Skip to main content Accessibility help
×
Home

Trapped modes in open channels

  • D. V. Evans (a1) and C. M. Linton (a1)

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.

Copyright

References

Hide All
Evans, D. V. & McIver, P., 1984 Edge waves over a shelf: full linear theory. J. Fluid Mech. 142, 7995.
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, 668684.
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, 243255.
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, 11551160.
Stokes, G. G.: 1846 Report on recent researches in hydrodynamics. Brit. Assoc. Rep.
Ursell, F.: 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347358.
Ursell, F.: 1952 Edge waves over a sloping beach. Proc. R. Soc. Land. A 214, 7997.
Ursell, F.: 1987 Mathematical aspects of trapping modes in the theory of surface waves. J. Fluid Mech. 183, 42137.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Trapped modes in open channels

  • D. V. Evans (a1) and C. M. Linton (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed