Propagation of surface waves past asymmetric elastic plates

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28

References

  1. 1.

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

    MathSciNet  MATH  Google Scholar 

  2. 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

    MathSciNet  MATH  Google Scholar 

  3. 3.

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

    MATH  Google Scholar 

  4. 4.

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

    MATH  Google Scholar 

  5. 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

    Google Scholar 

  6. 6.

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

    MathSciNet  MATH  Google Scholar 

  7. 7.

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

  8. 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

    MathSciNet  MATH  Google Scholar 

  9. 9.

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

    Google Scholar 

  10. 10.

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

    MATH  Google Scholar 

  11. 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

    MathSciNet  MATH  Google Scholar 

  12. 12.

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

    MathSciNet  MATH  Google Scholar 

  13. 13.

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

    Google Scholar 

  14. 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

    MathSciNet  MATH  Google Scholar 

  15. 15.

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

    Google Scholar 

  16. 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

    Google Scholar 

  17. 17.

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

    MATH  Google Scholar 

  18. 18.

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

    MATH  Google Scholar 

  19. 19.

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

    MathSciNet  MATH  Google Scholar 

  20. 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. 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. 22.

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

    MATH  Google Scholar 

  23. 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

    MATH  Google Scholar 

  24. 24.

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

    MATH  Google Scholar 

  25. 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

    Google Scholar 

  26. 26.

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

    MathSciNet  MATH  Google Scholar 

  27. 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

    MathSciNet  MATH  Google Scholar 

  28. 28.

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

    MATH  Google Scholar 

  29. 29.

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

    MathSciNet  MATH  Google Scholar 

  30. 30.

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

    Google Scholar 

  31. 31.

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

    MathSciNet  MATH  Google Scholar 

  32. 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

    MathSciNet  MATH  Google Scholar 

  33. 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

    MathSciNet  MATH  Google Scholar 

  34. 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

    MathSciNet  MATH  Google Scholar 

  35. 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

    MATH  Google Scholar 

  36. 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

    MathSciNet  MATH  Google Scholar 

  37. 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

    MathSciNet  Google Scholar 

  38. 38.

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

    Google Scholar 

  39. 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

    Google Scholar 

  40. 40.

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

    Google Scholar 

  41. 41.

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

    Google Scholar 

  42. 42.

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

    MathSciNet  MATH  Google Scholar 

  43. 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

    Google Scholar 

  44. 44.

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

    MathSciNet  Google Scholar 

  45. 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

    Google Scholar 

  46. 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

    MathSciNet  Google Scholar 

  47. 47.

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

    MathSciNet  MATH  Google Scholar 

  48. 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

    MathSciNet  MATH  Google Scholar 

  49. 49.

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

  50. 50.

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

    MathSciNet  MATH  Google Scholar 

Download references

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

Authors

Corresponding author

Correspondence to R. Gayen.

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

$$\begin{aligned} g_{i}, \frac{\partial g_{i}}{\partial {Y_i}}, \frac{\partial ^2 g_i}{\partial {Y_i}^2} \quad ~\text{ to } \text{ be } \text{ continuous } \text{ at }~ {Y_i} = \overline{Y_i}, \end{aligned}$$
(A.1)

and

$$\begin{aligned} \Bigg (\frac{\partial ^3 g_i}{\partial {Y_i}^3}\Bigg )_{{Y_i}=\overline{Y_i}+0} - \Bigg (\frac{\partial ^3 g_i}{\partial {Y_i}^3}\Bigg )_{{Y_i}=\overline{Y_i}-0} = -1. \end{aligned}$$
(A.2)

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

$$\begin{aligned} g_i(Y_i,\overline{Y_i})= \left\{ \begin{array}{lcl} \displaystyle A_{i1}\mathrm {e}^{\mathrm {i}\lambda _i {Y_i}}+B_{i1}\mathrm {e}^{-\mathrm {i}\lambda _i {Y_i}}+C_{i1}\mathrm {e}^{\lambda _i {Y_i}}+D_{i1}\mathrm {e}^{-\lambda _i {Y_i}}, ~-b_i< {Y_i}< \overline{Y_i}<b_i, \\ \displaystyle A_{i2}\mathrm {e}^{\mathrm {i}\lambda _i {Y_i}}+B_{i2}\mathrm {e}^{-\mathrm {i}\lambda _i {Y_i}}+C_{i2}\mathrm {e}^{\lambda _i {Y}}+D_{i2}\mathrm {e}^{-\lambda _i {Y_i}}, ~-b_i< \overline{Y_i}< {Y_i}< b_i, \end{array} \right. \end{aligned}$$
(A.3)

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

$$\begin{aligned} {\mathcal {U}}_{i}{\mathcal {V}}_{i}={\mathcal {W}}_{i}, \quad i=1,2. \end{aligned}$$
(A.4)

Here

$$\begin{aligned} {\mathcal {U}}_{i}= & {} \left[ \begin{array}{l} A_{1i}\\ B_{1i}\\ C_{1i}\\ D_{1i} \end{array}\right] ,\quad {\mathcal {V}}_{i}=\left[ \begin{array}{cccc} \mathrm {e}^{-\mathrm {i}\lambda _i b_i} &{} \mathrm {e}^{\mathrm {i}\lambda _i b_i} &{} \mathrm {e}^{-\lambda _i b_i} &{} \mathrm {e}^{\lambda _i b_i}\\ \mathrm {i}\mathrm {e}^{-\mathrm {i}\lambda _i b_i} &{} -\mathrm {i}\mathrm {e}^{\mathrm {i}\lambda _i b_i} &{} \mathrm {e}^{-\lambda _i b_i} &{} -\mathrm {e}^{\lambda _i b_i}\\ -\mathrm {e}^{\mathrm {i}\lambda _i b_i} &{} -\mathrm {e}^{-\mathrm {i}\lambda _i b_i} &{} \mathrm {e}^{\lambda _i b_i} &{} \mathrm {e}^{-\lambda _i b_i}\\ -(\mathrm {i}\lambda _i^{3}+M_{u_{i}}) \mathrm {e}^{\mathrm {i}\lambda _i b_i} &{} (\mathrm {i}\lambda _i^{3}-M_{u_{i}})\mathrm {e}^{-\mathrm {i}\lambda _i b_i} &{} (\lambda _i^{3}-M_{u_{i}})\mathrm {e}^{\lambda _i b_i} &{} -(\lambda _i^{3}+M_{u_{i}})\mathrm {e}^{-\lambda _i b_i} \end{array} \right] , \\ {\mathcal {W}}_{i}= & {} \frac{1}{4\lambda _i^3} \left[ \begin{array}{c} 0\\ 0\\ -\mathrm {i}\mathrm {e}^{\mathrm {i}\lambda _i \gamma _i}+\mathrm {i}\mathrm {e}^{-\mathrm {i}\lambda _i \gamma _i}+\mathrm {e}^{\lambda _i \gamma _i}-\mathrm {e}^{-\lambda _i \gamma _i}\\ -\mathrm {i}(\mathrm {i}\lambda _i^{3}+M_{u_{i}})\mathrm {e}^{\mathrm {i}\lambda _i \gamma _i}-\mathrm {i}(\mathrm {i}\lambda _i^{3}-M_{u_{i}})\mathrm {e}^{-\mathrm {i}\lambda _i \gamma _i}+(\lambda _i^{3}-M_{u_{i}})\mathrm {e}^{\lambda _i \gamma _i}+(\lambda _i^{3}+M_{u_{i}})\mathrm {e}^{-\lambda _i \gamma _i} \end{array} \right] , \end{aligned}$$

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

$$\begin{aligned} A_{2i}=A_{1i}-\frac{\mathrm {i}}{4\lambda _i^3}\mathrm {e}^{-\mathrm {i}\lambda _i \overline{Y_i}},~ B_{2i}=B_{1i}+\frac{\mathrm {i}}{4\lambda _i^3}\mathrm {e}^{\mathrm {i}\lambda _i \overline{Y_i}},~ C_{2i}=C_{1i}-\frac{1}{4\lambda _i^3}\mathrm {e}^{-\lambda _i \overline{Y_i}},~ D_{2i}=D_{1i}+\frac{1}{4\lambda _i^3}\mathrm {e}^{\lambda _i \overline{Y_i}}. \end{aligned}$$
(A.5)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

Keywords

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

Mathematics Subject Classification

  • 76B15