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An Efficient Integral Equation Approach to Study Wave Reflection by a Discontinuity in the Impedance-Type Surface Boundary Conditions

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Abstract

We investigate the interaction of obliquely incident waves by a discontinuity in the surface of water whose bottom is composed of non-dissipative porous medium. The discontinuity arises when two inertial surfaces with different surface densities float on the surface of the water, thereby giving rise to two different impedance-type surface boundary conditions. The two-dimensional problem is solved by employing Green’s second identity which leads to two coupled Fredholm integral equations in the potential function and the velocity across the vertical line below the point of discontinuity. Expressions for the reflection and the transmission coefficients are determined in terms of the solutions of the integral equations. The numerical results for the reflection coefficient are validated against the well known results for the normal incidence of the surface waves and rigid bottom available in the literature. The reflection coefficient is also analyzed for different values of the surface densities of the inertial surfaces, porous effect parameter and the angle of incidence of the surface waves. It is observed that the porous bottom causes resonance in the reflection coefficient for a particular wave frequency. Graphs for the surface profiles are also presented which reveal that the presence of inertial surface reduces the heights of the waves in a significant amount.

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Acknowledgments

The authors are thankful to the reviewers for their comments and suggestions to revise the paper in the present form. Srikumar Panda gratefully acknowledges the research support through UGC-Dr. D.S. Kothari Post Doctoral Fellowship (No: F.4-2/2006 (BSR)/MA/14-15/0041), Govt. of India. S. Panda also thanks to the CTS, IIT KGP for providing the visiting fellowship during which the work was initiated.

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Jadavpur, Kolkata, 700032, India

    Srikumar Panda

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

    Arpita Mondal & R Gayen

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  1. Srikumar Panda
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  2. Arpita Mondal
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  3. R Gayen
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Correspondence to Arpita Mondal.

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Panda, S., Mondal, A. & Gayen, R. An Efficient Integral Equation Approach to Study Wave Reflection by a Discontinuity in the Impedance-Type Surface Boundary Conditions. Int. J. Appl. Comput. Math 3, 1037–1051 (2017). https://doi.org/10.1007/s40819-016-0153-z

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