Recently, the authors have derived a new approximate model for the nonlinear water waves, the Irrotational Green-Naghdi (IGN) model. In this paper, we first derive the IGN equations applicable to variable water depth, and then perform numerical tests to show whether and how fast the solution of the IGN model converges to the true solution as its level increases. The first example given is the steady solution of progressive waves of permanent form, which includes the small-amplitude sinusoidal wave, the solitary wave and the nonlinear Stokes wave. The second example is the run-up of a solitary wave on a vertical wall. The last example is the shoaling of a wave train over a sloping beach. In each numerical test, the self-convergence of the IGN model is shown first. Then the converged solution is compared to the known analytic solutions and/or solutions of other approximate models such as the KdV and the Boussinesq equations.
A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth—The Irrotational Green-Naghdi Model
Contributed by the OOAE Division of the American Society of Mechanical Engineers for publication in the ASME JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received by the ASME OOAE Division, February 2002; final revision, September 2002. Guest Associate Editor: S. Chakrabarti.
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Kim, J. W., Bai, K. J., Ertekin, R. C., and Webster, W. C. (February 28, 2003). "A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth—The Irrotational Green-Naghdi Model ." ASME. J. Offshore Mech. Arct. Eng. February 2003; 125(1): 25–32. https://doi.org/10.1115/1.1537722
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