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A third-order Lagrangian solutions is found directly by the full Lagrangian parametric governing equations, with the parameters of identifying each water particle which is obtained in mass conservation from taking the wavelength-averaged level of the vertical displacement of water particles along the direction of the wave propagation and is just the position of particle in the original still water, for the irrotational free surface progressive gravity waves propagating on uniform current in the three dimension. It is found that the effect of wave-current interaction is occurred in Lagrangian velocity potential, so that the pressure is not affected by the uniform current in the wave-current field. The drift velocity, Lagrangian mean level of particle and its motion period longer than the wave period which are exponentially decreasing with the depth of particle's level and the pathlines of particle and the streakline are then all presented, these are to be excluded in Eulerian solution. Furthermore, the forms of the helical pathlines of particles varying with the direction and speed of the uniform current are appeared to be turned from the wave velocity and current velocity component in co-direction to in anti-direction, even become into a closed orbit. The special case when the wave-current field is at the steady motion is also interpreted its consequential charateristics. The present solution can be reduced to that of the pure progressive waves as no current exists.
[1] Longuet-Higgins M S, Stewart R W 1960 J. Fluid Mech. 8 565
[2] Longuet-Higgins M S, Stewart R W 1961 J. Fluid Mech. 10 529
[3] Jonsson I G, Skougaard C, Wang J D 1970 Proc. 12th Coastal Eng. Conf.(New York: ASCE)1 489
[4] Jonsson I G 1978 J. Hydraul. Res. 16(3)223
[5] Jonsson I G, Brink-Kjaer O, Thomas G P 1978 J. Fluid Mech. 87 401
[6] Peregrine D H 1976 Adv. Appl. Mech. 16 9
[7] Thomas G P 1981 J. Fluid Mech. 110 457
[8] Thomas G P 1990 J. Fluid Mech. 216 505
[9] Chen Y Y, Juang W J 1990 Proceeding of the 12th Ocean Engineering Conference in Taiwan(Taiwan: The Taiwan Society of Ocean Engineering)p248(in Chinese)[陈阳益, 庄文杰 1990 第十二届海洋工程研讨会论文集(台湾:台湾海洋工程学会)第248页]
[10] Baddour R E, Song S W 1990 Ocean Engng. 17 551
[11] Chang H K, Chen Y Y 1993 Harbour Technology 8 24(in Chinese)[张宪国, 陈阳益 1993 港湾技术 8 24]
[12] Groeneweg J, Battjes J 2003 J. Fluid Mech. 478 325
[13] Musumeci R E, Cavallo L, Foti E, Scandura P 2006 J. Geophys. Res. 111 c07019
[14] Olabarrieta M, Medina P, Castanedo S 2010 Coastal Engineering 57 643
[15] Hsu H C, Chen Y Y, Li M S, Tseng W J 2009 Acta Oceanol. Sin. 28 89
[16] Chen Y Y, Hsu H C 2005 Proceeding of the 27th Ocean Engineering Conference in Taiwan.(The Taiwan Society of Ocean Engineering)p63(in Chinese)[陈阳益, 许弘莒 2005 第二十七届海洋工程研讨会论文集(台湾:台湾海洋工程学会)第63页]
[17] Chen Y Y, Hsu H C 2009a Acta Physica Sinica 58 40(in Chinese)[陈阳益, 许弘莒 2009a 58 40]
[18] Chen Y Y, Hsu H C 2009b Acta Physica Sinica 58 3657 [陈阳益, 许弘莒 2009b 58 3637]
[19] Chen Y Y, Hsu H C 2009 Ocean Engineering 36 747
[20] Chen Y Y 1994a Proceeding of the 12th Ocean Engineering Conference in Taiwan(Taiwan: The Taiwan Society of Ocean Engineering)A1(in Chinese)[陈阳益 1994a 第十六届海洋工程研讨会论文集(台湾:台湾海洋工程学会)A1]
[21] Chen Y Y 1994b Proceeding of the 12th Ocean Engineering Conference in Taiwan.(The Taiwan society of Ocean Engineering.)(in Chinese)A30 [陈阳益 1994b 第十六届海洋工程研讨会论文集(台湾:台湾海洋工程学会)A30]
[22] Chen Y Y, Hsu H C 2009d China Physics B 18 1
[23] Chen Y Y, Hsu H C, Chen G Y 2010 Fluid Dyn. Res. 42 1
[24] Lamb H 1932 Hydrodynamics(6th Ed.)(London: Cambridge University Press)
[25] Pierson W J 1962 J. Geophy. Res. 67 3151
[26] Fenton J D 1985 J. Waterways Port Coastal Ocean Eng. 3 216
[27] Longuet-Higgins M S 1987 J. Fluid Mech. 179 547
[28] Longuet-Higgins M S 1986 J. Fluid Mech. 173 683
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[1] Longuet-Higgins M S, Stewart R W 1960 J. Fluid Mech. 8 565
[2] Longuet-Higgins M S, Stewart R W 1961 J. Fluid Mech. 10 529
[3] Jonsson I G, Skougaard C, Wang J D 1970 Proc. 12th Coastal Eng. Conf.(New York: ASCE)1 489
[4] Jonsson I G 1978 J. Hydraul. Res. 16(3)223
[5] Jonsson I G, Brink-Kjaer O, Thomas G P 1978 J. Fluid Mech. 87 401
[6] Peregrine D H 1976 Adv. Appl. Mech. 16 9
[7] Thomas G P 1981 J. Fluid Mech. 110 457
[8] Thomas G P 1990 J. Fluid Mech. 216 505
[9] Chen Y Y, Juang W J 1990 Proceeding of the 12th Ocean Engineering Conference in Taiwan(Taiwan: The Taiwan Society of Ocean Engineering)p248(in Chinese)[陈阳益, 庄文杰 1990 第十二届海洋工程研讨会论文集(台湾:台湾海洋工程学会)第248页]
[10] Baddour R E, Song S W 1990 Ocean Engng. 17 551
[11] Chang H K, Chen Y Y 1993 Harbour Technology 8 24(in Chinese)[张宪国, 陈阳益 1993 港湾技术 8 24]
[12] Groeneweg J, Battjes J 2003 J. Fluid Mech. 478 325
[13] Musumeci R E, Cavallo L, Foti E, Scandura P 2006 J. Geophys. Res. 111 c07019
[14] Olabarrieta M, Medina P, Castanedo S 2010 Coastal Engineering 57 643
[15] Hsu H C, Chen Y Y, Li M S, Tseng W J 2009 Acta Oceanol. Sin. 28 89
[16] Chen Y Y, Hsu H C 2005 Proceeding of the 27th Ocean Engineering Conference in Taiwan.(The Taiwan Society of Ocean Engineering)p63(in Chinese)[陈阳益, 许弘莒 2005 第二十七届海洋工程研讨会论文集(台湾:台湾海洋工程学会)第63页]
[17] Chen Y Y, Hsu H C 2009a Acta Physica Sinica 58 40(in Chinese)[陈阳益, 许弘莒 2009a 58 40]
[18] Chen Y Y, Hsu H C 2009b Acta Physica Sinica 58 3657 [陈阳益, 许弘莒 2009b 58 3637]
[19] Chen Y Y, Hsu H C 2009 Ocean Engineering 36 747
[20] Chen Y Y 1994a Proceeding of the 12th Ocean Engineering Conference in Taiwan(Taiwan: The Taiwan Society of Ocean Engineering)A1(in Chinese)[陈阳益 1994a 第十六届海洋工程研讨会论文集(台湾:台湾海洋工程学会)A1]
[21] Chen Y Y 1994b Proceeding of the 12th Ocean Engineering Conference in Taiwan.(The Taiwan society of Ocean Engineering.)(in Chinese)A30 [陈阳益 1994b 第十六届海洋工程研讨会论文集(台湾:台湾海洋工程学会)A30]
[22] Chen Y Y, Hsu H C 2009d China Physics B 18 1
[23] Chen Y Y, Hsu H C, Chen G Y 2010 Fluid Dyn. Res. 42 1
[24] Lamb H 1932 Hydrodynamics(6th Ed.)(London: Cambridge University Press)
[25] Pierson W J 1962 J. Geophy. Res. 67 3151
[26] Fenton J D 1985 J. Waterways Port Coastal Ocean Eng. 3 216
[27] Longuet-Higgins M S 1987 J. Fluid Mech. 179 547
[28] Longuet-Higgins M S 1986 J. Fluid Mech. 173 683
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