Journal ArticleDOI

# Scattering of water waves by a submerged thin vertical wall with a gap

01 Jan 1998-The Journal of The Australian Mathematical Society. Series B. Applied Mathematics (Cambridge University Press)-Vol. 39, Iss: 03, pp 318-331
TL;DR: In this article, two different integral equation formulations of the problem are presented using Havelock's expansion of water wave potential, and the reflection coefficient is evaluated by both the methods.
Abstract: A train of surface water waves normally incident on a thin vertical wall completely submerged in deep water and having a gap, experiences reflection by the wall and transmission through the gaps above and in the wall. Using Havelock's expansion of water wave potential, two different integral equation formulations of the problem are presented. While the first formulation involves multiple integral equations which are solved here by reducing them to a singular integral equation with Cauchy kernel in a double interval, the second formulation involves a first-kind singular integral equation in a double interval with a combination of logarithmic and Cauchy kernel, the solution of which is obtained by utilizing the solution of a singular integral equation with Cauchy kernel in (0, ∞) and also in a double interval. The reflection coefficient is evaluated by both the methods.
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Journal ArticleDOI
TL;DR: A function theoretic method is used to solve a singular integral equation with logarithmic kernel in two disjoint finite intervals to solve the problem of scattering of time harmonic surface water waves by a fully submerged thin vertical barrier with a single gap.
Abstract: We used function theoretic method to solve a singular integral equation with logarithmic kernel in two disjoint finite intervals where the unknown function satisfying the integral equation may be bounded or unbounded at the nonzero finite endpoints of the interval concerned. An appropriate solution of this integral equation is then applied to solve the problem of scattering of time harmonic surface water waves by a fully submerged thin vertical barrier with a single gap.

2 citations

### Cites methods from "Scattering of water waves by a subm..."

• ...[2] S. Banerjea and B. N. Mandal, Scattering of water waves by a submerged thin vertical wall with a gap, Journal of Australian Mathematical Society....

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• ...This problem was solved earlier by Chakrabarti and Vijaya Bharathi [4] and Banerjea and Mandal [2]....

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• ...Chakrabarti and Vijaya Bharathi [4] used complex variable technique while Banerjea and Mandal [2] used two types of singular integral equation method to solve the problem....

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• ...In both methods in [2], the weakly singular kernel was converted to strong singular kernel....

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• ...The singular integral equations arising in [2] consist of Cauchy kernel and a combination of logarithmic and Cauchy kernel....

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Proceedings ArticleDOI
02 Jun 1998
TL;DR: The Galerkin methods of obtaining approximate solutions of integral equations and their applications to problems of scattering of electromagnetic and surface water waves are examined in this paper, where two typical problems, one occurring in electromagnetic wave propagation and the other in the propagation of two dimensional surface water wave, are taken up as illustrative examples of the methods.
Abstract: The Galerkin methods of obtaining approximate solutions of integral equations and their applications to problems of scattering of electromagnetic and surface water waves are examined. Two typical problems, one occurring in electromagnetic wave propagation and the other in the propagation of two dimensional surface water waves, are taken up as illustrative examples of the methods.
Book ChapterDOI
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.
Journal ArticleDOI
TL;DR: In this article , the authors investigated scattering of obliquely incident surface water waves on a hurdle as a form of thick symmetric wall with a gap immersed in finite depth water body having cover of thin ice sheet.
Abstract: In this paper, we investigate scattering of obliquely incident surface water waves on a hurdle as a form of thick symmetric wall with a gap immersed in finite depth water body having cover of thin ice sheet. In context of linear theory of water waves, this two-dimensional problem is formulated as a first kind integral equation by splitting the velocity potential in symmetric and antisymmetric part. The integral equation is tackled by using two numerical methods. First method is Boundary element method where the range of integration is divided into finite number of small line elements and choosing the unknown function of the integral equation as constant in each line interval we reduce the integral equation into a linear system of algebraic equation . Second method is multi-term Galerkin Approximation method where the basis functions are chosen as ultraspherical Gegenbauer polynomials to reduce the integral equation to a system of algebraic equation. These system of equations are then solved to obtain the unknown function of the integral equation in both the methods. Here a Very accurate numerical estimates are obtained for reflection coefficient by both the methods which are depicted graphically against the wave number for different parameters involving this problem. Also there is a good agreement of results of reflection coefficient by the two methods.
##### References
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BookDOI
01 Jan 1977

4,382 citations

Journal ArticleDOI
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

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: In this article, a thin vertical plate makes small, simple harmonic rolling oscillations beneath the surface of an incompressible, irrotational liquid, and a train of plane waves of frequency equal to the frequency of oscillation of the plate, is normally incident on the plate.
Abstract: A thin vertical plate makes small, simple harmonic rolling oscillations beneath the surface of an incompressible, irrotational liquid. The plate is assumed to be so wide that the resulting equations may be regarded as two-dimensional. In addition, a train of plane waves of frequency equal to the frequency of oscillation of the plate, is normally incident on the plate. The resulting linearized boundary-value problem is solved in closed form for the velocity potential everywhere in the fluid and on the plate. Expressions are derived for the first- and second-order forces and moments on the plate, and for the wave amplitudes at a large distance either side of the plate. Numerical results are obtained for the case of the plate held fixed in an incident wave-train. It is shown how these results, in the special case when the plate intersects the free surface, agree, with one exception, with results obtained by Ursell (1947) and Haskind (1959) for this problem.

118 citations

Journal ArticleDOI
01 Mar 1972
TL;DR: In this article, the velocity potential of the resulting fluid motion is determined by a reduction procedure and also by an integral equation formulation, and it is shown that the two methods lead to the same Riemann-Hilbert problem.
Abstract: The two-dimensional configuration is considered of a fixed, semi-infinite, vertical barrier extending downwards from a fluid surface and having, at some depth, a gap of arbitrary width. A train of surface waves, incident on the barrier, is partly transmitted and partly reflected. The velocity potential of the resulting fluid motion is determined by a reduction procedure and also by an integral equation formulation. It is shown that the two methods lead to the same Riemann–Hilbert problem. Transmission and reflexion coefficients are calculated for several values of the ratio gap width/mean gap depth.

76 citations