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Nonlinear surface waves in closed basins

Published online by Cambridge University Press:  29 March 2006

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

The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics. This requires a preliminary description, in terms of q, of the fluid motion beneath the free surface, which kinematical boundary-value problem is solved through a variational formulation and the truncation and inversion of an infinite matrix. The results are applied to weakly coupled oscillations, using the time-averaged Lagrangian, and to resonantly coupled oscillations, using Poincaré's action—angle formulation. The general formulation provides for excitation through either horizontal or vertical translation of the basin and for dissipation. Detailed results are given for free and forced oscillations of two, resonantly coupled degrees of freedom.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 75 , Issue 3 , 11 June 1976 , pp. 419 - 448
Copyright
© 1976 Cambridge University Press

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References

Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. Roy. Soc A 225, 505–515.Google Scholar
Beth, H. J. E. 1913 The oscillations about a position of equilibrium where a simple linear relation exists between the frequencies of the principal vibrations Phil. Mag. 26, 268–324.Google Scholar
Brillouin, L. 1960 Poincaré and the shortcomings of the Hamilton—Jacobi method for classical or quantized mechanics Archiv. Rat. Mech. Anal. 5, 76–94.Google Scholar
Bryant, P. J. 1973 Periodic waves in shallow water J. Fluid Mech. 59, 625–644.Google Scholar
Chester, W. 1968 Resonant oscillations of water waves. I. Theory. Proc. Roy. Soc A 306, 5–22.Google Scholar
Chester, W. & Bones, J. A. 1968 Resonant oscillations of water waves. II. Experiment. Proc. Roy. Soc A 306, 23–39.Google Scholar
Hutton, R. E. 1963 An investigation of resonant, nonlinear, nonplanar free surface oscillations of a fluid. N.A.S.A. Tech. Note, D-1870. (See also Ph.D. dissertation, University of California, Los Angeles, 1962.)Google Scholar
Korteweg, D. J. 1897 Over zekere trillingen van hooger orde van abnormale intensiteit (relatietrillingen) bij mechanismen met meerdere graden van vrijheid. Verhandelingen der Koninklyke Akademie van Wettenschappen, 5; Archives Neérlandaises, 1 (2), 229–810 (cited by Beth 1913).Google Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface J. Fluid Mech. 27, 395–397.Google Scholar
Mcgoldrick, L. F. 1970 On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193–810.Google Scholar
Mcintyre, M. E. 1972 On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow J. Fluid Mech. 52, 209–243.Google Scholar
Mack, L. R. 1962 Periodic, finite-amplitude, axisymmetric gravity waves J. Geophys. Res. 67, 829–843.Google Scholar
Mettler, E. 1963 Kleine Schwingungen und Methode der säkularen Störungen Z. angew. Math. Mech. 43, T81–85.Google Scholar
Miles, J. W. 1962 Stability of forced oscillations of a spherical pendulum Quart. Appl. Math. 20, 21–32.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. Roy. Soc A 297, 459–475.Google Scholar
Minorsky, N. 1947 Non-Linear Mechanics. Ann Arbor: J. W. Edwards.
Poincaré, H. 1982 1982–810/1957 1892–810/1957 Les Méthodes Nouvelle de la Méchanique Céleste (three volumes). Dover.
Rayleigh, Lord 1915 Deep water waves, progressive or stationary, to the third order of approximation. Proc. Roy. Soc A 91, 345–353. (See also Scientific Papers, vol. 6, pp. 306–314. Cambridge University Press.)Google Scholar
Rott, N. 1970 A multiple pendulum for the demonstration of non-linear coupling Z. angew. Math. Phys. 21, 570–582.Google Scholar
Serrin, J. 1959 Mathematical principles of classical fluid mechanics. Encyclopedia Phys. 8, $24. Springer.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. Roy. Soc A 309, 551–575.Google Scholar
Struble, R. A. & Heinbockel, J. H. 1963 Resonant oscillations of a beam—pendulum system J. Appl. Mech. 30, 181–188.Google Scholar
Tdjbakhsh, I. & Keller, J. B. 1960 Standing surface waves of finite amplitude J. Fluid Mech. 8, 442–451.Google Scholar
Verma, G. H. & Keller, J. B. 1962 Three-dimensional standing surface waves of finite amplitude Phys. Fluids, 5, 52–56.Google Scholar
Wehausen, J. V. & Laitone, E. T. 1960 Surface waves Encyclopedia Phys. 9, 446–778.Google Scholar
Whitham, G. B. 1967 Variational methods and applications to water waves. Proc. Roy. Soc A 299, 6–25.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves Interscience.
Whittaker, E. T. 1944 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Dover.