# 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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01 Oct 196620 citations

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TL;DR: In this article, the problem of water-wave scattering by two thin vertical plates of unequal lengths submerged beneath the free surface of an infinitely deep water is studied assuming linear theory, and the problem is reduced to a pair of vector integral equations of first kind which are solved approximately by using single-term Galerkin approximation.

Abstract: The problem of water-wave scattering by two thin vertical plates of unequal lengths submerged beneath the free surface of an infinitely deep water is studied here assuming linear theory. The problem is reduced to a pair of vector integral equations of first kind which are solved approximately by using single-term Galerkin approximation. Very accurate numerical estimates for the reflection and transmission coefficients for different values of the wave number and other parameters are obtained. The numerical results for the reflection coefficient are plotted against the wave number in a number of figures for different configurations of the two plates. It is observed from these figures that the reflection coefficient vanishes for a sequence of values of the wave number only for two identical submerged plates. However, for two non-identical plates, the reflection coefficient never becomes zero, although there exists a few wave numbers at which this becomes small for some particular configurations of the plates. When the two plates become very close to each other, known numerical results for a single plate are deduced.

15 citations

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TL;DR: In this article, the problem of scattering of surface waves obliquely incident on a submerged fixed vertical plate is solved approximately for a small angle of incidence by reducing it to the solution of an integral equation.

Abstract: The problem of scattering of surface waves obliquely incident on a submerged fixed vertical plate is solved approximately for a small angle of incidence by reducing it to the solution of an integral equation The correction to the reflection and transmission coefficients over their normal incidence values for a small angle of incidence are obtained For different values of the incident angle these coefficients are evaluated numerically, taking particular values of the wave number and the depth of the plate, and represented graphically

15 citations

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TL;DR: In this paper, die problem der beugung einer Oberflachenwelle, die schief auf ein vollig versenktes, ebenes Hindernis stosst, wobei das ebene hindernis senkrecht zur ruhenden Oberflache steht, wird auf die Losung der Helmholtz-Gleichung reduziert.

Abstract: Das Problem der Beugung einer Oberflachenwelle, die schief auf ein vollig versenktes, ebenes Hindernis stosst, wobei das ebene Hindernis senkrecht zur ruhenden Oberflache steht, wird auf die Losung der Helmholtz-Gleichung reduziert Die Losung wird nach dem Wiener-Hopf-Verfahren durchgefuhrt und ergibt die Reflexionsbeiwerte

14 citations

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TL;DR: In this article, the diffraction of surface waves, obliquely incident on a partially immersed fixed vertical barrier in deep water, is solved approximately by reducing it to the solution of an integral equation, for small angle of incidence of the incident wave.

Abstract: The problem of the diffraction of surface waves, obliquely incident on a partially immersed fixed vertical barrier in deep water, is solved approximately by reducing it to the solution of an integral equation, for small angle of incidence of the incident wave. The corrections to the reflection and transmission coefficients over their normal incidence values for small angle of incidence are obtained and presented graphically for some intermediate values of wave numbers.

13 citations