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 paper, the effect of a breakwater in the form of two thin vertical barriers with symmetrical gaps submerged in water of uniform finite depth, on an obliquely incident train of surface water waves is considered.
Abstract: The effect of a breakwater in the form of two thin vertical barriers with symmetrical gaps submerged in water of uniform finite depth, on an obliquely incident train of surface water waves is considered in this paper. Galerkin approximation is utilized to obtain the upper and lower bounds of the reflection coefficient. As these bounds are seen to be very close numerically, their average produces very good approximations to the reflection coefficient. The behaviour of the reflection coefficient against the wave number for various values of angle of incidence, lengths of separation and gaps, is depicted in a number of figures. It is observed that increase in the separation length between the barriers results in the occurrence of a number of peaks and dips in the reflection curves. This is attributed to multiple reflections by the two barriers. It is also observed that the peak values of the reflection coefficient do not differ much from unity so that most of the energy is reflected back for certain frequencies and as such, the present model may act as an efficient breakwater.
11 citations
11 citations
TL;DR: In this paper, the scattering of obliquely incident water waves by two thin vertical barriers with gaps at different depths has been studied assuming linear theory, and the problem is reduced to two pairs of integral equations of the first kind, one pair involving a horizontal component of velocity across the gaps and the other pair involving the difference of potentials across each wall.
Abstract: The scattering of obliquely incident water waves by two thin vertical barriers with gaps at different depths has been studied assuming linear theory. Using Havelock’s expansion of water wave potential, the problem is reduced to two pairs of integral equations of the first kind, one pair involving a horizontal component of velocity across the gaps and the other pair involving the difference of potentials across each wall. These two pairs of integral equations can be solved approximately by employing a Galerkin single-term approximation technique to obtain numerical estimates for the reflection and transmission coefficients. These estimates for the reflection and transmission coefficients thus obtained are seen to satisfy the energy identity. The reflection coefficient is plotted against wave number in a number of figures for different values of various parameters involved in the problem. It is observed that the reflection coefficient vanishes at discrete frequencies when the vertical barriers are identical. For nonidentical vertical barriers the reflection coefficient never vanishes, though at some wave number it becomes close to zero. The results for a single barrier and fully submerged two barriers, and for a single barrier with a narrow gap, are also recovered as special cases.
9 citations
TL;DR: In this paper, the authors investigated the problem of water wave scattering by a wall with multiple gaps by using the solution of a singular integral equation with a combination of logarithmic and power kernels in disjoint multiple intervals.
Abstract: This paper is concerned with a reinvestigation of the problem of water wave scattering by a wall with multiple gaps by using the solution of a singular integral equation with a combination of logarithmic and power (Cauchy-type) kernels in disjoint multiple intervals. Use of Havelock's expansion of water wave potential reduces the problem to such an integral equation in the horizontal velocity across the gaps. The solution of the integral equation is obtained by utilizing the solutions of Cauchy-type integral equations in (0, oo) and also in multiple disjoint intervals. An explicit expression for the reflection coefficient is obtained for a wall with n gaps and supplemented by numerical results for up to three gaps.
9 citations