Hostname: page-component-7dd5485656-jtdwj Total loading time: 0 Render date: 2025-10-25T11:59:55.064Z Has data issue: false hasContentIssue false
  • English
  • Français

WAVE INTERACTION WITH AN H-SHAPED SUBMERGED BREAKWATER

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  15 September 2025

R. B. KALIGATLA Open the ORCID record for R. B. KALIGATLA [Opens in a new window] ,
P. PANJA Open the ORCID record for P. PANJA [Opens in a new window]  and
B. N. MANDAL Open the ORCID record for B. N. MANDAL [Opens in a new window]
R. B. KALIGATLA*
Affiliation:
Department of Mathematics and Computing, Indian Institute of Technology (ISM) Dhanbad , Dhanbad 826004, India; e-mail: panjapapita1997@gmail.com
P. PANJA
Affiliation:
Department of Mathematics and Computing, Indian Institute of Technology (ISM) Dhanbad , Dhanbad 826004, India; e-mail: panjapapita1997@gmail.com
B. N. MANDAL
Affiliation:
Physics and Applied Mathematics Unit, Indian Statistical Institute , 203, B.T. Road, Kolkata 700108, India; e-mail: bnm2006@rediffmail.com
*
e-mail: ramana@iitism.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a study of oblique-wave scattering by an H-shaped breakwater submerged in deep water. The H-shaped breakwater is designed using two thin vertical plates connected by a thin horizontal plate. The velocity potentials that describe the wave motion in different regions are expressed using the Havelock expansion. Two first-kind Fredholm-type integral equations are derived by applying the continuity of fluid velocity and pressure at the interface for the horizontal component of fluid velocity across the gap below and above the breakwater. The coefficients of wave reflection and transmission are derived in explicit forms that require the solution of integral equations. The solutions of integral equations are obtained by employing the Galerkin approximation that involves simple polynomials. In some limiting cases, wave scattering by submerged $\large \boldsymbol {\sqcap }$ and $\large \boldsymbol {\sqcup }$-shaped breakwaters is also studied. The correctness of the results is verified by comparing them with existing results reproduced in a limiting case and checking the wave energy balance equation. The results reveal that the H-, $\large \boldsymbol {\sqcap }$- and $\large \boldsymbol {\sqcup }$-shaped breakwaters show low wave transmission for most incident wavelengths. The horizontal plate between the vertical plates accounts for this low wave transmission. Wave forces and the overturning moment are also calculated. These breakwaters may be used in water regions that require low wave transmission.

Keywords

H-shaped breakwaterintegral equationswave scatteringGalerkin approximationwave forceoverturning moment

MSC classification

Primary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e32
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of the Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Chakrabarti, A., Banerjea, S., Mandal, B. N. and Sahoo, T., “A unified approach to problems of scattering of surface water waves by vertical barriers”, J. Aust. Math. Soc. Ser. B 39 (1997) 93–103; doi:10.1017/S0334270000009231.CrossRefGoogle Scholar
Christensen, E. D., Bingham, H. B., Friis, A. P. S., Larsen, A. K. and Jensen, K. L., “An experimental and numerical study of floating breakwaters”, Coastal Eng. 137 (2018) 43–58; doi:10.1016/j.coastaleng.2018.03.002.CrossRefGoogle Scholar
De, S., Mandal, B. N. and Chakrabarti, A., “Water-wave scattering by two submerged plane vertical barriers—Abel integral-equation approach”, J. Engrg. Math. 65 (2009) 75–87; doi:10.1007/s10665-009-9265-3.CrossRefGoogle Scholar
De, S., Mandal, B. N. and Chakrabarti, A., “Use of Abel integral equations in water wave scattering by two surface-piercing barriers”, Wave Motion 47 (2010) 279–288; doi:10.1016/j.wavemoti.2009.12.002.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., “Diffraction of water waves by a submerged vertical plate”, J. Fluid Mech. 40 (1970) 433–451; doi:10.1017/S0022112070000253.CrossRefGoogle Scholar
Evans, D. V. and Morris, C. A. N., “Complementary approximations to the solution of a problem in water waves”, IMA J. Appl. Math. 10 (1972) 1–9; doi:10.1093/imamat/10.1.1.CrossRefGoogle Scholar
Gesraha, M. R., “Analysis of $\varPi$ 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
Günaydın, K. and Kabdaşlı, M. S., “Performance of solid and perforated U-type breakwaters under regular and irregular waves”, Ocean Eng. 31 (11–12) (2004) 1377–1405; doi:10.1016/j.oceaneng.2004.02.002.CrossRefGoogle Scholar
Havelock, T. H., “Forced surface-waves on water”, London, Edinburgh, Dublin Philos. Mag. J. Sci. 8(51) (1929) 569–576; doi:10.1080/14786441008564913.Google Scholar
Jarvis, R. J., “The scattering of surface waves by two vertical plane barriers”, J. Inst. Math. Appl. 7 (1971) 207–215.Google Scholar
Kaligatla, R. B., Singh, S. and Mandal, B. N., “Wave scattering by Pi-type breakwater floating in deep water”, J. Engrg. Math. 143 (2023) article ID: 7; doi:10.1007/s10665-023-10301-7.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
Levine, H. and Rodemich, E., “Scattering of surface waves on an ideal fluid”, Appl. Math. Stat. Lab., Stanford Univ. 49 (1958) 65–74.Google Scholar
Linton, C. M. and McIver, P., Handbook of mathematical techniques for wave/structure interactions (CRC Press, Boca Raton, FL, 2001); doi:10.1201/9781420036060.CrossRefGoogle Scholar
Mandal, B. N. and Chakrabarti, A., Water wave scattering by barriers, 4th edn (WIT Press, Southampton, UK, 2000); https://api.semanticscholar.org/CorpusID:92876695.Google Scholar
Masoudi, E. and Gan, L., “Diffraction waves on general two-legged rectangular floating breakwaters”, Ocean Eng. 235 (2021) article ID: 109420; doi:10.1016/j.oceaneng.2021.109420.CrossRefGoogle Scholar
McIver, P., “Scattering of water waves by two surface-piercing vertical barriers”, IMA J. Appl. Math. 35(3) (1985) 339–355; doi:10.1093/imamat/35.3.339.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
Midya, C. H., Kanoria, M. and Mandal, B. N., “Scattering of water waves by inclined thin plate submerged in finite-depth water”, Arch. Appl. Mech. 71 (2001) 827–840; doi:10.1007/s004190100187.CrossRefGoogle Scholar
Mohapatra, S. C., da Silva Bispo, I. B., Guo, Y. and Guedes Soares, C., “Analysis of wave-induced forces on a floating rectangular box with analytical and numerical approaches”, J. Marine Sci. Appl. 23(1) (2024) 113–126; doi:10.1007/s11804024003857.CrossRefGoogle Scholar
Naftzger, R. A. and Chakrabarti, S. K., “Scattering of waves by two-dimensional circular obstacles in finite water depths”, J. Ship Res. 23 (1979) 32–42; doi:10.5957/jsr.1979.23.1.32.CrossRefGoogle Scholar
Nederstigt, N. J., “The deep water breakwater”, Offshore Engineering Thesis, TU Delft, 1996; https://resolver.tudelft.nl/uuid:0382c8c1-c10a-474a-9576-02632d3eaa02.Google Scholar
Neelamani, S. and Rajendran, R., “Wave interaction with T-type breakwaters”, Ocean Eng. 29 (2002) 151–175; doi: 10.1016/S0029-8018(00)00060-3.CrossRefGoogle Scholar
Neelamani, S. and Vedagiri, M., “Wave interaction with partially immersed twin vertical barriers”, Ocean Eng. 29 (2002) 215–238; doi:10.1016/S0029-8018(00)00061-5.CrossRefGoogle Scholar
Nishad, C. S., Vijay, K. G., Neelamani, S. and Chen, J. T., “Dual BEM for wave scattering by an H-type porous barrier with nonlinear pressure drop”, Eng. Anal. Bound. Elem. 131 (2021) 280–294; doi:10.1016/j.enganabound.2021.06.011.CrossRefGoogle Scholar
Panda, A., Karmakar, D. and Rao, M., “Hydrodynamic analysis of an H-shaped pile-restrained floating breakwater combined with a pair of vertical barriers”, Ocean Eng. 298 (2024) article ID: 117152; doi:10.1016/j.oceaneng.2024.117152.CrossRefGoogle Scholar
Parsons, N. F. and Martin, P. A., “Scattering of water waves by submerged plates using hypersingular integral equations”, Appl. Ocean Res. 14 (1992) 313–321; doi:10.1016/0141-1187(92)90035-I.CrossRefGoogle Scholar
Porter, D., “The transmission of surface waves through a gap in a vertical barrier”, Proc. Cambridge Philos. Soc. 71 (1972) 411–421; doi:10.1017/S0305004100050647.CrossRefGoogle Scholar
Porter, R. and Evans, D. V., “Complementary approximations to wave scattering by vertical barriers”, J. Fluid Mech. 294 (1995) 155–180; doi:10.1017/S0022112095002849.CrossRefGoogle Scholar
Roy, R., Basu, U. and Mandal, B. N., “Oblique water wave scattering by two unequal vertical barriers”, J. Engrg. Math. 97 (2016) 119–133; doi:10.1007/s10665-015-9800-3.CrossRefGoogle Scholar
Ruol, P., Martinelli, L. and Pezzutto, P., “Formula to predict transmission for $\pi$ -type floating breakwaters”, J. Waterway Port Coastal Ocean Eng. 139 (2013) 1–8; doi:10.1061/(ASCE)WW.1943-5460.0000153.CrossRefGoogle Scholar
Singh, S., Kaligatla, R. B. and Mandal, B. N., “Wave scattering by Pi-shaped breakwaters in finite depth water”, Appl. Ocean Res. 148 (2024) article ID: 104014; doi:10.1016/j.apor.2024.104014.CrossRefGoogle Scholar
Teh, H. M.,“Hydraulic performance of free surface breakwaters: a review”, Sains Malays. 42 (2013) 1301–1310; https://khub.utp.edu.my/scholars/id/eprint/3490.Google Scholar
Tuck, E. O., “Matching problems involving flow through small holes”, Adv. Appl. Mech. 15 (1975) 89–158; doi:10.1016/S0065-2156(08)70056-5.CrossRefGoogle Scholar
Ursell, F., “The effect of a fixed vertical barrier on surface waves in deep water”, Proc. Cambridge Philos. Soc. 43 (1947) 374–382; doi:10.1017/S0305004100023604.CrossRefGoogle Scholar
Wang, C. M. and Nguyen, H. P., “Floating breakwaters: sustainable solution for creating sheltered sea space”, in: Proceedings of the second international conference on sustainable civil engineering and architecture ICSCEA 2021, Singapore (Springer Nature, Singapore, 2022); doi:10.1007/978-981-19-3303-5-1.Google Scholar