Abstract
A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton’s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that (1) the set-up and the set-down are both non-monotonic quantities with the wave steepness, and (2) the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.
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Foundation item: The Jiangsu Province Natural Science Foundation for the Young Scholar under contract No. BK20130827; the Fundamental Research Funds for the Central Universities of China under contract No. 2010B02614; the National Natural Science Foundation of China under contract Nos 41076008 and 51009059; the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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Zhao, H., Song, Z., Li, L. et al. On the Fourier approximation method for steady water waves. Acta Oceanol. Sin. 33, 37–47 (2014). https://doi.org/10.1007/s13131-014-0470-1
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DOI: https://doi.org/10.1007/s13131-014-0470-1