# Water wave scattering by three thin vertical barriers with middle one partially immersed and outer two submerged

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## Abstract

The problem of oblique water wave scattering by three thin vertical barriers in deep water is investigated here assuming linear theory. The geometrical configurations of the three barriers are such the inner(middle) barrier is partially immersed and the two outer barriers are completely submerged and extend infinitely downwards. A system of three simultaneous integral equations of first kind involving differences of velocity potentials across the barriers has been obtained in the mathematical analysis by employing Havelock’s expansion of water wave potential alongwith Havelock’s inversion formulae. Approximate numerical solutions of these integral equations are obtained by using single-term Galerkin approximations where the single term is chosen to be the exact solution of the integral equation obtained for a single vertical barrier in deep water for normal incidence of surface waves. Fairely accurate numerical estimates for the reflection and transmission coefficients are obtained by solving this system of linear equations. These estimates satisfy the energy identity, thereby justifying the validity of the present method. Due to energy identity, the behaviour of reflection coefficient is discussed by depicting it graphically against wavenumber. Existence of zeros of reflection coefficient is observed only when the two outer barriers are identical i.e. they are submerged from the same depth below the mean free surface. Some earlier results for two completely submerged barriers and a single barrier are recovered as special cases for a particular set of values of the parameters involved in the problem. This provides another check on the correctness of the present method.

## Keywords

Water wave scattering Three thin vertical barriers Havelock’s expansion of water wave potential Integral equations Single-term Galerkin approximation Reflection coefficient## Mathematical Subject Classification

76B15## Notes

### Acknowledgements

The authors thank the Reviewer for his comments and suggestions to revise the paper in the present form. This work is partially supported by CSIR through a research project 25(0253)/16/EMR-II.

### Compliance with ethical standards

### Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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