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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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
so that
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
where \(_{1}F_{1}(.; .; .)\) is degenerate hypergeometric function (cf. Gradshteyn and Ryzhik [26, p. 1058]) and is given by
This function is obviously convergent as N becomes large. Again
after using the asymptotic form of gamma function for large N. Thus
Hence
so that
This shows that \(\left\{ C_{N}^{s,a}\right\} _{N=0}^{\infty }\) is convergent.
Type I barrier
From Eq. (4.10) we get
so that
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
Hence as before we can show that
so that
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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DOI: https://doi.org/10.1007/s10665-020-10049-4