线性长波越过水下理想防波堤反射效应的准确解析解和逼近解析解
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摘要
在港口、海岸、近海与海洋工程及水动力学领域,水波方程的解析解在验证数值模型时处于与实验解同等重要的地位,且由于它的经济、高效、准确、易重复性及在理论分析中的强大作用而超越实验解更受欢迎.本硕士学位论文致力于寻求线性长波越过水下理想防波堤反射效应的准确解析解和逼近解析解.
首先,我们推导了线性长波越过理想防波堤反射效应的准确解析解,所谓的理想防波堤是指防波堤前后坡相应的水深函数均为幂函数,其中前后坡相应幂函数的幂次可以不同,防波堤前后的水深也可以不同.通过技巧性的变量替换及函数变换,将海洋波传播的控制方程–长波方程转化为经典的欧拉方程和贝赛尔方程,从而成功求得闭合形式的解析解.本文的解析解同样适用于水下理想陷坑地形,因为它们本质上都是求解同一类型的二阶常微分方程.
其次,我们推导出了本文所得解析解在几个特殊地形下的简化表达式,与其中一些特殊地形下的经典解析解做了比较,确认了Lamb于1932年给出的无限长台阶地形的长波解析解, Kajiura于1961给出的无限长台阶后的抛物斜坡地形的长波解析解,美国工程院院士Dean于1964给出的线性斜坡后无限长台阶地形的长波解析解,美国科学院与工程院院士Mei于1989给出的矩形防波堤地形的长波解析解, Lin和Liu于2005年给出的广义梯形防波堤地形的长波解析解及Liu和Lin于2005年给出的广义梯形陷坑地形的长波解析解均为本文任意理想防波堤地形的长波解析解的特例.
利用本文给出的解析解,我们对长波越过一些实际海底地形引起的反射效应进行了计算,分析了防波堤形状对周期性反射系数和零反射系数现象的影响,验证了Xie,Liu和Lin于2011年首次揭示的有关带冲刷槽矩形防波堤零反射系数现象只在海底地形关于防波堤对称时才会发生的结论,同时也验证了Xie, Liu和Lin于2011年首次揭示的反射系数只对矩形防波堤才会具有周期性的结论.这些结论在理想防波堤情形的再次验证与确认在防波堤的设计及优化省材和海啸预防中无疑具有重要的指导意义.例如为加强防波堤的防波防浪效果,我们应避免建造对称形状的防波堤.另外我们还分析了防波堤前后坡宽度对反射效应的影响.
最后我们利用同伦摄动法求解了水下理想防波堤反射效应的逼近解析解.虽然跟本文前面部分已经给出的闭合形式解析解相比显得多余,但对以后更复杂水波方程近似解析求解的探索却具有启发性.
In waterway, port, coastal and ocean engineering as well as hydrodynamics, ana-lytical solutions to wave equations are as significant as experimental dada. They areeven more favorable than experimental dada because of their economy, efficiency, accu-racy and the powerful function in theoretical analysis. This thesis focuses on seekinganalytical solutions and approximate analytical solutions for long-wave reflected by asubmerged idealized breakwater.
Firstly, analytical solutions for linear long-wave reflected by a submerged idealizedbreakwater are given, where the idealization means that the water depths in both regionsof front and back slopes are power functions with different exponents. By skillfullyusing variable transform and function transform, the wave governing equation, i.e.,long-wave equation, can be transformed into the classical Euler equation and Besselequation, therefore analytical solutions in the closed-form can be successfully found. Itis worthwhile indicating that the analytical solutions given in this thesis are also validfor idealized trenches since the same type of ordinary equations are need to be solved.
Secondly, simplified expressions of the analytical solutions for a series of particularbathymetries are derived. Some of them are compared with classical analytical solutions,for example, the analytical solution for long-wave reflected by an infinite step given byLamb in 1932, the analytical solution for long-wave reflected by a parabolic slope behindan infinite step given by Kajiura in 1961, the analytical solution for long-wave reflectedby a linear slope joined with an infinite step given by Dean in 1964, the analyticalsolution for long-wave reflected by a rectangular breakwater given by Mei in 1989, theanalytical solution for long-wave reflected by a general trapezoidal breakwater given byLin and Liu in 2005 and the analytical solution for long-wave reflected by a generaltrapezoidal trench given by Liu and Lin in 2005.
Based on these analytical solutions, re?ection coe?cients for long-wave reflected bysome practical sea beds are calculated and the in?uence of the shape of breakwaters tore?ection e?ect is analyzed. Both the conclusions revealed by Xie, Liu and Lin in 2011 that zero reflection occurs for a rectangular breakwater with two scour trenches if andonly if the bathymetry is symmetrical with respect to the rectangular breakwater andperiodicity of reflection coefficient occurs if and only the breakwater is a rectangularbreakwater are confirmed. These confirmations in the case of idealized breakwaters aresignificant in the design of breakwaters. For example, to enhance the reflection effect,symmetrical breakwater should be avoid. In addition, we also analyze the influence ofboth slope widths of the breakwater with parabolic slope to reflection effect.
Finally, by using homotopy perturbation method, approximate analytical solutionsfor long-wave reflected by a submerged idealized breakwater are given. Though it seemsthat these approximate analytical solutions are unnecessary compared with exact ana-lytical solutions given in this thesis, they may illuminate us to find some approximateanalytical solutions to more complicated wave equations.
首先,我们推导了线性长波越过理想防波堤反射效应的准确解析解,所谓的理想防波堤是指防波堤前后坡相应的水深函数均为幂函数,其中前后坡相应幂函数的幂次可以不同,防波堤前后的水深也可以不同.通过技巧性的变量替换及函数变换,将海洋波传播的控制方程–长波方程转化为经典的欧拉方程和贝赛尔方程,从而成功求得闭合形式的解析解.本文的解析解同样适用于水下理想陷坑地形,因为它们本质上都是求解同一类型的二阶常微分方程.
其次,我们推导出了本文所得解析解在几个特殊地形下的简化表达式,与其中一些特殊地形下的经典解析解做了比较,确认了Lamb于1932年给出的无限长台阶地形的长波解析解, Kajiura于1961给出的无限长台阶后的抛物斜坡地形的长波解析解,美国工程院院士Dean于1964给出的线性斜坡后无限长台阶地形的长波解析解,美国科学院与工程院院士Mei于1989给出的矩形防波堤地形的长波解析解, Lin和Liu于2005年给出的广义梯形防波堤地形的长波解析解及Liu和Lin于2005年给出的广义梯形陷坑地形的长波解析解均为本文任意理想防波堤地形的长波解析解的特例.
利用本文给出的解析解,我们对长波越过一些实际海底地形引起的反射效应进行了计算,分析了防波堤形状对周期性反射系数和零反射系数现象的影响,验证了Xie,Liu和Lin于2011年首次揭示的有关带冲刷槽矩形防波堤零反射系数现象只在海底地形关于防波堤对称时才会发生的结论,同时也验证了Xie, Liu和Lin于2011年首次揭示的反射系数只对矩形防波堤才会具有周期性的结论.这些结论在理想防波堤情形的再次验证与确认在防波堤的设计及优化省材和海啸预防中无疑具有重要的指导意义.例如为加强防波堤的防波防浪效果,我们应避免建造对称形状的防波堤.另外我们还分析了防波堤前后坡宽度对反射效应的影响.
最后我们利用同伦摄动法求解了水下理想防波堤反射效应的逼近解析解.虽然跟本文前面部分已经给出的闭合形式解析解相比显得多余,但对以后更复杂水波方程近似解析求解的探索却具有启发性.
In waterway, port, coastal and ocean engineering as well as hydrodynamics, ana-lytical solutions to wave equations are as significant as experimental dada. They areeven more favorable than experimental dada because of their economy, efficiency, accu-racy and the powerful function in theoretical analysis. This thesis focuses on seekinganalytical solutions and approximate analytical solutions for long-wave reflected by asubmerged idealized breakwater.
Firstly, analytical solutions for linear long-wave reflected by a submerged idealizedbreakwater are given, where the idealization means that the water depths in both regionsof front and back slopes are power functions with different exponents. By skillfullyusing variable transform and function transform, the wave governing equation, i.e.,long-wave equation, can be transformed into the classical Euler equation and Besselequation, therefore analytical solutions in the closed-form can be successfully found. Itis worthwhile indicating that the analytical solutions given in this thesis are also validfor idealized trenches since the same type of ordinary equations are need to be solved.
Secondly, simplified expressions of the analytical solutions for a series of particularbathymetries are derived. Some of them are compared with classical analytical solutions,for example, the analytical solution for long-wave reflected by an infinite step given byLamb in 1932, the analytical solution for long-wave reflected by a parabolic slope behindan infinite step given by Kajiura in 1961, the analytical solution for long-wave reflectedby a linear slope joined with an infinite step given by Dean in 1964, the analyticalsolution for long-wave reflected by a rectangular breakwater given by Mei in 1989, theanalytical solution for long-wave reflected by a general trapezoidal breakwater given byLin and Liu in 2005 and the analytical solution for long-wave reflected by a generaltrapezoidal trench given by Liu and Lin in 2005.
Based on these analytical solutions, re?ection coe?cients for long-wave reflected bysome practical sea beds are calculated and the in?uence of the shape of breakwaters tore?ection e?ect is analyzed. Both the conclusions revealed by Xie, Liu and Lin in 2011 that zero reflection occurs for a rectangular breakwater with two scour trenches if andonly if the bathymetry is symmetrical with respect to the rectangular breakwater andperiodicity of reflection coefficient occurs if and only the breakwater is a rectangularbreakwater are confirmed. These confirmations in the case of idealized breakwaters aresignificant in the design of breakwaters. For example, to enhance the reflection effect,symmetrical breakwater should be avoid. In addition, we also analyze the influence ofboth slope widths of the breakwater with parabolic slope to reflection effect.
Finally, by using homotopy perturbation method, approximate analytical solutionsfor long-wave reflected by a submerged idealized breakwater are given. Though it seemsthat these approximate analytical solutions are unnecessary compared with exact ana-lytical solutions given in this thesis, they may illuminate us to find some approximateanalytical solutions to more complicated wave equations.
引文
[1] Lamb H., Hydrodynamics(M). Dover, New York, 6th Edith, 1932.
[2] Rey V., Belzons M., Guazzelli E., Propagation of surface gravity waves over a rectangularsubmerged bar[J]. Journal of Fluid Mechanics 235(1992): 453-479.
[3] Stamos D.G., Hajj M.R., Re?ection and transmission of waves over submerged break-waters[J]. Journal of Engineering Mechanics, ASCE, 127(2)(2001): 99-105.
[4] Liu P.L.F., Abbaspour M., An integral equation method for the di?raction of obliquewaves by an infinite cylinder[J]. International Journal for Numerical Methods in Engi-neering 18(10)(1982): 1497-1504.
[5] Dalrymple R.A., Kirby J.T., Water waves over ripples[J]. Journal of Waterway, Port,Coastal and Ocean Engineering 112(2)(1986): 309-319.
[6] Yu X.P., Chwang A.T., Wave Motion Through Porous Structures[J]. Journal of Engi-neering Mechanics, ASCE, 120(5)(1994): 989-1008.
[7] Williams A.N., Wang K.H., Flexible porous wave barrier for enhanced wetlands habitatrestoration[J]. Journal of Engineering Mechanics, ASCE, 129(1)(2003):1-8.
[8] Bartholomeusz E.F., The re?ection of long waves at a step[J]. Mathematical Proceedingsof the Cambridge Philosophical Society 54(1958): 106-118.
[9] Newman J.N., Propagation of water waves past long two dimensional obstacles[J]. Jour-nal of Fluid Mechanics 23(1965): 23-29.
[10] Newman J.N., Propagation of water waves over an infinite step[J]. Journal of FluidMechanics 23(1965): 399-415.
[11] Miles J.W., Surface-wave scattering matrix for a shelf[J]. Journal of Fluid Mechanics28(1967): 755-767.
[12] Mei C.C., Black J.L., Scattering of surface wave by rectangular obstacles in waters offinite depth[J]. Journal of Fluid Mechanics 38(1969): 499-511.
[13] Roseau M., Contribution a` la the′orie des ondes liquides de gravite′en profundeur vari-able[J]. Publication Scientifique et Technique du Ministe′re de l,Air, 1952, No.275, Paris(in French).
[14] Takano K., E?ects d’un obstacle parallelepipedique sur la propagation de la houle[J].Houille blanche 15(1960): 247-267, (in French).
[15] Kirby J., Dalrymple R.A., Propagation of oblique incident water waves over a trench[J].Journal of Fluid Mechanics 133(1983): 47-63.
[16] Liu P.L.F., Cho Y.S., Kostense J.K., Dingemans M.W., Propagation and trapping ofobliquely incident wave groups over a trench with currents[J]. Applied Ocean Research14(3)(1992): 201-213.
[17] Cho Y.S., Lee C., Resonant re?ection of waves over sinusoidally varying topographies[J].Journal of Coastal Research 16(2000): 870-876.
[18] Devillard P., Dunlop F., Souillard B., Localization of gravity waves on a channel withrandom bottom[J]. Journal of Fluid Mechanics 186(1998): 521-538.
[19] Lin P.Z., Liu H.W., Analytical study of linear long-wave re?ection by a two-dimensionalobstacle of general trapezoidal shape[J]. Journal of Engineering Mechanics, ASCE,131(8)(2005): 822-830.
[20] Liu H.W., Lin P.Z., Discussion of”Wave transformation by two-dimensional bathymetricanomalies with sloped transitions”[Coast. Eng. 50(2003)61-84][J]. Coastal Engineering52(2005): 197-200.
[21] Je?reys H., Motion of waves in shallow water. Note on the o?shore bar problem andre?exion from a bar[J]. London: Ministry of Supply Wave Report 3, 1944.
[22] Mei C.C. , The Applied Dynamics of Ocean Surface Waves[M]. World Scientific, Singa-pore, 1989.
[23] Epstein P.S., Re?ection of waves in an inhomogeneous absorbing medium[J]. Proceedingsof the National Academy of Sciences of the United States of America 16(10)(1930): 627-637.
[24] Yoshida K., On the partial re?ection of long waves[J]. Geophysical Notes GeophysicalInstitute, Tokyo University, 1948, No.31.
[25] Dean R.G., Long wave modification by linear transitions[J]. Journal of the Waterwaysand Harbors Division, Proceedings, ASCE, 90(1)(1964): 1-30.
[26] Kajiura K., On the partial re?ection of water waves passing over a bottom of variabledepth[J]. International Cooperation in Geodesy and Geophysics 24(1964): 206-230.
[27] Dingemans M.W., Water Wave Propagation over Uneven Bottoms[M]. Part 1-linearWave Propagation, World Scientific, Singapore, 1997.
[28] Bender C.J., Dean R.G., Wave transformation by two-dimensional bathymetric anoma-lies with sloped transitions[J]. Coastal Engineering 50(2003): 61-84.
[29] Xie J.J., Liu H.W., Lin P., Analytical solution for long wave re?ection by a rectangu-lar obstacle with two scour trenches[J]. Journal of Engineering Mechanics 137(2011),accepted.
[30]王高雄,周之铭,朱思铭,常微分方程.高等教育出版社, 2006年7月,第三版.
[31] He J.H., Homotopy perturbation technique[J]. Computer Methods in Applied Mechanicsand Engineering 178(1999): 257-262.
[32] He J.H., A coupling method homotopy technique and perturbation technique for non-linear problems[J]. International Journal of Non-Linear Mechanics 35(1)(2000): 37-43.
[33] He J.H., Homotopy perturbation method: A new nonlinear analytical technique[J]. Ap-plied Mathematics and Computation 135(2003): 73–79.
[34] He J.H., Some asymptotic methods for strongly nonlinear equations[J]. InternationalJournal of Modern Physics B 20(2006): 1141-1199.
[35] Chun C.B., Integration using He’s homotopy perturbation method[J]. Chaos SolitonsFractals 34(2007): 1130-1134.
[36] Gorji M., Ganji D.D., Soleimani S., New application of He’s homotopy perturba-tion method[J]. International Journal of Nonlinear Sciences and Numerical Simulation8(2007): 319-328.
[2] Rey V., Belzons M., Guazzelli E., Propagation of surface gravity waves over a rectangularsubmerged bar[J]. Journal of Fluid Mechanics 235(1992): 453-479.
[3] Stamos D.G., Hajj M.R., Re?ection and transmission of waves over submerged break-waters[J]. Journal of Engineering Mechanics, ASCE, 127(2)(2001): 99-105.
[4] Liu P.L.F., Abbaspour M., An integral equation method for the di?raction of obliquewaves by an infinite cylinder[J]. International Journal for Numerical Methods in Engi-neering 18(10)(1982): 1497-1504.
[5] Dalrymple R.A., Kirby J.T., Water waves over ripples[J]. Journal of Waterway, Port,Coastal and Ocean Engineering 112(2)(1986): 309-319.
[6] Yu X.P., Chwang A.T., Wave Motion Through Porous Structures[J]. Journal of Engi-neering Mechanics, ASCE, 120(5)(1994): 989-1008.
[7] Williams A.N., Wang K.H., Flexible porous wave barrier for enhanced wetlands habitatrestoration[J]. Journal of Engineering Mechanics, ASCE, 129(1)(2003):1-8.
[8] Bartholomeusz E.F., The re?ection of long waves at a step[J]. Mathematical Proceedingsof the Cambridge Philosophical Society 54(1958): 106-118.
[9] Newman J.N., Propagation of water waves past long two dimensional obstacles[J]. Jour-nal of Fluid Mechanics 23(1965): 23-29.
[10] Newman J.N., Propagation of water waves over an infinite step[J]. Journal of FluidMechanics 23(1965): 399-415.
[11] Miles J.W., Surface-wave scattering matrix for a shelf[J]. Journal of Fluid Mechanics28(1967): 755-767.
[12] Mei C.C., Black J.L., Scattering of surface wave by rectangular obstacles in waters offinite depth[J]. Journal of Fluid Mechanics 38(1969): 499-511.
[13] Roseau M., Contribution a` la the′orie des ondes liquides de gravite′en profundeur vari-able[J]. Publication Scientifique et Technique du Ministe′re de l,Air, 1952, No.275, Paris(in French).
[14] Takano K., E?ects d’un obstacle parallelepipedique sur la propagation de la houle[J].Houille blanche 15(1960): 247-267, (in French).
[15] Kirby J., Dalrymple R.A., Propagation of oblique incident water waves over a trench[J].Journal of Fluid Mechanics 133(1983): 47-63.
[16] Liu P.L.F., Cho Y.S., Kostense J.K., Dingemans M.W., Propagation and trapping ofobliquely incident wave groups over a trench with currents[J]. Applied Ocean Research14(3)(1992): 201-213.
[17] Cho Y.S., Lee C., Resonant re?ection of waves over sinusoidally varying topographies[J].Journal of Coastal Research 16(2000): 870-876.
[18] Devillard P., Dunlop F., Souillard B., Localization of gravity waves on a channel withrandom bottom[J]. Journal of Fluid Mechanics 186(1998): 521-538.
[19] Lin P.Z., Liu H.W., Analytical study of linear long-wave re?ection by a two-dimensionalobstacle of general trapezoidal shape[J]. Journal of Engineering Mechanics, ASCE,131(8)(2005): 822-830.
[20] Liu H.W., Lin P.Z., Discussion of”Wave transformation by two-dimensional bathymetricanomalies with sloped transitions”[Coast. Eng. 50(2003)61-84][J]. Coastal Engineering52(2005): 197-200.
[21] Je?reys H., Motion of waves in shallow water. Note on the o?shore bar problem andre?exion from a bar[J]. London: Ministry of Supply Wave Report 3, 1944.
[22] Mei C.C. , The Applied Dynamics of Ocean Surface Waves[M]. World Scientific, Singa-pore, 1989.
[23] Epstein P.S., Re?ection of waves in an inhomogeneous absorbing medium[J]. Proceedingsof the National Academy of Sciences of the United States of America 16(10)(1930): 627-637.
[24] Yoshida K., On the partial re?ection of long waves[J]. Geophysical Notes GeophysicalInstitute, Tokyo University, 1948, No.31.
[25] Dean R.G., Long wave modification by linear transitions[J]. Journal of the Waterwaysand Harbors Division, Proceedings, ASCE, 90(1)(1964): 1-30.
[26] Kajiura K., On the partial re?ection of water waves passing over a bottom of variabledepth[J]. International Cooperation in Geodesy and Geophysics 24(1964): 206-230.
[27] Dingemans M.W., Water Wave Propagation over Uneven Bottoms[M]. Part 1-linearWave Propagation, World Scientific, Singapore, 1997.
[28] Bender C.J., Dean R.G., Wave transformation by two-dimensional bathymetric anoma-lies with sloped transitions[J]. Coastal Engineering 50(2003): 61-84.
[29] Xie J.J., Liu H.W., Lin P., Analytical solution for long wave re?ection by a rectangu-lar obstacle with two scour trenches[J]. Journal of Engineering Mechanics 137(2011),accepted.
[30]王高雄,周之铭,朱思铭,常微分方程.高等教育出版社, 2006年7月,第三版.
[31] He J.H., Homotopy perturbation technique[J]. Computer Methods in Applied Mechanicsand Engineering 178(1999): 257-262.
[32] He J.H., A coupling method homotopy technique and perturbation technique for non-linear problems[J]. International Journal of Non-Linear Mechanics 35(1)(2000): 37-43.
[33] He J.H., Homotopy perturbation method: A new nonlinear analytical technique[J]. Ap-plied Mathematics and Computation 135(2003): 73–79.
[34] He J.H., Some asymptotic methods for strongly nonlinear equations[J]. InternationalJournal of Modern Physics B 20(2006): 1141-1199.
[35] Chun C.B., Integration using He’s homotopy perturbation method[J]. Chaos SolitonsFractals 34(2007): 1130-1134.
[36] Gorji M., Ganji D.D., Soleimani S., New application of He’s homotopy perturba-tion method[J]. International Journal of Nonlinear Sciences and Numerical Simulation8(2007): 319-328.