Water wave scattering by a thin vertical submerged permeable plate
Abstract
An alternative approach is proposed here to investigate the problem of scattering of surface water waves by a vertical permeable plate submerged in deep water within the framework of linear water wave theory. Using Havelock’s expansion of water wave potential, the associated boundary value problem is reduced to a second kind hypersingular integral equation of order 2. The unknown function of the hypersingular integral equation is expressed as a product of a suitable weight function and an unknown polynomial. The associated hypersingular integral of order 2 is evaluated by representing it as the derivative of a singular integral of the Cauchy type which is computed by employing an idea explained in Gakhov’s book [7]. The values of the reflection coefficient computed with the help of present method match exactly with the previous results available in the literature. The energy identity is derived using the Havelock’s theorems.
Keyword : water wave scattering, permeable plate, Havelocks theorems, hypersingular integral equation, reflection coefficient
How to Cite
Gayen, R., Gupta, S., & Chakrabarti, A. (2021). Water wave scattering by a thin vertical submerged permeable plate. Mathematical Modelling and Analysis, 26(2), 223-235. https://doi.org/10.3846/mma.2021.13207
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L. Farina and P.A. Martin. Scattering of water waves by a submerged disc using a hypersingular integral equation. Appl. Ocean Res., 20(3):121–134, 1998.
F.D. Gakhov. Chapter II - Riemann boundary value problem. In F.D. Gakhov(Ed.), Boundary Value Problems, volume 85 of International Series of Monographs on Pure and Applied Mathematics, pp. 85–142. Pergamon, 1966. https://doi.org/10.1016/B978-0-08-010067-8.50007-4
R. Gayen, S. Gupta and A. Chakrabarti. Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier. Applied Mathematics Letters, 58:19–25, 2016. https://doi.org/10.1016/j.aml.2016.01.018
R. Gayen and B.N. Mandal. Scattering of surface water waves by a floating elastic plate in two dimensions. SIAM Journal on Applied Mathematics, 69(6):1520– 1541, 2009. https://doi.org/10.1137/070685580
R. Gayen, B.N. Mandal and A. Chakrabarti. Water wave scattering by two sharp discontinuities in the surface boundary conditions. IMA Journal of Applied Mathematics, 71(6):811–831, 2006. https://doi.org/10.1093/imamat/hxl020
R. Gayen, B.N. Mandal and A. Chakrabarti. Water wave diffraction by a surface strip. J. Fluid Mech., 571:419–438, 2007. https://doi.org/10.1017/S0022112006003363
R. Gayen and A. Mondal. A hypersingular integral equation approach to the porous plate problem. Applied Ocean Research, 46:70–78, 2014. https://doi.org/10.1016/j.apor.2014.01.006
S. Gupta and R. Gayen. Scattering of oblique water waves by two thin unequal barriers with non-uniform permeability. Journal of Engineering Mathematics, 112(1):37–61, 2018. https://doi.org/10.1007/s10665-018-9964-8
S. Keuchel, N.C. Hagelstein, O. Zaleski and O. von Estorff. Evaluation of hypersingular and nearly singular integrals in the isogeometric boundary blement bethod for acoustics. Computer Methods in Applied Mechanics and Engineering, 325:488–504, 2017.
S. Kundu and R. Gayen. Surface wave scattering by an elastic plate submerged in water with uneven bottom. Mathematical Modelling and Analysis, 25(3):323– 337, 2020. https://doi.org/10.3846/mma.2020.10315
S. Kundu, R. Gayen and R. Datta. Scattering of water waves by an inclined elastic plate in deep water. cean Engineering, 167:221–228, 2018. https://doi.org/10.1016/j.oceaneng.2018.07.054
C. Macaskill. Reflexion of water waves by a permeable barrier. Journal of Fluid Mechanics, 95(01):141–157, 1979. https://doi.org/10.1017/S0022112079001385
Y. Mahmoudi. A new modified Adomian decomposition method for solving a class of hypersingular integral equations of second kind. Journal of Computational and Applied Mathematics, 255:737–742, 2014. https://doi.org/10.1016/j.cam.2013.06.026
S.R. Manam and M. Sivanesan. A note on the explicit solutions for wave scattering by vertical porous barriers. Wave Motion, 69:81–90, 2017. https://doi.org/10.1016/j.wavemoti.2016.11.010
B.N. Mandal and G.H. Bera. Approximate solution of a class of singular integral equations of second kind. Journal of Computational and Applied Mathematics, 206(1):189–195, 2007. https://doi.org/10.1016/j.cam.2006.06.011
P. McIver. Diffraction of water waves by a segmented permeable breakwater. Journal of Waterway, Port, Coastal, and Ocean Engineering, 131(2):69–76, 2005. https://doi.org/10.1061/(ASCE)0733-950X(2005)131:2(69)
A. Mondal, S. Panda and R. Gayen. Flexural-gravity wave scattering by a circular-arc-shaped porous plate. Studies in Applied Mathematics, 138(1):77– 102, 2017. https://doi.org/10.1111/sapm.12137
R. Porter and D.V. Evans. Complementary approximations to wave scattering by vertical barriers. Journal of Fluid Mechanics, 294:155–180, 1995. https://doi.org/10.1017/S0022112095002849
R.A. Rafar, N.M.A. Nik Long, N. Senu and N.A. Noda. Stress intensity factor for multiple inclined or curved cracks problem in circular positions in plane elasticity. ZAMM – Journal of Applied Mathematics and Mechanics/ Zeitschrift fr Angewandte Mathematik und Mechanik, 97(11):1482–1494, 2017. https://doi.org/10.1002/zamm.201600290
M. Sivanesan and S.R. Manam. Water wave scattering by a vertical porous barrier with two gaps. The ANZIAM Journal, 61(1):47–63, 2019. https://doi.org/10.1017/S1446181118000299
C.K. Sollitt and R.H. Cross. Wave transmission through permeable breakwaters. Coastal Engineering Proceedings, 1(13):1827–1846, 1972. https://doi.org/10.9753/icce.v13.99
C.H. Tsai and D.L. Young. The method of fundamental solutions for water-wave diffraction by thin porous breakwater. Journal of Mechanics, 27(1):149–155, 2011. https://doi.org/10.1017/jmech.2011.16
F. Ursell. The effect of a fixed vertical barrier on surface waves in deep water. Mathematical Proceedings of the Cambridge Philosophical Society, 43(3):374– 382, 1947. https://doi.org/10.1017/S0305004100023604
X. Yu and A.T. Chwang. Wave-induced oscillation in harbor with porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 120(2):125–144, 1994. https://doi.org/10.1061/(ASCE)0733-950X(1994)120:2(125)
Y.S. Chan, A.C. Fannjiang and G.H. Paulino. Integral equations with hypersingular kernels–theory and applications to fracture mechanics. International Journal of Engineering Science, 41(7):683–720, 2003. https://doi.org/10.1016/S0020-7225(02)00134-9
A. Chanda and S.N. Bora. Effect of a porous sea-bed on water wave scattering by two thin vertical submerged porous plates. European Journal of Mechanics - B/Fluids, 84:250–261, 2020. https://doi.org/10.1016/j.euromechflu.2020.06.009
R.A. Dalrymple, M.A. Losada and P.A. Martin. Reflection and transmission from porous structures under oblique wave attack. Journal of Fluid Mechanics, 224:625–644, 1991. https://doi.org/10.1017/S0022112091001908
L. Dragos. A collocation method for the integration of Prandtl’s equation. ZAMM – Journal of Applied Mathematics and Mechanics/ Zeitschrift fr Angewandte Mathematik und Mechanik, 74(7):289–290, 1994. https://doi.org/10.1002/zamm.19940740716
L. Farina and P.A. Martin. Scattering of water waves by a submerged disc using a hypersingular integral equation. Appl. Ocean Res., 20(3):121–134, 1998.
F.D. Gakhov. Chapter II - Riemann boundary value problem. In F.D. Gakhov(Ed.), Boundary Value Problems, volume 85 of International Series of Monographs on Pure and Applied Mathematics, pp. 85–142. Pergamon, 1966. https://doi.org/10.1016/B978-0-08-010067-8.50007-4
R. Gayen, S. Gupta and A. Chakrabarti. Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier. Applied Mathematics Letters, 58:19–25, 2016. https://doi.org/10.1016/j.aml.2016.01.018
R. Gayen and B.N. Mandal. Scattering of surface water waves by a floating elastic plate in two dimensions. SIAM Journal on Applied Mathematics, 69(6):1520– 1541, 2009. https://doi.org/10.1137/070685580
R. Gayen, B.N. Mandal and A. Chakrabarti. Water wave scattering by two sharp discontinuities in the surface boundary conditions. IMA Journal of Applied Mathematics, 71(6):811–831, 2006. https://doi.org/10.1093/imamat/hxl020
R. Gayen, B.N. Mandal and A. Chakrabarti. Water wave diffraction by a surface strip. J. Fluid Mech., 571:419–438, 2007. https://doi.org/10.1017/S0022112006003363
R. Gayen and A. Mondal. A hypersingular integral equation approach to the porous plate problem. Applied Ocean Research, 46:70–78, 2014. https://doi.org/10.1016/j.apor.2014.01.006
S. Gupta and R. Gayen. Scattering of oblique water waves by two thin unequal barriers with non-uniform permeability. Journal of Engineering Mathematics, 112(1):37–61, 2018. https://doi.org/10.1007/s10665-018-9964-8
S. Keuchel, N.C. Hagelstein, O. Zaleski and O. von Estorff. Evaluation of hypersingular and nearly singular integrals in the isogeometric boundary blement bethod for acoustics. Computer Methods in Applied Mechanics and Engineering, 325:488–504, 2017.
S. Kundu and R. Gayen. Surface wave scattering by an elastic plate submerged in water with uneven bottom. Mathematical Modelling and Analysis, 25(3):323– 337, 2020. https://doi.org/10.3846/mma.2020.10315
S. Kundu, R. Gayen and R. Datta. Scattering of water waves by an inclined elastic plate in deep water. cean Engineering, 167:221–228, 2018. https://doi.org/10.1016/j.oceaneng.2018.07.054
C. Macaskill. Reflexion of water waves by a permeable barrier. Journal of Fluid Mechanics, 95(01):141–157, 1979. https://doi.org/10.1017/S0022112079001385
Y. Mahmoudi. A new modified Adomian decomposition method for solving a class of hypersingular integral equations of second kind. Journal of Computational and Applied Mathematics, 255:737–742, 2014. https://doi.org/10.1016/j.cam.2013.06.026
S.R. Manam and M. Sivanesan. A note on the explicit solutions for wave scattering by vertical porous barriers. Wave Motion, 69:81–90, 2017. https://doi.org/10.1016/j.wavemoti.2016.11.010
B.N. Mandal and G.H. Bera. Approximate solution of a class of singular integral equations of second kind. Journal of Computational and Applied Mathematics, 206(1):189–195, 2007. https://doi.org/10.1016/j.cam.2006.06.011
P. McIver. Diffraction of water waves by a segmented permeable breakwater. Journal of Waterway, Port, Coastal, and Ocean Engineering, 131(2):69–76, 2005. https://doi.org/10.1061/(ASCE)0733-950X(2005)131:2(69)
A. Mondal, S. Panda and R. Gayen. Flexural-gravity wave scattering by a circular-arc-shaped porous plate. Studies in Applied Mathematics, 138(1):77– 102, 2017. https://doi.org/10.1111/sapm.12137
R. Porter and D.V. Evans. Complementary approximations to wave scattering by vertical barriers. Journal of Fluid Mechanics, 294:155–180, 1995. https://doi.org/10.1017/S0022112095002849
R.A. Rafar, N.M.A. Nik Long, N. Senu and N.A. Noda. Stress intensity factor for multiple inclined or curved cracks problem in circular positions in plane elasticity. ZAMM – Journal of Applied Mathematics and Mechanics/ Zeitschrift fr Angewandte Mathematik und Mechanik, 97(11):1482–1494, 2017. https://doi.org/10.1002/zamm.201600290
M. Sivanesan and S.R. Manam. Water wave scattering by a vertical porous barrier with two gaps. The ANZIAM Journal, 61(1):47–63, 2019. https://doi.org/10.1017/S1446181118000299
C.K. Sollitt and R.H. Cross. Wave transmission through permeable breakwaters. Coastal Engineering Proceedings, 1(13):1827–1846, 1972. https://doi.org/10.9753/icce.v13.99
C.H. Tsai and D.L. Young. The method of fundamental solutions for water-wave diffraction by thin porous breakwater. Journal of Mechanics, 27(1):149–155, 2011. https://doi.org/10.1017/jmech.2011.16
F. Ursell. The effect of a fixed vertical barrier on surface waves in deep water. Mathematical Proceedings of the Cambridge Philosophical Society, 43(3):374– 382, 1947. https://doi.org/10.1017/S0305004100023604
X. Yu and A.T. Chwang. Wave-induced oscillation in harbor with porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 120(2):125–144, 1994. https://doi.org/10.1061/(ASCE)0733-950X(1994)120:2(125)