Short-Wave and Wave Group Scattering by Submerged Porous Plate (original) (raw)
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Oblique Wave Scattering by a Vertical Flexible Porous Plate
Studies in Applied Mathematics, 2015
In the present study, oblique surface wave scattering by a submerged vertical flexible porous plate is investigated in both the cases of water of finite and infinite depths. Using Green's function technique, the boundary value problem is converted into a system of three Fredholm type integral equations. Various integrals associated with the integral equations are evaluated using appropriate Gauss quadrature formulae and the system of integral equations are converted into a system of algebraic equations. Further, using Green's second identity, expressions for the reflection and transmission coefficients are obtained in terms of the velocity potential and its normal derivative. Energy balance relations for wave scattering by flexible porous plates and permeable membrane barriers are derived using Green's identity and used to check the correctness of the computational results. From the general formulation of the submerged plate, wave scattering by partial plates such as (i) surface-piercing and (ii) bottom-standing plates are studied as special cases. Further, oblique wave scattering by bottom-standing and surface-piercing porous membrane barriers are studied in finite water depth as particular cases of the flexible plate problem. Various numerical results are presented to study the effect of structural rigidity, angle of incidence, membrane tension, structural length, porosity and water depth on wave scattering. It is found that wave reflection is more for a surface-piercing flexible porous plate in infinite water depth compared to finite water depth and opposite trend is observed for a submerged flexible porous plate. For a surface-piercing nonpermeable membrane, zeros in transmission coefficient are observed for waves of intermediate water depth
Ocean Engineering, 2018
An analytical study is presented in this paper for oblique wave scattering by a floating elastic plate in a one or twolayer body of water over a porous seabed of infinite depth. The solution procedure adopted, under the assumption of small-amplitude surface waves and structural response, is the eigenfunction expansion method. The study aims to look into the interaction between oblique waves and a floating elastic plate that serves as an effective breakwater. Effects of three types of edge conditions, viz. (i) free edge, (ii) simple-supported edge, and (iii) clamped edge are analyzed. Numerical results for the reflection and transmission coefficients are computed and examined for various values of the wave, porous bed and structural parameters. Results for wave interaction with an elastic plate over a non-porous bed are obtained as special cases and compared with results available in the literature. The study reveals that for various combinations of wave and structural parameters, zero reflection and full transmission may occur in case of rigid bottom and real porous-effect parameter of the porous bed. However, for complex porous-effect parameter, zero reflection and full transmission do not occur. Moreover, due to the energy dissipation by porous bed, wave transmission decreases significantly with increase in length of the floating plate in case of complex porous-effect parameter. The results will be useful in the design of breakwaters for the protection of harbors, inlets and beaches against attacks from surface waves.
Water wave scattering by an elastic plate floating in an ocean with a porous bed
Applied Ocean Research, 2014
The problem of water wave scattering by a thin horizontal elastic plate (semi-infinite as well as finite) floating on an ocean of uniform finite depth in which the ocean bed is composed of porous material of a specific type is analyzed. The method of eigenfunction expansion is used in the mathematical analysis and the quantities of physical interest, namely the reflection and transmission coefficients, are obtained. Numerical estimates for these coefficients are obtained for different values of the parameter describing the porosity of the ocean bed and for different edge conditions of the elastic plate. The edge conditions considered here involve (i) a free edge, (ii) a simply supported edge and (iii) a built-in edge. From the numerical results it is observed that for free edge condition, the porosity of the ocean bed has little effect on the reflection and transmission coefficient for both the cases of semi-infinite and finite elastic plate. The energy identity related to reflection and transmission coefficients in a porous bed is derived and is used as a partial check on the correctness of the numerical results for the semi-infinite elastic plate.
Wave transmission by partial porous structures in two-layer fluid
Engineering Analysis with Boundary Elements, 2015
The present study deals with oblique surface gravity wave scattering and trapping by bottom-standing and surface-piercing porous structures of finite width in two-layer fluid. The problems are analyzed based on the linearized water wave theory in water of uniform depth. Both the cases of interface piercing and non-piercing structures are considered to analyze the effect of porosity in attenuating waves in surface and internal modes. Eigenfunction expansion method is used to deal with wave past porous structures in two-layer fluid assuming that the associated eigenvalues are distinct. Further, the problems are analyzed using boundary element method and results are compared with the analytic solution derived based on the eigenfunction expansion method. Efficiency of the structures of various configuration and geometry on scattering and trapping of surface waves are studied by analyzing the reflection and transmission coefficients for waves in surface and internal modes, free surface and interface elevations, wave loads on the structure and rigid wall. The present study will be of significant importance in the design of various types of coastal structures used in the marine environment for reflection and dissipation of wave energy at continental shelves dominated by stratified fluid which is modeled here as a two-layer fluid.
Water Wave Scattering by a Thin Vertical Submerged Permeable Plate
Mathematical Modelling and Analysis, 2021
An alternative approach is proposed here to investigate the problem of scattering of surface water waves by a vertical permeable plate submerged in deep water within the framework of linear water wave theory. Using Havelock’s expansion of water wave potential, the associated boundary value problem is reduced to a second kind hypersingular integral equation of order 2. The unknown function of the hypersingular integral equation is expressed as a product of a suitable weight function and an unknown polynomial. The associated hypersingular integral of order 2 is evaluated by representing it as the derivative of a singular integral of the Cauchy type which is computed by employing an idea explained in Gakhov’s book [7]. The values of the reflection coefficient computed with the help of present method match exactly with the previous results available in the literature. The energy identity is derived using the Havelock’s theorems.
Surface wave scattering by porous and flexible barrier over a permeable bed
THE 11TH NATIONAL CONFERENCE ON MATHEMATICAL TECHNIQUES AND APPLICATIONS
Using small amplitude wave theory, scattering of water waves by a vertical flexible barrier over a porous bed is studied. The boundary value problem in the form of Helmholtz equation is solved by the matched vertical eigenfunction expansion method. By exploiting the continuity of pressure and velocity at the interface along with Darcy's law for porous structure, the obtained coupled relation is solved by least-squares approximation method. The behavior of flexible barrier against the wave action for various physical quantities are studied and the numerical results are discussed. It is observed that due to the porous structure, a tranquility zone is created on the lee side of the barrier.
Scattering of water waves by inclined thin plate submerged in finite-depth water
Archive of Applied Mechanics (Ingenieur Archiv), 2001
The problem of water wave scattering by an inclined thin plate submerged in water of uniform ®nite depth is investigated here under the assumption of irrotational motion and linear theory. A hypersingular integral equation formulation of the problem is obtained by an appropriate use of Green's integral theorem followed by utilization of the boundary condition on the plate. This hypersingular integral equation involves the discontinuity in the potential function across the plate, which is approximated by a ®nite series involving Chebyshev polynomials. The coef®cients of this ®nite series are obtained numerically by collocation method. The quantities of physical interest, namely the re¯ection and transmission coef®cients, force and moment acting on the plate per unit width, are then obtained numerically for different values of various parameters, and are depicted graphically against the wavenumber. Effects of ®nite-depth water, angle of inclination of the plate with the vertical over the deep water and vertical plate results for these quantities are shown. It is observed that the deep-water results effectively hold good if the depth of the mid-point of the submerged plate below the free surface is of the order of one-tenth of the depth of the bottom.
Water wave scattering by impermeable and perforated plates
Physics of Fluids, 2021
In the field of offshore renewable energy, impermeable plates are used as underwater lenses to amplify the wave amplitude, and perforated plates can harness wave energy as a power takeoff device. Within the framework of the linear potential-flow theory, the water wave scattering by impermeable and perforated horizontal plates is investigated in the present study, and both circular and elliptical plates are considered. The hypersingular integral equation is constructed to model the interaction between water waves and plates of small thickness. For wave scattering by impermeable plates with the focus on wave amplification, wave interference effects due to multiple plates can be utilized to achieve large wave amplification. For perforated plates used for harnessing wave energy, deploying an array of elliptical plates is promising if the deployment line coincides with the major axis and the incident wave propagates along the minor axis. The study gives an insight into harnessing energy from water waves by horizontal plates.
On waves propagating over a submerged poro-elastic structure
Ocean Engineering, 2010
In this paper, the problem of incident waves propagating over a submerged poro-elastic structure is studied theoretically. A linear wave theory is used to describe the wave motion. The submerged poro-elastic structure is modeled based on Biot's theory, in which the fluid motion is described using the potential wave theory of Sollitt and Cross (1972). In the present approach, the problem domain is divided into four subregions. Using general solutions for each region and matching dynamic and kinematic conditions for neighboring regions, analytic solutions are derived for the wave fields and poro-elastic structure. The present analytic solutions compare very well with simplified cases of impermeable, rigid structures, and with those of porous structures. Using the present analytic solution, the effects of a poro-elastic submerged structure on waves are studied. The results show that softer poro-elastic structures can induce higher reflection and lower transmission from incident waves. For low permeability conditions, the elasticity of the structure can induce resonance, while higher permeability can depress the resonant effects.
Journal of Waterway, Port, Coastal, and Ocean Engineering, 2009
This paper describes the development of an efficient scaled boundary finite-element model 5 (SBFEM) for the simulation of short-crested wave interaction with a concentric porous cylindri-6 cal structure. By weakening the governing differential equation in the circumferential direction, 7 the SBFEM is able to solve analytically the weakened equation in the radial direction. Only the 8 cylinder boundary on the circumference of the exterior porous cylinder is discretized with curved 9 surface finite-elements, while a complete analytical representation is obtained for the radial differ-10 ential equation. Comparisons of the numerical results on wave diffraction forces and surface wave 11 elevations at the cylinder to available analytical solutions demonstrate that excellent accuracy can 12 be achieved by the SBFEM with a very small number of surface finite-elements. The influence 13 of varying the wave parameters as well as the system configuration on the system hydrodynamics, 14 including the wave force, wave run-up and diffracted wave contour is examined and extensive re-15 sults on them are presented. This parametric study will help determine the various hydrodynamic 16 effects of a concentric porous cylindrical structure.