## Abstract

Water wave scattering by two asymmetric thin elastic plates with arbitrary inclinations is investigated using integral equations. The plates are submerged in finite depth water. The assumption of Euler–Bernoulli beam model for the plates, the use of the appropriate Euclidean transformations to handle the fifth-order plate conditions and the application of Green’s function technique allow us to obtain the expressions of normal velocities at arbitrary points over the plates. On the other hand, an application of Green’s integral theorem on the scattered potential and the source potential functions gives us the alternative expressions of the above-mentioned normal velocities. The comparison of these alternative forms provides two coupled integral equations involving the unknown potential differences across the plates. Kernels of the integral equations have regular as well as hypersingular parts so that the resulting integral equations are hypersingular in nature. These are solved numerically and the solutions are utilized to compute the numerical estimates for different physical quantities. Published results are recovered for different arrangements of the plates and new results are presented graphically for various parametric values.

This is a preview of subscription content, access via your institution.

## References

- 1.
Dean WR (1945) On the reflexion of surface waves by a submerged plane barrier. Proc Camb Philos Soc 41(3):231–238

- 2.
Ursell F (1947) The effect of a fixed vertical barrier on surface waves in deep water. Proc Camb Philos Soc 43(3):374–382

- 3.
Evans DV (1970) Diffraction of water waves by a submerged vertical plate. J Fluid Mech 40(3):433–451

- 4.
Porter D (1972) The transmission of surface waves through a gap in a vertical barrier. Proc Camb Philos Soc 71(2):411–421

- 5.
Kashiwagi M (2004) Transient responses of a VLFS during landing and take-off of an airplane. J Mar Sci Technol 9(1):14–23

- 6.
Korobkin AA, Khabakhpasheva TI (2006) Regular wave impact onto an elastic plate. J Eng Math 55(1–4):127–150

- 7.
Wang CM (2016) VLFS technology for land creation on the sea. In: Mech Struc Mat XXIV, pp 82–93. CRC Press

- 8.
Meylan MH (2019) The time-dependent vibration of forced floating elastic plates by eigenfunction matching in two and three dimensions. Wave Motion 88:21–33

- 9.
Meylan MH (1995) A flexible vertical sheet in waves. IJOPE 5(02):105–110

- 10.
Chakraborty R, Mandal BN (2014) Scattering of water waves by a submerged thin vertical elastic plate. Arch Appl Mech 84(2):207–217

- 11.
Chakraborty R, Mondal A, Gayen R (2016) Interaction of surface water waves with a vertical elastic plate: a hypersingular integral equation approach. Z Angew Math Phys 67(5):115

- 12.
John F (1948) Waves in the presence of an inclined barrier. Commun Pure Appl Math 1(2):149–200

- 13.
Parsons NF, Martin PA (1992) Scattering of water waves by submerged plates using hypersingular integral equations. Appl Ocean Res 14(5):313–321

- 14.
Parsons NF, McIver P (1999) Scattering of water waves by an inclined surface-piercing plate. Q J Mech Appl Math 52(4):513–524

- 15.
Gayen R, Mondal A (2016) Water wave interaction with two symmetric inclined permeable plates. Ocean Eng 124:180–191

- 16.
Kundu S, Gayen R, Datta R (2018) Scattering of water waves by an inclined elastic plate in deep water. Ocean Eng 167:221–228

- 17.
McIver M (1985) Diffraction of water waves by a moored, horizontal, flat plate. J Eng Math 19(4):297–319

- 18.
Ren X, Wang KH (1994) Mooring lines connected to floating porous breakwaters. Int J Eng Sci 32(10):1511–1530

- 19.
Newman JN (2008) Trapping of water waves by moored bodies. J Eng Math 62(4):303–314

- 20.
Korobkin AA, Khabakhpasheva TI, Malenica S (2016) Deformations of an elastic clamped plate in uniform flow and due to jet impact. In: Proc 31st int workshop water waves float bod, Plymouth, 3–6 April, pp 85–88

- 21.
Levine H, Rodemich E (1958) Scattering of surface waves on an ideal fluid. Tech rep, Stanford Univ CA Applied Mathematics and Statistics Labs

- 22.
Jarvis RJ (1971) The scattering of surface waves by two vertical plane barriers. J Inst Maths Appl 7:207–215

- 23.
Evans DV, Morris CAN (1972) Complementary approximations to the solution of a problem in water waves. IMA J Appl Math 10(1):1–9

- 24.
Newman JN (1974) Interaction of water waves with two closely spaced vertical obstacles. J Fluid Mech 66(1):97–106

- 25.
Das P, Dolai DP, Mandal BN (1997) Oblique wave diffraction by parallel thin vertical barriers with gaps. J Waterway Port Coast Ocean Eng 123(4):163–171

- 26.
Mandal BN, Gayen R (2002) Water-wave scattering by two symmetric circular-arc-shaped thin plates. J Eng Math 44(3):297–309

- 27.
De S, Mandal BN, Chakrabarti A (2009) Water-wave scattering by two submerged plane vertical barriers—Abel integral equation approach. J Eng Math 65(1):75–87

- 28.
Morris CAN (1975) A variational approach to an unsymmetric water-wave scattering problem. J Eng Math 9(4):291–300

- 29.
McIver P (1985) Scattering of water waves by two surface-piercing vertical barriers. IMA J Appl Math 35(3):339–355

- 30.
Neelamani S, Vedagiri M (2002) Wave interaction with partially immersed twin vertical barriers. Ocean Eng 29(2):215–238

- 31.
Roy R, Basu U, Mandal BN (2016) Oblique water wave scattering by two unequal vertical barriers. J Eng Math 97(1):119–133

- 32.
Gupta S, Gayen R (2018) Scattering of oblique water waves by two thin unequal barriers with non-uniform permeability. J Eng Math 112(1):37–61

- 33.
Gupta S, Gayen R (2019) Water wave interaction with dual asymmetric non-uniform permeable plates using integral equations. Appl Math Comput 346:436–451

- 34.
Gayen R, Gupta S (2020) Scattering of surface waves by a pair of asymmetric thin elliptic arc shaped plates with variable permeability. Eur J Mech-B 80:122–132

- 35.
Nik Long NMA, Eshkuvatov ZK (2009) Hypersingular integral equation for multiple curved cracks problem in plane elasticity. Int J Solids Struct 46(13):2611–2617

- 36.
Zakharov EV, Setukha AV, Bezobrazova EN (2015) Method of hypersingular integral equations in a three-dimensional problem of diffraction of electromagnetic waves on a piecewise homogeneous dielectric body. Differ Equ 51(9):1197–1210

- 37.
Wang X, Ang WT, Fan H (2018) Effective properties of magnetoelectroelastic interfaces weakened by micro-cracks. Z Angew Math Mech 98(5):727–748

- 38.
Lifanov IK, Poltavskii LN, Vainikko MGM (2003) Hypersingular integral equations and their applications, vol 4. CRC Press, Boca Raton

- 39.
Parsons NF, Martin PA (1994) Scattering of water waves by submerged curved plates and by surface-piercing flat plates. Appl Ocean Res 16(3):129–139

- 40.
McIver P (2005) Diffraction of water waves by a segmented permeable breakwater. J Water Port Coast Ocean Eng 131(2):69–76

- 41.
Gayen R, Mondal A (2014) A hypersingular integral equation approach to the porous plate problem. Appl Ocean Res 46:70–78

- 42.
Martin PA, Farina L (1997) Radiation of water waves by a heaving submerged horizontal disc. J Fluid Mech 337:365–379

- 43.
Farina L, Martin PA (1998) Scattering of water waves by a submerged disc using a hypersingular integral equation. Appl Ocean Res 20:121–134

- 44.
Nokob MH, Yeung RW (2015) Diffraction and radiation loads on open cylinders of thin and arbitrary shapes. J Fluid Mech 772:649–677

- 45.
Kundu S, Datta R, Gayen R, Islam N (2019) The interaction of flexural-gravity waves with a submerged rigid disc. Appl Ocean Res 92:101912

- 46.
Islam N, Kundu S, Gayen R (2019) Scattering and radiation of water waves by a submerged rigid disc in a two-layer fluid. Proc R Soc A 475(2232):20190331

- 47.
Golberg MA (1983) The convergence of several algorithms for solving integral equations with finite-part integrals. J Integr Equ 5(4):329–340

- 48.
Golberg MA (1985) The convergence of several algorithms for solving integral equations with finite part integrals. II. J Integr Equ 9(3):267–275

- 49.
Andrianov A (2005) Hydroelastic analysis of very large floating structures. Doctorale Thesis, Delft University of Technology

- 50.
Hassan M, Meylan MH, Peter MA (2009) Water-wave scattering by submerged elastic plates. Q J Mech Appl Math 62(3):321–344

## Acknowledgements

The authors are thankful to the reviewers for their valuable and insightful comments on the manuscript to improve the same into its present form.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix A: Determination of the expressions of \(g_i(Y_i,\overline{Y_i})\)

### Appendix A: Determination of the expressions of \(g_i(Y_i,\overline{Y_i})\)

Here we discuss the process of determining the Green’s functions satisfying the boundary value problems given by equations (15) and end conditions (14). Other inherent conditions to be satisfied by these Green’s functions are given as

and

The general solutions of Eq. (15) (for \(i=1,2\)) are of the form

where \(A_{i1},B_{i1}\ldots \) are to be determined. Using end conditions (14) and the inherent properties of the Green’s functions we get the system of equations

Here

where \(\gamma _i=b_i-\overline{Y_i}\). Also, the values of \(A_{2i},B_{2i},C_{2i},D_{2i}\) are found by the relations

## Rights and permissions

## About this article

### Cite this article

Kundu, S., Gayen, R. & Gupta, S. Propagation of surface waves past asymmetric elastic plates.
*J Eng Math* **126, **4 (2021). https://doi.org/10.1007/s10665-020-10076-1

Received:

Accepted:

Published:

### Keywords

- Coupled integral equations
- Hypersingular kernels
- Inclined thin elastic plates
- Wave scattering

### Mathematics Subject Classification

- 76B15