Mathematic Model of Green Function with Two-Dimensional Free Water Surface

ABSTRACT

Adopting complex number theory, a mathematic model of Green function is built for two dimension free water surface, and an analytic expression of Green function is obtained by introducing two parameters. The intrinsic properties of Green function are discussed on vertical line and horizontal line. At last, the derivation expression of Green function is obtained from the formula of Green function.

Cite this paper

S. Jin, X. Wang, J. Du, S. Zhang and S. Jin, "Mathematic Model of Green Function with Two-Dimensional Free Water Surface,"*Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 75-79. doi: 10.4236/am.2013.48A010.

S. Jin, X. Wang, J. Du, S. Zhang and S. Jin, "Mathematic Model of Green Function with Two-Dimensional Free Water Surface,"

References

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[2] J. V. Wehausen and E. V. Latoine, “Surface Waves,” In: S. Flügge, Ed., Encyclopedia of Physics, Springer, Berlin, 1960, pp. 446-778.

[3] A. H. Clement, “An Ordinary Differential Equation for the Green Function of Time-Domain Free-Surface Hydro dynamics,” Journal of Engineering Mathematics, Vol. 33, No. 2, 1998, pp. 201-217. doi:10.1023/A:1004376504969

[4] N. Kuznetsov, V. Maz’ya and B. Vainberg, “Linear Wa ter Waves: A Mathematical Approach,” Cambridge Uni versity Press, Cambridge, 2002.

[5] J. N. Newman, “Algorithm for the Free-Surface Green Function,” Journal of Engineering Mathematics, Vol. 19, No. 1, 1985, pp. 57-67.

[6] H. Shen, “Computational Method of Surface Green Func tion with No Numerical Integration,” Journal of Dalian institute of Technology, Vol. 17, No. 1, 1988, pp. 75-84.

[7] Q. B. Zhou, G. Zhang and L. S. Zhu, “The Fast Calcula tion of Free Surface Wave Green Function and Its Deri vatives,” Chinese Journal of Computational Physics, Vol. 16, No. 2, 1988, pp. 113-119.

[8] C. Yang, F. Noblesse and R. Lohner, “Comparison of Classical and Simple Free-Surface Green Functions,” Journal of Offshore and Polar Engineering, Vol. 14, No. 4, 2004, pp. 256-264.?

[9] L. Shen Liang, et al., “A Practical Numerical Method for Deep Water Time Domain Green Function,” Journal of Hydrodynamics A, Vol. 22, No. 3, 2007, pp. 380-386.

[10] F. John, “On the Motion of Floating Bodies I,” Commu nications on Pure and Applied Mathematics, Vol. 2, 1949, pp. 13-57.

[11] F. John, “On the Motion of Floating Bodies II,” Commu nications on Pure and Applied Mathematics, Vol. 3, 1950, pp. 45-101.

[12] Y. Z. Liu and G. P. Miu, “Theory of the Motion of Ships in Waves,” Shanghai Jiao Tong University Press, Shang hai, 1987.

[13] R. Hein, M. Duran and J.-C. Nedelec, “Explicit Represen tation for the Infinite-Depth Two-Dimensional Free-Sur face Green’s Function in Linear Water-Wave Theory,” SIAM Journal on Applied Mathematics, Vol. 70, No. 7, 2010, pp. 2353-2372. doi:10.1137/090764591

[14] W. D. Kim, “On the Harmonic Oscillations of a Rigid Body on a Free Surface,” Journal of Fluid Mechanics, Vol. 21, No. 3, 1965, pp. 427-451.

[15] M. D. Greenberg, “Application of Green’s Functions in Science and Engineering,” PrenticeHall, Englewood Cliffs, 1971.

[16] C. Macaskill, “Reflexion of Water Waves by a Permeable Barrier,” Journal of Fluid Mechanics, Vol. 95, No. 1, 1979, pp. 141-157.

[17] R. Dautray and J. L. Lions, “Analyse Mathématique et Calcul Numérique Pour les Scienceset les Techniques,” Vol. 2, Masson, Paris, 1987.

[18] C. F. Liu, et al., “New Convolution Algorithm of Time—Domain Green Function,” Journal of Hydrodynamics A, Vol. 25, No. 4, 2010, pp. 25-34.

[19] N. Kuznetsov, V. Maz’ya and B. Vainberg, “Linear Wa ter Waves: A Mathematical Approach,” Cambridge Uni versity Press, Cambridge, 2002.

[20] C. C. Mei, M. Stiassnie and D. K.-P. Yue, “Theory and Applications of Ocean Surface Waves, Part 1: Linear As pects,” World Scientific, Hackensack, 2005.

[21] J. V. Wehausen and E. V. Latoine, “Surface Waves,” In: S. Flügge, Ed., Encyclopedia of Physics, Vol. IX, Sprin ger, Berlin, 1960, pp. 446-778.

[22] R. Harter, I. D. Abrahams and M. J. Simon, “The Effect of Surface Tension on Trapped Modes in Water-Wave Problems,” Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sci ence, Vol. 463, No. 2, 2007, pp. 3131-3149.

[23] R. Harter, M. J. Simon and I. D. Abrahams, “The Effect of Surface Tension on Localized Free-Surface Oscilla tions about Surface-Piercing Bodies,” Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Science, Vol. 464, No. 2, 2008, pp. 3039-3054.

[24] O. V. Motygin and P. McIver, “On Uniqueness in the Problem of Gravity-Capillary Water Waves above Sub merged Bodies,” Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Science, Vol. 465, No. 3, 2009, pp. 1743-1761.

[1] P. Andersen and W. Z. He, “On the Calculation of Two Dimensional Added Mass and Damping Coefficients by Simple Green’s Function Technique,” Ocean Engineering, Vol. 12, No. 5, 1985, pp. 425-451. doi:10.1016/0029-8018(85)90003-4

[2] J. V. Wehausen and E. V. Latoine, “Surface Waves,” In: S. Flügge, Ed., Encyclopedia of Physics, Springer, Berlin, 1960, pp. 446-778.

[3] A. H. Clement, “An Ordinary Differential Equation for the Green Function of Time-Domain Free-Surface Hydro dynamics,” Journal of Engineering Mathematics, Vol. 33, No. 2, 1998, pp. 201-217. doi:10.1023/A:1004376504969

[4] N. Kuznetsov, V. Maz’ya and B. Vainberg, “Linear Wa ter Waves: A Mathematical Approach,” Cambridge Uni versity Press, Cambridge, 2002.

[5] J. N. Newman, “Algorithm for the Free-Surface Green Function,” Journal of Engineering Mathematics, Vol. 19, No. 1, 1985, pp. 57-67.

[6] H. Shen, “Computational Method of Surface Green Func tion with No Numerical Integration,” Journal of Dalian institute of Technology, Vol. 17, No. 1, 1988, pp. 75-84.

[7] Q. B. Zhou, G. Zhang and L. S. Zhu, “The Fast Calcula tion of Free Surface Wave Green Function and Its Deri vatives,” Chinese Journal of Computational Physics, Vol. 16, No. 2, 1988, pp. 113-119.

[8] C. Yang, F. Noblesse and R. Lohner, “Comparison of Classical and Simple Free-Surface Green Functions,” Journal of Offshore and Polar Engineering, Vol. 14, No. 4, 2004, pp. 256-264.?

[9] L. Shen Liang, et al., “A Practical Numerical Method for Deep Water Time Domain Green Function,” Journal of Hydrodynamics A, Vol. 22, No. 3, 2007, pp. 380-386.

[10] F. John, “On the Motion of Floating Bodies I,” Commu nications on Pure and Applied Mathematics, Vol. 2, 1949, pp. 13-57.

[11] F. John, “On the Motion of Floating Bodies II,” Commu nications on Pure and Applied Mathematics, Vol. 3, 1950, pp. 45-101.

[12] Y. Z. Liu and G. P. Miu, “Theory of the Motion of Ships in Waves,” Shanghai Jiao Tong University Press, Shang hai, 1987.

[13] R. Hein, M. Duran and J.-C. Nedelec, “Explicit Represen tation for the Infinite-Depth Two-Dimensional Free-Sur face Green’s Function in Linear Water-Wave Theory,” SIAM Journal on Applied Mathematics, Vol. 70, No. 7, 2010, pp. 2353-2372. doi:10.1137/090764591

[14] W. D. Kim, “On the Harmonic Oscillations of a Rigid Body on a Free Surface,” Journal of Fluid Mechanics, Vol. 21, No. 3, 1965, pp. 427-451.

[15] M. D. Greenberg, “Application of Green’s Functions in Science and Engineering,” PrenticeHall, Englewood Cliffs, 1971.

[16] C. Macaskill, “Reflexion of Water Waves by a Permeable Barrier,” Journal of Fluid Mechanics, Vol. 95, No. 1, 1979, pp. 141-157.

[17] R. Dautray and J. L. Lions, “Analyse Mathématique et Calcul Numérique Pour les Scienceset les Techniques,” Vol. 2, Masson, Paris, 1987.

[18] C. F. Liu, et al., “New Convolution Algorithm of Time—Domain Green Function,” Journal of Hydrodynamics A, Vol. 25, No. 4, 2010, pp. 25-34.

[19] N. Kuznetsov, V. Maz’ya and B. Vainberg, “Linear Wa ter Waves: A Mathematical Approach,” Cambridge Uni versity Press, Cambridge, 2002.

[20] C. C. Mei, M. Stiassnie and D. K.-P. Yue, “Theory and Applications of Ocean Surface Waves, Part 1: Linear As pects,” World Scientific, Hackensack, 2005.

[21] J. V. Wehausen and E. V. Latoine, “Surface Waves,” In: S. Flügge, Ed., Encyclopedia of Physics, Vol. IX, Sprin ger, Berlin, 1960, pp. 446-778.

[22] R. Harter, I. D. Abrahams and M. J. Simon, “The Effect of Surface Tension on Trapped Modes in Water-Wave Problems,” Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sci ence, Vol. 463, No. 2, 2007, pp. 3131-3149.

[23] R. Harter, M. J. Simon and I. D. Abrahams, “The Effect of Surface Tension on Localized Free-Surface Oscilla tions about Surface-Piercing Bodies,” Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Science, Vol. 464, No. 2, 2008, pp. 3039-3054.

[24] O. V. Motygin and P. McIver, “On Uniqueness in the Problem of Gravity-Capillary Water Waves above Sub merged Bodies,” Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Science, Vol. 465, No. 3, 2009, pp. 1743-1761.