# Diffraction of water waves by a submerged vertical plate

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.

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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.

195 citations

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TL;DR: In this article, a hypersingular integral equation for the discontinuity in the potential across the plate was formulated in terms of the hypersingularity integral equation, and a numerical solution was employed to find the point of discontinuity by a truncated series of orthogonal polynomials.

Abstract: The interaction of surface water waves with submerged plates is considered. The problem is formulated in terms of a hypersingular integral equation for the discontinuity in the potential across the plate. Once found, the discontinuity may be used for direct calculation of the reflection and transmission coefficients. A numerical solution is employed, whereby the discontinuity is approximated by a truncated series of orthogonal polynomials, multiplied by an appropriate weight function. The choice of polynomials is dictated by physical arguments. Published results are reproduced for horizontal and vertical plates. New results are presented for inclined plates, showing the variation of the reflection coefficient with angle of inclination and depth of submergence.

101 citations

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TL;DR: In this paper, the two-dimensional problems of scattering and radiation of small-amplitude water waves by thin vertical porous plates in finite water depth are considered using the linear water wave theory.

Abstract: The two-dimensional problems of scattering and radiation of small-amplitude water waves by thin vertical porous plates in finite water depth are considered using the linear water wave theory. Applying the method of eigenfunction expansion, these boundary value problems are converted to certain dual series relations. Solutions to these relations are then obtained by a suitable application of the least squares method. For the scattering problem, four different basic configurations of the barriers are investigated, namely, (I) a surface-piercing barrier, (II) a bottom-standing barrier, (III) a totally submerged barrier, and (IV) a barrier with a gap. The performance of these types of barriers as a breakwater are examined by studying the variation of their reflection and transmission coefficients, hydrodynamic forces and moments for different values of the porous effect parameter defined by Chwang [J. Fluid Mech. 132, 395–406 (1983)], or the Chwang parameter. For the radiation problem, three types of wavemakers, which resemble types (I), (II), and (III) of the above-mentioned configuration, are analyzed. The dependence of the amplitude to stroke ratio on other parameters is also investigated to study the features of these wavemakers.

90 citations

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TL;DR: In this article, the scattering of regular surface water waves by a single, flat, submerged plate is extended to consider the scattering by submerged, curved plates and also by surface-piercing, flat plates.

Abstract: Previous work on the scattering of regular surface water waves by a single, flat, submerged plate is extended to consider the scattering by submerged, curved plates and also by surface-piercing, flat plates. Problems are again formulated as hypersingular integral equations for the discontinuity in potential across the plate, which are then solved numerically using Chebyshev expansions and collocation. New results are given for submerged plates in the shape of a circular arc, and for surface-piercing plates at small angles of inclination to the horizontal. The latter configuration supports a hitherto unsuspected quasi-resonant behaviour, with a very spiky frequency response.

78 citations

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TL;DR: In this paper, the evanescent field structure over the wave front, as represented by equiphase planes, is identified as one of the most important and easily recognizable forms of surface wave.

Abstract: This paper calls attention to some of the most important and easily recognizable forms of surface wave, pointing out that their essential common characteristic is the evanescent field structure over the wave front, as represented by equiphase planes. The problems of launching and supporting surface waves must, in general, be distinguished from one another and it does not necessarily follow that because a particular surface is capable of supporting a surface wave that a given aperture distribution of radiation, e.g. a vertical dipole, can excite such a wave. The paper concludes with a discussion of the behavior of surface waves and their applications.

1,244 citations

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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.

299 citations