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Internal waves of finite amplitude and permanent form

  • T. Brooke Benjamin (a1)

Abstract

A theory is derived for the class of long two-dimensional waves, comprising solitary and periodic cnoidal waves, that can propagate with unchanging form in heterogeneous fluids. The treatment is generalized to the extent that the waves are supposed to arise on a horizontal stream of incompressible fluid whose density and velocity are arbitrary functions of height, and the upper surface of the fluid is allowed either to be free or to be fixed in a horizontal plane. Explicit formulae for the wave properties and a general interpretation of the physical conditions for the occurrence of the waves are achieved without need to specify particular physical models; but in a later part of the paper, §4, the results are applied to three examples that have been worked out by other means and so provide checks on the present theory. These general results are also shown to accord nicely with the principle of ‘conjugate-flow pairs’ which was explained by Benjamin (1962b) with reference to swirling flows along cylindrical ducts, but which is known to apply equally well to flow systems of the kind in question here.

The theory reveals certain physical peculiarities of a type of flow model often used in theoretical studies of internal-wave phenomena, being specified so as to make the equation for the stream-function linear. In an appendix, some observations are also made regarding the ‘Boussinesq approximation’, which too is often used as a simplifying assumption in this field. It is shown, adding to a recent discussion by Long (1965), that finite internal waves may depend crucially on small effects neglected in this approximation.

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References

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Benjamin, T. B. 1956 On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech. 1, 227.
Benjamin, T. B. 1962a The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97.
Benjamin, T. B. 1962b Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593.
Benjamin, T. B. 1965 Significance of the vortex breakdown phenomenon. Trans. Amer. Soc. Mech. Engrs, J. Basic Eng. 87, 518.
Benjamin, T. B. & Barnard, B. J. S. 1964 A study of the motion of a cavity in a rotating liquid. J. Fluid Mech. 19, 193.
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. Roy. Soc., A 224, 448.
Benney, D. J. 1966 Long non-linear waves in fluid flows. To appear in J. Math. Phys.
Dubreil-Jacotin, M. L. 1937 Sur les théorèmes d'existence relatifs aux ondes permanentes périodiques à deux dimensions dans les liquides hétérogènes. J. Math. Pures Appl. (9), 19, 43.
Friedrichs, K. O. & Hyers, D. H. 1954 The existence of solitary waves. Comm. Pure Appl. Math. 7, 517.
Howard, L. N. 1961 Note on a paper by John W. Miles. J. Fluid Mech. 10, 509.
Ince, E. L. 1926 Ordinary Differential Equations. London: Longmans. (Dover edition 1956.)
Jeffreys, H. & Jeffreys, B. S. 1946 Methods of Mathematical Physics. Cambridge University Press.
Keulegan, G. H. 1953 Characteristics of internal solitary waves. J. Res., Nat. Bureau of Standards, 51, 133.
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal and a new type of long stationary waves. Phil. Mag. (5), 39, 422.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.(Dover edition 1945.)
Littman, W. 1957 On the existence of periodic waves near critical speed. Comm. Pure Appl. Math. 10, 241.
Long, R. R. 1953 Some aspects of the flow of stratified fluids. Part I. A theoretical investigation. Tellus, 5, 42.
Long, R. R. 1956 Solitary waves in one- and two-fluid systems. Tellus, 8, 460.
Long, R. R. 1958 Tractable models of steady-state stratified flow with shear. Quart. J. Roy. Meteor. Soc. 84, 159.
Long, R. R. 1964 The initial-value problem for long waves of finite amplitude. J. Fluid Mech. 20, 161.
Long, R. R. 1965 On the Boussinesq approximation and its role in the theory of internal waves. Tellus, 17, 46.
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496.
Miles, J. W. 1963 On the stability of heterogeneous shear flows. Part 2. J. Fluid Mech. 16, 20.
Peters, A. S. & Stoker, J. J. 1960 Solitary waves in liquids having non-constant density. Comm. Pure Appl. Math. 13, 115.
Shen, M. C. 1964 Solitary waves in compressible media. New York Univ. Inst. Math. Sci., Rep. IMM-NYU 325.
Shen, M. C. 1965 Solitary waves in running gases. New York Univ. Inst. Math. Sci., Rep. IMM-NYU 341.
Ter-Krikorov, A. M. 1963 Théorie exacte des ondes longues stationnaires dans un liquide hétérogène. J. Mécanique, 2, 351.
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685.
Yanowitch, M. 1962 Gravity waves in a heterogeneous incompressible fluid. Comm. Pure Appl. Math. 15, 45.
Yih, C.-S. 1960a Gravity waves in a stratified fluid. J. Fluid Mech. 8, 481.
Yih, C.-S. 1960b Exact solutions for steady two-dimensional flow of a stratified fluid. J. Fluid Mech. 9, 161.
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