Quarter-plane problem of a floating elastic plate

An investigation is made into the hydro-elastic behavior of a floating elastic plate, which occupies a quarter plane to infinity and is excited by water waves. A boundary-integral equation based on the Green function for this problem is shown for the case of finite water depth, as well as for the case of shallow water. The solution of the quarter-plane problem is composed of the corner effect and the solution of the half-plane problem. The corner effect is divided into two parts. The first part is the end effect of the forcing term of the integral equation, which is analytically estimated and its asymptotic form is derived. The second part is the local contribution whose asymptotic form is also obtained. The asymptotic form of the corner effect is confirmed by a numerical evaluation.     

Nonlinear steady two-layer interfacial flow about a submerged point vortex

Two-dimensional, two-layer steady interfacial flow about a point vortex is studied in a uniform stream for each layer. The upper layer is of finite depth with a rigid lid on the upper surface, and the depth of the lower layer is assumed infinite. The point vortex is located in lower-layer fluid. We study this problem using not only a linear analytical method but also a nonlinear numerical method. A linear solution is derived in terms of a complex exponential integral function. The fully nonlinear problem is formulated by an integro-differential equation system. The equation system is solved using Newton’s method to determine the unknown steady interfacial surface. The numerical results of the downstream wave are provided by a linear solution and fully nonlinear solution. A comparison between linear solutions and nonlinear solutions shows that the nonlinear effect is apparent when the vortex strength increases. The effects of point vortex strengths, Froude numbers, and density ratios on the amplitudes of the downstream waves are studied. We analyze the effects of point vortex strengths, Froude numbers, and density ratios on the wavelengths of the downstream waves.     

The Wiener–Hopf and residue calculus solutions for a submerged semi-infinite elastic plate

We present a solution for the interaction of normally incident linear waves with a submerged elastic plate of semi-infinite extent, where the water has finite depth. While the problem has been solved previously by the eigenfunction-matching method, the present study shows that this problem is also amenable to the more analytical, and extremely efficient, Wiener–Hopf (WH) and residue calculus (RC) methods. We also show that the WH and RC solutions are actually equivalent for problems of this type, a result which applies to many other problems in linear wave theory. (e.g., the much-studied floating elastic plate scattering problem, or acoustic wave propagation in a duct where one wall has an abrupt change in properties.) We present numerical results and a detailed convergence study, and discuss as well the scattering by a submerged rigid dock, particularly the radiation condition beneath the dock.     

Expansion formulae for wave structure interaction problems with applications in hydroelasticity

Alternate derivations of the expansion formulae for wave structure interaction problems are obtained in case of water of infinite depth and utilized to analyze the hydroelastic behavior of large floating structures. Considering the boundary value problem associated with Laplace equation having higher order boundary condition on the horizontal boundary and a Dirichlet type boundary condition on the vertical boundary in a quarter plane, Fourier sine transform is applied in the horizontal direction to convert the problem to a Sturm-Liouville type boundary value problem associated with non-homogeneous ordinary differential equation (ODE) in the transformed variable. Finally, inverting the transformed functions and applying the regularity criterion of the transformed function, the required expansion formula is derived. The expansion formula thus derived is extended to deal with similar boundary value problems having Neumann type boundary condition. The expansion formulae are applied to (i) analyze oblique scattering of flexural gravity waves by an articulated floating elastic plate and (ii) study the effect of compression on the oblique scattering of flexural gravity waves by a line discontinuity in a large floating ice sheet in water of infinite depth, which find applications in marine technology and arctic engineering, respectively. The present derivations of the expansion formulae are very simple and straightforward and can be easily used to study a large class of problems in the area of fluids and structures in mathematical physics and engineering.     

The linear-wave response of a periodic array of floating elastic plates

The problem of an infinite periodic array of identical floating elastic plates subject to forcing from plane incident waves is considered. This study is motivated by the problem of trying to model wave propagation in the marginal ice zone, a region of ocean consisting of an arbitrary packing of floating ice sheets. It is shown that the problem considered can be formulated exactly in terms of the solution to an integral equation in a manner similar to that used for the problem of wave scattering by a single elastic floating plate, the key difference here being the use of a modified periodic Green function. The convergence of this Green function in its original form is poor, but can be accelerated by a transformation. It is shown that the results from the method satisfy energy conservation and that in the particular case of a fixed rigid rectangular plate which spans the periodicity uniformly the solution reduces to that for a two-dimensional rigid dock. Solutions for a range of elastic-plate geometries are also presented.     

Wave forces on a submerged vertical plate

Summary A vertical plate of finite length and depth is attacked by gravity waves in water of finite depth. The forces and moments acting on the plate are computed by using the theory of linearized waves. The forces depend on three dimensionless parameters combining the draft, length, water depth and wave length and on the angle of attack. The problem is reduced to the solution of two infinite linear systems of equations. Numerical solutions are presented for different particular combinations of the parameter values. In most of the cases the standing wave approximation yields sufficiently accurate results.     

Stress intensity factors for an embedded elliptical crack in a plate of finite thickness

Summary This paper deals with the three-dimensional analysis of the stress distribution in a plate of finite thickness containing an embedded elliptical crack subjected to a constant pressure. By using the douboe Fourier transform, the problem is reduced to the solution of the integro-differential equation which is solved iteratively. A numerical solution of the integro-differential equation is also obtained. These solutions are compared with the results in the published accounts.     

Wave interaction with a floating and submerged elastic plate system

Surface gravity wave interaction with a floating and submerged elastic plate system is analyzed under the assumption of small-amplitude surface water wave theory and structural response. The plane progressive wave solution associated with the plate system is analyzed to understand the characteristics of the flexural gravity waves in different modes. Further, linearized long-wave equations associated with the wave interaction with the elastic plate system are derived. The dispersion relations are derived based on small-amplitude wave theory and shallow-water approximation and are compared to ensure the correctness of the mathematical formulation. To deal with various types of problems associated with gravity wave interaction with a floating and submerged flexible plate system, Fourier-type expansion formulae are derived in the cases of water of both finite and infinite depths in two dimensions. Certain characteristics of the eigensystems of the developed expansion formulae are derived. Source potentials for surface wave interaction with a floating flexible structure in the presence of a submerged flexible structure are derived and used in Green’s identity to obtain the expansion formulae for flexural gravity wavemaker problems in the presence of submerged flexible plates. The utility of the expansion formulae and associated orthogonal mode-coupling relations is demonstrated by investigating the diffraction of surface waves by floating and submerged flexible structures of two different configurations. The accuracy of the computational results is checked using appropriate energy relations. The present study is likely to provide fruitful solutions to problems associated with floating and submerged flexible plate systems of various configurations and geometries arising in ocean engineering and other branches of mathematical physics and engineering including acoustic structure interaction problems.     

An integral equation method for the diffraction of oblique waves by an infinite cylinder

A hybrid integral equation method is formulated to study the diffraction of oblique waves by an infinite cylinder. The water depth and the geometry of the floating cylinder are assumed to be uniform in the y-direction (one of the horizontal axes). Numerical discretization and integrations are performed in the vertical plane. Analytical solutions are used in far fields such that radiation boundary conditions are satisfied. Numerical results are obtained for the case of wave scattering by a floating rectangular cylinder in a constant water depth. The accuracy and efficiency of present method are compared with those obtained by other numerical techniques.     

Unsteady three-dimensional sources for a two-layer fluid of finite depth and their applications

To our friend Ernie Tuck, in celebration of his multi-faceted talents. The velocity potentials of various unsteady point sources are derived in this paper for a two-layer fluid of finite depth. Two-layer fluids are often used to study effects of density stratification on hydrodynamics of marine systems. The sources here are restricted to the upper fluid layer and the potentials of the induced flows are given for the whole fluid domain. The velocity potentials of a transient source of arbitrary strength and in arbitrary three-dimensional motion are derived first. The potentials of a time-harmonic source without forward speed, and then with forward speed, are obtained from the transient source by specifying the appropriate source strength and motion. These potentials are fundamental to the analyses of various types of body motion in finite water depths under the influence of surface and interfacial waves. As a sample application, a numerical solution of the radiation and diffraction problem for a floating rectangular barge is presented. The results indicate that internal waves can have a strong effect on the motions of the floating barge over a wide range of incident-wave frequencies.     

On finite amplitude water waves

The modulation theory for finite amplitude water waves is developed using the variational technique. It is shown how Levi-Civita’s relation, Starr’s relation and the conservation equations all follow very simply and naturally from this approach. The present paper is limited to deep water waves, but the results can be extended to arbitrary depth. For deep water, the appropriate Lagrangian can be reduced to a single function, which can be taken from recent numerical calculations on periodic waves. This is used to discuss the stability of wavetrains to long modulations.     

The analytic form of Green’s function in elliptic coordinates

The purpose of this study is the derivation of a closed-form formula for Green’s function in elliptic coordinates that could be used for achieving an analytic solution for the second-order diffraction problem by elliptical cylinders subjected to monochromatic incident waves. In fact, Green’s function represents the solution of the so-called locked wave component of the second-order velocity potential. The mathematical analysis starts with a proper analytic formulation of the second-order diffraction potential that results in the inhomogeneous Helmholtz equation. The associated boundary-value problem is treated by applying Green’s theorem to obtain a closed-form solution for Green’s function. Green’s function is initially expressed in polar coordinates while its final elliptic form is produced through the proper employment of addition theorems.     

Free-surface wave interaction with a thick flexible dock or very large floating platform

In this paper the recently developed semi-analytic method to solve the free-surface wave interaction with a thin elastic plate is extended to the case of a plate of finite thickness. The method used is based on the reformulation of the differential–integral equation for this problem. The thickness of the plate is chosen such that the elastic behavior of the plate can be described by means of thin-plate theory, while the water pressure at the plate is applied at finite depth. The water depth is finite.     

柱形地形上单浮体的辐射和散射问题

在有限水深、同轴但半径大于或等于浮体半径的圆柱体障碍物地形条件下，基于特征函数展开法，推导了垂直放置的圆柱形浮体由于波的辐射和散射作用所表现的动力学和运动学特征表达式，涉及浮体做垂荡、横荡和横摇运动所产生的辐射势，以及在入射波的作用下，由于浮体固定不动而产生的散射势，并推导了激励力、附加质量和阻尼系数表达式。采用与同轴、同半径圆柱体障碍物地形上单浮体水动力学特性相比的方式和激励力计算两种方法验证了推导的表达式，最后分析了障碍物几何尺寸对浮体水动力学特性的特有影响。     

Hydroelastic behaviour of compound floating plate in waves

The paper deals with the plane problem of the hydroelastic behaviour of floating plates under the influence of periodic surface water waves. Analysis of this problem is based on hydroelasticity, in which the coupled hydrodynamics and structural dynamics problems are solved simultaneously. The plate is modeled by an Euler beam. The method of numerical solution of the floating-beam problem is based on expansions of the hydrodynamic pressure and the beam deflection with respect to different basic functions. This makes it possible to simplify the treatment of the hydrodynamic part of the problem and at the same time to satisfy accurately the beam boundary conditions. Two approaches aimed to reduce the beam vibrations are described. In the first approach, an auxiliary floating plate is added to the main structure. The size of the auxiliary plate and its elastic characteristics can be chosen in such a way that deflections of the main structure for a given frequency of incident wave are reduced. Within the second approach the floating beam is connected to the sea bottom with a spring, the rigidity of which can be selected in such a way that deflections in the main part of the floating beam are very small. The effect of the vibration reduction is quite pronounced and can be utilized at the design stage.     

Wave scattering by a thin elastic plate floating on a two-layer fluid

The hydroelastic interaction between an incident gravity wave and a thin elastic plate floating on a two-layer fluid of finite depth is analyzed with the aid of the method of matched eigenfunction expansions. The fluid is assumed to be inviscid and incompressible. A two-dimensional problem is formulated within the framework of linear potential theory for small-amplitude waves. The fluid domain is divided into two and three regions for semi-infinite and finite plates, respectively, with the matching relations representing the continuities of the pressure and velocity. A new inner product involving two single integrals is proposed, in which the vertical eigenfunctions in the open water region of the two-layer fluid are orthogonal. Then the orthogonality of the eigenfunctions with respect to the newly defined inner product is used to obtain a set of simultaneous equations for the expansion coefficients of the velocity potentials, and the edge conditions are included as a part of the equation system. The effects of the fluid density ratio and the position of interface on the wave reflection and transmission are discussed. Numerical analysis shows that the method proposed herein is effective with a higher rate of convergence.     

Subharmonic resonance of a trapped wave near a vertical cylinder by narrow-banded random incident waves

It is well-known that near an infinite linear array of periodically spaced cylinders trapped waves of certain eigenfrequencies can exist. If there are only a finite number of cylinders in an infinite sea, trapping is imperfect. Simple harmonic incident waves can excite a nearly trapped wave at one of the eigen frequencies through a linear mechanism. However, the maximum amplification ratio increases monotonically with the number of the cylinders; hence the solution is singular in the limit of infinitely many cylinders. Recently, a nonlinear theory of subharmonic resonance of perfectly trapped waves has been completed. In this article the theory is further extended to random incident waves with a narrow spectrum centered near twice the natural frequency of the trapped wave. The effects of detuning and bandwidth of the spectrum are examined. Dedicated to Professor J. N. Newman on his 70th birthday. We wish to express our profound admiration for Professor Newman’s scientific contributions and leadership in the ship-hydrodynamics discipline. The relation between this article and an early work of his reflects in part his impact on us.     

Wave drift force in a two-layer fluid of finite depth

Based on the momentum and energy conservation principles, a compact calculation formula is analytically derived for the wave-drift force on a 2-D body floating in a two-layer fluid of finite depth. In a two-layer fluid, two different wave modes (the surface-wave mode with longer wavelength and the internal-wave mode with shorter wavelength) exist not only in the incident wave but also in the body-scattered wave, and these wave characteristics are properly incorporated in the obtained formula. It is noted that, unlike the single-layer case, the wave-drift force can be negative in the incident wave of surface-wave mode, if the transmitted wave with internal-wave mode is large. Numerical computations are implemented for a Lewis-form body by means of the boundary-integral-equation method with Green’s function for the two-layer fluid problem. The effects of density ratio, interface position, and body motions on the wave-drift force are studied, and some important features are found for two-layer fluids.     

Orthotropic piezoelectric patches for control of symmetric laminates via an integral equation

Formulation of the problem for the feedback displacement control of a vibrating laminated plate with orthotropic piezoelectric sensors and actuators is given in terms of an integral equation. The objective is to develop a formulation which facilitates the numerical solution to obtain the eigenfrequencies and eigenfunctions of the piezo-controlled plate. The control is carried out via piezoelectric sensors and actuators which are of orthorhombic crystal class mm² with poling in the z direction. The initial formulation of the problem is given in terms of a differential equation which is the conventional formulation most often used in the literature. The conversion to an integral equation formulation is achieved by introducing an explicit Green’s function. Explicit expressions for the kernel of the integral equation are given and the method of solution using the new formulation is outlined. The solution technique involves approximating the integral equation with an infinite system of linear equations and using a finite number of these equations to obtain the numerical results.     

Interaction of hydro-elastic waves with a vertical wall

The linear two-dimensional problem of hydroelastic waves reflected by a vertical wall is analysed. The fluid is of finite depth and is covered by an ice sheet. The fluid is assumed incompressible and inviscid. The ice sheet is assumed thin compared with both the water depth and wavelength of the incident wave. The deflection of the ice sheet is described by linear elastic plate theory, and the fluid flow by using the potential-flow model. The ice sheet extends infinitely and is clamped to the vertical-walled structure. The incident hydroelastic wave is regular. An analytic solution is found by integral-transform methods. The ice deflection, the vertical and horizontal forces acting on the wall and the bending stresses in the ice caused by the incident wave are determined. The forces on the wall are analysed in detail, and relevant physical parameters are varied for comparison. The phase shift between the incident and reflected wave amplitudes is found as part of the complete solution. It is shown that the ice clamping condition leads to a specific effect on the ice deflection.