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Shoaling of finite-amplitude surface waves on water of slowly-varying depth

Published online by Cambridge University Press:  19 April 2006

M. Stiassnie  and
D. H. Peregrine
M. Stiassnie
Affiliation:
Department of Mathematics, University of Bristol, England Permanent address: Faculty of Civil Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
D. H. Peregrine
Affiliation:
Department of Mathematics, University of Bristol, England
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Abstract

Periodic wave trains propagating over water which varies in depth in the direction of wave propagation are studied by using accurate solutions for wave trains in constant depth of water. The accurate solutions are (i) Cokelet's (1977) extension of Stokes’ approximation and, for the longer waves, (ii) a solution for trains of solitary waves using the solitary-wave solution of Longuet-Higgins & Fenton (1974).

A representative selection of results is shown in diagrams. A feature which arises from the use of these accurate solutions is that near the highest wave two solutions are possible for a given incoming wave. Although the solutions cannot describe waves that break, it is shown that as depth is decreased a point is reached beyond which no solution can be found. This is taken to indicate the region in which waves break.

The limitations of the theory are discussed and analysed. Comparisons with experimental measurements of Hansen & Svendsen (1979) are included.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 97 , Issue 4 , 29 April 1980 , pp. 783 - 805
Copyright
© 1980 Cambridge University Press

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References

Baker, G. A. 1975 Essentials of Padé Approximants. Academic.
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. Roy. Soc. A 286, 183–230.Google Scholar
Crapper, G. D. 1979 Energy and momentum integrals for progressive capillary-gravity waves. J. Fluid Mech. 94, 13–24.Google Scholar
Dore, B. D. 1977 On mass transport velocity due to progressive waves. Quart. J. Mech. Appl. Math. 30, 157–173.Google Scholar
Hansen, J. B. & Svendsen, I. A. 1974 Laboratory generation of waves of constant form. Proc. 14th Coastal Engng Conf., Copenhagen, A.S.C.E., pp. 321–339.
Hansen, J. B. & Svendsen, I. A. 1979 Regular waves in shoaling water experimental data. Inst. of Hydrodyn. & Hydraul. Engng, Tech. Univ. of Denmark, Series Paper no. 21 (plus erratum correcting values of wave celerity, c).
Iwagaki, Y. 1968 Hyperbolic waves and their shoaling. Proc. 11th Conf. on Coastal Engng, vol. 1, pp. 124–144.
James, I. D. 1974a Nonlinear waves in the nearshore region: shoaling and set-up. Estuarine & Mar. Sci. 2, 207–234.Google Scholar
James, I. D. 1974b A nonlinear theory of longshore currents. Estuarine & Mar. Sci. 2, 235–249.Google Scholar
Jonsson, I. G. & Carlsen, N. A. 1976 Experimental and theoretical investigations in an oscillatory turbulent boundary layer. J. Hydraul. Res. 14, 45–60 (plus errata).Google Scholar
Kamphuis, J. W. 1975 Friction factor under oscillatory waves. Proc. Am. Soc. Civ. Eng., J. Waterways, Harbors and Coastal Engng. Div. 101, no. WW2, 135–144.Google Scholar
Lake, B. M. & Yuen, H. C. 1978 A new model for nonlinear wind waves. Part I. Physical model and experimental evidence. J. Fluid Mech. 88, 33–62.Google Scholar
Lax, P. D. 1968 Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. Roy. Soc. A 342, 157–174.Google Scholar
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. Roy. Soc. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. Roy. Soc. A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. 1978c Some new relations between Stokes's coefficients in the theory of gravity waves. J. Inst. Maths. Applics. 22, 261–273.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1978 The deformation of steep surface waves on water. II. Growth of normal mode instabilities. Proc. Roy. Soc. A 364, 1–28.Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum, energy and circulation of a solitary wave. II. Proc. Roy. Soc. A 340, 471–493.Google Scholar
Miles, J. W. 1976 Damping of weakly nonlinear shallow-water waves. J. Fluid Mech. 76, 251–257.Google Scholar
Miles, J. W. 1979 On the Korteweg-de Vries equation for a gradually varying channel. J. Fluid Mech. 91, 181–190.Google Scholar
Munk, W. H. 1949 The solitary wave and its application to surf problems. Ann. N.Y. Acad. Sci. 51 (3), 376–424.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Peregrine, D. H. 1976 Interactions of water waves and currents. Adv. Appl. Mech. 16, 9–117.Google Scholar
Peregrine, D. H. & Thomas, G. P. 1979 Finite-amplitude deep-water waves on currents. Phil. Trans. Roy. Soc. A 292, 371–390.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Russell, R. C. H. & Osorio, J. D. C. 1958 An experimental investigation of drift profiles in a closed channel. Proc. 6th Conf. on Coastal Engng, pp. 171–193.
Smith, R. 1976 Giant waves. J. Fluid Mech. 77, 417–431.Google Scholar
Stiassnie, M. & Peregrine, D. H. 1979 On averaged equations for finite-amplitude water waves. J. Fluid Mech. 94, 401–407.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455; Math. & Phys. Papers (1880) 1, 197–229.Google Scholar
Svendsen, I. A. & Brink-Kjaer, O. 1972 Shoaling of cnoidal waves. Proc. 13th Int. Conf. Coastal Engng, Vancouver, vol. 1, pp. 365–384.
Svendsen, I. A. & Hansen, J. B. 1976 Deformation up to breaking of periodic waves on a beach. 15th Int. Conf. Coastal Engng, A.S.C.E., vol. 1, pp. 477–496.
Svendsen, I. A. & Hansen, J. B. 1977 The wave height variation for regular waves in shoaling water. Coastal Engng 1, 261–284.Google Scholar
Svendsen, I. A. & Hansen, J. B. 1978 On the deformation of periodic long waves over a gently sloping bottom. J. Fluid Mech. 87, 433–448.Google Scholar
Whitham, G. B. 1967 Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399–412.Google Scholar
Whitham, G. B. 1974 Linear and Non-Linear Waves. Wiley-Interscience.
Yuen, H. C. & Lake, B. M. 1975 Nonlinear deep water waves: Theory and experiment. Phys. Fluids 18, 956–960.Google Scholar