# Oblique scattering by thin vertical barriers in deep water : solution by multi-term Galerkin technique using simple polynomials as basis

TL;DR: In this paper, a two-term Galerkin approximation involving simple polynomials as basis multiplied by appropriate weight function is used to solve the integral equations arising in the mathematical analysis of the oblique scattering problem.

Abstract: This paper is concerned with scattering of obliquely incident surface waves by a thin vertical barrier which may be either partially immersed or completely submerged extending infinitely downwards in deep water. Instead of one-term Galerkin approximation involving the known solution of the integral equation arising in the normal incidence problem, two-term Galerkin approximation involving simple polynomials as basis multiplied by appropriate weight function is used to solve the integral equations arising in the mathematical analysis of the oblique scattering problem. Very accurate numerical estimates for the reflection coefficient for each configuration of the barrier are obtained. The reflection coefficient is depicted graphically against the wavenumber and the incident angle for each configuration.

##### Citations

More filters

••

TL;DR: In this paper, the problem of oblique scattering of surface waves by a partially immersed rectangular barrier or a thick submerged rectangular barrier extending infinitely downwards in deep water is studied to obtain the reflection and transmission coefficients semi-analytically.

Abstract: The problem of oblique scattering of surface waves by a thick partially immersed rectangular barrier or a thick submerged rectangular barrier extending infinitely downwards in deep water is studied here to obtain the reflection and transmission coefficients semi-analytically. Use of Havelock’s expansion of water wave potential function reduces each problem to an integral equation of first kind on the horizontal component of velocity across the gap above or below the barrier. Multi-term Galerkin approximations involving polynomials as basis functions multiplied by appropriate weight functions are used to solve these equations numerically. Evaluated numerical results for the reflection coefficients are plotted graphically for both the barriers. The study reveals that the reflection coefficient depends significantly on the thickness of the barrier. The accuracy of the numerical results is checked by using energy identity and by obtaining results available in the literature as special cases.

3 citations

••

TL;DR: In this article, the mixed boundary value problem associated with scattering of obliquely incident water waves by a flexible porous barrier of different barrier configurations is considered and a novel connection is established between the solution potential of the converted problem and a resolvable potential in the quarter-plane.

2 citations

••

TL;DR: In this article, an integral equation method was developed to study the wave interaction with two symmetric permeable plates submerged in a two-layer fluid, where the plates are inclined and penetrate the common interface between the layers.

Abstract: An integral equation method is developed to study the wave interaction with two symmetric permeable plates submerged in a two-layer fluid. The plates are inclined and penetrate the common interface between the layers. The existence of two different wave modes for the incident wave gives rise to two problems. Both of these are tackled by reducing them to a set of coupled hypersingular integral equations of the second kind. Unknown functions of the integral equations are the discontinuities in the potential functions across portions of the plates. These are computed numerically by employing an expansion collocation method. New results for the reflection coefficients and the amount of energy loss are presented by varying several parameters such as porosity, angle of inclination, plate-length, separation between the plates, interface position and density ratio. Known results for two symmetric vertical permeable and impermeable plates, single vertical impermeable and horizontal permeable plates are recovered from the present analysis.

2 citations

••

01 Jan 2020

TL;DR: In this article, the Galerkin method with simple polynomials multiplied by appropriate weights was used to solve the problem of water wave scattering in a single thin plane vertical barrier partially immersed or completely submerged in water.

Abstract: The explicit solutions exist for normal incidence of the surface wave train or a single thin plane vertical barrier partially immersed or completely submerged in deep water. However, for oblique incidence of the wave train and/or for finite depth water, no such explicit solution is possible to obtain. Some approximate mathematical techniques are generally employed to solve them approximately in the sense that quantities of physical interest associated with each problem, namely the reflection and transmission coefficients, can be obtained approximately either analytically or numerically. The method of Galerkin approximations has been widely used to investigate such water wave scattering problems involving thin vertical barriers. Use of Galerkin method with basis functions involving somewhat complicated functions in solving these problems has been carried out in the literature. Choice of basis functions as simple polynomials multiplied by appropriate weights dictated by the edge conditions at the submerged end points of the barrier providing fairly good numerical estimates for the reflection and transmission coefficients have been demonstrated in this article.

##### References

More filters

••

01 Jul 1947

TL;DR: In this paper, it was shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined.

Abstract: In this paper the two-dimensional reflection of surface waves from a vertical barrier in deep water is studied theoretically.It can be shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined, apart from a motion consisting of simple standing waves. In the cases considered here the normal velocity is prescribed on a part of the vertical plane and is taken to be unknown elsewhere. From the condition of continuity of the motion above and below the barrier an integral equation for the normal velocity can be derived, which is of a simple type, in the case of deep water. We begin by considering in detail the reflection from a fixed vertical barrier extending from depth a to some point above the mean surface.

299 citations

••

TL;DR: In this paper, a Galerkin approximation method was proposed to solve the wave scattering problem in finite-depth water with respect to vertical barriers in a rectangular tank and a vertical barrier in a vertical pool.

Abstract: Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.

195 citations

••

01 Nov 1945

TL;DR: In this paper, the reflexion of waves on the surface of water by a thin plane vertical barrier is considered and the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is calculated.

Abstract: 1. The reflexion of waves on the surface of water by a thin plane vertical barrier is considered and the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is calculated. If the top edge is at a depth a below the surface, it is found that the coefficient of reflexion is about ¼ when where T is the period of the incident waves, so that the condition that the coefficient may exceed ¼ is a .

118 citations