Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T20:54:44.142Z Has data issue: false hasContentIssue false
  • English
  • Français

Trapped modes of internal waves in a channel spanned by a submerged cylinder

Published online by Cambridge University Press:  26 April 2006

Nikolay Kuznetsov
Nikolay Kuznetsov
Affiliation:
Laboratory on Mathematical Modelling in Mechanics, Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr., 61, VO, 199178, St. Petersburg, Russia
Get access Rights & Permissions [Opens in a new window]

Abstract

A horizontal channel of infinite length and depth and of constant width contains inviscid, incompressible, two-layer fluid under gravity. The upper layer has constant finite depth and is occupied by a fluid of constant density ρ*. The lower layer has infinite depth and is occupied by a fluid of constant density ρ > ρ*. The parameter ε = (ρ/ρ*)–1 is assumed to be small. The lower fluid is bounded internally by an immersed horizontal cylinder which extends right across the channel and has its generators normal to the sidewalls. The free, time-harmonic oscillations of fluid, which have finite kinetic and potential energy (such oscillations are called trapped modes), are investigated. Trapped modes in homogeneous fluid above submerged cylinders and other obstacles are well known. In the present paper it is shown that there are two sets of frequencies of trapped modes for the two-layer fluid. The frequencies of the first finite set are close to the frequencies of trapped modes in the homogeneous fluid (when ρ* = ρ). They correspond to the trapped modes of waves on the free surface of the upper fluid. The frequencies of the second finite set are proportional to ε, and hence, are small. These latter frequencies correspond to the trapped modes of internal waves on the interface between two fluids. To obtain these results the perturbation method for a quadratic operator family was applied. The quadratic operator family with bounded, symmetric, linear, integral operators in the space L2(−∞, +∞) arises as a result of two reductions of the original problem. The first reduction allows to consider the potential in the lower fluid only. The second reduction is the same as used by Ursell (1987).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 254 , September 1993 , pp. 113 - 126
Copyright
© 1993 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

Aranha, J. A. P. 1988 Existence and some properties of waves trapped by submerged cylinder. J. Fluid Mech. 192, 421–433.Google Scholar
Bonnet, A.-S. & Joly, P. 1990 Mathematical and numerical study of trapping waves. In Fifth Intl. Workshop on Water Waves and Floating Bodies, Manchester (ed. P. A. Martin), pp. 25–28.
Callan, M. A. 1990 Trapping modes above non-cylindrical bodies. In Fifth Intl Workshop on Water Waves and Floating Bodies, Manchester (ed. P. A. Martin), pp. 29–33.
Callan, M. A., Linton, C. M. & Evans, D. V. 1991 Trapped waves in two-dimensional waveguides. J. Fluid Mech. 229, 51–64.Google Scholar
Friedrichs, K. O. 1965 Perturbation of Spectra in Hilbert Space. Am. Math. Soc.Google Scholar
Friis, A., Grue, J. & Palm, E. 1991 Application of Fourier transform to the second order 2D wave diffraction problem. In M. P. Tulin's Festschrift: Mathematical Approaches in Hydrodynamics (ed. T. Miloh), pp. 209–227. SIAM.
Gradshtein, I. S. & Ryzhik, I. M. 1965 Tables of Integrals, Series, and Products. Academic.
Jones, D. S. 1953 The eigenvalues of ∇3u + λu = 0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684.Google Scholar
Kato, T. 1966 Perturbation Theory for Linear Operators. Springer.
Kuznetsov, N. G. 1991 Uniqueness of a solution of a linear problem for stationary oscillation of a liquid. Diff. Equat. 27, 187–194.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Martin, P. A. 1989 On the computation and excitation of trapping modes. In Fourth Intl Workshop on Water Waves and Floating Bodies, Univ. of Oslo, Dept. of Math. (ed. J. Grue), pp. 145–148.
McIver, P. 1991 Trapping of surface water waves by fixed bodies in a channel. Q. J. Mech. Appl. Maths 33, 193–208.Google Scholar
McIver, P. & Evans, D. V. 1985 The trapping of surface waves above a submerged horizontal cylinder. J. Fluid Mech. 151, 243–255.Google Scholar
Stokes, G. C. 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. 1987 Mathematical aspects of trapping modes in the theory of surface waves. J. Fluid Mech. 183, 421–437.Google Scholar
Vainberg, B. R. & Maz'ya, V. G. 1973 On the problem of the steady state oscillations of a fluid layer of variable depth. Trans. Moscow Math. Soc. 28, 56–73.Google Scholar
Vladimirov, V. S. 1971 Equations of Mathematical Physics. Marcel Dekker.