Journal ArticleDOI

# Scattering of surface water waves involving semi-infinite floating elastic plates on water of finite depth

07 Aug 2013-Journal of Marine Science and Application (Springer Berlin Heidelberg)-Vol. 12, Iss: 3, pp 325-333
TL;DR: In this paper, the results for the values of reflection and transmission coefficients obtained by using both the method of least squares as well as the singular value decomposition have been employed and tables of numerical values of the reflection coefficients are presented for specific choices of the parameters for modelling the elastic plates.
Abstract: Two problems of scattering of surface water waves involving a semi-infinite elastic plate and a pair of semi-infinite elastic plates, separated by a gap of finite width, floating horizontally on water of finite depth, are investigated in the present work for a two-dimensional time-harmonic case. Within the frame of linear water wave theory, the solutions of the two boundary value problems under consideration have been represented in the forms of eigenfunction expansions. Approximate values of the reflection and transmission coefficients are obtained by solving an over-determined system of linear algebraic equations in each problem. In both the problems, the method of least squares as well as the singular value decomposition have been employed and tables of numerical values of the reflection and transmission coefficients are presented for specific choices of the parameters for modelling the elastic plates. Our main aim is to check the energy balance relation in each problem which plays a very important role in the present approach of solutions of mixed boundary value problems involving Laplace equations. The main advantage of the present approach of solutions is that the results for the values of reflection and transmission coefficients obtained by using both the methods are found to satisfy the energy-balance relations associated with the respective scattering problems under consideration. The absolute values of the reflection and transmission coefficients are presented graphically against different values of the wave numbers.
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TL;DR: In this paper, the problem of radiation of surface and flexural-gravity waves by a submerged cylinder is investigated for two configurations, concerning; (i) a freely floating finite elastic plate modelling an ice floe, and (ii) two semi-infinite elastic plates separated by a region of open water (polynya).
Abstract: The problems of radiation (sway, heave and roll) of surface and flexural-gravity waves by a submerged cylinder are investigated for two configurations, concerning; (i) a freely floating finite elastic plate modelling an ice floe, and (ii) two semi-infinite elastic plates separated by a region of open water (polynya). The fluid of finite depth is assumed to be inviscid, incompressible and homogeneous. The linear two-dimensional problems are formulated within the framework of potential-flow theory. The method of mass sources distributed along the body contour is applied. The corresponding Green’s function is obtained by using matched eigenfunction expansions. The radiation load (added mass and damping coefficients) and the amplitudes of vertical displacements of the free surface and elastic plates are calculated. Reciprocity relations which demonstrate both symmetry of the radiation load coefficients and the relation of damping coefficients with the far-field form of the radiation potentials are found. It is shown that wave motion essentially depends on the position of the submerged body relative to the elastic plate edges. The results of solving the radiation problem are compared with the solution of the diffraction problem. It is noted that resonant frequencies in the radiation problem correlate with those frequencies at which the reflection coefficient in the diffraction problem has a local minimum.

46 citations

### Cites background from "Scattering of surface water waves i..."

• ...Previously, this problem was solved by Chung & Linton (2005) for identical plates and Chakrabarti & Mohapatra (2013) for non-identical ones....

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• ...Energy-balance relation for non-identical plates was given by Chakrabarti & Mohapatra (2013) in the form R2 +QT2 = 1, (B 6) where Q= p0 tanh p0H cosh 2 q0H q0 tanh q0H cosh2 p0H [ 2p0H(D2p40 + 1−ΩB2)+ (5D2p40 + 1−ΩB2) sinh 2p0H 2q0H(D1q40 + 1−ΩB1)+ (5D1q40 + 1−ΩB1) sinh 2q0H ] ....

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Journal ArticleDOI
TL;DR: In this article, the authors studied wave motions of a compressed elastic ice sheet caused by the motion of a two-dimensional dipole in the water beneath the sheet, where the fluid flow is described by the potential theory, while the ice sheet is modelled through a thin elastic plate floating on the water surface.
Abstract: In the linear approximation, we study wave motions of a compressed elastic ice sheet caused by the motion of a two-dimensional dipole in the water beneath the sheet. The fluid flow is described by the potential theory, while the ice sheet is modelled through a thin elastic plate floating on the water surface. The solution for the vertical displacement of the ice sheet is derived for a transient dipole undergoing arbitrary two-dimensional motion. Three cases are considered in detail when the dipole moves horizontally with a uniform speed at some depth or horizontally oscillates, or moves and oscillates. The formulae for the plate displacement are derived for the fluid of finite depth, but then analysed in detail for the infinitely deep case. We show that the character of the solutions is different in the different domains of the parameter plane and classify the possible cases. Then we calculate the wave patterns on the plate for the different regimes of dipole motion and typical values of plate parameters. The studied problem can be considered as the simplified model of motion of a circular cylinder in a water under an ice cover. In the last section we compare the characteristics of wave motions onsetting in the far-field zone of the flow around a circular cylinder and its dipole approximation and show that the difference in the wave characteristics and force loads for these two cases is small and quickly vanishes when the ice plate thickness increases. In conclusion, we present estimates of amplitudes and wavelengths of wave perturbations for the real oceanic conditions.

15 citations

### Cites background from "Scattering of surface water waves i..."

• ...Great attention was also given to the cases when ice only partially covers the ocean surface and contains cracks, polynyas and hummocks (see Liu & Mollo-Christensen (1988), Chakrabarti & Mohapatra (2013) and Li, Wu & Shi (2019) and references therein)....

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Journal ArticleDOI
TL;DR: In this paper, a hydroelastic model is considered to examine the proliferation of water waves over little deformation on a versatile seabed, where the Euler-Bernoulli beam equation is modelled as a thin large plate.

11 citations

Journal ArticleDOI
TL;DR: In this paper, a particular hydro-elastic model is considered to examine a radiation problem involving an immersed sphere in an infinitely extended ice-covered sea, where the lower surface is enveloped by a flexible base surface.
Abstract: A particular hydro-elastic model is considered to examine a radiation problem involving an immersed sphere in an infinitely extended ice-covered sea, where the lower surface is enveloped by a flexible base surface. Both the flexible base surface and floating ice-plate are modelled as thin elastic plates with different configurations and are based on the Euler–Bernoulli beam equation. The appearance of surface tension at the surface below the floating ice-plate is ignored. Under such circumstance, two different modes of propagating waves appear in the fluid for any particular frequency. One of the modes with lower wavenumber propagates along the surface beneath the ice-plate and the other with higher wavenumber propagates along the elastic base surface. The method of multipole expansions is used to calculate the solutions of the heave and sway radiation problems involving a submerged sphere in an ice-covered fluid. Furthermore, this procedure gives rise to an infinite system of linear equations, which can be solved computationally by any regular method. The added-mass as well as damping coefficients in case of heave as well as sway motions are calculated, and displayed graphically in various submergence depths of the oscillating sphere and elastic specifications of both the flexible base surface as well as the floating ice-plate.

11 citations

Journal ArticleDOI
TL;DR: Reflection and transmission phenomena of water waves due to undulating permeable bottom in a two-layer fluid system are investigated using two-dimensional linearized theory and the effect of surface tension on the free surface is included in this work.
Abstract: In the present paper, reflection and transmission phenomena of water waves due to undulating permeable bottom in a two-layer fluid system are investigated using two-dimensional linearized theory. The effect of surface tension on the free surface is included in this work. In two-layer fluid system, there exist waves with two different wave numbers (modes). When a wave of a particular wave number encounters the undulating bottom, reflection and transmission phenomena occur in both the layers. The reflection and transmission coefficients in both layers due to incident waves of both modes are analyzed with the aid of perturbation analysis along with Fourier transform technique. It is found that these coefficients are obtained in terms of integrals which depend on the shape function of the undulating bottom. Two different kinds of undulating bottoms are considered to determine these coefficients. For a particular undulating bottom, namely sinusoidal bottom undulation the effect of various physical para...

8 citations

### Cites background from "Scattering of surface water waves i..."

• ...The energy identity [6] relates the reflection as well as transmission coefficients associated with the scattering problem, which supports the theoretical validation of the present problem in the absence of experimental evidences....

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##### References
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Book
01 Jan 1993
TL;DR: The fifth edition of this hugely successful textbook retains the quality of earlier editions while at the same time seeing numerous minor improvements and major additions as mentioned in this paper, including a new chapter on singular values and singular vectors, including ways to analyze a matrix of data.
Abstract: Linear algebra is something all mathematics undergraduates and many other students, in subjects ranging from engineering to economics, have to learn. The fifth edition of this hugely successful textbook retains the quality of earlier editions while at the same time seeing numerous minor improvements and major additions. The latter include: a new chapter on singular values and singular vectors, including ways to analyze a matrix of data; a revised chapter on computing in linear algebra, with professional-level algorithms and code that can be downloaded for a variety of languages; a new section on linear algebra and cryptography; and a new chapter on linear algebra in probability and statistics. A dedicated and active website also offers solutions to exercises as well as new exercises from many different sources (e.g. practice problems, exams, development of textbook examples), plus codes in MATLAB, Julia, and Python.

1,913 citations

Journal ArticleDOI
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

Journal ArticleDOI
Neil Balmforth
TL;DR: In this paper, a complete analytical study of the reflection and transmission of surface gravity waves incident on an ice-covered ocean is presented, where the ice cover is idealized as a plate of elastic material for which flexural motions are described by the Timoshenko-Mindlin equation.
Abstract: A complete analytical study is presented of the reflection and transmission of surface gravity waves incident on ice-covered ocean. The ice cover is idealized as a plate of elastic material for which flexural motions are described by the Timoshenko–Mindlin equation. A suitable non-dimensionalization extracts parameters useful for the characterization of ocean-wave and ice-sheet interactions, and for scaled laboratory studies. The scattering problem is simplified using Fourier transforms and the Wiener–Hopf technique; the solution is eventually written down in terms of some easily evaluated quadratures. An important feature of this solution is that the physical conditions at the edge of the ice sheet are explicitly built into the analysis, and power-flow theorems provide verification of the results. Asymptotic results for large and small values of the non-dimensional parameters are extracted and approximations are given for general parameter values.

146 citations

Journal ArticleDOI
TL;DR: In this paper, a new inner product is developed based on the Fourier analysis to study the scattering of surface waves by a floating semi-infinite elastic plate in a two-dimensional water domain of finite depth.
Abstract: A new inner product is developed based on the Fourier analysis to study the scattering of surface waves by a floating semi-infinite elastic plate in a two-dimensional water domain of finite depth. The eigenfunctions for the plate-covered region are orthogonal with respect to this new inner product. The problem is studied for various wave and geometrical conditions. Especially, the influence of different edge conditions on the hydrodynamic behavior is investigated and compared. The edge conditions considered in the present study involve (i) a free edge, (ii) a simply supported edge, and (iii) a built-in edge. The hydrodynamic performance of an elastic plate is characterized for various conditions in terms of wave reflection and transmission, plate deflection, and surface strain. It is observed that the hydrodynamic behavior depends on the wave conditions, the geometrical settings, and the edge conditions. The built-in edge condition induces the maximum wave reflection and the minimum wave transmission. The free edge condition leads to the maximum plate deflection.

130 citations

Journal ArticleDOI
TL;DR: Theoretical and experimental results for the reflexion and transmission of water waves, passing over a step bottom between regions of finite and infinite depth, are presented in this article, with the wave crests parallel to the step, and linearized irrotational flow is assumed.
Abstract: Theoretical and experimental results are presented for the reflexion and transmission of water waves, passing over a step bottom between regions of finite and infinite depth. Two-dimensional motion is assumed, with the wave crests parallel to the step, and in the theory linearized irrotational flow is assumed. By matching ‘wavemaker’ solutions for the two regions at the cut above the step, an integral equation is derived for the horizontal velocity component on the cut. This integral equation is solved numerically and the reflexion and transmission coefficients and associated phase shifts are obtained. These results are compared with the long-wave theory and significant frequency effects are found, even for quite long waves. Experimental results are presented, which are in fair agreement with the theory.

124 citations

### "Scattering of surface water waves i..." refers background in this paper

• ...In this section, we consider the problem of scattering of water waves involving an ocean of finite depth having a flat rigid bottom surface, whereas the upper surface of the ocean is bounded above by a thin uniform semi-infinite elastic plate modelled as a thin elastic ice-cover (Ursell, 1947; Stoker, 1957; Weitz and Keller, 1950; Newman, 1965; Evans, 1985; Evans and Linton, 1994; Blamforth and Craster, 1999; Chakrabarti, 2000; Chung and Fox, 2002; Sahoo et al., 2001)....

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