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.     

Combined refraction and diffraction models for sea waves over complicated topography and with currents in shallow water

-Combined refraction and diffraction models in the form of linear parabolic approximation are derived through smallparameter method. More strictly theoretical basis and more accuracy in the models than Lozano's (1980) are obtained. Some theoretical defects in Liu's model (1985) with consideration of current are not only found but also eliminated. More strict and accurate models are, therefore, presented in this paper.The calculation results and analysis in applying the models to actual wave field with consideration of bottom friction will be given in the following paper.     

Combined refraction and diffraction of short waves using the finite element method

A two-dimensional finite element numerical model is presented that calculates combined refraction and diffraction of short waves. The wave equation solved governs the propagation of periodic, small amplitude surface gravity waves over a variable depth seabed of mild slope. An efficient computational scheme is employed that allows the solution of practical problems that typically require large computational grids. Comparisons are presented between the finite element model calculations and an analytical solution, a two-dimensional numerical solution, a three-dimensional numerical solution, and measurements from a hydraulic model.     

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 circular cylinder in water waves

This paper treats the problem in linearized water wave theory that arises when a wave travels across a submerged cylinder. This problem is classical and was first treated by Dean. The solution presented is novel and explicit. This is achieved through the use of certain recursive relations. The basic results are of course the same as in the older literature: (1) the coefficient of reflection is zero; (2) the only effect of the cylinder, is a phase shift relative to the undisturbed wave. However, the practical computation of the velocity potential and the phase shift is reduced almost to hand calculations. This is in contrast to the older literature which requires, in principle, the inversion of an infinite matrix. The work is motivated by the Institue's efforts to focus water waves. Cylinders are possible bodies to create the necessary phase shifts to achieve such focusing.     

Measurement of water particle velocities in waves

Water surface profiles and horizontal and vertical water particle velocity components have been measured to investigate the properties of intermediate depth waves generated in the laboratory. The data has been compared with linear wave theory. It was found that linear theory predicted the attenuation of velocity field with depth successfully and that it overestimates both components of velocity slightly.     

Kinematics of extreme waves in deep water

The velocity profiles under crest of a total of 62 different steep wave events in deep water are measured in laboratory using particle image velocimetry. The waves take place in the leading unsteady part of a wave train, focusing wave fields and random wave series. Complementary fully nonlinear theoretical/numerical wave computations are performed. The experimental velocities have been put on a nondimensional form in the following way: from the wave record (at a fixed point) the (local) trough-to-trough period, T_TT and the maximal elevation above mean water level, η_m of an individual steep wave event are identified. The local wavenumber, k and an estimate of the wave slope, ε are evaluated from ω²/(gk)=1+ε², where ω=2π/T_TT and g denotes the acceleration of gravity. A reference fluid velocity, is then defined. Deep water waves with a fluid velocity up to 75% of the estimated wave speed are measured. The corresponding kη_m is 0.62. A strong collapse of the nondimensional experimental velocity profiles is found. This is also true with the fully nonlinear computations of transient waves. There is excellent agreement between the present measurements and previously published Laser Doppler Anemometry data. A surprising result, obtained by comparison, is that the nondimensional experimental velocities fit with the exponential profile, i.e. e^ky, y the vertical coordinate, with y=0 in the mean water level.     

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.     

Steepness and asymmetry of extreme waves and the highest waves in deep water

Traditional wave steepness s=H/L does not define steep asymmetric waves in a random sea uniquey. Three additional parameters characterising single zero-downcross waves in a time series are crest front steepness, vertical asymmetry factor and horizontal asymmetry factor. Results for steepness and asymmetry from zero-downcross analysis of wave data obtained from full scale measurements in deep water on the Norwegian continental shelf in 58 time series are presented. The analysis demonstrates clearly the asymmetry of both “extreme waves” and the highest waves. The period and height of the highest waves are also given together with their correlation to spectral parameters. The measured maximum wave heights are also compared with predicted values of maximum wave heights showing good agreement.     

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.     

Long water waves and electromagnetic waves--An analogy

lt is shown that Maxwell's equations which describe the propagation of electromagnetic radiation are analogous to the kinematic relationships which govern the propagation of long water waves, if the latter equations are described in terms of a vector potential. The analogy between the two phenomena may be used to study, for example, seiching, or long wave resonance, in harbors by means of an equivalent microwave cavity. The plan geometry of the cavity would be geometrically similar to that of the harbor basin so that wave reflections were correctly reproduced, and the refraction caused by depth variations in the harbor would be modeled by varying the dielectric properties of the cavity interior. When supplied with an appropriate source of microwaves, the intensity of the electric field at a given point can be related to wave induced currents at a similar location in the harbor. Such information is an important factor in the siting of berthing and mooring facilties for shipping. It is also shown how the impedance concept can be used to study the transformation of long waves as they traverse a step change in water depth.     

Microwave scattering by surface waves on water

The electromagnetic backscattering from small-amplitude water waves has been studied using a microwave model (2.5 GHz). The first- and second-order backscatter Doppler spectra have been measured and compared with the theoretical models, and good agreement has been found. In this paper, the specific case of two water waves propagating at right angles to each other will be presented, and the level of contribution of both the hydrodynamic and the electromagnetic effects on the backscatter radar Doppler spectrum will be considered.     

Propagation of water waves over permeable rippled beds

Unsteady two-dimensional Navier-Stokes equations and Navier-Stokes type model equations for porous flow were solved numerically to simulate the propagation of water waves over a permeable rippled bed. A boundary-fitted coordinate system was adopted to make the computational meshes consistent with the rippled bed. The accuracy of the numerical scheme was confirmed by comparing the numerical results concerning the spatial distribution of wave amplitudes over impermeable and permeable rippled beds with the analytical solutions. For periodic incident waves, the flow field over the wavy wall is discussed in terms of the steady Eulerian streaming velocity. The trajectories of the fluid particles that are initially located close to the ripples were also determined. One of the main results herein is that under the action of periodic water waves, fluid particles on an impermeable rippled bed initially moved back and forth around the ripple crest, with increasing vertical distance from the rippled wall. After one or two wave periods, they are then lifted towards the next ripple crest. All of the marked particles on a permeable rippled bed were shifted onshore with a much larger displacement than those on an impermeable bed. Finally, the flow fields and the particle motions close to impermeable and permeable beds induced by a solitary wave are elucidated.     

Estimation of infragravity waves at intermediate water depth

The accuracy of nearshore infragravity wave height model predictions has been investigated using a combination of the spectral short wave evolution model SWAN and a linear 1D SurfBeat model (IDSB). Data recorded by a wave rider located approximately 3.5 km from the coast at 18 m water depth have been used to construct the short wave frequency-directional spectra that are subsequently translated to approximately 8 m water depth with the third generation short wave model SWAN. Next the SWAN-computed frequency-directional spectra are used as input for IDSB to compute the infragravity response in the 0.01 Hz–0.05 Hz frequency range, generated by the transformation of the grouped short waves through the surf zone including bound long waves, leaky waves and edge waves at this depth. Comparison of the computed and measured infragravity waves in 8 m water depth shows an average skill of approximately 80%. Using data from a directional buoy located approximately 70 km offshore as input for the SWAN model results in an average infragravity prediction skill of 47%. This difference in skill is in a large part related to the under prediction of the short wave directional spreading by SWAN. Accounting for the spreading mismatch increases the skill to 70%. Directional analyses of the infragravity waves shows that outgoing infragravity wave heights at 8 m depth are generally over predicted during storm conditions suggesting that dissipation mechanisms in addition to bottom friction such as non-linear energy transfer and long wave breaking may be important. Provided that the infragravity wave reflection at the beach is close to unity and tidal water level modulations are modest, a relatively small computational effort allows for the generation of long-term infragravity data sets at intermediate water depths. These data can subsequently be analyzed to establish infragravity wave height design criteria for engineering facilities exposed to the open ocean, such as nearshore tanker offloading terminals at coastal locations.     

Interaction of higher-order water waves with uniform currents

The interaction between current-free higher-order water waves with a wave-free uniform current normal to the wave crests is considered. The combined wave-current motion resulting from the interaction is assumed stable and irrotational. The velocity potential, dispersion relation, the particle kinematics and pressure distribution up to the third order in wave amplitude are developed. The conservation of mean mass, momentum and energy, together with the dispersion relation on the free surface are used to derive a set of four nonlinear equations, through which the relationship between wave-free current, current-free wave and the combined wave-current parameters is established. Numerical results for a range of current values are also presented.     

深水随机波列中畸形波统计特征的研究

本文基于Longuet-Higgins随机波浪模型和JONSWAP谱,进行了大量深水随机波的模拟,获取了畸形波发生概率稳定的随机波列,并对随机波列中的畸形波进行了分析。结果表明,畸形波发生的概率小于基于Rayleigh分布预测结果,且随谱宽的减小而增大。在固定时间段内,畸形波发生的频次服从泊松分布,时间间隔服从指数分布,且随着谱宽的增大,畸形波的发生频次减小,相邻畸形波的发生时间间隔增加。通过小波变换方法分离随机波中的波群,研究了出现畸形波的波群特征,发现一个波群中最多会出现4个畸形波,但是在发生畸形波的波群中,单个畸形波的概率最大。随着谱宽减小,一个波群中包含多个畸形波的概率增加。另外,出现畸形波的波群时间长度服从广义极值分布,随着谱宽减小,畸形波波群的时间跨度增加。     

不同介质底床上的波浪衰减

采用近年来的研究成果比较了良在不同介质底床上传播时的波浪衰减规律,对不同介质底床上的波浪衰减机理进行了总结,推荐了计算波浪衰减的公式和方法,根据实验和计算结果解释淤泥质海岸可能存在的波浪传播现象。     

Correction for water depth variations of marine crustal reflection seismic data using refraction statics

Water depth variations in marine reflection seismic profiling cause velocity push-down and, in regions of rapid fluctuations of the water-bottom slope, stack degradation. Static corrections are a very satisfactory and practical solution to these problems under typical survey conditions with water depths not exceeding a few hundred metres and relatively hard water bottoms. Static time shifts are best derived from a refraction analysis of first arrivals rather than from fathometer data, especially over underwater valleys where unconsolidated sediment of velocity close to that of water has been deposited unconformably onto the underlying lithified rocks. These points are illustrated by a field example from the GLIMPCE survey in Lake Superior. The availability of computer-effective algorithms such as the generalized linear inversion method allows the implementation of refraction statics during the initial processing of regional marine crustal surveys.LITHOPROBE Publication No. 263.