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 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.
276 citations
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TL;DR: The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined in this article, where the reflection and transmission coefficients are significantly altered and they are calculated using a plane-wave assumption.
Abstract: The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined. For normal wave incidence, the reflection and transmission from a porous breakwater has been studied many times using eigenfunction expansions in the water region in front of the structure, within the porous medium, and behind the structure in the down-wave water region. For oblique wave incidence, the reflection and transmission coefficients are significantly altered and they are calculated here. Using a plane-wave assumption, which involves neglecting the evanescent eigenmodes that exist near the structure boundaries (to satisfy matching conditions), the problem can be reduced from a matrix problem to one which is analytic. The plane-wave approximation provides an adequate solution for the case where the damping within the structure is not too great. An important parameter in this problem is Γ 2 = ω 2 h ( s - i f )/ g , where ω is the wave angular frequency, h the constant water depth, g the acceleration due to gravity, and s and f are parameters describing the porous medium. As the friction in the porous medium, f , becomes non-zero, the eigenfunctions differ from those in the fluid regions, largely owing to the change in the modal wavenumbers, which depend on Γ 2 . For an infinite number of values of ΓF 2 , there are no eigenfunction expansions in the porous medium, owing to the coalescence of two of the wavenumbers. These cases are shown to result in a non-separable mathematical problem and the appropriate wave modes are determined. As the two wavenumbers approach the critical value of Γ 2 , it is shown that the wave modes can swap their identity.
220 citations
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TL;DR: In this paper, a Galerkin approximation method was proposed to solve the wave scattering problem in finite-depth water with respect to vertical barriers in a rectangular tank and a vertical barrier in a vertical pool.
Abstract: Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.
163 citations
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TL;DR: In this paper, the diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory and a numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries; the resulting matrix equation is solved numerically.
Abstract: The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries ; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission ; however, reflection is not total owing to the finiteness of the obstacle. Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution. 1. Introduction The problem of the diffraction of incident waves by a finite obstacle in an otherwise infinite and uniform domain remains of general interest in linear wave theory. Several geometries of interest can be schematized by domains divided into separate regions by vertical geometrical boundaries, with the fluid depth being constant in each subdomain. Representative two-dimensional problems, with the boundary geometry uniform in the direction normal to the plane of interest, include those of elevated rectangular sills, fixed or floating rectangular obstacles at the water surface, and submerged trenches. The approach to the solution of problems of this type has typically been to construct solutions for each constant-depth subdomain in terms of eigenfunction expansions of the velocity potential ; the solutions are then matched at the vertical boundaries, resulting in sets of linear integral equations which must be truncated to a finite number of terms and solved numerically. One of the earliest solutions of this type was given by Takano (1960), who studied the cases of normal wave incidence on an elevated sill and fixed obstacle at the surface. In this study, we employ a modification of Takano’s method, discussed in $3. Newman (19653) also employed an integral-equation approach to study reflection and transmission of waves normally incident on a single step between finite- and infinite-depth regions. A variational approach, developed by Schwinger to study discontinuitiesin waveguides (see Schwinger & Saxon 1968) has been used by Miles (1967), to study Newman’s single-step problem, and by Mei & Black (1969), who studied the symmetric elevated sill and a floating rectangular cylinder. Lassiter (1 972), using the variational approach, studied waves normally incident on a rectangular trench where the water depths before and after the trench are constant but not necessarily equal, referred to here as the asymmetric case. Lee &
161 citations
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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.
114 citations