有限水深中垂直下潜弹性薄板的水波散射(英文)

The problem of water wave scattering by a thin vertical elastic plate submerged in uniform finite depth water is investigated here.The boundary condition on the elastic plate is derived from the Bernoulli-Euler equation of motion satisfied by the plate.Using the Green’s function technique,from this boundary condition,the normal velocity of the plate is expressed in terms of the difference between the velocity potentials(unknown)across the plate.The two ends of the plate are either clamped or free.The reflection and transmission coefficients are obtained in terms of the integrals’involving combinations of the unknown velocity potential on the two sides of the plate,which satisfy three simultaneous integral equations and are solved numerically.These coefficients are computed numerically for various values of different parameters and depicted graphically against the wave number in a number of figures.     

半无限惯性表面产生的波浪散射问题研究的另一种方法(英文)

A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green’s second identity to the potential functions and appropriate Green’s functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.     

多转角结构中转角处附加的阻振质量对结构波传递的抑制

应用波分析法研究在由平板构成的多转角结构中附加阻振质量对结构声传递的抑制作用.通过引入由半无限长板构成的转角处的透射及反射系数来求解在弯曲波及纵波入射条件下多转角结构中各板内不同形式波的波动表达式,避免了求解复杂的偏微分方程组,从而简化了分析,并在此基础上推导出附加两条阻振质量的双转角结构的能量透射效率.文中同时还进行了模拟船舶结构的简化模型的实验,验证了数值计算得出的-些规律.研究表明:附加在双转角结构的转角处的阻振质量不仅在高频区对结构声的传递有很好的抑制效果,而且还能显著增加两阻振质量之间的有限长板的-阶共振频率处的传递损失. 的多转角结构中附加阻振质量对结构声传递的抑制作用.通过引入由半无限长板构成的转角处的透射及反射系数来求解在弯曲波及纵波入射条件下多转角结构中各板内不同形式波的波动表达式,避免了求解复杂的偏微分方程组,从而简化了分析,并在此基础上推导出附加两条阻振质量的双转角结构的能量透射效率.文中同时还进行了模拟船舶结构的简化模型的实验,验证了数值计算得出的-些规律.研究表明:附加在双转角结构的转角处的阻振质量不仅在高频区对结构声的传递有很好的抑制效果,而且还能显著增加两阻振质量之间的有限长板的 阶共振频率处的传递损失. 的多转角结构中附加阻振质量对结构声传递的抑制作用.通过引入由半无限长板构成的转角处的透射及反射系数来求解在弯曲波及纵波入射条件下多转角结构中各板内不同形式波的波动表达式,避免了求解复杂的偏微分方程组,从而简化了分析,并在此基础上推导出附加两条阻振质量的双转角结构的能量透射效率.文中同时还进行了模拟船舶结构的简化模型的实验,验证了数值计算得出的-些规律.研究表明:附加在双转角结构的转角处的阻振质量不仅在高频区对结构声的传递有很好的抑制效果,而且还能显著增加两阻振质量之间的有限长板的 阶共振频率处的传递损失. 的多转角结构中附加阻振质量对结构声传递的抑制作用.通过引入由半无限长板构成的转角处的透射及反射系数来求解在弯曲波及纵波入射条件下多转角结构中各板内不同形式波的波动表达式,避免了求解复杂的偏微分方程组,从而简化了分析,并在此基础上推导出附加两条阻振质量的双转角结构的能量透射效率.文中同时还进行了模拟船舶结构的简化模型的实验,验证了数值计算得出的一些规律.研究表明:附加在双转角结构的转角处的阻振质量不仅在高频区对结构声的传递有很好的抑制效果,而且还能显著增加两阻振质量之间的有限长板的     

冰盖下两层流体水波与可渗透起伏海床的作用(英文)

The scattering problem involving water waves by small undulation on the porous ocean-bed in a two-layer fluid,is investigated within the framework of the two-dimensional linear water wave theory where the upper layer is covered by a thin uniform sheet of ice modeled as a thin elastic plate.In such a two-layer fluid there exist waves with two different modes,one with a lower wave number propagate along the ice-cover whilst those with a higher wave number propagate along the interface.An incident wave of a particular wave number gets reflected and transmitted over the bottom undulation into waves of both modes.Perturbation analysis in conjunction with the Fourier transform technique is used to derive the first-order corrections of reflection and transmission coefficients for both the modes due to incident waves of two different modes.One special type of bottom topography is considered as an example to evaluate the related coefficients in detail.These coefficients are depicted in graphical forms to demonstrate the transformation of wave energy between the two modes and also to illustrate the effects of the ice sheet and the porosity of the undulating bed.     

带空海底上小变形柱体边缘斜浪散射研究（英文）

The scattering of oblique incident surface waves by the edge of a small cylindrical deformation on a porous bed in an ocean of finite depth, is investigated here within the framework of linearized water wave theory. Using perturbation analysis, the corresponding problem governed by modified Helmholtz equation is reduced to a boundary value problem for the first-order correction of the potential function. The first-order potential and, hence, the reflection and transmission coefficients are obtained by a method based on Green's integral theorem with the introduction of appropriate Green's function. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number along x-direction and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the free-surface, and the reflection coefficient becomes a multiple of the number of ripples. Again, for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. It is also observed that the reflected energy is somewhat sensitive to the changes in the porosity of the ocean bed. From the derived results, the solutions for problems with impermeable ocean bed can be obtained as particular cases.     

The influence of water depth on the hydroelastic response of a very large floating platform

The hydroelastic response of a two-dimensional very large floating platform to plane incident wave is investigated for three different cases: infinite, finite and shallow water depth. An integro-differential equation is presented to describe the deflection of the platform due to incident waves. Reflection and transmission coefficients are obtained as well. We consider the case of a strip and a half-plane. Numerical results are obtained for various values of the parameters. The results for the strip and for the semi-infinite platform are compared for different values of depth.     

弹性浮板对水波的衍射与动弯矩分析

文章基于水波理论和Mindlin厚板动力学理论,采用Wiener-Hopf方法,分析研究了水面上弹性浮板对水波的衍射与动弯矩分布问题.为确定极点,首先分析求解了有限水深的导波问题.而后将原来混合边值问题的求解归结为求解无穷线性代数方程组的问题.文中将分析计算结果与前人的计算结果及实验数据进行了对比,并对结果进行了分析讨论.最后,分析了结构几何参数和物理参数挠度和动弯矩分布的影响.     

多阶底部上的垂直可渗透结构斜海洋波的阻尼研究(英文)

Oblique ocean wave damping by a vertical porous structure placed on a multi-step bottom topography is studied with the help of linear water wave theory. Some portion of the oblique wave, incident on the porous structure, gets reflected by the multi-step bottom and the porous structure, and the rest propagates into the water medium following the porous structure. Two cases are considered: first a solid vertical wall placed at a finite distance from the porous structure in the water medium following the porous structure and then a special case of an unbounded water medium following the porous structure. In both cases, boundary value problems are set up in three different media, the first medium being water, the second medium being the porous structure consisting of p vertical regions-one above each step and the third medium being water again. By using the matching conditions along the virtualvertical boundaries, a system of linear equations is deduced. The behavior of the reflection coefficient and the dimensionless amplitude of the transmitted progressive wave due to different relevant parameters are studied. Energy loss due to the propagation of oblique water wave through the porous structure is also carried out. The effects of various parameters, such as number of evanescent modes, porosity, friction factor, structure width, number of steps and angle of incidence, on the reflection coefficient and the dimensionless amplitude of the transmitted wave are studied graphically for both cases. Number of evanescent modes merely affects the scattering phenomenon. But higher values of porosity show relatively lower reflection than that for lower porosity. Oscillation in the reflection coefficient is observed for lower values of friction factor but it disappears with an increase in the value of friction factor. Amplitude of the transmitted progressive wave is independent of the porosity of the structure. But lower value of friction factor causes higher transmission. The investigation is then carried out for the second case, i.e., when the wall is absent. The significant difference between the two cases considered here is that the reflection due to a thin porous structure is very high when the solid wall exists as compared to the case when no wall is present. Energy loss due to different porosity, friction factor, structure width and angle of incidence is also examined. Validity of our model is ascertained by matching it with an available one.     

冰盖海洋中多孔海底结构上微小起伏边缘产生的波浪散射分析(英文)

Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investigated within the framework of linearized water wave theory. The effect of surface tension at the surface below the ice-cover is neglected. There exists only one wave number propagating at just below the ice-cover. A perturbation analysis is employed to solve the boundary value problem governed by Laplace's equation by a method based on Green's integral theorem with the introduction of appropriate Green's function and thereby evaluating the reflection and transmission coefficients approximately up to first order. A patch of sinusoidal ripples is considered as an example and the related coefficients are determined.     

Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean with ice-cover

Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investigated within the framework of linearized water wave theory. The effect of surface tension at the surface below the ice-cover is neglected. There exists only one wave number propagating at just below the ice-cover. A perturbation analysis is employed to solve the boundary value problem governed by Laplace＇s equation by a method based on Green＇s integral theorem with the introduction of appropriate Green＇s function and thereby evaluating the reflection and transmission coefficients approximately up to first order. A patch of sinusoidal ripples is considered as an example and the related coefficients are determined.     

支撑条件对浮动式厚弹性板水弹性行为的影响分析（英文）

The hydroelastic response of very large floating structures(VLFS) under the action of ocean waves is analysed considering the small amplitude wave theory. The very large floating structure is modelled as a floating thick elastic plate based on TimoshenkoMindlin plate theory, and the analysis for the hydroelastic response is performed considering different edge boundary conditions.The numerical study is performed to analyse the wave reflection and transmission characteristics of the floating plate under the influence of different support conditions using eigenfunction expansion method along with the orthogonal mode-coupling relation in the case of finite water depth. Further, the analysis is extended for shallow water depth, and the continuity of energy and mass flux is applied along the edges of the plate to obtain the solution for the problem. The hydroelastic behaviour in terms of reflection and transmission coefficients, plate deflection, strain, bending moment and shear force of the floating thick elastic plate with support conditions is analysed and compared for finite and shallow water depth. The study reveals an interesting aspect in the analysis of thick floating elastic plate with support condition due to the presence of the rotary inertia and transverse shear deformation. The present study will be helpful for the design and analysis of the VLFS in the case of finite and shallow water depth.     

Scattering of oblique surface water waves by thin vertical barrier over undulating bed topography

The present study deals with the scattering of oblique surface water waves by small undulation on the bottom in the presence of a thin vertical barrier. Here, three different configurations of vertical barriers are investigated. Perturbation analysis is employed to determine the physical quantities, namely, the reflection and transmission coefficients. In this analysis, many different Boundary Value Problems (BVPs) are obtained out of which the first two bvps are considered. The zeroth order bvp is solved with the aid of eigenfunction expansion method. The first order reflection and transmission coefficients are derived in terms of the integrals by the method of the Green’s integral theorem. The variation of these coefficients is plotted and analyzed for different physical parameters. Furthermore, the energy balance relation, an important relation in the study of water wave scattering, is derived and checked for assuring the correctness of the numerical results for the present problem.     

A note on the solution of water wave scattering problem involving small deformation on a porous channel-bed

The solution of water wave scattering problem involving small deformation on a porous bed in a channel, where the upper surface is bounded above by an infinitely extent rigid horizontal surface, is studied here within the framework of linearized water wave theory. In such a situation, there exists only one mode of waves propagating on the porous surface. A simplified perturbation analysis, involving a small parameter ε ( ? 1), which measures the smallness of the deformation, is employed to reduce the governing Boundary Value Problem (BVP) to a simpler BVP for the first-order correction of the potential function. The first-order potential function and, hence, the first-order reflection and transmission coefficients are obtained by the method based on Fourier transform technique as well as Green’s integral theorem with the introduction of appropriate Green’s function. Two special examples of bottom deformation: the exponentially damped deformation and the sinusoidal ripple bed, are considered to validate the results. For the particular example of a patch of sinusoidal ripples, the resonant interaction between the bed and the upper surface of the fluid is attained in the neighborhood of a singularity, when the ripples wavenumbers of the bottom deformation become approximately twice the components of the incident field wavenumber along the positive x-direction. Also, the main advantage of the present study is that the results for the values of reflection and transmission coefficients are found to satisfy the energy-balance relation almost accurately.     

水波与双圆型透水板的相互作用（英文）

In this paper we have investigated the reflection and the transmission of a system of two symmetric circular-arc-shaped thin porous plates submerged in deep water within the context of linear theory. The hypersingular integral equation technique has been used to analyze the problem mathematically. The integral equations are formulated by applying Green's integral theorem to the fundamental potential function and the scattered potential function into a suitable fluid region, and then using the boundary condition on the porous plate surface. These are solved approximately using an expansion-cum-collocation method where the behaviour of the potential functions at the tips of the plates have been used. This method ultimately produces a very good numerical approximation for the reflection and the transmission coefficients and hydrodynamic force components. The numerical results are depicted graphically against the wave number for a variety of layouts of the arc. Some results are compared with known results for similar configurations of dual rigid plate systems available in the literature with good agreement.     

采用有限差分方法模拟穿孔式防波堤规则波中水动力性能研究（英文）

Using a discretized finite difference method, a numerical model was developed to study the interaction of regular waves with a perforated breakwater. Considering a non-viscous, non-rotational fluid, the governing equations of Laplacian velocity potential were developed, and specific conditions for every single boundary were defined. The final developed model was evaluated based on an existing experimental result. The evaluated model was used to simulate the condition for various wave periods from 0.6 to2 s. The reflection coefficient and transmission coefficient of waves were examined with different breakwater porosities, wave steepnesses, and angular frequencies. The results show that the developed model can suitably present the effect of the structural and hydraulic parameters on the reflection and transmission coefficients. It was also found that with the increase in wave steepness, the reflection coefficient increased logarithmically, while the transmission coefficient decreased logarithmically.     

Diffraction of oblique water waves by small uneven channel-bed in a two-layer fluid

Obliquely incident water wave scattering by an uneven channel-bed in the form of a small bottom undulation in a two-layer fluid is investigated within the frame work of three-dimensional linear water wave theory. The upper fluid is assumed to be bounded above by a rigid lid, while the lower one is bounded below by a bottom surface having a small deformation and the channel is unbounded in the horizontal directions. Assuming irrotational motion, perturbation technique is employed to calculate the first-order corrections to the velocity potentials in the two fluids by using Fourier transform approximately, and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that the reflection coefficient is an oscillatory function of the ratio of twice the component of the wave number along x-axis and the ripple wave number. When this ratio approaches one, the theory predicts a resonant interaction between the bed and interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of incident wave energy occurs if this number is large.     

The interaction of oblique flexural gravity waves with a small bottom deformation on a porous ocean-bed: Green’s function approach

The interaction of oblique incident water waves with a small bottom deformation on a porous ocean-bed is examined analytically here within the framework of linear water wave theory. The upper surface of the ocean is assumed to be covered by an infinitely extended thin uniform elastic plate, while the lower surface is bounded by a porous bottom surface having a small deformation. By employing a simplified perturbation analysis, involving a small parameter δ(=1), which measures the smallness of the deformation, the governing Boundary Value Problem (BVP) is reduced to a simpler BVP for the first-order correction of the potential function. This BVP is solved using a method based on Green’s integral theorem with the introduction of suitable Green’s function to obtain the first-order potential, and this potential function is then utilized to calculate the first-order reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number propagating just below the elastic plate and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the surface below the elastic plate. Again, for small angles of incidence, the reflected wave energy is more as compared to the other angles of incidence. It is also observed that the reflected wave energy is somewhat sensitive to the changes in the flexural rigidity of the elastic plate, the porosity of the bed and the ripple wave numbers. The main advantage of the present study is that the results for the values of reflection and transmission coefficients obtained are found to satisfy the energy-balance relation almost accurately.     

Water wave scattering by a sphere submerged in uniform finite depth water with an ice-cover

Using the multipole method, we formulate the problem of water wave scattering by a submerged sphere in uniform finite water depth with an ice-cover, the ice-cover being modelled as an elastic plate of very small thickness. This leads to an infinite system of linear equations which are solved numerically by standard techniques. The vertical and horizontal forces on the sphere are obtained and depicted graphically against the wave number for various values of the depth of water and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover and also the effect of varying depth of water on these quantities. When the flexural rigidity is taken to be zero, the numerical results exactly coincide with the curves of the vertical and horizontal forces on the sphere for the cases of uniform finite depth water with a free surface.     

梁结构耦合点的振动功率传播

本文详细讨论了多根梁耦合点的振动功率传播特性.耦合点采用刚体质量来模拟.采用波动分析法研究纵向波或弯曲波入射半无限梁结构耦合点的传播.首先根据耦合点的平衡条件和连续条件,得到了波动方程中的幅值.然后建立了波动幅值和振动功率之间的关系,并得到振动功率反射和传播系数.反射和传播系数确定为振动频率和耦台角度的函数.算例计算了二叉梁的反射系数和传播系数,分析了耦合角度和振动频率以及是否计及耦合点质量对反射系数和传播系数的影响,得到了一些有价值的结论,对于振动控制和结构优化设计具有重要的意义.     

起伏河床地形上薄的垂直物体对倾斜水面波的散射（英文）

The present study deals with the scattering of oblique surface water waves by small undulation on the bottom in the presence of a thin vertical barrier. Here, three different configurations of vertical barriers are investigated. Perturbation analysis is employed to determine the physical quantities, namely, the reflection and transmission coefficients. In this analysis, many different Boundary Value Problems(BVPs) are obtained out of which the first two bvps are considered. The zeroth order bvp is solved with the aid of eigenfunction expansion method. The first order reflection and transmission coefficients are derived in terms of the integrals by the method of the Green's integral theorem. The variation of these coefficients is plotted and analyzed for different physical parameters. Furthermore, the energy balance relation, an important relation in the study of water wave scattering, is derived and checked for assuring the correctness of the numerical results for the present problem.