spectrum. Q. J. Mech. Appl. Maths (to be published).

Jones, D. S. 1953 The eigenvalues of V2uu = 0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668684.

Linton, C. M. & Evans, D. V. 1992 Integral equations for a class of problems concerning obstacles in waveguides. J. Fluid Mech. 245, 349365.

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Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347358.

Ursell, F.1991Trapped modes in a circular cylindrical acoustic waveguide. Proc. R. Soc. Lond. A 435, 575589.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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