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WATER WAVE SCATTERING BY $\large \boldsymbol {\sqcap }$-SHAPED AND INVERTED $\large \boldsymbol {\sqcap }$-SHAPED POROUS BREAKWATERS

Published online by Cambridge University Press:  05 September 2025

PRIYA SHARMA Open the ORCID record for PRIYA SHARMA [Opens in a new window] ,
RANITA ROY  and
SOUMEN DE Open the ORCID record for SOUMEN DE [Opens in a new window]
PRIYA SHARMA
Affiliation:
Department of Applied Mathematics, https://ror.org/01e7v7w47 University of Calcutta , 92, A.P.C. Road, Kolkata 700009, India; e-mail: priya3physics@gmail.com
RANITA ROY
Affiliation:
Department of Mathematics, Serampore College, Serampore, West Bengal 712201, India; e-mail: roy.ranita1@gmail.com
SOUMEN DE*
Affiliation:
Department of Applied Mathematics, https://ror.org/01e7v7w47 University of Calcutta , 92, A.P.C. Road, Kolkata 700009, India; e-mail: priya3physics@gmail.com
*
e-mail: soumenisi@gmail.com
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Abstract

A semi-analytical study of oblique wave interaction with two $\boldsymbol {\sqcap }$-shaped breakwater designs—floating and bottom-fixed structures—incorporating two thin porous plates is presented using linearized theory. Wave potential for both configurations is developed using the eigenfunction expansion method, considering both progressive and evanescent wave modes. The problem of oblique wave scattering by $\boldsymbol {\sqcap }$-shaped breakwaters is reduced to a set of coupled integral equations of first kind, based on horizontal velocity components. These equations are solved using the multi-term Galerkin approximation with appropriate basis functions to handle the square-root singularities at sharp edges of the porous barriers. The performance of the models is evaluated by examining reflection, transmission and energy dissipation coefficients, along with free surface elevation and horizontal drift force. We observe that increasing the plate length of the breakwaters attenuates the incident waves more effectively than increasing the width. Additionally, the floating $\boldsymbol {\sqcap }$-shaped breakwater significantly reduces the free surface elevation in the transmitted region. The results from the developed model can provide valuable insights for the design of wave–structure systems in shallow waters.

Keywords

⊓-shaped breakwaterporous side platesintegral equationsmulti-term Galerkin approximationfree surface elevationhorizontal drift force

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Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e31
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of the Australian Mathematical Publishing Association Inc.

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References

Bai, K. J., “Diffraction of oblique waves by an infinite cylinder”, J. Fluid Mech. 68 (1975) 513–535; doi:10.1017/S0022112075001802.CrossRefGoogle Scholar
Behera, H., Mandal, S. and Sahoo, T., “Oblique wave trapping by porous and flexible structures in a two-layer fluid”, Phys. Fluids 25 (2013) Article ID: 112110; doi:10.1063/1.4832375.CrossRefGoogle Scholar
Behera, H. and Ng, C.-O., “Interaction between oblique waves and multiple bottom-standing flexible porous barriers near a rigid wall”, Meccanica 53 (2018) 871–885; doi:10.1007/s11012-017-0789-8.CrossRefGoogle Scholar
Black, J. L., Mei, C. C. and Bray, M. C. G., “Radiation and scattering of water waves by rigid bodies”, J. Fluid Mech. 46 (1971) 151–164; doi:10.1017/S0022112071000454.CrossRefGoogle Scholar
Burcharth, H. F. and Hughes, S. A., “Types and functions of coastal structures”, Coastal Eng. Manual 6 (2003) VI-2-i–VI-2-44.Google Scholar
Cho, I.-H., “Transmission coefficients of a floating rectangular breakwater with porous side plates”, Int. J. Nav. Archit. Ocean Eng. 8 (2016) 53–65; doi:10.1016/j.ijnaoe.2015.10.002.CrossRefGoogle Scholar
Chwang, A. T., “A porous-wavemaker theory”, J. Fluid Mech. 132 (1983) 395–406; doi:10.1017/S0022112083001676.CrossRefGoogle Scholar
Chwang, A. T. and Chan, A. T., “Interaction between porous media and wave motion”, Annu. Rev. Fluid Mech. 30 (1998) 53–84; doi:10.1146/annurev.fluid.30.1.53.CrossRefGoogle Scholar
Dai, J., Wang, C. M., Utsunomiya, T. and Duan, W., “Review of recent research and developments on floating breakwaters”, Ocean Eng. 158 (2018) 132–151; doi:10.1016/j.oceaneng.2018.03.083.CrossRefGoogle Scholar
Das, P., Dolai, D. P. and Mandal, B. N., “Oblique wave diffraction by parallel thin vertical barriers with gaps”, J. Waterway Port Coast. Ocean Eng. 123 (1997) 163–171; doi:10.1061/(ASCE)0733-950X(1997)123:4(163).CrossRefGoogle Scholar
Deng, Z., Wang, L., Zhao, X. and Huang, Z., “Hydrodynamic performance of a T-shaped floating breakwater”, Appl. Ocean Res. 82 (2019) 325–336; doi:10.1016/j.apor.2018.11.002.CrossRefGoogle Scholar
Evans, D. V. and Porter, R., “Complementary methods for scattering by thin barriers”, Math. Techniques Water Waves 8 (1997) 1–43.Google Scholar
Fouladi, M. Q., Bahmanpouri, F., Rezazadeh, S., Kollolemad, F., Mashayekhi, M. and Viccione, G., “Investigating the sidewall’s effects on $\sqcap$ -shaped floating breakwaters interacting with water waves by the scaled boundary FEM”, Ocean Eng. 284 (2023) Article ID: 115200; doi:10.1016/j.oceaneng.2023.115200.CrossRefGoogle Scholar
Gesraha, M. R., “Analysis of $\sqcap$ shaped floating breakwater in oblique waves: I. Impervious rigid wave boards”, Appl. Ocean Res. 28 (2006) 327–338; doi:10.1016/j.apor.2007.01.002.CrossRefGoogle Scholar
Guo, Y. C., Mohapatra, S. C. and Soares, C. G., “Composite breakwater of a submerged horizontal flexible porous membrane with a lower rubble mound”, Appl. Ocean Res. 104 (2020) Article ID: 102371; doi:10.1016/j.apor.2020.102371.CrossRefGoogle Scholar
Havelock, T. H., “LIX. Forced surface-waves on water”, Lond., Edinb., Dublin Philos. Mag. J. Sci. 8 (1929) 569–576; doi:10.1080/14786441008564913.CrossRefGoogle Scholar
Karmakar, D. and Guedes Soares, C., “Wave transformation due to multiple bottom-standing porous barriers”, Ocean Eng. 80 (2014) 50–63; doi:10.1016/j.oceaneng.2014.01.012.CrossRefGoogle Scholar
Khan, M. B. M. and Behera, H., “Analysis of wave action through multiple submerged porous structures”, J. Offshore Mech. Arct. Eng. 142 (2020) Article ID: 011101; doi:10.1115/1.4044360.CrossRefGoogle Scholar
Khan, M. B. M., Behera, H., Sahoo, T. and Neelamani, S., “Boundary element method for wave trapping by a multi-layered trapezoidal breakwater near a sloping rigid wall”, Meccanica 56 (2021) 317–334; doi:10.1007/s11012-020-01286-z.CrossRefGoogle Scholar
Kirby, J. T. and Dalrymple, R. A., “Propagation of obliquely incident water waves over a trench”, J. Fluid Mech. 133 (1983) 47–63; doi:10.1017/S0022112083001780.CrossRefGoogle Scholar
Koutandos, E., Prinos, P. and Gironella, X., “Floating breakwaters under regular and irregular wave forcing: reflection and transmission characteristics”, J. Hydraul. Res. 43 (2005) 174–188; doi:10.1080/00221686.2005.9641234.CrossRefGoogle Scholar
Lee, M. M. and Chwang, A. T., “Scattering and radiation of water waves by permeable barriers”, Phys. Fluids 12 (2000) 54–65; doi:10.1063/1.870284.CrossRefGoogle Scholar
Longuet-Higgins, M. S., “The mean forces exerted by waves on floating or submerged bodies with applications to sand bars and wave power machines”, Proc. R. Soc. Lond. A Math. Phys. Sci. 352 (1977) 463–480; doi:10.1098/rspa.1977.0011.Google Scholar
Lundström, J., “Wave transmission analysis of marine structures: evaluating the accuracy of theoretical models for determining the transmission coefficient”, Master's thesis in Naval Architecture and Ocean Engineering, Chalmers University of Technology, Gothenburg, Sweden, 2021.Google Scholar
Mandal, B. N. and Chakrabarti, A., Water wave scattering by barriers, 1st edn (WIT Press, Southampton, 2000).Google Scholar
Mandal, B. N. and Kanoria, M., “Oblique wave-scattering by thick horizontal barriers”, J. Offshore Mech. Arct. Eng. 122 (2000) 100–108; doi:10.1115/1.533731.CrossRefGoogle Scholar
Mei, C. C. and Black, J. L., “Scattering of surface waves by rectangular obstacles in waters of finite depth”, J. Fluid Mech. 38 (1969) 499–511; doi:10.1017/S0022112069000309.CrossRefGoogle Scholar
Mohapatra, S. C., Sahoo, T. and Soares, C. G., “Surface gravity wave interaction with a submerged horizontal flexible porous plate”, Appl. Ocean Res. 78 (2018) 61–74; doi:10.1016/j.apor.2018.06.002.CrossRefGoogle Scholar
Mohapatra, S. C. and Soares, C. G., “Surface gravity wave interaction with a horizontal flexible floating plate and submerged flexible porous plate”, Ocean Eng. 237 (2021) Article ID: 109621; doi:10.1016/j.oceaneng.2021.109621.CrossRefGoogle Scholar
Mohapatra, S. C. and Soares, C. G., “Hydroelastic behaviour of a submerged horizontal flexible porous structure in three-dimensions”, J. Fluids Struct. 104 (2021) Article ID: 103319; doi:10.1016/j.jfluidstructs.2021.103319.CrossRefGoogle Scholar
Ouyang, H. T., Chen, K. H. and Tsai, C. M., “Wave characteristics of Bragg reflections from a train of submerged bottom breakwaters”, J. Hydro-Environ. Res. 11 (2016) 91–100; doi:10.1016/j.jher.2015.06.004.CrossRefGoogle Scholar
Ruol, P., Martinelli, L. and Pezzutto, P., “Formula to predict transmission for $\sqcap$ -type floating breakwaters”, J. Waterway Port Coastal Ocean Eng. 139 (2013) 1–8; doi:10.1061/(ASCE)ww.1943-5460.0000153.CrossRefGoogle Scholar
Sarkar, B., De, S. and Gayen, R., “Water wave scattering by pair of porous barriers in the presence of a bottom-standing rectangular obstacle”, Ships Offshore Struct. 19 (2024) 1442–1464; doi:10.1080/17445302.2023.2247194.CrossRefGoogle Scholar
Sarkar, B., Roy, R. and De, S., “Oblique wave interaction by two thin vertical barriers over an asymmetric trench”, Math. Methods Appl. Sci. 45 (2022) 11667–11682; doi:10.1002/mma.8473.CrossRefGoogle Scholar
Sharma, P., Sarkar, B. and De, S., “Oblique wave scattering by a pair of asymmetric inverse $\sqcap$ -shaped breakwater”, J. Offshore Mech. Arct. Eng. 147 (2025) Article ID: 041201; doi:10.1115/1.4066492.CrossRefGoogle Scholar
Singh, S., Kaligatla, R. B. and Mandal, B. N., “Wave scattering by $\sqcap$ -shaped breakwaters in finite depth water”, Appl. Ocean Res. 148 (2024) 1–10; doi:10.1016/j.apor.2024.104014.CrossRefGoogle Scholar
Sollitt, C. K. and Cross, R. H., “Wave transmission through permeable breakwaters”, Coastal Eng. 1972 (1972) 1827–1846.Google Scholar
Söylemez, M. and Gören, Ö., “Diffraction of oblique waves by thick rectangular barriers”, Appl. Ocean Res. 25 (2003) 345–353; doi:10.1016/j.apor.2004.04.001.CrossRefGoogle Scholar
Venkateswarlu, V., Panduranga, K., Vijay, K. G. and Behera, H., “Evaluation of oscillating water column efficiency in the presence of multiple bottom-standing breakwaters under oblique waves”, Phys. Fluids 36 (2024) Article ID: 117175; doi:10.1063/5.0237370.CrossRefGoogle Scholar
Yu, X. and Chwang, A. T., “Wave motion through porous structures”, J. Eng. Mech. 120 (1994) 989–1008; doi:10.1061/(ASCE)0733-9399(1994)120:5(989).Google Scholar