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Existence theorems for trapped modes

Published online by Cambridge University Press:  26 April 2006

D. V. Evans ,
M. Levitin  and
D. Vassiliev
D. V. Evans
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
M. Levitin
Affiliation:
Department of Mathematics, Heriot–Watt University, Riccarton, Edinburgh EH14 4AS, UK
D. Vassiliev
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, UK
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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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 261 , 25 February 1994 , pp. 21 - 31
Copyright
© 1994 Cambridge University Press

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References

Birman, M. S. & Solomjak, M. Z. 1987 Spectral Theory of Self-Adjoint Operators in Hilbert Spaces. D. Reidel.
Callan, M., Linton, C. M. & Evans, D. V. 1991 Trapped modes in two-dimensional waveguides. J. Fluid Mech. 229, 51–64.Google Scholar
Evans, D. V. 1992 Trapped acoustic modes. IMA J. Appl. Maths 49, 45–60.Google Scholar
Evans, D. V. & Linton, C. M. 1991 Trapped modes in open channels. J. Fluid Mech. 225, 153–175.Google Scholar
Evans, D. V., Linton, C. M. & Ursell, F. 1993 Trapped mode frequencies embedded in the continuous spectrum. Q. J. Mech. Appl. Maths (to be published).Google Scholar
Jones, D. S. 1953 The eigenvalues of V2u+Àu = 0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684.Google Scholar
Linton, C. M. & Evans, D. V. 1992 Integral equations for a class of problems concerning obstacles in waveguides. J. Fluid Mech. 245, 349–365.Google Scholar
Parker, R. 1966 Resonance effects in water shedding from parallel plates: some experimental observations. J. Sound Vib. 4, 62–72.Google Scholar
Parker, R. & Stoneman, S. A. T. 1989 The excitation and consequences of acoustic resonances in enclosed fluid flow around solid bodies. Proc. Inst. Mech. Engrs 203, 9–19.Google Scholar
Sanchez-Hubert, J. & Sanchez-Palencia, E. 1989 Vibration and Coupling of Continuous Systems. Springer.
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.Google Scholar
Ursell, F. 1991 Trapped modes in a circular cylindrical acoustic waveguide. Proc. R. Soc. Lond. A 435, 575–589.Google Scholar