Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T21:58:17.266Z Has data issue: false hasContentIssue false
  • English
  • Français

On direct methods in water-wave theory

Published online by Cambridge University Press:  21 April 2006

Jonathan J. Shields  and
William C. Webster
Jonathan J. Shields
Affiliation:
Department of Naval Architecture and Offshore Engineering, University of California at Berkeley, CA 94720, USA
William C. Webster
Affiliation:
Department of Naval Architecture and Offshore Engineering, University of California at Berkeley, CA 94720, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Model equations for three-dimensional, inviscid flow between two arbitrary, time-varying material surfaces are derived using a ‘direct’ or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to ‘the theory of directed fluid sheets’ of Green & Naghdi (1974, 1976).

The theory can be used to study the propagation of long waves in water of finite depth and, as such, competes with theories derived using the classical Rayleigh–Boussinesq perturbation methods. In order to demonstrate that there is an advantage to the present approach, we compare predictions for steady, two-dimensional waves over a horizontal bottom. Numerical solutions indicate that the direct theory converges more rapidly than the perturbation theories. Also, the equations of the higher-order direct theories contain singularities related to waves of limiting height, and indeed such waves can be predicted with relative accuracy. Finally, the range of applicability of the direct theory is far greater: waves as short as three times the water depth can be modelled. This is essentially a deep-water condition, well beyond the range of convergence of the Rayleigh–Boussinesq approach.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 197 , December 1988 , pp. 171 - 199
Copyright
© 1988 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

Antman, S. S.1972 The theory of rods. In Handbuch der Physik (ed. S. Flügge), vol. VIa/2, pp. 641–703. Springer.
Benjamin, T. B. & Lighthill, M. J.1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448–460.Google Scholar
Boussinesq, J.1871 Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Acad. Sci. Paris 72, 755–759.Google Scholar
Boussinesq, J.1877 Essai sur la théorie des eaux courantes. Mem. Divers Savants Acad. Sci. Inst. France, Sci. Math. Phys. Paris 23, 1–680; Additions et éclaircissements, 24, 1–64.Google Scholar
Byatt-Smith, J. G. B. & Longuet-Higgins, M. S. 1976 On the speed and profile of steep solitary waves. Proc. R. Soc. Lond. A 350, 175–189. (Numerical values for the curves presented in this paper provided by J. G. B. Byatt-Smith in a private communication, 1987.)Google Scholar
Cokelet, E. D.1977 Step gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
De, S. C.1955 Contributions to the theory of Stokes waves. Proc. Camb. Phil. Soc. 51, 713–736.Google Scholar
Ertekin, R. C.1984 Soliton generation by moving disturbances in shallow water. Ph.D. thesis. University of California, Berkeley. v + 352 pp.
Ertekin, R. C., Webster, W. C. & Wehausen, J. V.1984 Ship-generated solitons. Proc. 15th Symp. Naval Hydrodynam., Hamburg, pp. 347–361, disc. 361–364.
Ertekin, R.C., Webster, W. C. & Wehausen, J. V.1986 Waves caused by a moving disturbance in a shallow channel of finite width. J. Fluid Mech. 169, 275–292.Google Scholar
Fenton, J. D.1972 A ninth-order solution for the solitary wave. J. Fluid Mech. 53, 237–246.Google Scholar
Fenton, J. D.1979 A high-order cnoidal wave theory. J. Fluid Mech. 94, 129–161.Google Scholar
Fenton, J. D.1985 A fifth-order Stokes theory for steady waves. J. Waterway, Port, Coastal and Ocean Engng, ACSE 111, 216–234.Google Scholar
Green, A. E., Laws, N. & Naghdi, P. M.1974 On the theory of water waves. Proc. R. Soc. Lond. A 338, 43–55.Google Scholar
Green, A. E. & Naghdi, P. M.1976 Directed fluid sheets. Proc. R. Soc. Lond. A 347, 447–473.Google Scholar
Green, A. E. & Naghdi, P. M.1977 Water waves in a non-homogeneous incompressible fluid. Trans. ASME E: J. Appl. Mech. 44, 523–528.Google Scholar
Green, A. E. & Naghdi, P. M.1984 A direct theory of viscous fluid flow in channels. Arch. Rat. Mech. Anal 86, 39–63.Google Scholar
Grimshaw, R.1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 86, 415–431.Google Scholar
Kantorovich, L. V. & Krylov, V. I.1958 Approximate Methods of Higher Analysis. P. Noordhoff Ltd. (Groningen, The Netherlands 1964), 681 pp.
Laitone, E. V.1960 The second approximation to cnoidal and solitary waves. J. Fluid Mech. 9, 430–444.Google Scholar
Lamb, H.1932 Hydrodynamics, 6th edn. Cambridge University Press, 1932 (Dover, New York, 1945), 738 pp.
Levich, V. G. & Krylov, V. S.1969 Surface-tension-driven phenomena. Ann. Rev. Fluid Mech. 1, 293–316.Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum, energy and circulation of a solitary wave. II. Proc. R. Soc. Lond. A 340, 471–493.Google Scholar
Madsen, O. S. & Mei, C. C.1969 The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. 39, 781–791.Google Scholar
Miles, J. W.1979 On the Korteweg—de Vries equation for a gradually varrying channel. J. Fluid Mech. 91, 181–190.Google Scholar
Miles, J. W. & Salmon, R.1985 Weakly dispersive, nonlinear gravity waves. J. Fluid Mech. 157, 519–531.Google Scholar
Naghdi, P. M. & Rubin, M. B.1981a On the transition to planing of a boat. J. Fluid Mech. 103, 345–374.Google Scholar
Naghdi, P. M. & Rubin, M. B.1981b On inviscid flow in a waterfall. J. Fluid Mech. 103, 375–387.Google Scholar
Naghdi, P. M. & Vongsarnpigoon, L.1986 The downstream flow beyond an obstacle. J. Fluid Mech. 162, 223–236.Google Scholar
Peregrine, D. H.1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Rayleigh, Lord1876 On waves. Phil. Mag. 1, 257–279; Scientific Paper, vol. 1, pp. 251–271.Google Scholar
Schwartz, L. W.1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Shields, J. J.1986 A direct theory for waves approaching a beach. Ph.D. thesis, University of California, Berkeley, iii + 137 pp.
Shields, J. J. & Webster, W. C.1988 Conservation of mechanical energy and circulation in the theory of inviscid fluid sheets. J. Engng Maths (in press).Google Scholar
Skjelbreia, L. & Hendrickson, J.1961 Fifth order gravity wave theory. Proc. 7th Conf. Coastal Engng, pp. 184–196.
Stokes, G. G.1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Su, C. H. & Mirie, R. M.1980 On head-on collisions between two solitary waves. J. Fluid Mech. 98, 509–525.Google Scholar
Ursell, F.1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685–694.Google Scholar
Whitham, G. B.1967 Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399–412.Google Scholar
Wu, T. Y.1981 Long waves in ocean and coastal waters. J. Engng Mech. Div. ASCE 107, EM3, pp. 501–522.Google Scholar