Abstract
The effect of porosity on surface wave scattering by a vertical porous barrier over a rectangular trench is studied here under the assumption of linearized theory of water waves. The fluid region is divided into four subregions depending on the position of the barrier and the trench. Using the Havelock’s expansion of water wave potential in different regions along with suitable matching conditions at the interface of different regions, the problem is formulated in terms of three integral equations. Considering the edge conditions at the submerged end of the barrier and at the edges of the trench, these integral equations are solved using multi-term Galerkin approximation technique taking orthogonal Chebyshev’s polynomials and ultra-spherical Gegenbauer polynomial as its basis function and also simple polynomial as basis function. Using the solutions of the integral equations, the reflection coefficient, transmission coefficient, energy dissipation coefficient and horizontal wave force are determined and depicted graphically. It was observed that the rate of convergence of the Galerkin method in computing the reflection coefficient, considering special functions as basis function is more than the simple polynomial as basis function. The change of porous parameter of the barrier and variation of trench width and height significantly contribute to the change in the scattering coefficients and the hydrodynamic force. The present results are likely to play a crucial role in the analysis of surface wave propagation in oceans involving porous barrier over submarine trench.
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Abbreviations
- a :
-
Length of the partially immersed barrier
- 2b :
-
Width of the trench
- c :
-
Depth of the trench from the free surface
- h :
-
Depth of the fluid region
- l :
-
Distance of the barrier from the middle of the trench on the free surface
- σ :
-
Angular frequency of incoming wave train
- K :
-
Wavenumber
- φ inc :
-
Velocity potential of the incident wave
- φ 1, φ 2, φ 3, φ 4 :
-
Velocity potential of resultant motion in the fluid subregions
- r 1 :
-
Distance from the submerged left edge of the trench
- r 2 :
-
Distance from the submerged edge of the barrier
- r 3 :
-
Distance from the submerged right edge of the trench
- G :
-
Dimensionless porous parameter
- G r, G i :
-
Resistance force coefficient and inertial force coefficient of the porous material
- R, T :
-
Reflection and Transmission coefficient
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Acknowledgement
The authors thank Prof. Sudeshna Banerjea, Department of Mathematics, Jadavpur University, India, for providing helpful suggestions to improve the work. They also thank the referees for their valuable suggestions and comments that improved the presentation of the paper.
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Article Highlights
• Considering linear theory, problem of water wave scattering by porous thin submerged plate over rectangular trench is studied.
• Two different positions of the barrier are considered.
• A multi term Galerkin approximation technique together with taking orthogonal polynomials as well as simple polynomial as basis function. To test the rate of convergence compares the both results numerically.
• The reflection coefficient, transmission coefficient, energy dissipation coefficient and horizontal wave force are determined and depicted graphically.
• Porous parameter of the barrier, variation of trench width and height affect the energy coefficients significantly.
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Das, G., Chakraborty, R. Effect of Porosity on Wave Scattering by a Vertical Porous Barrier over a Rectangular Trench. J. Marine. Sci. Appl. 23, 85–100 (2024). https://doi.org/10.1007/s11804-024-00396-4
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DOI: https://doi.org/10.1007/s11804-024-00396-4