Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-18T18:53:25.985Z Has data issue: false hasContentIssue false
  • English
  • Français

Trapped modes in open channels

Published online by Cambridge University Press:  26 April 2006

D. V. Evans  and
C. M. Linton
D. V. Evans
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
C. M. Linton
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Trapped or edge-wave modes are well-known in linear water-wave theory. They occur at discrete frequencies below a certain cutoff frequency and consist of local oscillations trapped near a long horizontal submerged body in finite or infinite depth or over a sloping beach. Less well known is the existence of trapped modes in certain problems in acoustics where the governing equation is the Helmholtz equation. Jones (1953) has proved the existence of such modes which correspond to point-eigenvalues of the spectrum of the differential operator satisfying certain boundary conditions in a semi-infinite region. In this paper we describe a constructive method for determining point-eigenvalues or trapped-mode frequencies in two specific problems in which the two-imensional Helmholtz equation is satisfied.

The problems arise from a consideration of the fluid motion in a long narrow wave tank with a free water surface which contains a vertical cylinder of uniform horizontal cross-section extending throughout the water depth. Separation of the depth dependence results in Helmholtz's equation with Neumann boundary conditions. By seeking solutions which are antisymmetric with respect to the centreline of the channel, trapped modes are constructed for the case of a cylinder of rectangular cross-section placed symmetrically in the centre of the channel and also for the case of a symmetric rectangular indentation in the tank walls. These problems do not appear to be covered directly by Jones’ theory and whilst the method described provides convincing numerical evidence, it falls short of a rigorous existence proof. Extensions to other purely acoustic problems having no water-wave interpretation, including problems which are covered by the general theory of Jones, are also discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 225 , April 1991 , pp. 153 - 175
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Evans, D. V. & McIver, P., 1984 Edge waves over a shelf: full linear theory. J. Fluid Mech. 142, 79–95.Google Scholar
Jones, D. S.: 1953 The eigenvalues of V2u+λ = 0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684.Google Scholar
Le Blond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.
McIver, P. & Evans, D. V., 1985 The trapping of surface waves above a submerged horizontal cylinder. J. Fluid Mech. 151, 243–255.Google Scholar
Noblk, B.: 1958 Methods based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon.
Razavy, M.: 1989 An acoustic waveguide with variable cross section. J. Acoust. Soc. Am. 86, 1155–1160.Google Scholar
Stokes, G. G.: 1846 Report on recent researches in hydrodynamics. Brit. Assoc. Rep.Google Scholar
Ursell, F.: 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.Google Scholar
Ursell, F.: 1952 Edge waves over a sloping beach. Proc. R. Soc. Land. A 214, 79–97.Google Scholar
Ursell, F.: 1987 Mathematical aspects of trapping modes in the theory of surface waves. J. Fluid Mech. 183, 421–37.Google Scholar