海洋表面波越过非理想海底地形时的解析模拟
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摘要
海洋表面波的散射效应(包括波浪的反射、折射和衍射)在港口、码头、航道和防波堤的工程设计,以及海啸预报、海洋平台和船舶耐波性评估等方面具有重要的应用价值.由于实际问题的难度和复杂性,一般都结合计算机进行数值模拟.而数值解或者数值模型必须通过实验解和解析解的验证.相比于实验解,解析解具有开发成本低,精确性高,且容易发现物理规律等优点,因此最受欢迎.过去已经获得的线性或非线性水波问题的准确解析解,一般只能针对简单的方程或简单的地形.例如,海底地形几乎都不得不设定成理想情形,即变水深区域的水深函数为x或径向的幂函数,或者说水下浅滩的顶点落在自由液面上.因为这使得控制方程可以转化为经典微分方程,如Bessel方程和Euler方程,从而易于得到闭合形式解析解.
本学位论文的创新点在于,将这种理想地形推广到了非理想地形.所谓非理想地形是指,变水深区域的水深不再被假定为自变量的幂函数.正是这种非理想性,增加了我们求解控制方程的困难,主要体现在保证解析解在物理区间上的收敛性方面.
在本文中,首先,我们研究了线性长波越过带有冲刷槽的一维矩形障碍物时的反射效应;其次,又分别研究了线性长波越过水下浅滩、置于非理想浅滩上的圆柱岛以及非理想截顶浅滩三种二维非理想海底地形时的散射效应;最后,又研究了涵盖线性长波、中波和短波在内的全波谱波场越过一维非理想陷坑时的反射效应.
Refraction and di?raction e?ects of ocean surface waves have important researchvalue in design of the ports, docks, navigation channels, breakwater, forecast of thetsunami and evaluation of the seakeeping capability of platforms and ships in the sea,etc. For di?culty and complexity of the physical problems, numerical simulations aregenerally adopted. However, the numerical solutions or the numerical models must beverified by experimental data or analytic solutions. Compared with experimental data,the advantages of the low cost , the high accuracy and the ease to find laws make analyticsolutions to be the most popular. While the existing results are usually restricted tosimple equations or topographies. For example, the topographies are mostly assumed tobe idealized, which means the water depth function over the variable region is a powerfunction of the independent variable, i.e., the vertex of the shoal is located on the stillwater level. As under this assumption, the governing equation can be transformed intoclassical Euler or Bessel equation, in which the closed-form analytic solutions are easilyobtained.
The innovation of this thesis is generalizing the idealized topographies into un-idealized. The so-called un-idealized means that the water depth function over thevariable region is no longer a power function of the independent variable. It is thisgeneralization that increase the di?culty for us to guarantee the convergency of theanalytic solution.
In this text, firstly, we investigate the re?ection e?ects of linear long waves prop-agating over a one-dimensional rectangular obstacle with scour trenches. Secondly, wegive the di?raction e?ects of linear long waves scattering by a shoal, a circular cylinderisland mounted on an un-idealized shoal and an un-idealized truncated shoal, respec-tively. Finally, we study the re?ection e?ects of linear waves encompassing the wholewave range propagating over a one-dimensional un-idealized trench.
本学位论文的创新点在于,将这种理想地形推广到了非理想地形.所谓非理想地形是指,变水深区域的水深不再被假定为自变量的幂函数.正是这种非理想性,增加了我们求解控制方程的困难,主要体现在保证解析解在物理区间上的收敛性方面.
在本文中,首先,我们研究了线性长波越过带有冲刷槽的一维矩形障碍物时的反射效应;其次,又分别研究了线性长波越过水下浅滩、置于非理想浅滩上的圆柱岛以及非理想截顶浅滩三种二维非理想海底地形时的散射效应;最后,又研究了涵盖线性长波、中波和短波在内的全波谱波场越过一维非理想陷坑时的反射效应.
Refraction and di?raction e?ects of ocean surface waves have important researchvalue in design of the ports, docks, navigation channels, breakwater, forecast of thetsunami and evaluation of the seakeeping capability of platforms and ships in the sea,etc. For di?culty and complexity of the physical problems, numerical simulations aregenerally adopted. However, the numerical solutions or the numerical models must beverified by experimental data or analytic solutions. Compared with experimental data,the advantages of the low cost , the high accuracy and the ease to find laws make analyticsolutions to be the most popular. While the existing results are usually restricted tosimple equations or topographies. For example, the topographies are mostly assumed tobe idealized, which means the water depth function over the variable region is a powerfunction of the independent variable, i.e., the vertex of the shoal is located on the stillwater level. As under this assumption, the governing equation can be transformed intoclassical Euler or Bessel equation, in which the closed-form analytic solutions are easilyobtained.
The innovation of this thesis is generalizing the idealized topographies into un-idealized. The so-called un-idealized means that the water depth function over thevariable region is no longer a power function of the independent variable. It is thisgeneralization that increase the di?culty for us to guarantee the convergency of theanalytic solution.
In this text, firstly, we investigate the re?ection e?ects of linear long waves prop-agating over a one-dimensional rectangular obstacle with scour trenches. Secondly, wegive the di?raction e?ects of linear long waves scattering by a shoal, a circular cylinderisland mounted on an un-idealized shoal and an un-idealized truncated shoal, respec-tively. Finally, we study the re?ection e?ects of linear waves encompassing the wholewave range propagating over a one-dimensional un-idealized trench.
引文
[1] Agnon, Y., Pelinovsky, E., Accurate refraction- di?raction equations for water waves ona variable-depth rough bottom [J]. Journal of Fluid Mechanics, 449(2001): 301- 311.
[2] Batchelor, G.K., An Introduction to Fluid Dynamics [M]. Cambridge University Press,London, 1967.
[3] Berkho?, J.C.W., Computation of combined refraction-di?raction [C]. Proceedings of the13th International Coastal Engineering Conference, ASCE, Vancouver, Canada, 1972:471-490.
[4] Bettess, P., Zienkiewicz, O.C., Di?raction and refraction of surface waves using finiteand infinite elements [J]. International Journal for Numerical Methods in Engineering,11(1977): 1271-1290.
[5] Booij, N., A note on the accuracy of the mild-slope equation [J]. Coastal Engineering,7(1983): 191-203.
[6] Chamberlain, P.G., Porter, D., The modified mild-slope equation [J]. Journal of FluidMechanics, 291(1995): 393–407.
[7] Chandrasekera, C.N., Cheung, K.F., Extended linear refraction-di?raction model [J].Journal of Waterway, Port, Coastal and Ocean Engineering, 123(1997): 280-286.
[8] Chang, H.K., Liou, J.C., Long wave re?ection from submerged trapezoidal breakwaters[J]. Ocean Engineering, 34(2007): 185-191.
[9] Cheng, Y.-M., Analytic Solutions of the Mild-slope Equation [D]. Doctoral Thesis, Na-tional Cheng Kung University, Taiwan, 2007.
[10] Copeland, G.J.M., A practical alternative to the mild-slope wave equation [J]. CoastalEngineering, 9(1985): 125–149.
[11] Dean, R.G., Long wave modification by linear transitions [J]. Journal of WaterwaysHarbors Div, Proc ASCE, 90(1964): 1-29.
[12] Dingemans, M.W., Water wave propagation over uneven bottoms, Part 1-Linear wavepropagation [M]. World Scientific, Singapore, 1997.
[13] Eckart, C., The propagation of gravity waves from deep to shallow water [J]. GravityWaves, National Bureau of Standards, Circular, 521(1952): 165–173.
[14] Homma, S., On the behavior of seismic sea waves around circular island [J]. GeophysicalMagazine, 21(1950): 199-208.
[15] Houston, J.R., Combined refraction and di?raction of short waves using the finite ele-ment method [J]. Applied Ocean Research, 3(1981): 163-170.
[16] Hsu, T.W., Lin, T.Y., Wen, C.C., Ou, S.H., A complementary mild-slope equationderived using higher-order depth function for waves obliquely propagating on slopingbottom [J]. Physics of Fluids, 18(2006):1-14
[17] Hsiao, S.S., Chang, C.M., Wen, C.C., Solution for wave propagation through a circularcylinder mounted on di?erent topography ripple-bed profile shoals using DRBEM [J].Engineering Analysis with Boundary Elements, 33(2009): 1246-1257.
[18] Hsiao, S.S., Chang, C.M., Wen, C.C., An analytical solution to the modified mild-slopeequation for waves propagating around a circular conical island [J]. Journal of MarineScience and Technology, 18(2010): 520-529.
[19] Hunt, J.N., Direct solution of wave dispersion equation [J]. Journal of the WaterwayPort Coastal and Ocean Division, ASCE, 105(1979): 457-459.
[20] Jonsson, I.G., Skovgaard, O., Brink-Kjaer, O., Di?raction and refraction calculationsfor waves incident on an island [J]. Journal of Marine Research, 34(1976): 469-496.
[21] Jung, T.H., Cho, Y.-S., Analytical approach for long wave solution to an arbitrarilyvarying topography [J]. Journal of Coastal Research, 25(2009): 216-223.
[22] Jung, T.H., Lee, C., Cho, Y.-S., Analytical solutions for long waves over a circular island[J]. Coastal Engineering, 57(2010): 440-446.
[23] Jung, T.H., Suh, K.D., An analytic solution to the mild slope equation for waves prop-agating over an axi-symmetric pit [J]. Coastal Engineering, 54(2007): 865-877.
[24] Jung, T.H., Suh, K.D., An analytical solution to the extended mild-slope equation forlong waves propagating over an axi-symmetric pit [J]. Wave Motion, 45(2008): 835-845.
[25] Jung, T.H., Suh, K.D., Lee, S.O., Cho, Y.-S., Linear wave re?ection by trench withvarious shapes [J]. Ocean Engineering, 35(2008): 1226-1234.
[26] Kajiura, K., On the partial re?ection of water waves passing over a bottom of variabledepth [M]. Proc. Tsunami Meetings 10th Pacific Science Congress, I.U.G.G. Monograph,24(1961): 206-234.
[27] Kanoglu, U., Synolakis, C.E., Long wave runup on piecewise linear topographies [J].Journal of Fluid Mechanics, 374(1998): 1-28.
[28] Lamb, H., Hydrodynamics. 6th edn. Dover, 1932.
[29] Li, B., A generalized conjugate-gradient model for the mild slope equation [J]. CoastalEngineering, 23(1994): 215-225.
[30] Li, B., An evolution equation for water waves [J]. Coastal Engineering, 23(1994): 227-242.
[31] Li, B., Fleming, C. A., Modified mild-slope equations for wave propagation [J]. Pro-ceedings of the Institution of Civil Engineers - Water Maritime and Energy, 136(1999):43-60.
[32] Lin, P.Z., A compact numerical algorithm for solving the time-dependent mild slopeequation [J]. International Journal for Numerical Methods in Fluids , 45(2004): 625-642.
[33] Lin, P.Z., Liu, H.-W., Analytical study of linear long-wave re?ection by a two-dimensional obstacle of general trapezoidal shape [J]. Journal of Engineering Mechanics,ASCE, 131(2005): 822-830.
[34] Lin, P.Z., Liu, H.-W., Scattering and trapping of wave energy by a submerged t truncatedparaboloidal shoal [J]. Journal of Waterway, Port, Coastal and Ocean Engineering,ASCE, 133(2007): 94-103.
[35] Liu, H.-W., Numerical Modeling of the Propagation of Ocean Waves [D]. Doctoral thesis,University of Wollongong, Australia, 2001.
[36] Liu, H.-W., Li, Y.B., An analytical solution for long-wave scattering by a submergedcircular truncated shoal[J]. Journal of Engineering Mathematics, 57(2007): 133-144.
[37] Liu, H.-W., Lin, P.Z., Discussion of“Wave transformation by two-dimensional bathy-metric anomalies with sloped transitions”[Coast. Eng. 50 (2003) 61–84] [J]. CoastalEngineering, 52(2005): 197-200.
[38] Liu, H.-W., Lin, P.Z., Shankar, N.J., An analytical solution of the mild-slope equationfor waves around a circular island on a paraboloidal shoal [J]. Coastal Engineering,51(2004): 421-437.
[39] Liu, H.-W., Lin, P.Z., Shankar, N.J., An analytical solution of the mild-slope equationfor scattering by a truncated conical shoal. The Proceedings of the Second Asian andPacific Coastal Engineering Conference, Makuhari, Japan, 2004: 49-50.
[40] Liu, H.-W., Lin, P.Z., An analytic solution for wave scattering by a circular cylindermounted on a conical shoal [J]. Coastal Engineering Journal, 49(2007): 393-416.
[41] Liu, H.-W., Xie, J.J., Discussion of“Analytic solution of long wave propaga-tion over a submerged hump”by Niu and Yu (2011) [J]. Coastal Engineering,doi:10.1016/j.coastaleng.2011.05.005, in press.
[42] Liu, H.-W., Yang, J., Lin, P.Z., [J]. Journal of Fluid Mechanics, 2011, under view.
[43] Longuet-Higgins, M.S., On the trapping of wave energy round islands [J]. Journal ofFluid Mechanics, 29(1967): 781-821.
[44] Lozano, C.J., Meyer, R.E., Leakage and response of waves trapped round islands [J].Physics of Fluids, 19(1976): 1075-1088.
[45] MacCamy, R.C., Fuchs, R.A., Wave forces on piles: a di?raction theory. Tech. Memo.US Army Corps of Engineering, Beach Erosion Board, Washington, DC. No. 69, 1954.
[46] Massel, S.R., Extended refraction-di?raction equation for surface waves [J]. CoastalEngineering, 19(1993):97-126.
[47] Mei, C.C., The Applied Dynamics of Ocean Surface Waves [M]. World Scientific, Singa-pore, 1989.
[48] Miles, J.W., Chamberlain, P.G., Topographical scattering of gravity waves [J]. Journalof Fluid Mechanics, 361(1998): 175-188.
[49] Nardini, D., Brebbia, C.A., A new approach to free vibration analysis using boundaryelements [J]. Applied Mathematical Modelling , 7(1983): 157-162.
[50] Niu, X.J., Yu, X.P., Analytic solution of long wave propagation over a submerged hump[J]. Coastal Engineering, 58(2011): 143-150.
[51] Niu, X.J., Yu, X.P., Analytical study on long wave refraction over a dredge excavationpit [J]. Wave Motion , 48(2011): 259-267.
[52] Niu, X.J., Yu, X.P., Long wave scattering by a vertical cylinder with idealized scour pit[J]. Journal of Waterway, Port, Coastal and Ocean Engineering, to appear, 2011.
[53] Porter, D., The mild-slope equations [J]. Journal of Fluid Mechanics, 494(2003): 51-63.
[54] Porter, D., Staziker, D.J., Extensions of the mild-slope equation [J]. Journal of FluidMechanics, 300(1995): 367-382.
[55] Schonfeld, J.C., Propagation of Two Dimensional Short Waves[M]. Delft University ofTechnology Manuscript (in Dutch), 1972.
[56] Smith, R., Sprinks, T., Scattering of surface waves by a conical island [J]. Journal ofFluid Mechanics, 72(1975): 373.
[57] Spiegel, M.R., Applied Di?erential Equations [M]. Prentice-Hall, Englewood Cli?s, NJ,1981.
[58] Suh, K.-D., Jung, T.-H., Haller, M.C., Long waves propagating over a circular bowl pit[J]. Wave Motion, 42(2005): 143-154.
[59] Suh, K.D., Lee, C., Park, W.S., Time-dependent equations for wave propagation onrapidly varying topography [J]. Coastal Engineering, 32(1997): 91-117.
[60] Sumer, B.M., Whitehouse, R.J.S., and T?um, A., Scour around coastal structures: Asummary of recent research [J]. Coastal Engineering, 44(2001): 153-190.
[61] Tsay, T.-K., Liu, P.L.-F., A finite element model for wave refraction and di?raction [J].Applied Ocean Research, 5(1983): 30-37.
[62] Vastano, A.C., Reid, R.O., Tsunami response for islands: verification of a numericalprocedure [J]. Journal of Marine Research, 25 (1967): 129-39.
[63] Yu, X.P., Zhang, B.Y., An extended analytical solution for combined refraction anddi?raction of long waves over circular shoals [J]. Ocean Engineering, 30(2003): 1253-1267.
[64] Zhang, Y.L., Zhu, S.-P., New solutions for the propagation of long water waves overvariable depth [J]. Journal of Fluid Mechanics, 278 (1994): 391-406.
[65] Zhu, S.-P., A new DRBEM model for wave refraction and di?raction [J]. EngineeringAnalysis with Boundary Elements, 12(1993): 261-74.
[66] Zhu, S.-P., Harun, F.-N., An analytical solution for long wave refraction over a circularhump [J]. Journal of Applied Mathematics and Computing, 30 (2009): 315-333.
[67] Zhu, S.-P., Liu, H.-W., Chen, K., A general DRBEM model for wave refraction anddi?raction [J]. Engineering Analysis with Boundary Elements, 24 (2000): 377-390.
[68] Zhu, S.-P., Zhang, Y.L., Scattering of long waves around a circular island mounted ona conical shoal [J]. Wave Motion, 23 (1996): 353-362.
[69] Zill, D.G., Cullen, M.R., Di?erential Equations with Boundary-Value Problems[M]. Cen-gage Learning, Brooks/Cole, 2009.
[2] Batchelor, G.K., An Introduction to Fluid Dynamics [M]. Cambridge University Press,London, 1967.
[3] Berkho?, J.C.W., Computation of combined refraction-di?raction [C]. Proceedings of the13th International Coastal Engineering Conference, ASCE, Vancouver, Canada, 1972:471-490.
[4] Bettess, P., Zienkiewicz, O.C., Di?raction and refraction of surface waves using finiteand infinite elements [J]. International Journal for Numerical Methods in Engineering,11(1977): 1271-1290.
[5] Booij, N., A note on the accuracy of the mild-slope equation [J]. Coastal Engineering,7(1983): 191-203.
[6] Chamberlain, P.G., Porter, D., The modified mild-slope equation [J]. Journal of FluidMechanics, 291(1995): 393–407.
[7] Chandrasekera, C.N., Cheung, K.F., Extended linear refraction-di?raction model [J].Journal of Waterway, Port, Coastal and Ocean Engineering, 123(1997): 280-286.
[8] Chang, H.K., Liou, J.C., Long wave re?ection from submerged trapezoidal breakwaters[J]. Ocean Engineering, 34(2007): 185-191.
[9] Cheng, Y.-M., Analytic Solutions of the Mild-slope Equation [D]. Doctoral Thesis, Na-tional Cheng Kung University, Taiwan, 2007.
[10] Copeland, G.J.M., A practical alternative to the mild-slope wave equation [J]. CoastalEngineering, 9(1985): 125–149.
[11] Dean, R.G., Long wave modification by linear transitions [J]. Journal of WaterwaysHarbors Div, Proc ASCE, 90(1964): 1-29.
[12] Dingemans, M.W., Water wave propagation over uneven bottoms, Part 1-Linear wavepropagation [M]. World Scientific, Singapore, 1997.
[13] Eckart, C., The propagation of gravity waves from deep to shallow water [J]. GravityWaves, National Bureau of Standards, Circular, 521(1952): 165–173.
[14] Homma, S., On the behavior of seismic sea waves around circular island [J]. GeophysicalMagazine, 21(1950): 199-208.
[15] Houston, J.R., Combined refraction and di?raction of short waves using the finite ele-ment method [J]. Applied Ocean Research, 3(1981): 163-170.
[16] Hsu, T.W., Lin, T.Y., Wen, C.C., Ou, S.H., A complementary mild-slope equationderived using higher-order depth function for waves obliquely propagating on slopingbottom [J]. Physics of Fluids, 18(2006):1-14
[17] Hsiao, S.S., Chang, C.M., Wen, C.C., Solution for wave propagation through a circularcylinder mounted on di?erent topography ripple-bed profile shoals using DRBEM [J].Engineering Analysis with Boundary Elements, 33(2009): 1246-1257.
[18] Hsiao, S.S., Chang, C.M., Wen, C.C., An analytical solution to the modified mild-slopeequation for waves propagating around a circular conical island [J]. Journal of MarineScience and Technology, 18(2010): 520-529.
[19] Hunt, J.N., Direct solution of wave dispersion equation [J]. Journal of the WaterwayPort Coastal and Ocean Division, ASCE, 105(1979): 457-459.
[20] Jonsson, I.G., Skovgaard, O., Brink-Kjaer, O., Di?raction and refraction calculationsfor waves incident on an island [J]. Journal of Marine Research, 34(1976): 469-496.
[21] Jung, T.H., Cho, Y.-S., Analytical approach for long wave solution to an arbitrarilyvarying topography [J]. Journal of Coastal Research, 25(2009): 216-223.
[22] Jung, T.H., Lee, C., Cho, Y.-S., Analytical solutions for long waves over a circular island[J]. Coastal Engineering, 57(2010): 440-446.
[23] Jung, T.H., Suh, K.D., An analytic solution to the mild slope equation for waves prop-agating over an axi-symmetric pit [J]. Coastal Engineering, 54(2007): 865-877.
[24] Jung, T.H., Suh, K.D., An analytical solution to the extended mild-slope equation forlong waves propagating over an axi-symmetric pit [J]. Wave Motion, 45(2008): 835-845.
[25] Jung, T.H., Suh, K.D., Lee, S.O., Cho, Y.-S., Linear wave re?ection by trench withvarious shapes [J]. Ocean Engineering, 35(2008): 1226-1234.
[26] Kajiura, K., On the partial re?ection of water waves passing over a bottom of variabledepth [M]. Proc. Tsunami Meetings 10th Pacific Science Congress, I.U.G.G. Monograph,24(1961): 206-234.
[27] Kanoglu, U., Synolakis, C.E., Long wave runup on piecewise linear topographies [J].Journal of Fluid Mechanics, 374(1998): 1-28.
[28] Lamb, H., Hydrodynamics. 6th edn. Dover, 1932.
[29] Li, B., A generalized conjugate-gradient model for the mild slope equation [J]. CoastalEngineering, 23(1994): 215-225.
[30] Li, B., An evolution equation for water waves [J]. Coastal Engineering, 23(1994): 227-242.
[31] Li, B., Fleming, C. A., Modified mild-slope equations for wave propagation [J]. Pro-ceedings of the Institution of Civil Engineers - Water Maritime and Energy, 136(1999):43-60.
[32] Lin, P.Z., A compact numerical algorithm for solving the time-dependent mild slopeequation [J]. International Journal for Numerical Methods in Fluids , 45(2004): 625-642.
[33] Lin, P.Z., Liu, H.-W., Analytical study of linear long-wave re?ection by a two-dimensional obstacle of general trapezoidal shape [J]. Journal of Engineering Mechanics,ASCE, 131(2005): 822-830.
[34] Lin, P.Z., Liu, H.-W., Scattering and trapping of wave energy by a submerged t truncatedparaboloidal shoal [J]. Journal of Waterway, Port, Coastal and Ocean Engineering,ASCE, 133(2007): 94-103.
[35] Liu, H.-W., Numerical Modeling of the Propagation of Ocean Waves [D]. Doctoral thesis,University of Wollongong, Australia, 2001.
[36] Liu, H.-W., Li, Y.B., An analytical solution for long-wave scattering by a submergedcircular truncated shoal[J]. Journal of Engineering Mathematics, 57(2007): 133-144.
[37] Liu, H.-W., Lin, P.Z., Discussion of“Wave transformation by two-dimensional bathy-metric anomalies with sloped transitions”[Coast. Eng. 50 (2003) 61–84] [J]. CoastalEngineering, 52(2005): 197-200.
[38] Liu, H.-W., Lin, P.Z., Shankar, N.J., An analytical solution of the mild-slope equationfor waves around a circular island on a paraboloidal shoal [J]. Coastal Engineering,51(2004): 421-437.
[39] Liu, H.-W., Lin, P.Z., Shankar, N.J., An analytical solution of the mild-slope equationfor scattering by a truncated conical shoal. The Proceedings of the Second Asian andPacific Coastal Engineering Conference, Makuhari, Japan, 2004: 49-50.
[40] Liu, H.-W., Lin, P.Z., An analytic solution for wave scattering by a circular cylindermounted on a conical shoal [J]. Coastal Engineering Journal, 49(2007): 393-416.
[41] Liu, H.-W., Xie, J.J., Discussion of“Analytic solution of long wave propaga-tion over a submerged hump”by Niu and Yu (2011) [J]. Coastal Engineering,doi:10.1016/j.coastaleng.2011.05.005, in press.
[42] Liu, H.-W., Yang, J., Lin, P.Z., [J]. Journal of Fluid Mechanics, 2011, under view.
[43] Longuet-Higgins, M.S., On the trapping of wave energy round islands [J]. Journal ofFluid Mechanics, 29(1967): 781-821.
[44] Lozano, C.J., Meyer, R.E., Leakage and response of waves trapped round islands [J].Physics of Fluids, 19(1976): 1075-1088.
[45] MacCamy, R.C., Fuchs, R.A., Wave forces on piles: a di?raction theory. Tech. Memo.US Army Corps of Engineering, Beach Erosion Board, Washington, DC. No. 69, 1954.
[46] Massel, S.R., Extended refraction-di?raction equation for surface waves [J]. CoastalEngineering, 19(1993):97-126.
[47] Mei, C.C., The Applied Dynamics of Ocean Surface Waves [M]. World Scientific, Singa-pore, 1989.
[48] Miles, J.W., Chamberlain, P.G., Topographical scattering of gravity waves [J]. Journalof Fluid Mechanics, 361(1998): 175-188.
[49] Nardini, D., Brebbia, C.A., A new approach to free vibration analysis using boundaryelements [J]. Applied Mathematical Modelling , 7(1983): 157-162.
[50] Niu, X.J., Yu, X.P., Analytic solution of long wave propagation over a submerged hump[J]. Coastal Engineering, 58(2011): 143-150.
[51] Niu, X.J., Yu, X.P., Analytical study on long wave refraction over a dredge excavationpit [J]. Wave Motion , 48(2011): 259-267.
[52] Niu, X.J., Yu, X.P., Long wave scattering by a vertical cylinder with idealized scour pit[J]. Journal of Waterway, Port, Coastal and Ocean Engineering, to appear, 2011.
[53] Porter, D., The mild-slope equations [J]. Journal of Fluid Mechanics, 494(2003): 51-63.
[54] Porter, D., Staziker, D.J., Extensions of the mild-slope equation [J]. Journal of FluidMechanics, 300(1995): 367-382.
[55] Schonfeld, J.C., Propagation of Two Dimensional Short Waves[M]. Delft University ofTechnology Manuscript (in Dutch), 1972.
[56] Smith, R., Sprinks, T., Scattering of surface waves by a conical island [J]. Journal ofFluid Mechanics, 72(1975): 373.
[57] Spiegel, M.R., Applied Di?erential Equations [M]. Prentice-Hall, Englewood Cli?s, NJ,1981.
[58] Suh, K.-D., Jung, T.-H., Haller, M.C., Long waves propagating over a circular bowl pit[J]. Wave Motion, 42(2005): 143-154.
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