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An Integral Equation Method for Wave Scattering by a Pair of Horizontal Porous Plates

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Topics in Integral and Integro-Differential Equations

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 340))

Abstract

A mathematical method comprising of solutions to a pair of coupled hypersingular integral equations for studying the interaction of water waves with a pair of horizontal porous plates submerged at different depths in case of finite as well as infinite depth water domain is developed in the context of linear water wave theory. The porosities of the two plates are arbitrary but uniform across the plates. In practice, as the still seawater level may change significantly due to tide, the use of dual horizontal plates instead of a single horizontal plate has been proposed by researchers. So far two plates of equal length placed at different depths have been considered in such a manner that one plate fully overlaps the other one i.e. one of the plates lies exactly above or below the other plate. Mostly eigenfunction expansion and matching technique has been employed to study such problems. The purpose of the present work is to examine the effect of a pair of horizontal permeable plates of different orientations situated at different depths on the propagation of surface waves. Here we study the hydrodynamic performances of a dual porous plates system for the three cases viz. when the plates are kept in a step-like manner, when there is a partial overlap between the plates and when the plates are fully overlapped. This enables us to compare the effectiveness of different models mentioned above for the three different cases. The assumption of Darcy’s law as the boundary condition for fluid across the plates and the appropriate use of Green’s integral technique in the water domain reduce the corresponding boundary value problem into two coupled hypersingular integral equations involving the unknown potential difference functions across the plates. The coupled integral equations are solved numerically and with the help of these solutions, the numerical estimates for the reflection coefficient, the transmission coefficient, vertical wave forces acting on the plates and the energy-loss coefficient are computed. The correctness of the numerical results is checked through an energy identity relation for porous plates and by comparing the present computed results with the previous results available in the literature. New results related to different orientations of the plates, different submergence depths, varying lengths and different values of permeability of the plates are depicted.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Sanjib Naskar, Souvik Kundu & R. Gayen

Authors
  1. Sanjib Naskar
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  2. Souvik Kundu
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  3. R. Gayen
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Correspondence to R. Gayen .

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Editors and Affiliations

  1. Department of Mathematics, Post Graduate College, Ghazipur, Uttar Pradesh, India

    Harendra Singh

  2. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  3. Departamento de Matemática, Universidade Estadual de Maringá, Maringá, Brazil

    Marcelo M. Cavalcanti

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Naskar, S., Kundu, S., Gayen, R. (2021). An Integral Equation Method for Wave Scattering by a Pair of Horizontal Porous Plates. In: Singh, H., Dutta, H., Cavalcanti, M.M. (eds) Topics in Integral and Integro-Differential Equations. Studies in Systems, Decision and Control, vol 340. Springer, Cham. https://doi.org/10.1007/978-3-030-65509-9_9

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