The New Form of Time-Dependent Mild Slope Equation for Random Waves

The mild-slope equation derived by Berkhoff (1972), has widely been used in the numerical calculation of refraction and diffraction of regular waves. However, it is well known that the random sea waves has a significant effect in the refraction and diffraction problems. In this paper, a new form of time-dependent mild slope equation for irregular waves was derived with Fade approximation and Kubo's time series concept. The equation was simplified using WKB method, and simple and practical irregular mild slope equation was obtained. Results of numerical calculations are compared with those of laboratory experiments.

Hyperbolic Mild Slope Equations with Inclusion of Amplitude Dispersion Effect:Regular Waves

A new form of hyperbolic mild slope equations is derived with the inclusion of the amphtude dispersion of nonlinear waves. The effects of including the amplitude dispersion effect on the wave propagation are discussed. Wave breaking mechanism is incorporated into the present model to apply the new equations to surf zone. The equations are solved nu- merically for regular wave propagation over a shoal and in surf zone, and a comparison is made against measurements. It is found that the inclusion of the amplitude dispersion can also improve model＇ s performance on prediction of wave heights around breaking point for the wave motions in surf zone.

Hyperbolic Mild Slope Equations with Inclusion of Amplitude Dispersion Effect:Random Waves

New hyperbolic mild slope equations for random waves are developed with the inclusion of amplitude dispersion. The frequency perturbation around the peak frequency of random waves is adopted to extend the equations for regular waves to random waves. The nonlinear effect of amplitude dispersion is incorporated approximately into the model by only considering the nonlinear effect on the carrier waves of random waves, which is done by introducing a representative wave amplitude for the carrier waves. The computation time is gready saved by the introduction of the representative wave amplitude. The extension of the present model to breaking waves is also considered in order to apply the new equations to surf zone. The model is validated for random waves propagate over a shoal and in surf zone against measurements.

Numerical Simulation of Breaking Wave Based on Higher-Order Mild Slope Equation

The 'surface roller' to simulate wave energy dissipation of wave breaking is introduced into the random wave model based on approximate parabolic mild slope equation in this paper to simulate the random wave transportation in chiding diffraction, refraction and breaking in nearshore areas. The roller breaking random wave higher-order approximate parabolic equation model has been verified by the existing experimental data for a plane slope beach and a circular shoal, and the numerical results of random wave breaking model agree with the experimental data very well, This model can be applied to calculate random wave propagation from deep to shallow water in large areas near the shore over natural topography.

Efficient Numerical Solution of the Modified Mild-Slope Equation

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

Numerical Modeling of the Hyperbolic Mild-Slope Equation in Curvilinear Coordinates

The mild-slope equation is familiar to coastal engineers as it can effectively describe wave propagation in nearshore regions. However, its computational method in Cartesian coordinates often renders the model inaccurate in areas with irregular shorelines, such as estuaries and harbors. Based on the hyperbolic mild-slope equation in Cartesian coordinates, the numerical model in orthogonal curvilinear coordinates is developed. The transformed model is discretized by the finite difference method and solved by the ADI method with space-staggered grids. The numerical predictions in curvilinear co- ordinates show good agreemenl with the data obtained in three typical physical expedments, which demonstrates that the present model can be used to simulate wave propagation, for normal incidence and oblique incidence, in domains with complicated topography and boundary conditions.

Numerical Simulation of Wave-Induced Currents Combined with Parabolic Mild-Slope Equation in Curvilinear Coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.

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. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in dealing with the nonlinear waves in practice. In this paper, a modified form of mild-slope equation 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 solve nonlinear effect of waves. The model is tested against the laboratory measurement for the case of a submerged elliptical shoal on a slope beach given by Berkhoff et al. The present numerical results are also compared with those obtained through linear wave theory. Better agreement is obtained as the modified mild-slope equation is employed. And the modified mild-slope equation can reasonably simulate the weakly nonlinear effect 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.

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

A time-dependent mild-slope equation for the extension of the classic mild-slope equation of Berkhoff is developed for the interactions of large ambient currents and waves propagating over an uneven bottom, using a Hamiltonian formulation for irrotational motions. The bottom topography consists of two components the slowly varying component which satisfies the mild-slope approximation, and the fast varying component with wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter describing the mild-slope condition. The theory is more widely applicable and contains as special cases the following famous mild-slope type equations: the classical mild-Slope equation, Kirby's extended mild-slope equation with current, and Dingemans's mild-slope equation for rippled bed. Finally, good agreement between the classic experimental data concerning Bragg reflection and the present numerical results is observed.

Calculation of Wave Radiation Stress in Combination with Parabolic Mild Slope Equation

A new method for the calculation of wave radiation stress is proposed by linking the expressions for wave radiation stress with the variables in the parabolic mild slope equation. The governing equations are solved numerically by the finite difference method. Numerical results show that the new method is accurate enough, can be efficiently solved with little programming effort, and can be applied to the calculation of wave radiation stress for large coastal areas.

A Finite-Difference Approach to the Time-Dependent Mild-Slope Equation

A finite-difference approach is used to develop a time-dependent mild-slope equation incorporating the effects of bottom dissipation and nonlinearity. The Enler predietor-corrector method and the three-point finite-difference method with varying spatial steps are adopted to discretize the time derivatives and the two-dimensional horizontal ones, respectively, thus leading both the time and spatial derivatives to the second-order accuracy. The boundary conditions for the present model are treated on the basis of the general conditions for open and fixed boundaries with an arbitrary reflection coefficient and phase shift. Both the linear and nonlinear versions of the numerical model are applied to the wave propagation and transformation over an elliptic shoal on a sloping beach, respectively, and the linear version is applied to the simulation of wave propagation in a fully open rectangular harbor. From comparison of numerical results with theoretical or experimental ones, it is found that they are in reasonable agreement.

An Extended Mild-Slope Equation

On the assumption that the vortex and the vertical velocity component of the current are small, a mild-slope equation for wave propagation on non-uniform flows is deduced from the basic hydrodynamic equations, with the terms of (V(h)h)(2) and V(h)(2)h included in the equation. The terms of bottom friction, wind energy input and wave nonlinearity are also introduced into the equation. The wind energy input functions for wind waves and swells are separately considered by adopting Wen's (1989) empirical formula for wind waves and Snyder's observation results for swells. Thus, an extended mild-slope equation is obtained, in which the effects of refraction, diffraction, reflection, current, bottom friction, wind energy input and wave nonlinearity are considered synthetically.

Comparison between characteristics of mild slope equations and Boussinesq equations

Boussinesq-type equations and mild-slope equations are compared in terms of their basic forms and characteristics. It is concluded that linear mild-slope equations on dispersion relation are better than non-linear Boussinesq equations. In addition, Berkhoffexperiments are computed and compared by the two models, and agreement between model results and available experimental data is found to be quite reasonable, which demonstrates the two models＇ capacity to simulate wave transformation. However they can deal with different physical processes respectively, and they have their own characteristics.

Tide simulation using the mild-slope equation with Coriolis force and bottom friction

Since the mild-slope equation was derived by Berkhoff （1972）,the researchers considered various mechanism to simplify and improve the equation,which has been widely used for coastal wave field calculation.Recently,some scholars applied the mild-slope equation in simulating the tidal motion,which proves that the equation is capable to calculate the tide in actual terrain.But in their studies,they made a lot of simplifications,and did not consider the effects of Coriolis force and bottom friction on tidal wave.In this paper,the first-order linear mild-slope equations are deduced from Kirby mild-slope equation including wave and current interaction.Then,referring to the method of wave equations’ modification,the Coriolis force and bottom friction term are considered,and the effects of which have been performed with the radial sand ridges topography.Finally,the results show that the modified mild-slope equation can be used to simulate tidal motion,and the calculations agree well with the measurements,thus the applicability and validity of the mild-slope equation on tidal simulation are further proved.

High Order Numerical Code for Hyperbolic Mild-slope Equations with Nonlinear Dispersion Relation

Based on the hyperbolic mild-slope equations derived by Copeland (1985), a numerical model is established in unstag- gered grids. A composite 4 th-order Adam-Bashforth-Moulton (ABM) scheme is used to solve the model in the time domain. Terms involving the first order spatial derivatives are differenced to O ( Δx )4accuracy utilizing a five-point formula. The nonlinear dispersion relationship proposed by Kirby and Dalrymple (1986) is used to consider the nonlinear effect. A numerical test is performed upon wave propagating over a typical shoal. The agreement between the numerical and the experimental results validates the present model. Biodistribution and applications are also summarized.

EXTENDED MILD-SLOPE EQUATION

The Hamiltonian formalism for surface waves and the mild-slope approximation were empolyed in handling the case of slowly varying three-dimensional currents and an uneven bottom, thus leading to an extended mild-slope equation. The bottom topography consists of two components: the slowly varying component whose horizontal length scale is longer than the surface wave length, and the fast varying component with the amplitude being smaller than that of the surface wave. ne frequency of the fast varying depth component is, however, comparable to that of the surface waves. The extended mild-slope equation is more widely applicable and contains as special cases famous mild-slope equations below: the classical mild-slope equation of Berkhoff, Kirby's mild-slope equation with current, and Dingemans's mild-slope equation for rippled bed. The extended shallow water equations for ambient currents and rapidly varying topography are also obtained.

An extended time-dependent numerical model of the mild-slope equation with weakly nonlinear amplitude dispersion

In the present paper, by introducing the effective wave elevation, we transform the extended elliptic mild-slope equation with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simplest time-dependent hyperbolic equation. Based on this equation and the empirical nonlinear amplitude dispersion relation proposed by Li et al. （2003）, the numerical scheme is established. Error analysis by Taylor expansion method shows that the numerical stability of the present model succeeds the merits in Song et al. （2007）＇s model because of the introduced dissipation terms. For the purpose of verifying its performance on wave nonlinearity, rapidly varying topography and wave breaking, the present model is applied to study： （1） wave refraction and diffraction over a submerged elliptic shoal on a slope （Berkhoff et al., 1982）; （2） Bragg reflection of monochromatic waves from the sinusoidal ripples （Davies and Heathershaw, 1985）; （3） wave transformation near a shore attached breakwater （Watanabe and Maruyama, 1986）. Comparisons of the numerical solutions with the experimental or theoretical ones or with those of other models （REF/DIF model and FUNWAVE model） show good results, which indicate that the present model is capable of giving favorably predictions of wave refraction, diffraction, reflection, shoaling, bottom friction, breaking energy dissipation and weak nonlinearity in the near shore zone.

Numerical solutions for two nonlinear wave equations

The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is modified at its second step by avoiding transforming the wave height function into a frequency domain function. Thus, the periodic boundary condition is not required, and the new method is easy to implement. In order to validate its performance, the proposed method was used to solve the nonlinear parabolic mild-slope equation and the spatial modified nonlinear Schrodinger （MNLS） equation, which were used to model the wave propagation under different bathymetric conditions. Good agreement between the numerical and experimental results shows that the present method is effective and efficient in solving nonlinear wave eouations.

ANALYSIS OF SLOPING ELASTIC PILE UNDER ARBITRARY LOADS BY LINE-LOADED INTEGRAL EQUATION METHOD

For analysis of displacement and stress, an elastic sloping pile embedded in a homogeneous isotropic elastic half space under arbitrary loads at the top can be decomposed into two plane systems, i.e., the inclined plane xOz and its normal plane yOz . Let Mindlin's forces be the fundamental loads with unknown intensity function X(t),Y(t),Z(t) ,parallel to x,y,z_axis respectively, be distributed along the t axis of the pile in and concentrated forces Q x,Q y,Z ,couples M y,M x at top of the pile. Then, according to the boundary conditions of elastic pile, the problem is reduced to a set of Fredholm_Volterra type equations. Numerical solution is given and the accuracy of calculation can be checked by the reciprocal theorem of work.