Representing frequency spectra for shallow water waves

Wind waves recorded in water from 1.4 to 3.8 m deep near the southeastern shore of Lake Erie during 1981 were used to compare two methods for representing wave spectra in shallow water. The results show that the semi-theoretical Wallops model, which requires total energy, peak energy frequency, and depth as parameters, provides fair agreement with observed spectra at the deeper stations but only marginal agreement in very shallow water. The general empirical model, which requires average frequency and energy density at the spectral peak as additional parameters, provides closer agreement with observed wave spectra for all depths.     

Spectral characteristics of high shallow water waves

The spectral characteristics of shallow water waves with significant wave height more than 2 m based on the data collected along the Indian coast is examined. It was found that the value of Joint North Sea Wave Project (JONSWAP) parameters (α and γ) increases with significant wave height and mean wave period and decreases with spectral peak period. The estimated average value (0.0027 and 1.63) of the JONSWAP parameters, α and γ were less than the generally recommended values of 0.0081 and 3.3, respectively. By carrying out a multi-regression analysis, an empirical equation is arrived relating the JONSWAP parameters with significant wave height, peak wave period and mean wave period. It was found that the Scott spectra underestimate the maximum spectral energy of high waves. The study shows that the measured wave spectra can be represented by JONSWAP spectra with the JONSWAP parameters estimated based on the equation proposed in this paper.     

Nonlinear refraction-diffraction of surface waves in intermediate and shallow water

A numerical scheme for solving the nonlinear Boussinesq equations is introduced. The numerical model is used to investigate nonlinear refraction-diffraction of surface gravity waves over a semicircular shoal. Results are compared with experimental data (Whalin, 1971) and previous reported numerical results by Liu and Tsay (1984) and Liu, Yoon and Kirby (1985). The present calculations reproduce the earlier results for shallow water waves, but are superior in intermediate water depth.     

On spectral and statistical characteristics of shallow water waves

Spectral, zero up-crossing and Tucker's methods of analysis are examined for shallow water wave records. Among the wave height parameters H_s and are most reliable and consistent irrespective of the analysis technique. Tucker analysis, though simple, seems versatile for estimating these wave height parameters. The standard period parameters are less consistent. For practical purposes every period that might reasonably occur has to be considered along with their corresponding height estimate. Joint distribution of heights and periods is in agreement with the function proposed by CNEXO.     

Nonlinear crest distribution for shallow water Stokes waves

This article concerns the calculation of nonlinear crest distribution for shallow water Stokes waves. The calculations have been carried out by incorporating a second order nonlinear wave model into an asymptotic analysis method. This is a new approach to the calculation of wave crest distribution, and, as all of the calculations are performed in the probability domain, avoids the need for long time-domain simulations. The accuracy and efficiency of this new approach for calculating the wave crest distribution are validated by comparing the results predicted using it with those predicted by using the Monte Carlo simulation (MCS) method, by using a previous Transformed Rayleigh method, by using some existing wave crest distribution formulas, and by using the measured surface elevation data at the Poseidon platform in the Japan Sea.     

规则波在浅水中运动及破碎的数值计算

利用一维Boussinesq方程描述了在浅水中的波浪运动以及破碎情况。在方程中引入了表面翻滚的概念，认为翻滚的水体是以波速运动的，翻滚的作用表现在水平速度的垂直分布上，产生了附加迁移动量项。通过对Airy波在浅水中运动以及破碎情况的研究，得出的一些结论与Schaeffer和合田的成果吻合良好。     

Numerical simulation for the two-dimensional nonlinear shallow water waves

This study deals with the general numerical model to simulate the two-dimensional tidal flow, flooding wave (long wave) and shallow water waves (short wave). The foundational model is based on nonlinear Boussinesq equations. Numerical method for modelling the short waves is investigated in detail. The forces, such as Coriolis forces, wind stress, atmosphere and bottom friction, are considered. A two-dimensional implicit difference scheme of Boussinesq equations is proposed. The low-reflection outflow open boundary is suggested. By means of this model,both velocity fields of circulation current in a channel with step expansion and the wave diffraction behind a semi-infinite breakwater are computed, and the results are satisfactory.     

Parameterization of nonlinear shallow water waves over sloping bottoms

Investigation of the bottom slope effects on the nonlinear transformation of irregular waves, which are generated based on JONSWAP spectra, is carried out in a physical wave flume with three slopes (β = 1/15, 1/30, 1/45). The slope effects on the estimation of representative wave height are examined first. To obtain a better estimation of wave height, the slope effect should be considered when slope is larger than 1/30. The nonlinear parameters (bicoherence, skewness and asymmetry) are estimated by using the wavelet-based bispectrum, and the empirical formulae regarding these nonlinear parameters as a function of the local Ursell number are derived based on the present data measured on each slope. The results indicate that the slopes have a negligible effect on the variations of the skewness. The fitted coefficients of the formulae for the other parameters on slope β = 1/15 are clearly different from the results on the slopes β = 1/30 and 1/45, indicating that slope influence on the parameterization cannot be ignored when β > 1/30. Hence, new formulae considering the slope effect are presented. Furthermore, the empirical formulae for the data in surf zone are recommended.     

Refined seaward boundary conditions for modelling shallow water waves in semi-enclosed water bodies

Based on the theory of characteristics, this research elaborates on the numerical treatment of two types of seaward boundary conditions for modelling long-wave dynamics in truncated estuarine and coastal domains. These seaward boundary conditions are devised for the solution of the fully non-linear shallow water equations in the time domain. The first type is the clamped boundary, at which the water level variation is given and the velocity is computed along the characteristic line going out of the domain. The second type is the non-reflecting boundary, where the incident wave information is introduced and the reflected waves from inside the computational domain are allowed to escape at the same time. The essence of its numerical implementation is to distinguish the inward and outward characteristics and to disconnect the incoming characteristic relation from the actual flow inside the domain. Compared with previous techniques, the present method includes extra terms in the derivation to account for the effects of the uneven bed, bottom friction and shape of the characteristic lines. A shock-capturing finite difference method is used to solve the shallow water equations in the deviatoric format, but the seaward boundary algorithms constructed herein are generic and applicable to other solvers. The necessity of these refinements is highlighted by simulating the tidal oscillation in the Persian/Arabian Gulf, periodic wave runup on the coastline and the wave resonance in a narrow harbour. It is found that neglecting the bed slope at the boundary may result in biased mean water levels in the prediction.     

地形与流对水平无旋浅水波的影响

本文在水平无旋及Boussinesq假设之下,导出了水面变化与水平速度场耦合方程组,以及相应的压力与垂向速度解析表示式。通过数值方法求得水面变化及某一特定水深的水平速度分布之后,压力分布及其余水深的速度分布即可由简单计算得到。由色散关系式可知,不同水深的长波色散关系在O(ε)近似之下是等同的。粘性的存在会使波高随时间的增加而衰减,但粘性与底斜率的耦合又可能使波高增长,形成不稳定;计算及分析说明,当同向流增大时,波速增大,波长增大,波高会增加;而水深减少,使波速减少,波长缩短,振幅增加。     

浅水极限波浪几何特征的实验研究

该文通过物理模型实验,对浅水区域内的波浪在破碎前极限状态下的几何特征进行了研究。实验基于JONSWAP谱对不规则波浪进行模拟,通过对波群中出现的单体极限波浪进行捕捉并对波形进行测量而得到研究样本。为了考察底坡因素对极限波浪几何特征的影响,实验共考虑了3组大小分别为β=1/15、1/30以及1/45的地形坡度。统计结果表明,在实验所采用的坡度范围内,当地波高与水深对近岸极限波浪的影响最为显著,随着水深与波高因素变化,极限波浪的几何特征也出现明显的改变。坡度因素对极限波陡和偏度的影响很小,可以被忽略,但是对不对称度参数的影响相对比较明显,坡度越陡,不对称程度越剧烈。最后,通过参数化,本文给出了极限波浪几何特征变化的经验公式。     

Threshold of motion of coarse-grained sediment under waves in shallow water

This paper considers the threshold of motion of sediment in shallow coastal waters under breaking and non-breaking waves. A simple model, representing conditions at the fluid/sediment interface, is developed. The representation of a breaking wave is based on bore theory, and the shear stress at the bed is based on the friction formula of O'Connor and Yoo. A threshold formula is presented based on the use of energy dissipation. The threshold data is also related to the Shields threshold criterion.     

Interaction of flexural gravity waves with shear current in shallow water

In the present study, the effect of shear current on the propagation of flexural gravity waves is analyzed under the assumptions of linearized shallow-water theory. Explicit expressions for the reflection and transmission coefficients associated with flexural gravity wave scattering by a step discontinuity in both water depth and current speed are derived. Further, trapping and scattering of flexural gravity waves by a jet-like shear current with a top-hat profile are examined and certain limiting conditions for the waves to exist are derived. The effects of change in water depth, current speed, incident wavelength and the angle of incidence on the group and phase velocities as well as on the reflection and transmission characteristics are analyzed through different numerical results.     

Measurement of seabed sound speeds from head waves in shallow water

Acoustic signals from small explosive charges have been measured with sonobuoys on twelve tracks in Australian northern shallow waters with the aim of assessing whether useful geoacoustic information could be obtained. Using the frequency band from 14 to 70 Hz, travel times of head waves were monitored, and the sound speeds and depths of corresponding interfaces in the seabed were derived. The water sound speed varied a little with range, and its depth dependence was allowed for by using its average value. Head waves from interfaces indistinguishable from the seafloor (the water/seabed interface) were detected on only three of the tracks, with derived sound speeds of 2100 to 2300 m/s. The first sub-bottom interfaces were from 50 to 600 m beneath the seafloor, and their sound speeds ranged from around 2000 m/s to 6400 m/s. Thus the head waves were from chalk or limestone, cemented sediments in which sound-speed gradients would be small. The amount of data obtained for the seafloor was limited by incoherence of the signals and, for some tracks, by excessive spacing between shots. The incoherence is generally attributed to multiple head waves that are individually unresolvable, while on two tracks there were indications of medal ground waves. Occasional anomalous data were obtained, but generally the assumptions of the simple interpretation method were found to be valid. Since no curvature in the range-time lines was observed, there was no evidence of sub-bottom sound-speed gradients being significant     

Observations of cnoidal internal waves and their effect on acousticpropagation in shallow water

Distinctive packets of periodic internal waves were observed during an experiment in the Gulf of Mexico. There was a 65-m-deep mixed layer overlying a thin strong density interface. A layer of weaker density stratification extended below the interface to the bottom, at a depth of 185 m. The waves had 2-10-m amplitudes, narrow frequency bandwidths with central frequencies of 8.5 cph, and they propagated in the upslope direction. The wave packets were observed on three consecutive days. They lasted about 3 h and were always observed at the same time of day, clearly in response to tidal forcing. A model of the time/space structure of the waves was tuned to match that of the observations, showing that the data are consistent with a cnoidal wave hypothesis. Observations of low-frequency acoustic propagation along two baselines show fluctuations that we hypothesize are due to interactions with the cnoidal waves. The fluctuations have spatial correlation scales (in the slantwise direction) on the order of 76 m. We simulate these effects using a time-step PE approach. We find that a mode-coupling resonance with the internal wave field results in elevated acoustic variability along a set of discrete spokes, emanating from the acoustic source. While acoustic variability tends to increase with range and with internal wave amplitude, tangential and radial correlation scales do not show a systematic dependence. The patterns in tangential and radial correlation scales show strong anisotropic patterns in azimuth, but little systematic trend in range     

Wave kinematics for simulated shallow water storm waves - Analysis and experiments

Computation of wave kinematics at or near offshore structures is a vitally important consideration in the design of offshore structures. Design waves often include breaking and near-breaking storm waves in the presence of currents. It is important to predict the kinematics of these steep waves. Experiments were carried out in a wave tank with simulated steep waves with and without in-line current in which the wave profiles and the corresponding kinematics were simultaneously measured. The simulated waves represent shallow-water Gulf of Mexico storm waves. Many of these waves broke at or near the measuring instruments. Irregular stream-function theory was used to compute the wave kinematics and was found to generally predict the measured wave-current kinematics well. The differences found between the two are noted. Some of the noteworthy features of the breaking waves are also discussed.     

SWAN predictions of waves observed in shallow water onshore of complex bathymetry

SWAN model predictions, initialized with directional wave buoy observations in 550-m water depth offshore of a steep, submarine canyon, are compared with wave observations in 5.0-, 2.5-, and 1.0-m water depths. Although the model assumptions include small bottom slopes, the alongshore variations of the nearshore wave field caused by refraction over the steep canyon are predicted well over the 50 days of observations. For example, in 2.5-m water depth, the observed and predicted wave heights vary by up to a factor of 4 over about 1000 m alongshore, and wave directions vary by up to about 10°, sometimes changing from south to north of shore normal. Root-mean-square errors of the predicted wave heights, mean directions, periods, and radiation stresses (less than 0.13 m, 5°, 1 s, and 0.05 m³/s² respectively) are similar near and far from the canyon. Squared correlations between the observed and predicted wave heights usually are greater than 0.8 in all water depths. However, the correlations for mean directions and radiation stresses decrease with decreasing water depth as waves refract and become normally incident. Although mean wave properties observed in shallow water are predicted accurately, nonlinear energy transfers from near-resonant triads are not modeled well, and the observed and predicted wave energy spectra can differ significantly at frequencies greater than the spectral peak, especially for narrow-band swell.     

Parameterization and simulation of near bed orbital velocities under irregular waves in shallow water

A set of empirical formulations is derived that describe important wave properties in shallow water as functions of commonly used parameters such as wave height, wave period, local water depth and local bed slope. These wave properties include time varying near-bed orbital velocities and statistical properties such as the distribution of wave height and wave period. Empirical expressions of characteristic wave parameters are derived on the basis of extensive analysis of field data using recently developed evolutionary algorithms. The field data covered a wide range of wave conditions, though there were few conditions with wave periods greater than 15 s. Comparison with field measurements showed good agreement both on a time scale of a single wave period as well as time averaged velocity moments.     

Depth inversion in shallow water based on nonlinear properties of shoaling periodic waves

Two depth inversion algorithms (DIA) applicable to coastal waters are developed, calibrated, and validated based on results of computations of periodic waves shoaling over mild slopes, in a two-dimensional numerical wave tank based on fully nonlinear potential flow (FNPF) theory. In actual field situations, these algorithms would be used to predict the cross-shore depth variation h based on sets of values of wave celerity c and length L, and either wave height H or left–right asymmetry s₂/s₁, simultaneously measured at a number of locations in the direction of wave propagation, e.g., using video or radar remote sensing techniques. In these DIAs, an empirical relationship, calibrated for a series of computations in the numerical wave tank, is used to express c as a function of relative depth k_oh and deep water steepness k_oH_o. To carry out depth inversion, wave period is first predicted as the mean of observed L/c values, and H_o is then predicted, either based on observed H or s₂/s₁ values. The celerity relationship is finally inverted to predict depth h. The algorithms are validated by applying them to results of computations for cases with more complex bottom topography and different incident waves than in the original calibration computations. In all cases, root-mean-square (rms)-errors for the depth predictions are found to be less than a few percent, whereas depth predictions based on the linear dispersion relationship—which is still the basis for many state-of-the-art DIAs—have rms-errors 5 to 10 times larger.