Use of Galerkin technique to the rolling of a plate in deep water
Abstract
The classical problems of surface water waves produced by small oscillations of a thin vertical plate partially immersed as well as submerged in deep water are reinvestigated here. Each problem is reduced to an integral equation involving horizontal component of velocity across the vertical line outside the plate. The integral equations are solved numerically using Galerkin approximation in terms of simple polynomials multiplied by an appropriate weight function whose form is dictated by the behaviour of the fluid velocity near the edge(s) of the plate. Fairly accurate numerical estimates for the amplitude of the radiated wave at infinity due to rolling and also for swaying of the pate in each case are obtained and these are depicted graphically against the wave number for various cases.
Keyword : rolling motion, partially immersed and submerged plate, integral equation, Galerkin approximation, amplitude of radiated wave
How to Cite
Ray, S., De, S., & Mandal, B. N. (2021). Use of Galerkin technique to the rolling of a plate in deep water. Mathematical Modelling and Analysis, 26(2), 209-222. https://doi.org/10.3846/mma.2021.12767
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S. Banerjea and B.N. Mandal. On waves due to rolling of a vertical plate. Indian J. Pure Appl. Math., 23:753–761, 1992.
H. Cheng, M.J. Peng and Y.M. Cheng. The hybrid complex variable elementfree Galerkin method for 3D elasticity problems. Engg. Structures, 219:110835, 2020. https://doi.org/10.1016/j.engstruct.2020.110835
B.C. Das, S. De and B.N. Mandal. Oblique scattering by thin vertical barriers in deep water: solution by multiterm Galerkin technique using simple polynomials as basis. J. Marine Sc. Tech., 23:915–925, 2018. https://doi.org/10.1007/s00773-017-0520-4
B.C. Das, S.De and B.N. Mandal. Oblique water waves scattering by a thick barrier with rectangular cross section in deep water. J. Engg. Math., 2020. https://doi.org/10.1007/s10665-020-10049-4
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D.V. Evans and M. Fernyhough. Edge waves along periodic coastlines. Part 2. Mathl. Comput. Modelling, 297:307–325, 1995. https://doi.org/10.1017/S0022112095003119
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D.V. Evans and R. Porter. Hydrodynamic characteristics of a thin rolling plate in finite depth of water. Appl. Ocean Res., 18(4):215–228, 1996. https://doi.org/10.1016/S0141-1187(96)00026-0
S. Gupta and R. Gayen. Water wave interaction with dual asymmetric non-uniform permeable plates using integral equations. Appl. Math. Comp., 346(1):436–451, 2019. https://doi.org/10.1016/j.amc.2018.10.062
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M. Kanoria and B.N. Mandal. Water wave scattering by a submerged circular-arc-shaped plate. Fluid Dyn. Res., 31(5):317–331, 2002. https://doi.org/10.1016/S0169-5983(02)00136-3
A. Kaur, S.C. Martha and A. Chakrabarti. Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches. Appl. Ocean Res, 93(101946), 2019. https://doi.org/10.1016/j.apor.2019.101946
A.-J. Li, Y. Liu and H.-J. Li. Accurate solutions to water wave scattering by vertical thin porous barriers. Mathematical Problems in Engineering, 2015(985731), 2015. https://doi.org/10.1155/2015/985731
F.B. Liu and Y.M. Cheng. The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels. Int. J. Appl. Mech., 10(4):1850047, 2018. https://doi.org/10.1142/S1758825118500473
F.B. Liu, Q. Wu and Y.M. Cheng. A meshless method based on the nonsingular weight functions for elastoplastic large deformation problems. Int. J. Appl. Mech., 11(1):1950006, 2019. https://doi.org/10.1142/S1758825119500066
P. Maiti, P. Rakshit and S. Banerjea. Wave motion in an ice covered ocean due to small oscillations of a submerged thin vertical plate. J. Marine Sc. Appl., 14:355–365, 2015. https://doi.org/10.1007/s11804-015-1326-6
B.N. Mandal. On waves due to small oscillations of a vertical plate submerged in deep water. J. Austral. Math. Soc. Ser., 32(3):296–303, 1991. https://doi.org/10.1017/S0334270000006871
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B.N. Mandal and G.H. Bera. Approximate solution for a class of hypersingular integral equations. Appl. Math. Letter, 19(11):1286–1290, 2006. https://doi.org/10.1016/j.aml.2006.01.013
B.N. Mandal and G.H. Bera. Approximate solution of a class of singular integral equations of second kind. J. Comp. Appl. Math., 206(1):189–195, 2007. https://doi.org/10.1016/j.cam.2006.06.011
B.N. Mandal and A. Chakrabarti. Water wave scattering by barriers. WIT Press, Southampton, 2000.
Z.J. Meng, H. Cheng, L.D. Ma and Y.M. Cheng. The hybrid elementfree Galerkin method for threedimensional wave propagation problems. International Journal for Numerical Methods in Engineering, 117(1):15–37, 2018. https://doi.org/10.1002/nme.5944
R. Porter and D.V. Evans. Complementary approximations to wave scattering by vertical barriers. J Fluid Mech., 294:155–180, 1995. https://doi.org/10.1017/S0022112095002849
P. Rakshit and S. Banerjea. Effect of bottom undulation on the waves generated due to rolling of a plate. J. Marine Sc. Tech., 10:7–16, 2011. https://doi.org/10.1007/s11804-011-1035-8
R. Roy, U. Basu and B.N. Mandal. Water-wave scattering by two submerged thin vertical unequal plates. Arch. Appl. Mech., 86:1681–1692, 2016. https://doi.org/10.1007/s00419-016-1143-7
R. Roy, U. Basu and B.N. Mandal. Water wave scattering by a pair of thin vertical barriers with submerged gaps. J. Engg. Math., 105:85–97, 2017. https://doi.org/10.1007/s10665-016-9884-4
R. Roy, S. De and B.N. Mandal. Water wave scattering by multiple thin vertical barriers. Appl. Math. Comp., 355:458–481, 2019. https://doi.org/10.1016/j.amc.2019.03.004
R. Roy and B.N. Mandal. Water wave scattering by three thin vertical barriers with middle one partially immersed and outer two submerged. Meccanica, 54:71– 84, 2019. https://doi.org/10.1007/s11012-018-0922-3
F. Ursell. On waves due to rolling motion of a ship. Q. J. Mech. Appl. Math., 1:246–252, 1947. https://doi.org/10.1093/qjmam/1.1.246
S. Banerjea and B.N. Mandal. On waves due to rolling of a vertical plate. Indian J. Pure Appl. Math., 23:753–761, 1992.
H. Cheng, M.J. Peng and Y.M. Cheng. The hybrid complex variable elementfree Galerkin method for 3D elasticity problems. Engg. Structures, 219:110835, 2020. https://doi.org/10.1016/j.engstruct.2020.110835
B.C. Das, S. De and B.N. Mandal. Oblique scattering by thin vertical barriers in deep water: solution by multiterm Galerkin technique using simple polynomials as basis. J. Marine Sc. Tech., 23:915–925, 2018. https://doi.org/10.1007/s00773-017-0520-4
B.C. Das, S.De and B.N. Mandal. Oblique water waves scattering by a thick barrier with rectangular cross section in deep water. J. Engg. Math., 2020. https://doi.org/10.1007/s10665-020-10049-4
D.V. Evans. A note on the waves produced by the small oscillations of a partially immersed vertical plate. J. Inst. Maths. Applics., 17:135–140, 1976. https://doi.org/10.1093/imamat/17.2.135
D.V. Evans and M. Fernyhough. Edge waves along periodic coastlines. Part 2. Mathl. Comput. Modelling, 297:307–325, 1995. https://doi.org/10.1017/S0022112095003119
D.V. Evans and C.A.N. Morris. The effect of a fixed vertical barrier on obliquely incident surface waves in deep water. J. Inst. Math. Appl, 9:198–204, 1972. https://doi.org/10.1093/imamat/9.2.198
D.V. Evans and R. Porter. Hydrodynamic characteristics of a thin rolling plate in finite depth of water. Appl. Ocean Res., 18(4):215–228, 1996. https://doi.org/10.1016/S0141-1187(96)00026-0
S. Gupta and R. Gayen. Water wave interaction with dual asymmetric non-uniform permeable plates using integral equations. Appl. Math. Comp., 346(1):436–451, 2019. https://doi.org/10.1016/j.amc.2018.10.062
M.D. Haskind. Radiation and diffraction of surface waves by a flat plate floating vertically. Prikl. Mat. Mech, 23(3):546–556, 1959. https://doi.org/10.1016/0021-8928(59)90168-6 (In Russian).
T.H. Havelock. Forced surface waves on water. Phillos. Mag, 8:569–576, 1929. https://doi.org/10.1080/14786441008564913
M. Kanoria and B.N. Mandal. Water wave scattering by a submerged circular-arc-shaped plate. Fluid Dyn. Res., 31(5):317–331, 2002. https://doi.org/10.1016/S0169-5983(02)00136-3
A. Kaur, S.C. Martha and A. Chakrabarti. Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches. Appl. Ocean Res, 93(101946), 2019. https://doi.org/10.1016/j.apor.2019.101946
A.-J. Li, Y. Liu and H.-J. Li. Accurate solutions to water wave scattering by vertical thin porous barriers. Mathematical Problems in Engineering, 2015(985731), 2015. https://doi.org/10.1155/2015/985731
F.B. Liu and Y.M. Cheng. The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels. Int. J. Appl. Mech., 10(4):1850047, 2018. https://doi.org/10.1142/S1758825118500473
F.B. Liu, Q. Wu and Y.M. Cheng. A meshless method based on the nonsingular weight functions for elastoplastic large deformation problems. Int. J. Appl. Mech., 11(1):1950006, 2019. https://doi.org/10.1142/S1758825119500066
P. Maiti, P. Rakshit and S. Banerjea. Wave motion in an ice covered ocean due to small oscillations of a submerged thin vertical plate. J. Marine Sc. Appl., 14:355–365, 2015. https://doi.org/10.1007/s11804-015-1326-6
B.N. Mandal. On waves due to small oscillations of a vertical plate submerged in deep water. J. Austral. Math. Soc. Ser., 32(3):296–303, 1991. https://doi.org/10.1017/S0334270000006871
B.N. Mandal, S. Banerjea and M. Kanoria. The rolling ship problem-revisited. Mathl. Comput. Modelling, 25(1):11–18, 1997. https://doi.org/10.1016/S0895-7177(96)00181-1
B.N. Mandal and G.H. Bera. Approximate solution for a class of hypersingular integral equations. Appl. Math. Letter, 19(11):1286–1290, 2006. https://doi.org/10.1016/j.aml.2006.01.013
B.N. Mandal and G.H. Bera. Approximate solution of a class of singular integral equations of second kind. J. Comp. Appl. Math., 206(1):189–195, 2007. https://doi.org/10.1016/j.cam.2006.06.011
B.N. Mandal and A. Chakrabarti. Water wave scattering by barriers. WIT Press, Southampton, 2000.
Z.J. Meng, H. Cheng, L.D. Ma and Y.M. Cheng. The hybrid elementfree Galerkin method for threedimensional wave propagation problems. International Journal for Numerical Methods in Engineering, 117(1):15–37, 2018. https://doi.org/10.1002/nme.5944
R. Porter and D.V. Evans. Complementary approximations to wave scattering by vertical barriers. J Fluid Mech., 294:155–180, 1995. https://doi.org/10.1017/S0022112095002849
P. Rakshit and S. Banerjea. Effect of bottom undulation on the waves generated due to rolling of a plate. J. Marine Sc. Tech., 10:7–16, 2011. https://doi.org/10.1007/s11804-011-1035-8
R. Roy, U. Basu and B.N. Mandal. Water-wave scattering by two submerged thin vertical unequal plates. Arch. Appl. Mech., 86:1681–1692, 2016. https://doi.org/10.1007/s00419-016-1143-7
R. Roy, U. Basu and B.N. Mandal. Water wave scattering by a pair of thin vertical barriers with submerged gaps. J. Engg. Math., 105:85–97, 2017. https://doi.org/10.1007/s10665-016-9884-4
R. Roy, S. De and B.N. Mandal. Water wave scattering by multiple thin vertical barriers. Appl. Math. Comp., 355:458–481, 2019. https://doi.org/10.1016/j.amc.2019.03.004
R. Roy and B.N. Mandal. Water wave scattering by three thin vertical barriers with middle one partially immersed and outer two submerged. Meccanica, 54:71– 84, 2019. https://doi.org/10.1007/s11012-018-0922-3
F. Ursell. On waves due to rolling motion of a ship. Q. J. Mech. Appl. Math., 1:246–252, 1947. https://doi.org/10.1093/qjmam/1.1.246