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.

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.

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.

WATER WAVE SIMULATION IN CURVILINEAR COORDINATES USING A TIME-DEPENDENT MILD SLOPE EQUATION

The purpose of this article is to model the detailed progress of wave propagation in curvilinear coordinates with an effective time-dependent mild slope equation. This was achieved in the following approach, firstly deriving the numerical model of the equation, i.e., Copeland＇s hyperbolic mild-slope equation, in orthogonal curvilinear coordinates based on principal of coordinate transformation, and then finding the numerical solution of the transformed model by use of the Alternative Directions Implicit （ADI） method with a space-staggered grid. To test the curvilinear model, two cases of a channel with varying cross section and a semi-circular channel were studied with corresponding analytical solutions. The model was further investigated through a numerical simulation in Ponce de Leon Inlet, USA. Good agreement is reached and therefore, the use of the present model is valid to calculate the progress of wave propagation in areas with curved shorelines, nearshore breakwaters and other complicated geometries.

PARALLEL SIMULATION OF THE ELLIPTIC MILD SLOPE EQUATION WITH A PERSONAL COMPUTER CLUSTER

With the massage passing interface, a parallel solution method was proposedfor the simulation of the elliptic mild slope equation, and implemented numerically on a parallelsystem based on a personal computer cluster, which was constructed by the authors. The wavetransformations over two typical topographies with mild slopes were simulated. Numerical resultsshow that the parallel solution method presented in this paper can not only increase thecomputational efficiency, but also decrease very much the memory storage on a single computer, sothe parallel system based on a PCC can be used to simulate wave transformations over much largerareas.

Numerical Simulation of Non-Linear Wave Propagation in Waters of Mildly Varying Topography with Complicated Boundary

In this paper, the characteristics of different forms of mild slope equations for non-linear wave are analyzed, and new non-linear theoretic models for wave propagation are presented, with non-linear terms added to the mild slope equations for non-stationary linear waves and dissipative effects considered. Numerical simulation models are developed of non-linear wave propagation for waters of mildly varying topography with complicated boundary, and the effects are studied of different non-linear corrections on calculation results of extended mild slope equations. Systematical numerical simulation tests show that the present models can effectively reflect non-linear effects.

Numerical Simulation for Refraction-Diffraction of Waves in Water of Slowly Varying Current and Topography

A new numerical finite difference iteration method for refraction-diffraction of waves ia water of slowly varying current and topography is developed in this paper. And corresponding theoretical model including the dissipation term is briefly described, together with some analysis and comparison of computational results of the model with measurements in a hydraulic scale model (Berkhoff et al., 1982). An example of practical use of the method is given, showing that the present model is useful to engineering practice.

Nonlinear Dispersion Effect on Wave Transformation

A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better approximation to Hedges' empirical relation than the modified relations by Hedges (1987). Kirby and Dalrymple (1987) for shallow waters. The new dispersion relation is simple in form, thus it can be used easily in practice. Meanwhile, a general explicit approximation to the new dispersion and other and other nonlinear dispersion relations is given. By use of the explicit approximation to the new dispersion relation along with the mild slope equation taking into account weakly nonlinenr effect, a mathematical model is obtained, and it is applied to laboratory data. The results show that the model developed with the new dispersion relation predicts wave transformation over complicated topography quite well.

A composite numerical model for wave diffraction in a harbor with varying water depth

A composite numerical model is presented for computing the wave field in a harbor. The mild slope equation is discretized by a finite element method in the domain concerned. Out of the computational domain, the water depth is assumed to be constant. The boundary element method is applied to the outer boundary for dealing with the infinite boundary condition. Because the model satisfies strictly the infinite boundary condition, more accurate results can be obtained. The model is firstly applied to compute the wave diffraction in a narrow rectangular bay and the wave diffraction from a porous cylinder. The numerical results are compared with the analytical solution, experimental data and other numerical results. Good agreements are obtained. Then the model is applied to computing the wave diffraction in a square harbor with varying water depth. The effects of the water depth in the harbor and the incoming wave direction on the wave height distribution are discussed.

Nonlinear Dispersion Relation in Wave Transformation

A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over prediction of both Hedges′ modified relation and Kirby and Dalrymple′s modified relation in the region of 1< kh <1 5 for small wave steepness and maintains the monotonicity in phase speed variation for large wave steepness. And it has a simple form. By use of the new nonlinear dispersion relation along with the mild slope equation taking into account weak nonlinearity, a mathematical model of wave transformation is developed and applied to laboratory data. The results show that the model with the new dispersion relation can predict wave transformation over complicated bathymetry satisfactorily.

Verification of A Numerical Harbour Wave Model

A numerical model for wave propagation in a harbour is verified by use of physical models. The extended time-dependent mild slope equation is employed as the governing equation, and the model is solved by use of ADI method containing the relaxation factor. Firstly, the reflection coefficient of waves in front of rubble-mound breakwaters under oblique incident waves is determined through physical model tests, and it is regarded as the basis for simulating partial reflection boundaries of the numerical model. Then model tests on refraction, diffraction and reflection of waves in a harbour are performed to measure wave height distribution. Comparative results between physical and numerical model tests show that the present numerical model can satisfactorily simulate the propagation of regular and irregular waves in a harbour with complex topography and boundary conditions.

A WEAKLY NONLINEAR WATER WAVE MODEL TAKING INTO ACCOUNT DISPERSION OF WAVE PHASE VELOCITY

This paper presents a weakly nonlinear water wave model using a mild slope equation and a new explicit formulation which takes into account dispersion of wave phase velocity, approximates Hedges’ (1987) nonlinear dispersion relationship, and accords well with the original empirical formula. Comparison of the calculating results with those obtained from the experimental data and those obtained from linear wave theory showed that the present water wave model considering the dispersion of phase velocity is rational and in good agreement with experiment data.

Parabolic Wave Propagation Modelling in Orthogonal Coordinate Systems

This paper presents a numerical model study of the propagation of water waves using the parabolic approximation of the mild slope equation in the orthogonal coordinate system. Two types of coordinate systems are studied: (a) a general form of orthogonal coordinate system and (b) the conformal system, a special form of orthogonal coordinate system. Two typical examples, namely, expanded breakwaters and a circular channel, are studied to validate the model. First, the examples are studied by use of the general orthogonal coordinates. Then the same examples are computed by use of the conformal system. The computational results show that the conformal coordinate system generally gives better predictions than the general orthogonal system. A numerical technique for generating the conformal grid is combined with the numerical model to improve the practicability of the model. The comparison between the result from the numerical grid system and that from the analytical grid system shows that reliable computational results can be obtained by use of the numerical conformal grid system.

A LINEAR HYBRID MODEL OF MSE AND BEM FOR FLOATING STRUCTURES IN COASTAL ZONES

A linear hybrid model of Mild Slope Equation （MSE） and Boundary Element Method （BEM） is developed to study the wave propagation around floating structures in coastal zones. Both the wave refraction under the influence of topography and the wave diffraction by floating structures are considered. Hence, the model provides wave properties around the coastal floating structures of arbitrary shape but also the wave forces on and the hydrodynamic characteristics of the structures. Different approaches are compared to demonstrate the validity of the present hybrid model. Several numerical tests are carried out for the cases of pontoons under different circumstances. The results show that the influence of topography on the hydrodynamic characteristics of floating structures in coastal regions is important and must not be ignored in the most wave period range with practical interests.