# Use of Galerkin Technique in Some Water Wave Scattering Problems Involving Plane Vertical Barriers

01 Jan 2020-Vol. 177, pp 405-432

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.

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TL;DR: In this article, the problem of oblique water wave diffraction by two equal thin, parallel, fixed vertical barriers with gaps present in uniform finite-depth water is investigated, and three types of barrier configurations are considered.

Abstract: The problem of oblique water wave diffraction by two equal thin, parallel, fixed vertical barriers with gaps present in uniform finite-depth water is investigated here. Three types of barrier configurations are considered. A one-term Galerkin approximation is used to evaluate upper and lower bounds for reflection and transmission coefficients for each configuration. These bounds are seen to be very close numerically for all wave numbers and as such their averages produce good numerical estimates for these coefficients. Only the bounds for the reflection coefficient are numerically computed. These are also numerically compared with the results obtained by using multiterm Galerkin approximations involving Chebyshev polynomials for a wide range of parameters. Numerical results for the reflection coefficients for the three barrier configurations are presented graphically. It is seen that total reflection occurs only for the surface-piercing barriers while total transmission occurs for all the three configurations considered here. It is also observed that the introduction of an equal second barrier to a submerged barrier increases the reflection coefficient considerably in some frequency bands and as such submerged double barrier configurations are preferable to a submerged single barrier for the purpose of reflecting more wave energy into the open sea.

57 citations

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21 Jan 2000

TL;DR: In this article, the basic equations of the wave-scattering problem are discussed. But they do not specify a solution to the boundary value problem, which is a special case of boundary value problems.

Abstract: Contents: Introduction The Basic Equations Some Important Mathematical Concepts and Results Explicit Solutions to Some Barrier Problems Vertical Wall with a Narrow Gap Approximate Solution Oblique Wave Scattering by Barriers Nearly Vertical Barriers and Special Boundary Value Problems Thin Vertical Barriers in Finite Depth Water Thick Rectangular Barriers in Finite Depth Water Interface Wave Scattering by Barrier Incoming Water Waves Against a Vertical Cliff Second-Order Wave Scattering Appendices Bibliography Index.

56 citations

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01 Sep 1969

TL;DR: In this paper, the authors used a method due to Williams to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.

Abstract: In this paper we use a method due to Williams(1) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.

53 citations

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01 Jul 1966

TL;DR: In this article, an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water.

Abstract: Introduction. In this note an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water. This problem was first solved by Ursell ((1)) and generalizations of the problem have been considered by John ((2)) and Lewin ((3)).

47 citations