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Surface waves in basins of variable depth

Published online by Cambridge University Press:  20 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla, California 92093
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Abstract

The linearized boundary-value problem for surface waves of frequency ω in a closed basin of variable depth is reduced to a non-self-adjoint partial differential equation in the plane of the free surface. The corresponding variational form (which does not provide a definite upper or lower bound) for the eigenvalue κ = ω2/g is constructed. A self-adjoint partial differential equation, for which the variational form is the Rayleigh quotient (which provides an upper bound to κ), also is constructed; it offers significant advantages vis-à-vis the non-self-adjoint formulation, but at the expense of a more complicated operator. Three relatively simple variational approximations are constructed, two for a class of basins with sloping sides and the third for basins for which the variation of the depth relative to its mean is small. These general results are illustrated by comparison with Rayleigh's (1899) results for a semicircular channel, Sen's (1927) inverse results for a family of circular basins, and Lamb's (1932) results for a shallow circular paraboloid. The eigenvalue for the dominant mode in the paraboloid is determined through O(δ5), where δ = depth/radius.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 152 , March 1985 , pp. 379 - 389
Copyright
© 1985 Cambridge University Press

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References

Ball, F. K. 1963 An exact theory of simple finite shallow water oscillations on a rotating Earth. In Proc. 1st Austral. Conf. on Hydraulics and Fluid Mech., 1962, pp. 293–305. Pergamon.
Chrystal, G. 1905 On the hydrodynamical theory of seiches. Trans. R. Soc. Edinb. 41, 599–649.Google Scholar
Finlayson, B. A. 1972 The Method of Weighted Residuals and Variational Principles, p. 188. Academic Press.
Goldsbrough, G. R. 1930 The tidal oscillations in an elliptic basin of variable depth. Proc. R. Soc. Lond. A 130, 157–167.Google Scholar
Hutchinson, G. E. 1975 A Treatise on Limnology, vol. 1, part 1, pp. 299–331. Wiley.
Kirchhoff, G. 1879/1880 Über stehende Schwingungen einer schweren Flüssigkeit. Mber. Akad. Wiss. Berlin 1879, 395–410; Ann. Phys. Chem., N.F. 10 (246), 34–46 (1880) (cited by Wehausen & Laitone 1960).
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lundberg, P. 1984 Nonhydrostatic waves in a parabolic basin. Z. angew. Math. Mech. 64, 1–6.Google Scholar
Poisson, S. 1820 Sur l'intégration de quelques équations linéaires aux différences partielles, et particulièrement de l’équation générate du mouvement des fluides élastiques. Mém. de l'Institut 3, 121 (cited by Rayleigh 1877).Google Scholar
Proudman, J. 1915 Free and forced longitudinal tidal motion in a lake. Proc. Lond. Math. Soc. (2) 14, 240–250.Google Scholar
Rayleigh, Lord 1873 Some general theorems relating to vibrations. Proc. Lond. Math. Soc. 4, 357–368 (Scientific Papers, vol. 1, pp. 170–181).Google Scholar
Rayleigh, Lord 1878 Theory of Sound, vol. 2, 273. Macmillan.
Rayleigh, Lord 1899 On the calculation of the frequency of vibration of a system in its gravest mode, with an example from hydrodynamics. Phil. Mag. 47, 566–572 (Scientific Papers, vol. 4, pp. 407–412).Google Scholar
Sen, B. M. 1927 Waves in canals and basins. Proc. Lond. Math. Soc. (2) 26, 363–376.Google Scholar
Storchi, E. 1949 Legame fra la forma del pelo libero e quella recipiente nelle oscillazioni di un liquido. 1st Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat. (3) 13(82), 95–112 (cited by Wehausen & Laitone 1960).Google Scholar
Storchi, E. 1952 Piccole oscillazioni dell'acqua contenuta da pareti piane. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8) 12, 544–552 (cited by Wehausen & Laitone 1960).Google Scholar
Vint, J. 1923 Surface waves on limited sheets of water. Proc. Lond. Math. Soc. (2) 22, 1–14.Google Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface waves. Encyclopedia of Physics 9, 624–626.Google Scholar