An Analytical Solution for Wave Transformation Over Axi-Symmetric Topography

The present study considers wave scattering phenomena around a cylindrical island mounted on a general axisymmetric topography or a general submerged truncated axi-symmetric shoal based on the mild-slope equation. The method of separation of variables and Taylor series expansion are invoked to find the approximate solution to the variable water depth region which varies proportionally to an arbitrary power of radial distance. Validations against the solutions for the combined wave refraction and diffraction around a cylindrical island mounted on a paraboloidal shoal of Liu et al. in 2004 and the scattering and trapping of wave energy by a submerged truncated paraboloidal shoal of Lin and Liu in 2007 show excellent agreements as the power of radial distance being equal to two. For the solutions of wave refraction and diffraction around a cylindrical island mounted on a shoal with depth proportionally to an arbitrary power of radial distance, good agreements with Zhai et al.'s(2013) solutions are demonstrated. Since the robustness of the assumption of a general axi-symmetric geometry based on an arbitrary power variability of the radial distance, the present solution can be very conveniently employed to investigate the effects of bottom topography on wave scattering and trapping patterns.     

An extended analytic solution for combined refraction and diffraction of long waves over circular shoals

A new analytic solution of the mild-slope long wave equation is derived for studying the effects of bottom topography on combined refraction and diffraction. The solution is essentially of a series form involving the Bessel functions of real orders but is found to be singular as the bottom tends to be parabolic. Numerical evaluation of the solution nearby the singularity requires some special considerations. The particular solution under the singular condition is also given. Study on combined refraction and diffraction for waves around a circular island on the top of a shoal, which is radially described by a power function with two independent parameters, indicates that there exist an extremely high wave zone near the top of a hyperparabolic shoal. It is also found that the intensity of wave ray focusing increases significantly as the mean slope decreases. A direct consequence of the wave ray focusing is the concentration of wave energy and an increase of the maximal wave runup height around the island.     

A Modified Form of Mild-Slope Equation with Weakly Nonlinear Effect

Nonlinear effect is of importance to waves propagating from deep water to shallow water.Thenon-linearity of waves is widely discussed due to its high precision in application.But there are still someproblems in dealing with the nonlinear waves in practice.In this paper,a modified form of mild-slope equa-tion with weakly nonlinear effect is derived by use of the nonlinear dispersion relation and the steady mild-slope equation containing energy dissipation.The modified form of mild-slope equation is convenient to solvenonlinear effect of waves.The model is tested against the laboratory measurement for the case of a submergedelliptical shoal on a slope beach given by Berkhoff et al,The present numerical results are also comparedwith those obtained through linear wave theory.Better agreement is obtained as the modified mild-slope e-quation is employed.And the modified mild-slope equation can reasonably simulate the weakly nonlinear ef-fect of wave propagation from deep water to coast.     

New Numerical Scheme for Simulation of Hyperbolic Mild-Slope Equation

The original hyperbolic mild-slope equation can effectively take into account the combined effects of wave shoaling, refraction, diffraction and reflection, but does not consider the nonlinear effect of waves, and the existing numerical schemes for it show some deficiencies. Based on the original hyperbolic mild-slope equation, a nonlinear dispersion relation is introduced in present paper to effectively take the nonlinear effect of waves into account and a new numerical scheme is proposed. The weakly nonlinear dispersion relation and the improved numerical scheme are applied to the simulation of wave transformation over an elliptic shoal. Numerical tests show that the improvement of the numerical scheme makes efficient the solution to the hyperbolic mild-slope equation. A comparison of numerical results with experimental data indicates that the results obtained by use of the new scheme are satisfactory.     

Linear wave reflection by trench with various shapes

Two types of analytical solutions for waves propagating over an asymmetric trench are derived. One is a long-wave solution and the other is a mild-slope solution, which is applicable to deeper water. The water depth inside the trench varies in proportion to a power of the distance from the center of the trench (which is the deepest water depth point and the origin of x-coordinate in this study). The mild-slope equation is transformed into a second-order ordinary differential equation with variable coefficients based on the longwave assumption [Hunt's, 1979. Direct solution of wave dispersion equation. Journal of Waterway, Port, Coast. and Ocean Engineering 105, 457–459] as approximate solution for wave dispersion. The analytical solutions are then obtained by using the power series technique. The analytical solutions are compared with the numerical solution of the hyperbolic mild-slope equations. After obtaining the analytical solutions under various conditions, the results are analyzed.     

Experimental verification of horizontal two-dimensional modified mild-slope equation model

In order to verify modified mild-slope equation models in a horizontal two-dimensional space, a hydraulic experiment is made for surface wave propagation over a circular shoal on which water depth varies substantially. A horizontal two-dimensional numerical model is also constructed based on the hyperbolic equations that have been developed from the modified mild-slope equation to account for the substantial depth variation. Comparison between experimental measurements and numerical results shows that the modified mild-slope equation model is capable of producing accurate results for wave propagation in a region where water depth varies substantially, while the conventional mild-slope equation model gives large errors as the mild-slope assumption is violated.     

Wave refraction–diffraction effect in the wind wave model WWM

Based on the extended mild-slope equation, the wind wave model (WWM; Hsu et al., 2005) is modified to account for wave refraction, diffraction and reflection for wind waves propagating over a rapidly varying seabed in the presence of current. The combined effect of the higher-order bottom effect terms is incorporated into the wave action balance equation through the correction of the wavenumber and propagation velocities using a refraction–diffraction correction parameter. The relative importance of additional terms including higher-order bottom components, the wave–bottom interaction source term and wave–current interaction that influence the refraction–diffraction correction parameter is discussed. The applicability of the proposed model to calculate a wave transformation over an elliptic shoal, a series of parallel submerged breakwater induced Bragg scattering and wave–current interaction is evaluated. Numerical results show that the present model provides better predictions of the wave amplitude as compared with the phase-decoupled model of Holthuijsen et al. (2003).     

A finite element model for wave refraction and diffraction

A two-dimensional hybrid finite element method is developed to study the scattering of water waves by an island and to calculate wave forces and moments on offshore structures. The offshore structure, which could be either semi-submerged or fully extended in the water, is assumed to be stationary. The numerical model is based on the mild-slope equation. It can be applied to both long-wave and short-wave problems. A special treatment for the problem with the semi-submerged structure is introduced. Comparisons are given with existing analytical solutions and other numerical results. The present model is shown to be an efficient and accurate method for the solution of wave refraction and diffraction problems.     

A numerical model for combined refraction-diffraction of short waves on an island

A combined wave refraction-diffraction numerical model was developed to predict wave conditions around an arbitrary island. The methodology was based on the mild-slope equation, solved using a finite difference scheme with a marching procedure. The new model reduced the computer's memory demand considerably in comparison with finite-element, parabolic, error vector propagation and other finite difference approaches, and could therefore predict wave conditions for a large coastal area under given offshore boundary-wave conditions. Laboratory data on wave conditions under submerged circular and elliptical shoal conditions were selected to validate the numerical results. Good agreement was observed in all cases. Wave characteristics around an island were predicted using this model with the given deep-water wave condition. The model can predict wave conditions for any island with a mild-slope coastline.     

A note on the accuracy of the mild-slope equation

The mild-slope equation is a vertically integrated refraction-diffraction equation, used to predict wave propagation in a region with uneven bottom. As its name indicates, it is based on the assumption of a mild bottom slope. The purpose of this paper is to examine the accuracy of this equation as a function of the bottom slope. To this end a number of numerical experiments is carried out comparing solutions of the three-dimensional wave equation with solutions of the mild-slope equation.For waves propagating parallel to the depth contours it turns out that the mild-slope equation produces accurate results even if the bottom slope is of order 1. For waves propagating normal to the depth contours the mild-slope equation is less accurate. The equation can be used for a bottom inclination up to 1:3.     

波浪在三维圆形岛地形上的绕射研究

波浪在传播过程中遇到岛屿就会发生绕射。本文使用混合元方法对修正型缓坡方程进行了数值求解,并与KUO et al的解析解进行了比较验证。在此基础上研究了工程尺度背景下,波浪在三维圆形岛地形上的绕射,计算了不同入射波浪周期、浅滩形状参数和岛屿尺寸情况下,沿波浪传播方向断面上和岛屿岸线上的相对波高大小。计算结果表明：随着入射波周期的减小、浅滩形状参数的增大和岛屿尺寸的减小,圆形岛迎浪侧的相对波高振荡幅度、圆形岛背浪侧的相对波高大小以及岛屿岸线上的相对波高振幅和大小均随之增大。不同情况下,岛屿岸线上的相对波高最大值大多数发生在迎浪点,个别发生在迎浪点两侧20°~25°处;最小值发生在背浪点两侧30°附近。     

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.     

Numerical implementation of the harmonic modified mild-slope equation

A numerical solver is presented of the modified time-independent mild-slope equation, which incorporates energy dissipation. Using a second-order parabolic approximation, the following external boundary conditions are modelled: open and fully transmitting to both incoming and outgoing waves; partially reflecting, and; fully absorbing. Discretisation of the governing equation and boundary conditions is by means of a second-order accurate central difference scheme. The resulting sparse-banded matrix is solved using an inexpensive banded solver with Gaussian elimination. The numerical predictions are in excellent agreement with the analytical solution for the interaction of non-breaking waves with an array of vertical surface-piercing circular cylinders on a horizontal bed. Results are compared with those for the same array on various seabed topographies. The model is robust and can be used for wave propagation in complex geometries. It has fewer restrictions associated with wave obliqueness at boundaries than traditional models based on the mild-slope equation.     

A time-dependent numerical model of the mild-slope equation

On the basis of the previous studies, the simplest hyperbolic mild-slope equation has been gained and the linear time-dependent numerical model for the water wave propagation has been established combined with different boundary conditions. Through computing the effective surface displacement and transforming into the real transient wave motion, related wave factors will be calculated. Compared with Lin’s model, analysis shows that calculation stability of the present model is enhanced efficiently, because the truncation errors of this model are only contributed by the dissipation terms, but those of Lin’s model are induced by the convection terms, dissipation terms and source terms. The tests show that the present model succeeds the merit in Lin’s model and the computational program is simpler, the computational time is shorter, and the computational stability is enhanced efficiently. The present model has the capability of simulating transient wave motion by correctly predicting at the speed of wave propagation, which is important for the real-time forecast of the arrival time of surface waves generated in the deep sea. The model is validated against analytical solution for wave diffraction and experimental data for combined wave refraction and diffraction over a submerged elliptic shoal on a slope. Good agreements are obtained. The model can be applied to the theory research an d engineering applications about the wave propagation in a biggish area.     

Effect of higher-order bottom variation terms on the refraction of water waves in the extended mild-slope equations

This study investigates how the refraction of water waves is affected by the higher-order bottom effect terms proportional to the square of bottom slope and to the bottom curvature in the extended mild-slope equations. Numerical analyses are performed on two cases of waves propagating over a circular shoal and over a circular hollow. Numerical results are analyzed using the eikonal equation derived from the wave equations and the wave ray tracing technique. It is found that the higher-order bottom effect terms change the wavelength and, in turn, change the refraction of waves over a variable depth. In the case of waves over a circular shoal, the higher-order bottom effects increase the wavelength along the rim of shoal more than near the center of shoal, and intensify the degree of wave refraction. However, the discontinuity of higher-order bottom effects along the rim of shoal disperses the foci of wave rays. As a result, the amplification of wave energy behind the shoal is reduced. Conversely, in the case of waves over a circular hollow, the higher-order bottom effects decrease the wavelength near the center of the hollow in comparison with the case of neglecting higher-order bottom effects. Consequently, the degree of wave refraction is decreased, and the spreading of wave energy behind the hollow is reduced.     

Efficient Numerical Solution of the Modified Mild-Slope Equation

An efficient numerical model for wave refraction,diffraction and reflection is presented in thispaper.In the model,the modified time-dependent mild-slope equation is transformed into an evolutionequation and an improved ADI method involving a relaxation factor is adopted to solve it.The methodhas the advantage of improving the numerical stability and convergence rate by properly determining therelaxation factor.The range of the relaxation factor making the differential scheme unconditionally stableis determined by stability analysis.Several verifications are performed to examine the accuracy of the pres-ent model.The numerical results coincide with the analytic solutions or experimental data very well,andthe computer time is reduced.     

An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.     

修正型缓坡方程的有限元模型

与缓坡方程相比,修正型缓坡方程增加了地形曲率项和坡度平方项,从而提高了数值求解的复杂性。本文将计算域划分为内域和外域,内域为水深变化区域,使用修正型缓坡方程,其中的地形曲率项和坡度平方项可用有限单元各节点的水深信息和单元插值函数表示,外域为水深恒定区,速度势满足Helmholtz方程,通过内外域的边界匹配建立有限元方程,并用高斯消去法求解。进而分别模拟了波浪传过Homma岛和圆形浅滩的变形,其结果与相关的解析解和实验数据吻合良好,证明了本文有限元模型的正确性。同时,通过与实验数据的对比也明显看出,在地形坡度较陡的情况下,修正型缓坡方程较缓坡方程具有更高的计算精度。     

抛物型缓坡方程的变分及数值模拟

对线性水波的折射一绕射问题应用变分原理,对非等深、具有缓坡和不连续的底被导出了一种修改的抛物型缓坡方程近似模型,可预测三维地形上波浪的折射一绕射。同抛物型缓坡方程的线性方程进行了对比。通过数值模拟方法进行数值求解,表明本方法可用于地形条件下的波浪折射一绕射问题。     

Wave transformation over submerged permeable breakwater on porous bottom

A numerical model is presented in this study to investigate the wave transformation over a submerged permeable breakwater on a porous slope seabed. For this purpose, the time-dependent mild-slope equation is newly derived for waves propagating over two layers of porous medium. This new mild-slope equation involves the parameters of the porous medium, and it is a type of hyperbolic differential equation, therefore numerically efficient. The validity of the present model is verified based on the comparisons with the previous experiments. The effects of the permeable properties of both the porous seabed and the submerged permeable breakwater are discussed in detail. The geometry of the submerged permeable breakwater to the wave transformation is also investigated based on the numerical solutions.