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.     

关于带有小变形的多孔河床水波散射问题研究（英文）

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 ?(7)(28)1(8), 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.     

经过海峡的双层流体由底部波动产生的斜向波散射问题研究(英文)

The problem of oblique wave(internal wave) propagation over a small deformation in a channel flow consisting of two layers was considered.The upper fluid was assumed to be bounded above by a rigid lid,which is an approximation for the free surface,and the lower one was bounded below by an impermeable bottom surface having a small deformation;the channel was unbounded in the horizontal directions.Assuming irrotational motion,the perturbation technique was employed to calculate the first-order corrections of the velocity potential in the two fluids by using Green’s integral theorem suitably with the introduction of appropriate Green’s functions.Those functions help in calculating the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation.Three-dimensional linear water wave theory was utilized for formulating the relevant boundary value problem.Two special examples of bottom deformation were considered to validate the results.Consideration of a patch of sinusoidal ripples(having the same wave number) shows that the reflection coefficient is an oscillatory function of the ratio of twice the x-component of the wave number to the ripple wave number.When this ratio approaches one,the theory predicts a resonant interaction between the bed and the 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.Similar results were observed for a patch of sinusoidal ripples having different wave numbers.It was also observed that for small angles of incidence,the reflected energy is greater compared to other angles of incidence up to.These theoretical observations are supported by graphical results.     

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.     

双层流中由小型不平坦河床引起的斜向入射水波绕射研究（英文）

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.     

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.     

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.     

双层通道中可渗透底部起伏的波浪散射(英文)

The scattering of plane surface waves by bottom undulations in channel flow consisting of two layers is investigated by assuming that the bed of the channel is composed of porous material. The upper surface of the fluid is bounded by a rigid lid and the channel is unbounded in the horizontal directions. There exists only one wave mode corresponding to an internal wave. For small undulations, a simplified perturbation analysis is used to obtain first order reflection and transmission coefficients in terms of integrals involving the shape function describing the bottom. For sinusoidal bottom undulations and exponentially decaying bottom topography, the first order coefficients are computed. In the case of sinusoidal bottom the first order transmission coefficient is found to vanish identically. The numerical results are depicted graphically in a number of figures.     

Wave scattering by undulating bed topography in a two-layer ocean

The problem of wave scattering by undulating bed topography in a two-layer ocean is investigated on the basis of linear theory. In a two-layer fluid with the upper layer having a free surface, there exist two modes of waves propagating at both the free surface of the upper layer and the interface between the two layers. Due to a wave train of a particular mode incident on an obstacle which is bottom-standing on the lower layer, reflected and transmitted waves of both modes are created by the obstacle. For small undulations on the bottom of the lower layer, a perturbation method is employed to obtain first-order reflection and transmission coefficients of both modes for incident wave trains of again both modes in terms of integrals involving the bed-shape function. For sinusoidal undulations, numerical results are presented graphically to illustrate the energy transfer between the waves of different modes by the undulating bed. U. BASU was born in 1949. She is a professor in the Department Applied Mathematics, Calcutta University, India. Her current research interests include water wave problems, continuum mechanics, etc.     

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.     

水底地形变化对摇板造波的影响

In the present paper,the effect of a small bottom undulation of the sea bed in the form of periodic bed form on the surface waves generated due to a rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of non uniform finite depth is investigated.A simplified perturbation technique involving a non dimensional parameter characterizing the smallness of the bottom deformation is applied to reduce the given boundary value problem to two independent boundary value problems upto first order.The first boundary value problem corresponds to the problem of water wave generation due to rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of uniform finite depth.This is a well known problem whose solution is available in the literature.From the second boundary value problem,the first order correction to the wave amplitude at infinity is evaluated in terms of the shape function characterizing the bottom undulation,by employing Green’s integral theorem.For a patch of sinusoidal ripples at the sea bottom,the first order correction to the wave amplitude at infinity for both the configuration of the barrier is then evaluated numerically and illustrated graphically for various values of the wave number.It is observed that resonant interaction of the wave generated,with the sinusoidal bottom undulation occurs when the ratio of twice the wavelength of the sinusoidal ripple to the wave length of waves generated,approaches unity.Also it is found that the resonance increases as the length of the barrier increases.     

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.     

波浪作用下沙纹床面形态及底摩阻系数研究

波浪作用形成的沙纹在近海区域普遍存在,准确预测沙纹床面形态并计算对应的底摩阻系数是研究近岸波浪变形、泥沙输运及岸滩演变的基础。利用国外公开发表的室内试验和现场观测数据对已有计算沙纹形态的典型公式进行分析评价。考虑床面形态与水流条件的互相适应,提出新的平衡状态下沙纹长度、高度及波陡计算公式;引入临界Shields参数判别函数来考虑底床泥沙运动状态对沙纹形态的影响;通过理论推导得到波浪摩阻系数计算公式,并利用沙纹形态计算公式改进粗糙高度的计算方法。结果表明,提出的沙纹形态计算公式能够较好地刻画不同底沙运动状态下沙纹床面几何特征,临界Shields参数对于沙纹形态的计算具有重要影响;新得到的沙纹长度和高度公式可以有效地应用于波浪摩阻系数计算。     

Experimental study on reflection coefficient of curved perforated plate

A set of experiments is carried out in a towing tank to study the effects of the curvature of perforated plates on the wave reflection coefficient (C _r ). The curvature of a perforated plate can be changed by rotating a reference perforated plate aboutits origin according to the parabolic equation y=?x ². A plunger-type wave maker is used to generate regular waves. The reflection coefficients are calculated using Goda and Suzuki’s (1976) method. The results are compared with those of vertical or sloped passive wave absorbers. The comparison shows that a perforated plate with a curved profile is highly efficient in terms of reducing the wave reflection coefficient. A correlation is established to estimate the reflection coefficient of curved perforated plates as a function of both flow and geometry characteristics.     

存在底部起伏影响的近壁面处双层多孔栅栏陷波研究（英文）

Trapping of oblique surface gravity waves by dual porous barriers near a wall is studied in the presence of step type varying bottom bed that is connected on both sides by water of uniform depths. The porous barriers are assumed to be fixed at a certain distance in front of a vertical rigid wall. Using linear water wave theory and Darcy's law for flow past porous structure, the physical problem is converted into a boundary value problem. Using eigenfunction expansion in the uniform bottom bed region and modified mild-slope equation in the varying bottom bed region, the mathematical problem is handled for solution. Moreover, certain jump conditions are used to account for mass conservation at slope discontinuities in the bottom bed profile. To understand the effect of dual porous barriers in creating tranquility zone and minimum load on the sea wall, reflection coefficient, wave forces acting on the barrier and the wall, and surface wave elevation are computed and analyzed for different values of depth ratio, porous-effect parameter, incident wave angle, gap between the barriers and wall and slope length of undulated bottom. The study reveals that with moderate porosity and suitable gap between barriers and sea wall, using dual barriers an effective wave trapping system can be developed which will exert less wave force on the barriers and the rigid wall. The proposed wave trapping system is likely to be of immense help for protecting various facilities/infrastructures in coastal environment.     

筲箕背卵石急滩碍航特征以及卵石沙波形态分析

根据筲箕背滩的多年实测资料、现场踏勘资料，结合历年整治方案，分析筲箕背滩卵石沙波形态以及碍航特征。分析表明：1996年整治方案实施后，练家槽上段的“浅”得到了治理，中段的“急”得到了缓解，下段弯险的碍航特征还比较突出；筲箕背滩卵石运动强烈，卵石呈三维沙波形式向下游输移；扯船槽、新槽、老槽目前为卵石沙波的波谷，但极有可能发展成波峰；卵石沙波运动较为缓慢；练家槽以上的主航槽较为稳定。练家槽是筲箕背卵石沙波向鹭鸶碛输移的主要通道，其地形变化较大。     

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.     

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.     

波浪作用下沙波的运动速度和沙波床面的底沙输移

根据沙波床面底沙运动特点,引用二阶Stokes波浪理论,推导出波浪作用下沙波运行速度计算公式,并给出沙波床面上底沙输移的计算公式.公式计算结果与水槽试验结果吻合很好.     

对称波形底箱型水面防波堤性能试验研究

为了提高箱型水面防波堤的消波性能,将箱型水面防波堤的主体底部外形进行了改造,即将主体底部由平底改造为对称波形底,提出了一种对称波形底水面防波堤。对平底和对称波形底箱型防波堤分别进行了物理模型试验研究。试验时,将平底和对称波形底的方箱型消波主体通过支架固定在小型波浪水槽中,保持水槽水深不变,改变箱体吃水、波陡、波长,进行了系列试验。给出两种模型的透射系数、反射系数和能量损耗系数随相对宽度BL的变化规律。通过比较发现:对称波形底的消波性能比平底的优良,增加的消波效果高达60%左右;两种模型的反射系数相差无几,说明改变箱型底部结构对于防波堤反射性能的影响不大;对称波形底对于波能的损耗作用要强于平底。