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Complete analytical solution for the normally incident water wave scattering by a porous flexible vertical elastic plate or tensioned membrane is found. The physical problem in a half-plane is reduced to a couple of equivalent quarter-plane problems by allowing incident waves from either direction of the structure. In the same way, quarter-plane boundary value problems are posed for solid wave potentials that are solutions of the scattering problem involving a rigid structure of the same geometric configuration. Then, two novel integral relations are introduced to establish a link between the required solution wave potentials and few resolvable solid wave potentials. Explicit expressions for the scattering quantities such as the reflection and the transmission coefficients are obtained. Also, the deflection of the flexible vertical structure and the solution potentials are determined analytically. Numerical results for the explicitly derived scattering quantities and structure deflections are presented. © 2020 Elsevier Ltd

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Department of Mathematics

Journal of Fluids and Structures

Explicit solutions of the scattering problems involving vertical flexible porous structures

Explicit solutions are rarely available for water wave scattering problems. An analytical procedure is presented here to solve the boundary value problem associated with wave scattering by a complete vertical porous barrier with two gaps in it. The original problem is decomposed into four problems involving vertical solid barriers. The decomposed problems are solved analytically by using a weakly singular integral equation. Explicit expressions are obtained for the scattering amplitudes and numerical results are presented. The results obtained can be used as a benchmark for other wave scattering problems involving complex geometrical structures. © 2019 Australian Mathematical SocietyA.

ANZIAM Journal

WATER WAVE SCATTERING by A VERTICAL POROUS BARRIER with TWO GAPS

Complete analytical solutions are obtained here for normally incident water wave scattering by vertical porous barriers of various configurations. By allowing incident waves from either direction separately, two quarter-plane problems are posed for porous potentials associated with the physical problem as well as for solid potentials associated with the scattering problem that involves the vertical solid barrier of same configuration. Then, two integral relations are introduced to establish a link between the porous and the solid wave potentials. They involve auxiliary wave potentials that describe radiation of bi-directional waves by the solid barrier. Moreover, they aid to decompose the original problem into four analytically solvable problems in a quarter-plane and to determine the porous potentials explicitly. © 2016 Elsevier B.V.

Wave Motion

A note on the explicit solutions for wave scattering by vertical porous barriers

An analytical approach is proposed here to study scattering of deep water waves by a submerged or a surface piercing vertical porous barrier. It involves a connection between two wave potentials of which one is the solution of a boundary value problem associated with wave scattering by the porous barrier and the other is the solution of a complementary type problem where barrier and gap positions are interchanged and solid barrier takes the position of the porous barrier. The connection also involves an auxiliary or a connection wave potential. The potential for the solid barrier problem involves incident wave forcing while the auxiliary potential describes a solid barrier type problem that involves a non-physical forcing. The solution procedure of Ursell (Ursell, 1947) is chosen to solve these boundary value problems explicitly in the case of normal wave incidence as it also determines necessary exact behavior of the potential at the barrier edge. The reflection coefficients are also connected and the reflection amplitudes of the normally incident wave against the vertical porous barriers are obtained analytically. Numerical results for reflection and transmission coefficients are presented. © 2016 Elsevier B.V.

Scattering of water waves by vertical porous barriers: An analytical approach

Two-dimensional linear flexural gravity wave scattering by a nearly vertical porous wall is analyzed through a simplified perturbational analysis. A continuous semi-infinite ice sheet of uniform thickness is assumed to be floating over water of infinite depth. The ice sheet, with inclusion or exclusion of compressive stress, has either a free edge or a clamped edge at the porous wall. The first-order correction to the reflected flexural gravity wave amplitude is obtained by two different methods. The first method involves an application of Green’s theorem, and the second method involves a first kind integral equation. The integral equation method proves to be robust as it provides a complete solution in all cases of the problem, whereas the first method fails to produce the same when the ice sheet with a free edge is under compressive stress. The strain in the ice sheet and shear force along the ice sheet are computed and explained graphically for suitable parameters and a particular wall shape function. © 2014, Springer Science+Business Media Dordrecht.

Journal of Engineering Mathematics

Flexural gravity wave scattering by a nearly vertical porous wall

We study the reflection of membrane-coupled gravity waves in deep water against a vertical barrier with a gap. A floating membrane is attached on both sides of the barrier. The associated mixed boundary value problem, which is not particularly well posed, is analysed. We utilize an orthogonal mode-coupling relation to reduce the problem to solving a set of dual integral equations with trigonometric kernel. We solve these by using a weakly singular integral equation. The reflection coefficient is determined explicitly, while having freedom to clamp the membrane with a spring of a certain stiffness on only one side of the vertical barrier. The physical problem is of capillary-gravity wave scattering by a vertical barrier with a gap, when the membrane density is neglected. In this case, the reflection coefficient is known up to an undetermined edge slope on either side of the barrier. The scattering quantity is computed and presented graphically against a wave parameter for different values of nondimensional parameters pertaining to the structures involved in the problem. © 2014 Australian Mathematical Society.

Membrane-coupled gravity wave scattering by a vertical barrier with a GAP

An explicit solution is provided for the scattering of flexural and membrane-coupled gravity waves in deep water against a complete vertical porous barrier by employing certain mode-coupling relations. The floating structure is considered to be attached with the vertical porous barrier in three different ways, giving rise to edge conditions: (i) free edge, (ii) simply supported edge, (iii) clamped or built-in edge in the case of a floating ice-sheet and to fixed or clamped edge condition in the case of a floating membrane. The effect of the porous barrier on these structure-coupled gravity waves is discussed numerically by plotting reflection coefficient against a wave parameter. Also, important physical quantities such as strain in the ice-sheet, shear force in the ice-sheet and ice-sheet deflections are computed and plotted against relevant parameters. The effect of compressive force on flexural gravity wave scattering is also included in the numerical study. © IMechE 2013.

Proceedings of the Institution of Mechanical Engineers Part M: Journal of Engineering for the Maritime Environment

Structure-coupled gravity waves past a vertical porous barrier

A depth averaged equation is derived via variational principle to study the effect of varying bottom on membrane-coupled gravity waves caused by a floating membrane with spatially varying material properties. Variation of gradients for bottom topography, membrane tension and membrane mass density is assumed to be small. The surface energy generated by the membrane deflection contributes to the total energy for membrane-coupled gravity waves. An alternative derivation is also provided by making use of integration by parts. The model equation for the uniformly tensional membrane with uniform mass density has been solved numerically, for the membrane-coupled gravity wave scattering in two dimensions, using mass conserving jump conditions applied at the locations of possible bottom slope discontinuities. This model also describes the reflection characteristics for capillary-gravity waves when the membrane mass density alone is neglected. Reflected amplitudes of membrane-coupled as well as capillary-gravity waves by different bottom topographies are shown graphically for the physical parameters of surface tension, membrane tension and uniform mass of membrane. Moreover, the phenomenon of Bragg resonance is discussed numerically for sinusoidal bottom variations. Further, the model has been utilized to study the effect of depth variations beneath a floating membrane of finite length on the free-surface gravity waves. © 2012 Elsevier Ltd.

A mild-slope model for membrane-coupled gravity waves

A mixed boundary value problem for Laplace's equation involving a higher order boundary condition, associated with scattering (radiation) of capillary-gravity waves in deep water by submerged as well as surface piercing vertical barrier (wave-maker) is considered for its complete solution. Utilizing recently developed mode-coupling relations for eigenfunctions in the expansion formula of the potential function, the boundary value problem has been reduced to solving dual integral equations with kernels comprised of trigonometric functions. A fully analytical solution is derived by the aid of a weakly singular integral equation whose solution has bounded behavior at the end points. The reflection and transmission coefficients, for an incident wave, have been obtained analytically in terms of modified Bessel functions. Numerical results are computed and presented graphically for a surface tension parameter, plotted against a non-dimensional wave parameter. The present method of solution is essentially an extension of the reduction method originally described by Williams [20].

Fulltext

Journal of Integral Equations and Applications

A dual integral equation method for capillary-gravity wave scattering

A quick method of solution for multiple integral equations which are defined over a partition consisting of three intervals of the positive axis and whose kernel is the combination of trigonometric functions has been explained. The solution procedure can be extended to deal with similar integral equations defined over any finite partition of the positive axis. To represent the solution uniquely, certain solvability criteria are obtained in terms of the forcing functions involved. Limiting cases of dual integral equations over two disjoint intervals are discussed. © 2011 Elsevier Ltd. All rights reserved.

Applied Mathematics Letters

Multiple integral equations arising in the theory of water waves

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. By using a streamfunction formulation instead of a velocity potential one, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact linear theory compared to other MS-type equations. The main goal of this work is to extend the CMSE model for solving two-layer flow with a free-surface. In order to allow for an exact reference, an analytical solution for a two-layer fluid over a sloping plane beach is derived. This analytical solution is used for validating the results of the approximated MS-type models. It is found that the two-layer CMSE model performs better than the potential based one. In addition, the new model is used for investigating the scattering of linear surface water waves and interfacial ones over variable bathymetry. © 2010 Elsevier B.V.

Complementary mild-slope equations in a two-layer fluid

An explicit solution is provided for the scattering of flexural gravity waves by a rigid vertical barrier submerged in an infinite depth of water. By applying recently developed mode-coupling relation for eigenfunctions, the mixed boundary value problem has been converted to solve dual integral equations with kernel consisting of trigonometric functions. And then complete analytical solutions are derived with an aid of singular integral equations whose solutions are bounded at the end points. The important hydrodynamical scattering quantities such as reflection and transmission coefficients associated with the flexural gravity wave scattering have been obtained analytically in terms of modified Bessel functions and Struve functions. It is observed that these quantities are sensitive to both combined as well as individual effect of plate thickness and barrier depth of submergence. Numerical results are computed and explained graphically for different parameters such as time period and non-dimensional wave length. Further, the effect of compressive force and plate thickness on the flexural gravity waves against a submerged vertical barrier is studied. © 2011 Elsevier Ltd. All rights reserved.

International Journal of Engineering Science

Effect of a submerged vertical barrier on flexural gravity waves

Scattering of membrane coupled gravity waves in deep water by partial vertical barriers is investigated by the recently developed expansion formulae for wave structure interaction problems. The horizontal thin membrane is considered to be under uniform tension and is covering the free surface. The analysis is based on the linearized theory of water waves, and by combining the kinematic and dynamic conditions at the membrane covered surface, one may derive a not so well-posed mixed boundary value problem for Laplaces equation with third-order boundary condition. The flexible membrane is attached by a spring to the surface piercing barrier, giving suitable edge conditions for the unique solution. The boundary value problem has been converted into dual integral equations with kernels composed of trigonometric functions, which are then solved analytically. The important physical quantities such as reflection and transmission coefficients for both cases of submerged and surface piercing barriers are obtained analytically in terms of modified Bessel functions. It is found that complete reflection or transmission is possible at certain resonant frequencies for the incident membrane coupled waves. Numerical results are plotted and discussed for different values of the nondimensional membrane tension parameter. © 2010 Australian Mathematical Society.

Scattering of membrane coupled gravity waves by partial vertical barriers

A quick method of solution of dual integral equations involving a kernel comprised of trigonometric functions is explained. Certain solvability criteria are obtained in terms of forcing functions for the unique solution of the dual integral equations.&nbsp;

pplied Mathematics Letters

On the solution of dual integral equations

The scattering of water waves by a flexible porous membrane barrier in a two-layer fluid having a free surface is analysed in two dimensions. The membrane barrier is extended over the entire water depth in a two-layer fluid, each fluid being of finite depth. In the present analysis, linear wave theory and small amplitude membrane response are assumed. The porous membrane barrier is tensioned and pinned at both the free surface and the seabed. The associated mixed boundary value problem is reduced to a linear system of equations by utilizing a general orthogonality relation along with least-squares approximation method. Because of the flow discontinuity at the interface, the eigenfunctions involved have a discontinuity at the interface and the orthogonality relation used is a generalization of the classical one corresponding to a single-layer fluid. The reflection and transmission coefficients for the surface and internal modes, the free surface and interface elevations and the nondimensional membrane deflection are computed for various physical parameters like the nondimensional tension parameter, porous-effect parameter, fluid density ratio, ratio of water depths of the two fluids to analyse the efficiency of a porous membrane as a wave barrier in the two-layer fluid. © 2006 Elsevier Ltd. All rights reserved.

Wave scattering by flexible porous vertical membrane barrier in a two-layer fluid

A large class of problems in the field of fluid-structure interaction involves higher-order boundary conditions for the governing partial differential equation and the eigenfunctions associated with these problems are not orthogonal in the usual sense. In the present study, mode-coupling relations are derived by utilizing the Fourier integral theorem for the solutions of the Laplace equation with higher-order derivatives in the boundary conditions in both the cases of a semi-infinite strip and a semi-infinite domain in two dimensions. The expansion for the velocity potential is derived in terms of the corresponding eigenfunctions of the boundary-value problem. Utilizing such an expansion of the velocity potential, the symmetric wave source potentials or the so-called Green's function for the boundary-value problem of the flexural gravity wave maker is derived. Alternatively, utilizing the integral form of the wave source potential, the expansion formulae for the velocity potentials are recovered, which justifies the completeness of the eigenfunctions involved. As an application of the wave maker problem, oblique water wave scattering caused by cracks in a floating ice-sheet is analysed in the case of infinite depth. © 2005 The Royal Society.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Expansion formulae in wave structure interaction problems

Havelock’s type of expansion theorems, for an integrable function having a single discontinuity point in the domain where it is defined, are utilized to derive analytical solutions for the radiation or scattering of oblique water waves by a fully extended porous barrier in both the cases of finite and infinite depths of water in two-layer fluid with constant densities. Also, complete analytical solutions are obtained for the boundary-value problems dealing with the generation or scattering of axi-symmetric water waves by a system of permeable and impermeable co-axial cylinders. Various results concerning the generation and reflection of the axisymmetric surface or interfacial waves are derived in terms of Bessel functions. The resonance conditions within the trapped region are obtained in various cases. Further, expansions for multipole-line-source oblique-wave potentials are derived for both the cases of finite and infinite depth depending on the existence of the source point in a two-layered fluid.

Waves Past Porous Structures in a Two-layer Fluid

A quick method of solution of a singular integral equation of the first kind involving both logarithmic singularity as well as Cauchy-type singularity is explained. © Institute of Mathematics and its Applications 2004; all rights reserved.

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)

A note on a singular integral equation arising in water wave scattering

A class of boundary value problems involving propagation of two-dimensional surface water waves, associated with water of uniform finite depth, against a plane vertical wave maker is investigated under the assumption that the surface is covered by a thin sheet of ice. It is assumed that the ice-cover behaves like a thin isotropic elastic plate. Then the problems under consideration lead to those of solving the two-dimensional Laplace equation in a semi-infinite strip, under Neumann boundary conditions on the vertical boundary as well as on one of the horizontal boundaries, representing the bottom of the fluid region, and a condition involving upto fifth order derivatives of the unknown function on the top horizontal ice-covered boundary, along with the two appropriate edge-conditions, at the ice-covered corner, ensuring the uniqueness of the solutions. The mixed boundary value problems are solved completely, by exploiting the regularity property of the Fourier cosine transform.

Indian Journal of Pure and Applied Mathematics

A note on surface water waves for finite depth in the presence of an ice-cover

A class of boundary value problems involving propagation of two-dimensional surface water waves, associated with deep water and a plane vertical rigid barrier is investigated under the assumption that the surface is covered by a thin sheet of ice. Assuming that the ice-cover behaves like a thin isotropic elastic plate, the problems under consideration lead to those of solving the two-dimensional Laplace equation in a quarter-plane, under a Neumann boundary condition on the vertical boundary and a condition involving up to fifth order derivatives of the unknown function on the horizontal ice-covered boundary, along with two appropriate edge conditions, ensuring the uniqueness of the solutions. Two different methods are employed to solve the mixed boundary value problems completely, by determining the unique solution of a special type of integral equation of the first kind in the first method and by exploiting the analyticity property of the Fourier cosine transform in the second method. © 2003 Elsevier Science Ltd. All rights reserved.