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.     

Flexural gravity wave over a floating ice sheet near a vertical wall

The behavior of flexural gravity waves propagating over a semi-infinite floating ice sheet is studied under the assumptions of small amplitude linear wave theory. The vertical wall is assumed to be either fixed or harmonically oscillating with constant horizontal displacement, in which case the problem is analogous with a harmonically oscillating plane vertical wavemaker. The potential flow approach is adhered to and the higher-order mode–coupling relations are applied to determine the unknown coefficients present in the Fourier expansion formula of the potential functions. The ice sheet is modeled as a thin semi-infinite elastic beam. Three different edge conditions are applied at the finite edge of the floating ice sheet. The effects of different edge conditions, the thickness of the ice sheet and the water depth on the surface strain, the shear force along the ice sheet, the horizontal force on the vertical wall, and the flexural gravity wave profile are analyzed in detail.     

Transformation of flexural gravity waves by heterogeneous boundaries

The transformation of flexural gravity waves due to wave scattering by heterogeneous boundaries is investigated under the assumption of the linearized water-wave theory. The heterogeneous boundaries include step-type bottom topography as well as heterogeneity in the material property of a floating ice-sheet. By applying the generalized expansion formulae along with the corresponding orthogonal mode-coupling relations, the boundary-value problem (BVP) is reduced to linear system of algebraic equations. The system of equations is solved numerically to determine the full solution of the problem under consideration. Energy relations are derived and used to check the accuracy of the computational results of the scattering problem. Explicit relations for the shoaling and scattering coefficients due to the change in water depth and heterogeneous ice-sheet are derived. These derivations are based on the law of conservation of energy flux under the assumptions of the linearized shallow-water theory. The change in water depth and the structural characteristics of the medium significantly contribute to the change in the scattering and shoaling coefficients and the deflection of the structure. The present results are likely to play a significant role in the analysis of flexural gravity-wave propagation in problems of variable topography for which a direct computational approach is being utilized.     

On capillary gravity-wave motion in two-layer fluids

Generalized expansion formulae for the velocity potentials associated with plane gravity-wave problems in the presence of surface tension and interfacial tension are derived in both the cases of finite and infinite water depths in two-layer fluids. As a part of the expansion formulae, orthogonal mode-coupling relations, associated with the eigenfunctions of the velocity potential, are derived. The dispersion relations are analyzed to determine the characteristics of the two propagating modes in the presence of surface and interfacial tension in both the cases of deep-water and shallow-water waves. The expansion formulae are then generalized to deal with boundary-value problems satisfying higher-order boundary conditions at the free surface and interface. As applications of the expansion formulae, the solutions associated with the source potential, forced oscillation and reflection of capillary–gravity waves in the presence of interfacial tension are derived.     

Effect of a submerged vertical barrier on flexural gravity waves

An explicit solution is provided for the scattering of flexural gravity waves by a rigid vertical barrier submerged in an infinite depth of water. By applying recently developed mode-coupling relation for eigenfunctions, the mixed boundary value problem has been converted to solve dual integral equations with kernel consisting of trigonometric functions. And then complete analytical solutions are derived with an aid of singular integral equations whose solutions are bounded at the end points. The important hydrodynamical scattering quantities such as reflection and transmission coefficients associated with the flexural gravity wave scattering have been obtained analytically in terms of modified Bessel functions and Struve functions. It is observed that these quantities are sensitive to both combined as well as individual effect of plate thickness and barrier depth of submergence. Numerical results are computed and explained graphically for different parameters such as time period and non-dimensional wave length. Further, the effect of compressive force and plate thickness on the flexural gravity waves against a submerged vertical barrier is studied.     

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.     

Transmission and Reflection of Water-Wave on a Floating Ship in Vast Oceans

In this paper, we study the water-wave flow under a floating body of an incident wave in a fluid. This model simulates the phenomenon of waves abording a floating ship in a vast ocean. The same model, also simulates the phenomenon of fluid-structure interaction of a large ice sheet in waves. According to this method. We divide the region of the problem into three subregions. Solutions, satisfying the equation in the fluid mass and a part of the boundary conditions in each subregion, are given. We obtain such solutions as infinite series including unknown coefficients. We consider a limited number only of the coefficients by truncating the infinite series and satisfy the remaining boundary conditions approximately. Numerical experiments show that the results are acceptable. Tables are given along with the graph of the system of the resulting streamlines and the dynamical pressure acting on the obstacle. The drawn system of streamlines shows the correctness of the solution and the pressure is maximum on the side facing the upstream extremity of the channel. The same procedure can be adequately applied for problems with more complicated geometry and other phenomenon can thus be simulated.     

Absorbing boundary condition for floating two-dimensional objects in current and waves

In this paper an absorbing boundary condition for floating two-dimensional objects in current and waves is studied. A numerical algorithm has been developed, which computes the velocity potential in the physical time domain, by using an artificial boundary to split the infinite fluid domain into a computational part and a residual part. A special Green's function has been developed in the residual part. The condition on the artificial boundary is independent of wave frequency, hence not restricted to harmonic waves. Because of the smaller computational domain and the independence of frequency, the time to compute the hydrodynamic coefficients of floating objects decreases.     

Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth

Wave motion due to line, point and ring sources submerged in an inviscid fluid are analytically investigated. The initially quiescent fluid of finite depth, covered by a thin elastic plate or by an inertial surface with the capillary effect, is assumed to be incompressible and homogenous. The strengths of the sources are time-dependent. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The perturbed flow is decomposed into the regular and the singular components. An image system is introduced for the singular part to meet the boundary condition at the flat bottom. The solutions in integral form for the velocity potentials and the surface deflexions due to various singularities are obtained by means of a joint Laplace-Fourier transform. To analyze the dynamic characteristics of the flexural- and capillary-gravity waves due to unsteady disturbances, the asymptotic representations of the wave motion are explicitly derived for large time with a fixed distance-to-time ratio by virtue of the Stokes and Scorer methods of stationary phase. It is found that the generated waves consist of three wave systems, namely the steady-state gravity waves, the transient gravity waves and the transient flexural/capillary waves. The transient wave system observed depends on the moving speed of the observer in relation to the minimal and maximal group velocities. There exists a minimal depth of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. Furthermore, the results for the pure gravity and capillary-gravity waves in a clean surface can also be recovered as the flexural and inertial parameters tend to zero.     

Surface water waves involving a vertical barrier in the presence of an ice-cover

A class of boundary value problems involving propagation of two-dimensional surface water waves, associated with deep water and a plane vertical rigid barrier is investigated under the assumption that the surface is covered by a thin sheet of ice. Assuming that the ice-cover behaves like a thin isotropic elastic plate, the problems under consideration lead to those of solving the two-dimensional Laplace equation in a quarter-plane, under a Neumann boundary condition on the vertical boundary and a condition involving up to fifth order derivatives of the unknown function on the horizontal ice-covered boundary, along with two appropriate edge conditions, ensuring the uniqueness of the solutions. Two different methods are employed to solve the mixed boundary value problems completely, by determining the unique solution of a special type of integral equation of the first kind in the first method and by exploiting the analyticity property of the Fourier cosine transform in the second method.     

地面谐运动激励下任意外形水坝稳态响应的双边界元解

本文基于有限水深带形域势流问题的基本解和二维线弹性力学问题的Kelvin解,建立了坝库系统在谐激励下稳态响应的双边界积分方程.推导过程中,利用了Nardini和Brebbia方法将分布惯性力项的体积分化为相应的边界积分.然后通过边界元离散技术,针对两个不同型式的坝体计算了作用在界面上的水动压力分布,其中一个算例的结果和已有的有限元解作了比较.     

A new approximate formulation of periodic gravity waves

Periodic gravity waves travelling in irrotational deep water flows are examined. A new method of implicit approximation to the travelling waves along the free water surface rather than the explicit Stokes wave expansions along the calm water surface is formulated. Therefore, the approximate waves satisfy exactly the dynamic free surface boundary condition along the free water surface, while the corresponding Stokes waves satisfy the dynamic free surface boundary condition approximately along the calm water surface based on the Taylor expansion. The distinction between the proposed wave and the corresponding Stokes wave can be ignored at small wave steepness but becomes clear with the increment of the wave steepness due to the nonphysical form of the Stokes wave at large wave steepness. Approximation to the Stokes highest wave is demonstrated.     

Waves Past Porous Structures in a Two-layer Fluid

Havelock’s type of expansion theorems, for an integrable function having a single discontinuity point in the domain where it is defined, are utilized to derive analytical solutions for the radiation or scattering of oblique water waves by a fully extended porous barrier in both the cases of finite and infinite depths of water in two-layer fluid with constant densities. Also, complete analytical solutions are obtained for the boundary-value problems dealing with the generation or scattering of axi-symmetric water waves by a system of permeable and impermeable co-axial cylinders. Various results concerning the generation and reflection of the axisymmetric surface or interfacial waves are derived in terms of Bessel functions. The resonance conditions within the trapped region are obtained in various cases. Further, expansions for multipole-line-source oblique-wave potentials are derived for both the cases of finite and infinite depth depending on the existence of the source point in a two-layered fluid.     

Electroelastic fields and effective moduli of a medium containing cavities or rigid inclusions of arbitrary shape under anti-plane mechanical and in-plane electric fields

Summary We examine the two-dimensional problem of an infinite piezoelectric medium containing a solitary cavity or rigid inclusion of arbitrary shape, subjected to a coupled anti-plane mechanical and in-plane electric load at the remote boundary of the matrix. Conformal mapping techniques are employed to analyze the boundary value problems. Specific results are given for elliptical, polygonal and star-shape inclusions. Local fields of this type are used to estimate the overall moduli of a medium containing voids or rigid inclusions. This is accomplished with the help of an extension of Eshelby's formula which evaluates the total electric enthalpy by a particular line integral. Explicit estimates of the effective moduli are derived for dilute as well as for moderate area fractions of inclusions. The formulae depend solely on the cross area of the inclusion, area fraction and one particular coefficient of the mapping function. In addition, the stress and electric displacement singularities around the sharp corners of the inclusion are examined. The existence of uniform fields inside the inclusion is also envisaged. The present results, with appropriate modifications, apply equally well to those of thermoelectric and magnetoelectric effects.     

Flexural gravity wave scattering by a nearly vertical porous wall

Two-dimensional linear flexural gravity wave scattering by a nearly vertical porous wall is analyzed through a simplified perturbational analysis. A continuous semi-infinite ice sheet of uniform thickness is assumed to be floating over water of infinite depth. The ice sheet, with inclusion or exclusion of compressive stress, has either a free edge or a clamped edge at the porous wall. The first-order correction to the reflected flexural gravity wave amplitude is obtained by two different methods. The first method involves an application of Green’s theorem, and the second method involves a first kind integral equation. The integral equation method proves to be robust as it provides a complete solution in all cases of the problem, whereas the first method fails to produce the same when the ice sheet with a free edge is under compressive stress. The strain in the ice sheet and shear force along the ice sheet are computed and explained graphically for suitable parameters and a particular wall shape function.     

竖向成层介质中标量波传播问题的高精度人工边界条件

为了实现含竖向成层介质以及表面不规则地形场地中标量波传播问题的高效且高精度求解,该文基于连分式展开和扩展的一致边界,建立了一种频域下折线形高精度人工边界条件。通过在每个竖向地层内引入独立的斜角坐标变换,新的人工边界条件可以用于多起伏地表地形条件。新的折线形人工边界在频域下推导,仅含有连分式阶数一个待定实参数,用于调整计算精度,该参数不随外行波的频率和传播角度改变。人工边界条件可以与内域有限元方程无缝耦合,应用简单方便。由于新边界条件的高精度,内域尺寸可以取较小甚至可以直接将人工边界加在结构周围或者地表,从而极大提高计算效率。通过典型数值算例,将人工边界计算模型与有限元大模型的解进行了对比分析,验证了该文提出的折线形人工边界条件的有效性和高精度。     

Elastodynamic guided wave scattering in infinite plates

The scattering of elastic guided waves by defects in two‐dimensional infinite plates is analysed in the plane and antiplane cases, corresponding, respectively, to Lamb and SH modes. A hybrid boundary element–finite element technique is used, where the defect neighbourhood is discretized with quadratic boundary elements and the radiation condition in the plate is satisfied through a normal mode expansion. A semi‐analytical finite element technique is applied in the infinite plate to calculate its dispersion curves and normal modes. This hybrid technique, which showed excellent performance in the solution of Lamb wave reflection at the edge of semi‐infinite plates, is extended in this paper to a wider range of problems, such as Lamb mode scattering by delaminations and surface defects and SH mode interaction with step discontinuities. Copyright © 2003 John Wiley & Sons, Ltd.     

An improved boundary element formulation for wave propagation problems

In this paper the dual reciprocity formulation for scalar wave equations and elastodynamic problems developed by Nardini & Brebbia is extended to the problem of waves propagating in an infinite domain by applying the Sommerfeld's radiation condition on a suitable artificial boundary. The free surface condition of first order can also be taken into consideration. To validate the present scheme, some examples have been worked out and compared with analytical solutions.     

Infinite elements in the time domain using a prolate spheroidal multipole expansion

Many phenomena in acoustically loaded structural vibrations are better understood in the time domain, particularly transient radiation, shock, and problems involving non-linearities, cavitation, and bulk structural motion. In addition, the geometric complexity of structures of interest drives the analyst toward domain-discretized solution methods, such as finite elements or finite differences, and large numbers of degrees of freedom. In such methods, efficient numerical enforcement of the Sommerfeld radiation condition in the time domain becomes difficult. Although a great many methodologies for doing so have been demonstrated, there seems to exist no consensus on the optimal numerical implementation of this boundary condition in the time domain. Here, we present theoretical development of several new boundary operators for conventional finite element codes. Each proceeds from successful domain-discretized, projected field-type harmonic solutions, in contrast to boundary integral equation operators or those derived from analyses of outgoing waves. We exploit the separable prolate-spheroidal co-ordinate system, which is sufficiently general for a large variety of problems of naval interest, to obtain finite element-like operators (matrices) for the boundary points. Use of this co-ordinate system results in element matrices that can be analytically inverse transformed from the frequency to the time domain, without imposing continuity requirements on the solution above those imposed by the underlying partial differential equation. In addition, use of element-like boundary operators does not alter the banded structure of assembled system matrices. Results presented here include theoretical derivation of the infinite elements, resolution of the Fourier inversion issues, and element matrices for the boundary operators which introduce no new continuity requirements on the fluid field variable. The simplest infinite elements are verified in a coupled three-dimensional context against DAA2 and Helmholtz integral equation results. © 1998 John Wiley & Sons, Ltd.     

An infinite element and a formula for numerical quadrature over an infinite interval

In this publication we present a new infinite element and discuss a formula for numerical quadrature over infinite regions. Both the element and the formula are based on the properties of a set of orthonormal functions, obtained by mapping the Legendre polynomials on the infinite interval. The element can represent any (physical) function over the entire infinite region, and the formula can integrate most integrands over the infinite interval. They are thus very convenient for implementation in a general computer program, which could then be used to solve different problems with no restrictions concerning the asymptotic behaviour of the solutions or the form of integrands to be integrated over the infinite interval. However, the element and the formula will perform at their best when used in conjunction, and for problems of electrostatic, magnetostatic, etc., potential. The formula was compared with other existing formulae and found to be either superior or satisfactory in practically all cases, as shown by the results of many numerical experiments. Expressions of shape functions are given for the element. A test problem was solved using the element and the formula, and the results are shown to be more accurate than those obtained when solving the same problem using infinite elements with exponential decay and Gauss-Laguerre numerical quadrature.