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TECHNICAL PAPERS

A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth—The Irrotational Green-Naghdi Model

[+] Author and Article Information
J. W. Kim

American Bureau of Shipping, Houston, TX 77079

K. J. Bai

Seoul National University, Seoul, Republic of Korea

R. C. Ertekin

University of Hawaii, Honolulu, HI 96822

W. C. Webster

University of California at Berkeley, Berkeley, CA 94720

J. Offshore Mech. Arct. Eng 125(1), 25-32 (Feb 28, 2003) (8 pages) doi:10.1115/1.1537722 History: Received February 01, 2002; Revised September 01, 2002; Online February 28, 2003
Copyright © 2003 by ASME
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References

Ertekin,  R. C., Webster,  W. C., and Wehausen,  J. V., 1986, “Waves Caused by a Moving Disturbance in a Shallow Channel of Finite Width,” J. Fluid Mech., 169, pp. 272–292.
Webster, W. C., and Kim, D. Y., 1990, “The Dispersion of Large-Amplitude Gravity Waves in Deep Water,” Proc. 18th Symp. On Naval Hydrodynamics, pp. 397–415.
Xu,  Q., Pawlowski,  J. W., and Baddour,  R. E., 1997, “Development of Green-Naghdi models with a wave-absorbing beach for nonlinear, irregular wave propagation,” J. of Marine Sci. & Tech., pp. 21–34.
Kim,  J. W., Bai,  K. J., Ertekin,  R. C., and Webster,  W. C., 2001, “A derivation of the Green-Naghdi equations for irrotational flows,” J. Eng. Math., 40(1), pp. 17–34.
Kim,  J. W., and Ertekin,  R. C., 2000, “A Numerical Study of Nonlinear Wave Interaction in Irregular Seas: Irrotational Green-Naghdi Model,” Mar. Struct., 13(4-5), pp. 331–347.
Kim, J. W., and Bai, K. J., 2003, “A New Complementary Mild-Slope Equation,” Submitted to J. Fluid Mech.
Shields,  J. J., and Webster,  W. C., 1988, “On Direct Methods in Water-Wave Theory,” J. Fluid Mech., 197, pp. 171–199.
Goldstein, H., 1980, Classical Dynamics, Addison-Wesley Pub. Co., 2nd Ed.
Bai,  K. J., and Kim,  J. W., 1995, “A Finite-Element Method for Free-Surface Flow Problems,” Journal of Theoretical and Applied Mechanics, 1(1), pp. 1–27.
Longuet-Higgins,  M. S., and Fenton,  J. D., 1974, “On the Mass, Momentum, Energy and Circulation of a Solitary Wave II,” Proc. R. Soc. London, Ser. A, A340, 471–493.
Tanaka,  M., 1986, “The Stability of Solitary Waves,” Phys. Fluids, 29(3), pp. 650–655.
Longuet-Higgins,  M. S., and Fox,  M. J. H., 1978, “Theory of the Almost-Highest Wave Part 2. Matching and Analytic Extension,” J. Fluid Mech., 85, pp. 769–786.
Longuet-Higgins,  M. S., and Fox,  M. J. H., 1996, “Asymptotic Theory for the Almost-Highest Solitary Wave,” J. Fluid Mech., 317, pp. 1–19.
Hunter,  J. K., and Vanden-Broeck,  J.-M., 1983, “Accurate Computations for Steep Solitary Waves,” J. Fluid Mech., 136, pp. 63–71.
Cokelet,  E. D., 1977, “Steep Gravity Waves in Water of Arbitrary Uniform Depth,” Philos. Trans. R. Soc. London, Ser. A, 286, pp. 183–230.
Su,  C. H., and Mirie,  R. M., 1980, “On Head-On Collision Between Solitary Waves,” J. Fluid Mech., 98(3), pp. 509–525.
Ertekin, R. C., and Wehausen, J. V., 1986, “Some Soliton Calculations,” Proc. 16th Symp. Naval Hydrodynamics, Berkeley, CA, pp. 167–184.
Chan, R. K.-C., and Street, R. L., 1970, “The Shoaling of Finite-Amplitude Water Waves on Sloping Beaches,” Twelfth Conference on Coastal Engineering, ASCE, pp. 345–362.
Kennedy,  A. B., Kirby,  J. T., Chen,  Q., and Dalrymple,  R. A., 2001, “Boussinesq-type Equations with Improved Nonlinear Performance,” Wave Motion, 33(3), pp. 225–243.

Figures

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Dispersion relation of the IGN equations. Numbers on the curves denote the Level K. Note that the negative horizontal axis denotes the imaginary axis of the wave number k.
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Dispersion relation of the linear IGN equation compared with the exact and the perturbation results. Numbers on the curves for the IGN equations denote the Level K.
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The speed-amplitude relation of solitary waves
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Characteristic quantities of solitary waves calculated by the IGN Level 3 equations: (a) celerity, (b) mass, (c) kinetic energy, (d) potential energy
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Surface profiles of solitary waves for ϖ=0.9
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Nonlinear dispersion relation of the Stokes waves in finite depth  
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Evolution of the wave profiles during reflection
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Maximum run-up of solitary wave on a vertical wall
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Initial wave elevation and bottom topography for unsteady shoaling
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Crest and trough envelopes for unsteady shoaling

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