# 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: Etude du probleme de la reflexion ou de la transmission d'ondes d'eau profonde sur une paire de barrieres traversant la surface, a l'aide d'une methode precise basee sur le developpement de fonctions propres.

Abstract: Etude du probleme de la reflexion ou de la transmission d'ondes d'eau profonde sur une paire de barrieres traversant la surface, a l'aide d'une methode precise basee sur le developpement de fonctions propres

46 citations

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TL;DR: In this paper, the problem of scattering an obliquely incident wave train by two non-identical thin vertical barriers either partially immersed or fully submerged in infinitely deep water was studied by employing Havelock's expansion of water wave potential.

Abstract: Scattering of an obliquely incident wave train by two non-identical thin vertical barriers either partially immersed or fully submerged in infinitely deep water was studied by employing Havelock’s expansion of water wave potential and reducing the problem ultimately to the solution of a pair of vector integral equations of the first kind. A one-term Galerkin approximation in terms of a known exact solution of the integral equation corresponding to a single vertical barrier is used to obtain very accurate numerical estimates for the reflection and transmission coefficients. The reflection coefficient is depicted graphically for two different arrangements of the vertical barriers. It is observed that total reflection is possible for some discrete values of the wavenumber only when the barriers are identical, either partially immersed or completely submerged. As the separation length between the two vertical barriers increases, the reflection coefficient becomes oscillatory as a function of the wavenumber, which is due to multiple reflections by the barriers. Also, as the separation length becomes very small, the known results for a single barrier are obtained for normal incidence of the wave train.

28 citations

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TL;DR: In this paper, a plane surface wave train on infinitely deep water is incident upon a pair of fixed thin vertical barriers, one of which is in the surface, the other submerged, and the relation between the input and output amplitudes is obtained via a variational approximation for large barrier separations.

Abstract: A plane surface wave train on infinitely deep water is incident upon a pair of fixed thin vertical barriers, one of which is in the surface, the other submerged. The relation between the input and output amplitudes is obtained via a variational approximation for large barrier separations. It is shown that, within this approximation, infinite spectra of totally reflected and totally transmitted waves exist if the barriers overlap, but for non-overlapping barriers this is not the case.

27 citations

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TL;DR: In this paper, a train of small-amplitude surface waves is obliquely incident on a fixed, thin, vertical plate submerged in deep water, and an appropriate one-term Galerkin approximation is employed to calculate very accurate upper and lower bounds for the reflection and transmission coefficients for any angle of incidence and any wave number.

Abstract: A train of small-amplitude surface waves is obliquely incident on a fixed, thin, vertical plate submerged in deep water. The plate is infinitely long in the horizontal direction. An appropriate one-term Galerkin approximation is employed to calculate very accurate upper and lower bounds for the reflection and transmission coefficients for any angle of incidence and any wave number thereby producing very accurate numerical results.

22 citations