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Oblique water waves scattering by a thick barrier with rectangular cross section in deep water

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Abstract

The problem of oblique scattering of surface waves by a thick partially immersed rectangular barrier or a thick submerged rectangular barrier extending infinitely downwards in deep water is studied here to obtain the reflection and transmission coefficients semi-analytically. Use of Havelock’s expansion of water wave potential function reduces each problem to an integral equation of first kind on the horizontal component of velocity across the gap above or below the barrier. Multi-term Galerkin approximations involving polynomials as basis functions multiplied by appropriate weight functions are used to solve these equations numerically. Evaluated numerical results for the reflection coefficients are plotted graphically for both the barriers. The study reveals that the reflection coefficient depends significantly on the thickness of the barrier. The accuracy of the numerical results is checked by using energy identity and by obtaining results available in the literature as special cases.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Levine H, Rodemich E (1958) Scattering of surface waves on an ideal fluid. Math Stat Lab Tech Rep 78:1–64

    Google Scholar 

  3. Lewin M (1963) The effect of vertical barrier on progressive waves. J Math Phys 42:287–300

    Article  MathSciNet  Google Scholar 

  4. Mei CC (1966) Rodiation and scattering of transient gravity waves by vertical plates. Q J Mech Appl Math 19:417–440

    Article  Google Scholar 

  5. Williams WE (1966) Note on the scattering of water waves by a vertical barriers. Proc Camb Philos Soc 62:507–509

    Article  MathSciNet  Google Scholar 

  6. Evans DE (1970) Diffraction of surface waves by a submerged vertical plate. J Fluid Mech 40:433–451

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Banerjea S (1996) Scattering of water waves by a vertical wall with gaps. J Aust Math Ser B 37:512–529

    Article  MathSciNet  Google Scholar 

  9. Porter R, Evans DV (1995) Complementary approximation to wave scattering by vertical barriers. J Fluid Mech 294:155–180

    Article  MathSciNet  Google Scholar 

  10. Evans DV, Morris ACN (1972) The effect of a fixed vertical barrier on oblique incident surface waves in deep water. J Inst Math Appl 9:198–204

    Article  Google Scholar 

  11. Das BC, De S, Mandal BN (2018) Oblique scattering by thin vertical barriers in deep water: solution by multi-term Galerkin technique using simple polynomials as basis. J Mar Sci Technol 23:915–925

    Article  Google Scholar 

  12. Mandal BN, Dolai DP (1994) Oblique water wave diffraction by thin vertical barriers in water of uniform finite depth. Appl Ocean Res 16:195–203

    Article  Google Scholar 

  13. Mandal BN, Chakrabarti A (2000) Water wave scattering by barrier. WIT Press, Southampton

    MATH  Google Scholar 

  14. Newman JN (1965) Propagation of water waves past long two-dimensional obstacles. J Fluid Mech 23:23–29

    Article  Google Scholar 

  15. Mei CC, Black JL (1969) Scattering of surface waves by rectangular obstacles in water of finite depth. J Fluid Mech 38:499–511

    Article  Google Scholar 

  16. Guiney DC, Noye BJ, Tuck EO (1972) Transmission of water waves through small apertures. J Fluid Mech 55:149–161

    Article  Google Scholar 

  17. Tuck EO (1971) Transmission of water wave through small apertures. J Fluid Mech 49:481–491

    Article  Google Scholar 

  18. Owen D, Bhatt BS (1985) Transmission of water wave through a small aperture in a vertical thick barrier. Q J Mech Appl Math 38:379–409

    Article  MathSciNet  Google Scholar 

  19. Liu PLF, Wu J (1986) Transmission of oblique waves through submerged apertures. Appl Ocean Res 8:144–150

    Article  Google Scholar 

  20. Liu PLF, Wu J (1987) Wave transmission through submerged apertures. J Wtry Port Coast Ocean Eng 113:660–671

    Article  Google Scholar 

  21. Kanoria M, Dolai DP, Mandal BN (1999) Water wave scattering by thick vertical barriers. J Eng Math 35:361–384

    Article  MathSciNet  Google Scholar 

  22. Havelock TH (1929) Forced surface waves on water. Philos Mag 8:569–576

    Article  Google Scholar 

  23. Kirby JT, Dalrymple RA (1983) Propagation of obliquely incident water waves over a trance. J Fluid Mech 133:47–63

    Article  Google Scholar 

  24. Stiassnie M, Nahecr E, Boguslavsky I (1984) Energy losses due to vortex shedding from the lower edge of a vertical plate attacked by surface waves. Pro R Soc Lond A396:131–142

    Google Scholar 

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

    Article  Google Scholar 

  26. Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products. Academic Press, New York

    MATH  Google Scholar 

  27. Yong L, Hua-Jun L, Yu-Cheng L (2012) A new analytical solution for wave scattering by a submerged horizontal porous plate with finite thickness. Ocean Eng 42:83–92

    Article  Google Scholar 

  28. Hu J, Liu PL (2018) A unified coupled-mode method for wave scattering by rectangular-shaped objects. Appl Ocean Res 79:88–100

    Article  Google Scholar 

  29. Hu J, Yang Z, Liu PL (2019) A model for obliquely incident wave interacting with a multi-layered object. Appl Ocean Res 87:211–222

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the three anonymous reviewers for their comments and suggestions to revise the paper in the present form and particularly to one reviewer for drawing their attention to the papers under references [27,28,29]. B. C. Das thanks the UGC, India, for providing financial support (file number: 22/122013(ii)EU-V), as a PhD research scholar of the University of Calcutta, India. This work is also supported by SERB through the research project No. EMR/2016/005315. An abridged version of this paper was presented at 34th IWWWFB held during 7–10th April 2019 in Newcastle, Australia.

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Appendix A

Appendix A

1.1 Proof of convergence of the Galerkin approximations

Here we will prove convergence of the reflection (|R|) and transmission (|T|) coefficients in the Galerkin approximation involving simple polynomials multiplied by suitable weight function (Eqs. (4.1) and (5.1)) for type I and type II barriers. Since |R| and |T| are expressed in terms of \(C^{s,a}\) given in the equations (3.34) and (3.35), it is sufficient to prove convergence of the sequence \(\left\{ C_{N}^{s,a}\right\} _{N=0}^{\infty }\) (given by (4.10) for type I barrier and (5.8) for type II barrier). We will prove convergence of \(\left\{ C_{N}^{s,a}\right\} _{N=0}^{\infty }\) for type II barrier first and then these results will be utilized to prove the convergence of \(\left\{ C_{N}^{s,a}\right\} _{N=0}^{\infty }\) for type I barrier.

Type II barrier

From Eq. (5.8) we get

$$\begin{aligned} C_{N}^{s,a}=\sum _{l=0}^{N}\beta _{l}^{s,a}G_{l}, \end{aligned}$$
(A1)

so that

$$\begin{aligned} C_{N}^{s,a}-C_{N-1}^{s,a}=\beta _{N}^{s,a}G_{N}, \end{aligned}$$
(A2)

where \(\beta _{N}^{s,a}\) is unknown constant coefficient of the system of equations (5.1).

We may regard \(\beta _{N}^{s,a}\) to be bounded. From Eq. (5.7) we get

$$\begin{aligned} G_{N}=cB\left( N+1,\frac{2}{3}\right) _{1}F_{1}\left( N+1; N+\frac{5}{3}; -Kc\right) , \end{aligned}$$
(A3)

where \(_{1}F_{1}(.; .; .)\) is degenerate hypergeometric function (cf. Gradshteyn and Ryzhik [26, p. 1058]) and is given by

$$\begin{aligned} _{1}F_{1}\left( N+1; N+\frac{5}{3}; -Kc\right)&=1-\frac{N+1}{N+\frac{5}{3}}\frac{Kc}{1!}+\frac{(N+1)(N+2)}{\left( N+\frac{5}{3}\right) \left( N+\frac{8}{3}\right) }\frac{(Kc)^{2}}{2!}\nonumber \\&\quad -\frac{(N+1)(N+2)(N+3)}{\left( N+\frac{5}{3}\right) \left( N+\frac{8}{3}\right) \left( N+\frac{11}{3}\right) }\frac{(Kc)^{3}}{3!}+\cdots \end{aligned}$$
(A4)

This function is obviously convergent as N becomes large. Again

$$\begin{aligned} B\left( N+1, \frac{2}{3}\right) =\frac{\Gamma (N+1)\Gamma \left( \frac{2}{3}\right) }{\Gamma \left( N+\frac{5}{3}\right) }\approx \frac{(N+1)^{N+\frac{1}{2}}}{\left( N+\frac{5}{3}\right) ^{N+\frac{7}{6}}}O(1). \end{aligned}$$

after using the asymptotic form of gamma function for large N. Thus

$$\begin{aligned} B\left( N+1, \frac{2}{3}\right) \rightarrow 0 \quad ~\hbox {as}~N\rightarrow \infty . \end{aligned}$$
(A5)

Hence

$$\begin{aligned} G_{N}\rightarrow 0 \quad ~\hbox {as}~N\rightarrow \infty . \end{aligned}$$
(A6)

so that

$$\begin{aligned} C_{N}^{s,a}-C_{N-1}^{s,a}\rightarrow 0 \quad ~\hbox {as}~N\rightarrow \infty . \end{aligned}$$
(A7)

This shows that \(\left\{ C_{N}^{s,a}\right\} _{N=0}^{\infty }\) is convergent.

Type I barrier

From Eq. (4.10) we get

$$\begin{aligned} C_{N}^{s,a}=\sum _{l=0}^{N}\alpha _{l}^{s,a}G_{l} \end{aligned}$$
(A8)

so that

$$\begin{aligned} C_{N}^{s,a}-C_{N-1}^{s,a}=\alpha _{N}^{s,a}G_{N}, \end{aligned}$$
(A9)

where \(\alpha _{N}^{s,a}\) is unknown constant coefficient of the system of equations (4.4). We may regard \(\alpha _{N}^{s,a}\) to be bounded. From Eq. (4.9) we get

$$\begin{aligned} G_{N}=a\pi \sec \pi \left( N+\frac{1}{6}\right) \left[ \frac{_{1}F_{1}\left( \frac{1}{3};\frac{1}{3}-N; -(N+1)Ka\right) }{\left\{ (N+1)Ka\right\} ^{N+\frac{2}{3}}}- \frac{\mathrm{e}^{-Ka}}{2}\frac{_{1}F_{1}\left( \frac{1}{3};\frac{1}{3}-N; -NKa\right) }{\left\{ NKa\right\} ^{N+\frac{2}{3}}}\right] . \end{aligned}$$
(A10)

Hence as before we can show that

$$\begin{aligned} G_{N}\rightarrow 0 \quad ~\hbox {as}~N\rightarrow \infty . \end{aligned}$$
(A11)

so that

$$\begin{aligned} C_{N}^{s,a}-C_{N-1}^{s,a}\rightarrow 0~\hbox {as}~N\rightarrow \infty . \end{aligned}$$
(A12)

This shows that \(\left\{ C_{N}^{s,a}\right\} _{N=0}^{\infty }\) is convergent.

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Das, B.C., De, S. & Mandal, B.N. Oblique water waves scattering by a thick barrier with rectangular cross section in deep water. J Eng Math 122, 81–99 (2020). https://doi.org/10.1007/s10665-020-10049-4

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