A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking—Reply to discussion by T.S. Hedges

The equations of Hedges [Hedges, T.S., 2009. Discussion of “A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking” by J.P. Le Roux, Coastal Eng.], although yielding similar wavelengths, are not consistent with the fact that the horizontal water particle velocity in the wave crest should equal the wave celerity at breaking over a nearly horizontal bottom.     

Response to reply by J.P. Le Roux

In the response given by Le Roux [Le Roux, J.P., 2008. A simple method to determine breaker height and depth for different deepwater wave height/length ratios and sea floor slopes — Reply to discussion by M.C. Haller and P.C. Catalan, Coast. Eng. 55, 185–188] to the discussion of Haller and Catalán [Haller, M.C., Catalan, P.A., 2008. Discussion of “A simple method to determine breaker height and depth for different deepwater wave height/length ratios and sea floor slopes”, by J.P. Le Roux [Coastal Engineering 54 (2007) 271–277], Coast. Eng. 55, 181–184], the author presents a defense of the large number of inconsistencies/errors that we pointed out in regards to the earlier work of Le Roux [Le, Roux, J.P., 2007. A simple method to determine breaker height and depth for different wave height/length ratios and sea floor slopes, Coast. Eng. 54, 271–277]. We appreciate the response for the fact that it further clarifies the lines of reasoning used in the previous work. Unfortunately, we are not convinced by the defenses offered and still posit that the original work contains many inconsistencies and downright calculation errors. We try to avoid repetition herein, and instead of rehashing all of the points made in our previous discussion, we will concentrate on a few fundamental problems that undermine the whole premise of the original paper. We feel it is important to make these clear to the readers of Coastal Engineering.     

WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

The generation and growth of waves in deep water is controlled by winds blowing over the sea surface. In fully developed sea states, where winds and waves are in equilibrium, wave parameters may be calculated directly from the wind velocity. We provide an Excel spreadsheet to compute the wave period, length, height and celerity, as well as horizontal and vertical particle velocities for any water depth, bottom slope, and distance below the reference water level. The wave profile and propagation can also be visualized for any water depth, modeling the sea surface change from sinusoidal to trochoidal and finally cnoidal profiles into shallow water. Bedload entrainment is estimated under both the wave crest and the trough, using the horizontal water particle velocity at the top of the boundary layer. The calculations are programmed in an Excel file called WAVECALC, which is available online to authorized users. Although many of the recently published formulas are based on theoretical arguments, the values agree well with several existing theories and limited field and laboratory observations. WAVECALC is a user-friendly program intended for sedimentologists, coastal engineers and oceanographers, as well as marine ecologists and biologists. It provides a rapid means to calculate many wave characteristics required in coastal and shallow marine studies, and can also serve as an educational tool.     

Interaction of cnoidal waves with an array of vertical concentric porous cylinders

An array of large concentric porous cylinder arrays is mounted in shallow water exposed to cnoidal waves. The interactions between waves and cylinders are studied theoretically using an eigenfunction expansion approach. Semi-analytical solutions of hydrodynamic loads and wave run-up on each cylinder are obtained using first approximation to cnoidal waves. The square array configuration of four-legged identical concentric porous cylinder is investigated in present study. Numerical results reveal the variation of dimensionless wave force and wave run-up on individual cylinder with angle of incidence, porosity parameter, spacing between outer and inner cylinders, spacing between concentric porous cylinders and wave parameter. Different mechanism of wave force is found under different range of scattering parameter.     

The Nonlinear Wave Force on a Circular Cylinder in Shallow Water

Based on the 1st order cnoidal wave theory, the nonlinear wave diffraction around a circular cylinder in shallow water is studied in this paper. The equation of the wave surface around the cylinder is formulated and by using this formula the wave surface elevation on the cylinder surface can be obtained. In this paper, the formula for calculating the cnoidal wave force on a circular cylinder is also derived. For the wave conditions which are often encountered in practical engineering designs, the ratios of the nonlinear wave forces to the linear wave forces are calculated, and the results are plotted in this paper for design purposes. In order to verify the theoretical results, model tests are conducted. After comparing the test results with the theoretical ones, it is concluded that, in shallow water, for the case of T g / d~(1/2) > 8-10 and H / d > 0.3, the cnoidal wave theory should be used to calculate the wave action on a cylindrical pier.     

Unsteady ship waves in shallow water of varying depth based on Green–Naghdi equation

Based on Green–Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green–Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green–Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green–Naghdi equation. The numerical examples of a Series 60 (C_B=0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green–Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green–Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow , and the Green–Naghdi equation and the shallow water equations predict almost the same wave resistance when , the frequency dispersion can be neglected in supercritical flows.     

Generalized boussinesq model for periodic non-linear shallow-water waves

This paper presents the development of a generalized Boussinesq (gB) model for the periodic non-linear shallow-water waves. An incident cnoidal wave solution for the gB model is derived and applied to the wave simulation. A set of radiation boundary conditions is also established to transmit effectively the cnoidal waves out of the computational domain. The classical solutions of the second-order cnoidal waves are discussed within the content of the KdV equation and the generalized Boussinesq equations. An Euler's predictor-corrector finite-difference algorithm is used for numerical computation. The propagation of normally incident cnoidal waves in a channel is studied. The simulated wave profiles agree well with the analytical results. The temporal and spatial evolution of an obliquely incident cnoidal wave is also modelled. The phenomenon of Mach reflection is discussed.     

一种新型二阶全非线性Boussinesq方程及其数值验证

通过改进二阶全非线性 Boussinesq 波浪方程中的色散项,得到了一组没有改变原方程的数学形式但适用于更大变化水深的新方程,其色散性能和变浅性能都比原方程有了很大改进,所适用的水深范围更大,能更好地描述从深水到近岸浅水处的波浪传播;并基于新方程建立了波浪数值模型,通过模拟波浪从浅水到深水的传播变形来验证新方程的有效性.     

Numerical simulation of wave-induced laminar boundary layers

The three-dimensional numerical model with σ-coordinate transformation in the vertical direction is applied to the simulation of surface water waves and wave-induced laminar boundary layers. Unlike most of the previous investigations that solved the simplified one-dimensional boundary layer equation of motion and neglected the interaction between boundary layer and outside flow, the present model solves the full Navier–Stokes equations (NSE) in the entire domain from bottom to free surface. A non-uniform mesh system is used in the vertical direction to resolve the thin boundary layer. Linear wave, Stokes wave, cnoidal wave and solitary wave are considered. The numerical results are compared to analytical solutions and available experimental data. The numerical results agree favorably to all of the experimental data. It is found that the analytical solutions are accurate for both linear wave and Stokes wave but inadequate for cnoidal wave or solitary wave. The possible reason is that the existing analytical solutions for cnoidal and solitary waves adopt the first-order approximation for free stream velocity and thus overestimate the near bottom velocity. Besides velocity, the present model also provides accurate results for wave-induced bed shear stress.     

驱动非线性浅水波的行波特征研究

采用带有外界强迫效应的浅水动力学模式研究非线性波动、获得了依赖于外界输入形式的驱动水波的行波解。研究结果表明,驱动水波仍具有非线性波动的一般性质,而当外界强迫波速与水波固有速度一致时,水波出现共振效应,并且外界强迫孤立子将导致驱动水波孤立子产生。     

Tsunami Wave Interaction with Data Buoys

To plan for proper mitigation measures, one should have an advanced knowledge of the phenomenon of tsunami propagation from the deep ocean to coastal waters. There are a few methods to predict tsunamis in the ocean waters; one method is the effective use of data buoy measurements. Although data buoys have been used along the Indian waters there has been a tremendous growth in the number of buoy deployment recently. Under the National Data Buoy Programme (NDBP) of India, the 2.2 m diameter discus data buoys were deployed along the east and west coasts of India for measuring meteorological and ocean parameters. It would be advantageous if these buoys could be efficiently used to measure rare events such as tsunamis. Understanding the dynamic behavior of the buoy is of prime importance if a tsunami warning system is to be successful. This may be accomplished through experimental or numerical studies. A comprehensive experimental study has been conducted to understand the dynamic behavior of a wave rider buoy exposed to a variety of waves. It is common that tsunami waves are represented in terms of shallow water waves, namely solitary and cnoidal waves. Hence, in the present study, the discus type data buoy is scale modeled and tested under the action of solitary and cnoidal waves in the laboratory. The time histories of wave elevations, as well as heave and pitch motions of the buoy model, were analyzed through a spectral approach as well as through wavelet transformations. The wavelet approach gives more detailed insight into the spectral characteristics of the buoy motion in the time scale. The harmonic analyses were performed for the cnoidal wave elevations and subsequent motion characteristics that give an insight into the energy variations. The details of the model, instrumentation, testing conditions and the results are presented in this paper.     

Numerical Model of Cnoidal Wave Flume

The volume of fluid(VOF)method is used to set up a wave flume with an absorbing wavemaker of cnoidal waves.Based on the transfer function between wave surface and paddle velocity obtained bythe shallow water wave theory,the velocity boundary condition of an absorbing wave maker is introduced toabsorb reflected waves that reach the numerical wave maker.For H/d ranging from 0.1 to 0.59 and T(g/d)~(1/2)from 7.9 to 18.3,the parametric studies have been carried out and compared with experiments.     

基于破碎临界区的新型拍岸浪统计计算模型研究

When waves propagate from deep water to shallow water, wave heights and steepness increase and then waves roll back and break. This phenomenon is called surf. Currently, the present statistical calcula...     

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.     

V形防波堤与圆弧型防波堤之浅水波绕射作用的比较

应用椭圆余弦波的绕射理论,推导了V形防波堤的浅水波浪绕射解析解,从而对现有的Airy微幅波理论进行了有效拓展。据此理论对V形防波堤的浅水波绕射作用进行了解析计算,并与几何形状相近的圆弧型防波堤结果加以了对比。结果表明：椭圆余弦波理论计算的V形防波堤最大波浪力和最大绕射波面明显大于微幅波理论的对应值。本方法适用于张角180°的有限长直立薄壁防波堤的浅水波绕射作用计算,从而将无限长直立薄壁堤的反射波理论加以有效拓展。张角同为120°的V形堤与圆弧堤的堤后防浪效果相近,而180°圆弧堤的堤后防浪效果优于张角90°的V形堤。     

A hybrid model for numerical wave forecasting and its implementation——Ⅰ. The wind wave model

The authors make an endeavor to explain why a new hybrid wave model is here proposed when several such models have already been in operation and the so- called third generation wave modej is proving attractive. This part of the paper is devoted to the wind wave model. Both deep and shallow water models have been developed, the former being actually a special case of the latter when water depth is great. The deep water model is exceptionally simple in form. Significant wave height is the only prognostic variable. In comparison with the usual methods to compute the energy input and dissipations empirically or by "tuning", the proposed model has the merit that the effects of all source terms are combined into one term which is computed through empirical growth relations for significant waves, these relations being, relatively speaking, easier and more reliable to obtain than those for the source terms in the spectral energy balance equation. The discrete part of the model and the implementation of the mode     

An explicit approximation to the wavelength of nonlinear waves

An explicit and concise approximation to the wavelength in which the effect of nonlinearity is involved and presented in terms of wave height, wave period, water depth and gravitational acceleration. The present approximation is in a rational form of which Fenton and Mckee's (1990, Coastal Engng 14, 499–513) approximation is reserved in the numerator and the wave steepness is involved in the denominator. The rational form of this approximation can be converted to an alternative form of a power-series polynomial which indicates that the wavelength increases with wave height and decreases with water depth. If the determined coefficients in the present approximation are fixed, the approximating formula can provide a good agreement with the wavelengths numerically obtained by Rienecker and Fenton's (1981, J. Fluid Mech. 104, 119–137) Fourier series method, but has large deviations when waves of small amplitude are in deep water or all waves are in shallow water. The present approximation with variable coefficients can provide excellent predictions of the wavelengths for both long and short waves even, for high waves.     

海底边坡稳定性分析中波浪力的求解

针对浅水区波浪的非线性特性,提出了在海底边坡稳定性分析中应用椭圆余弦波理论来研究波浪力的问题,利用非线性弥散关系建立了新的适用于整个水深范围的椭圆余弦波的近似求解方法.结合工程实例,确立了海底边坡波浪力的计算步骤,并编制了计算程序.     

Numerical study on cnoidal wave run-up around a vertical circular cylinder

A finite element model of Boussinesq-type equations was set up, and a direct numerical method is proposed so that the full reflection boundary condition is exactly satisfied at a curved wall surface. The accuracy of the model was verified in tests. The present model was used to further examine cnoidal wave propagation and run-up around the cylinder. The results showed that the Ursell number is a nonlinear parameter that indicates the normalized profile of cnoidal waves and has a significant effect on the wave run-up. Cnoidal waves with the same Ursell number have the same normalized profile, but a difference in the relative wave height can still cause differences in the wave run-up between these waves. The maximum dimensionless run-up was predicted under various conditions. Cnoidal waves hold entirely distinct properties from Stokes waves under the influence of the water depth, and the nonlinearity of cnoidal waves enhances rather than weakens with increasing wavelength. Thus, the variations in the maximum run-up with the wavelength for cnoidal waves are completely different from those for Stokes waves, and there are even significant differences in the variation between different cnoidal waves.