Depth decay rate for surface gravity wave pressure and velocity

Linear governing equations are formulated for the depth decay of the pressure and velocity variations associated with propagating surface gravity waves. These governing equations come from combining Bernoulli’s equation for steady frictionless flow along a streamline and the crossstream force balance involving gravity, the centrifugal force and a pressure gradient. Qualitative solutions show that the pressure decreases downward faster than the velocity does and at a rate that is probably not the normal exponential decrease, which does not agree with the classical result. The radius of curvature of the streamlines is a non-constant coefficient in these equations and it needs to be supplied, either from measurements or another theory, in order to complete the solution of the derived governing equations. There is no sensitivity of the solution to the exact path the radius of curvature takes between its minimum value at the surface of a crest and trough and infinity at great depth. In the future measurements, perhaps streak photographs, will be needed to distinguish between the new and old theories.

Wave-Induced Loads on Very Large FPSOs at Restricted Water Depth

The effects of water depth on the wave-induced vertical bending moment and shearing force on a very large FPSO are studied by experiments and computations for regular and irregular waves. The restricted water depth composite Green function is employed to develop a program for the computation of the hydrodynamic coefficients of the very large FPSO at shallow water. A three-segment model with 1∶100 scale is tested in the State Key Laboratory of Ocean Engineering at Shanghai Jiao Tong University for the verification of the numerical method. The experimental and computational results show that the water depth has a substantial effect on wave-induced loads. The wave-induced vertical loads increase with the decrease of water depth for shallow water. Especially, for ultra-shallow water these loads increase very evidently with the decrease of water depth. The long-term prediction values of wave-induced vertical loads increase with the decrease of the ratio of water depth to draught. The long-term prediction values of wave-induced vertical loads are about 8% larger than those for deep water when the ratio of water depth to draught is 3.0. However, water depth hardly affects the long-term prediction values of wave-induced loads when the ratio of water depth to draught is larger than 5.0.

Modified Rayleigh Distribution of Wave Heights in Transitional Water Depths

This paper concerns the calculation of wave height exceedance probabilities for nonlinear irregular waves in transitional water depths, and a Transformed Rayleigh method is first proposed for carrying out the calculation. In the proposed Transformed Rayleigh method, the transformation model is chosen to be a monotonic exponential function, calibrated such that the first three moments of the transformed model match the moments of the true process. The proposed new method has been applied for calculating the wave height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform. It is demonstrated in this case that the proposed new method can offer better predictions than those by using the conventional Rayleigh wave height distribution model. The proposed new method has been further applied for calculating the total horizontal loads on a generic jacket, and its accuracy has once again been substantiated. The research findings gained from this study demonstrate that the proposed Transformed Rayleigh model can be utilized as a promising alternative to the well-established nonlinear wave height distribution models.

Oceanic pycnocline depth retrieval from SAR imagery in the existence of solitary internal waves

Oceanic pycnocline depth is usually inferred from in situ measurements. It is attempted to estimate the depth remotely. As solitary internal waves occur on oceanic pycnocline and propagate along it, it is possible to retrieve the depth indirectly in virtue of the solitary internal waves. A numerical model is presented for retrieving the pycnocline depth from synthetic aperture radar （SAR） images where the solitary internal waves are visible and when ocean waters are fully stratified. This numerical model is constructed by combining the solitary internal wave model and a two-layer ocean model. It is also assumed that the observed groups of solitary internal wave packets on the SAR imagery are generated by local semidiurnal tides. A case study in the East China Sea shows a good agreement with in situ CTD （conductivity-temperature-depth） data.

EFFECT OF WATER DEPTH ON WIND-WAVE FREQUENCY SPECTRUM Ⅱ. VERIFICATION AND COMPARISON

The spectrum derived in Part 1 of the presert paper is here systematically verified with field data andcompared at some length with that obtained by multiplying the deep-water spectrum with theKitaigorodskii factor.

A study on the uncertainties of the centroid depth of the 2013 Lushan earthquake from teleseimic body wave data

Centroid depth of earthquakes is essential for seismic hazard mitigation. But, various studies provided different solutions for the centroid depth of the damaging 2013 Lushan earthquake, thus hindering further studies of the earthquake processes. To resolve its centroid depth and assess the uncertainties, we apply the teleseismic cut and paste method to invert for centroid depth with teleseismic body waves in the epicentral distance of 300-90~. We performed the inversion for P waves only as well the case of both P and SH waves and found that both cases lead to depth solutions with difference less than 0.5 km. We also investigated the effects on depth inversion from azimuth gap of seismic stations, source duration, and comer fre- quency of filter. These various tests show that even azi- muthal distribution of seismic stations is helpful for accurate depth inversion. It is also found that estimate of centroid depth is sensitive to source duration. Moreover, the depth is biased to larger values when corner frequency of low-pass filter is very low. The uncertainty in the velocity model can also generate some error in the depth estimation （- 1.0 km）.With all the above factors consid- ered, the centroid depth of Lushan earthquake is proposed to be around 12 km, with uncertainty about 2 km.

Numerical Calculation for Nonlinear Waves in Water of Arbitrarily Varying Depth with Boussinesq Equations

Based on the high order nonlinear and dispersive wave equation with a dissipative term, a numerical model for nonlinear waves is developed, It is suitable to calculate wave propagation in water areas with an arbitrarily varying bottom slope and a relative depth h/L(0)less than or equal to1. By the application of the completely implicit stagger grid and central difference algorithm, discrete governing equations are obtained. Although the central difference algorithm of second-order accuracy both in time and space domains is used to yield the difference equations, the order of truncation error in the difference equation is the same as that of the third-order derivatives of the Boussinesq equation. In this paper, the correction to the first-order derivative is made, and the accuracy of the difference equation is improved. The verifications of accuracy show that the results of the numerical model are in good agreement with those of analytical Solutions and physical models.

Statistical Distribution of Depth-Integrated Local Horizontal Momentum for Second-Order Random Ocean Waves in Finite Water Depth

Based on the second order random wave solutions of water wave equations in finite water depth, statistical distributions of the depth integrated local horizontal momentum components are derived by use of the characteristic function expansion method. The parameters involved in the distributions can be all determined by the water depth and the wave number spectrum of ocean waves. As an illustrative example, a fully developed wind generated sea is considered and the parameters are calculated for typical wind speeds and water depths by means of the Donelan and Pierson spectrum. The effects of nonlinearity and water depth on the distributions are also investigated.

LINEAR GRAVITY WAVES ON MAXWELL FLUIDS OF FINITE DEPTH

Linear surface gravity waves on Maxwell viscoelastic fluids with finite depth are studied in this paper.A dispersion equation describing the spatial decay of the gravity wave in finite depth is derived.A dimensionless memory(time)number θ is introduced.The dispersion equation for the pure viscous fluid will be a specific case of the dispersion equation for the viscoelastic fluid as θ=0.The complex dispersion equation is numerically solved to investigate the dispersion relation.The influences of θ and water depth on the dispersion characteristics and wave decay are discussed.It is found that the role of elasticity for the Maxwell fluid is to make the surface gravity wave on the Maxwell fluid behave more like the surface gravity wave on the inviscid fluid.

Capillary Wave’s Depth Decay

Depth decay rates for pressure and velocity variations of a propagating capillary wave are found to be significantly different from each other, and neither one is expected to have the classical exponential character. To obtain these results Bernoulli’s equation along streamlines in the steady reference frame is combined with the force balance on fluid particles in the cross-stream direction: a pressure gradient offsets the centrifugal force on particles moving along a curved path. The two starting equations for pressure and velocity are nonlinear, but two linear first order ordinary differential equations are produced from them, one for each variable, and they can be integrated immediately. A full solution awaits further information on the non-constant coefficient, the radius of curvature function for the streamlines, either from observations or another theory.

Numerical simulation and inversion of offshore area depth based on x-band microwave radar

A detection method of offshore area depth utilizing the x-band microwave radar is proposed. The method is based on the sea clutter imaging mechanism of microwave radar, and combined with dispersion equation of the liner wave theorem and least square method （LSM）, consequently get the inversion results of water depth in the detected region. The wave monitoring system OSMAR-X exploited by the Ocean State Laborato-ry, Wuhan University, based on a microwave radar has proven to be a powerful tool to monitor ocean waves in time and space. Numerical simulation and inversion of offshore area depth are carried out here; since JONSWAP model can give description of stormy waves in different growth phase, it is suitable for simulation. Besides, some results from measured data detected by OSMAR-X x-band radar located at Longhai of Fujian Province, China, validates this method. The tendency of the average water depths inferred from the radar images is in good agreement with the tide level detected by Xiamen tide station. These promising results suggest the possibility of using OSMAR-X to monitor operationally morphodynamics in coastal zones. This method can be applied to both shore-based and shipborne x-band microwave radar.

High-Order Models of Nonlinear and Dispersive Wave in Water of Varying Depth with Arbitrary Sloping Bottom

High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).

Water Wave Scattering by an Elastic Thin Vertical Plate Submerged in Finite Depth Water

The problem of water wave scattering by a thin vertical elastic plate submerged in uniform finite depth water is investigated here.The boundary condition on the elastic plate is derived from the Bernoulli-Euler equation of motion satisfied by the plate.Using the Green’s function technique,from this boundary condition,the normal velocity of the plate is expressed in terms of the difference between the velocity potentials(unknown)across the plate.The two ends of the plate are either clamped or free.The reflection and transmission coefficients are obtained in terms of the integrals’involving combinations of the unknown velocity potential on the two sides of the plate,which satisfy three simultaneous integral equations and are solved numerically.These coefficients are computed numerically for various values of different parameters and depicted graphically against the wave number in a number of figures.

QUASI-PERIODIC WAVES AND QUASI-SOLITARY WAVES IN STRATIFIED FLUID OF SLOWLY VARYING DEPTH

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper.The model considered here consists of a two-layer incompressible constant-density inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall.The Korteweg-de Vries(KdV)equation with varying coefficients is derived with the aid of the reductive perturbation method.By using the method of multiple scales,the approximate solutions of this equation are obtained.It is found that the unevenness of bottom may lead to the generation of socalled quasi-periodic waves and quasi-solitary waves,whose periods,propagation velocities and wave profiles vary slowly.The relations of the period of quasi-periodic waves and of the amplitude,propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented.The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.

Numerical method for wave height distribution within the artificial harbor with water depth of steep variation

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湖北襄阳地区土体剪切波波速与深度的相关性研究

为给出湖北襄阳地区典型粘性土、粉土、砂土及碎石土剪切波波速与深度之间的统计公式,基于收集到的102个实测钻孔波速资料,利用多种一元回归模型对襄阳地区常见土层剪切波波速与深度之间的关系进行拟合分析和相关性研究;同时,讨论土体状态对二者关系的影响;最后,利用实测钻孔数据验证统计公式的合理性和可靠性。研究结果表明:1)除人工填土外,襄阳地区其他常规土类剪切波波速与深度之间存在较强的相关性,且离散程度随深度的增加而增大;2)同一土类随密实度或塑性状态的增大,其平均剪切波波速也变大;3)考虑土体状态可以使回归模型的预测结果更加具体、更接近真实值,但对模型拟合优度的变化具有不确定性;4)区域性对土层剪切波波速与深度的统计关系有较大影响。本文研究的襄阳地区各土类剪切波波速回归模型较全国土类模型具有更好的预测精度和误差平稳性,可为南襄盆地及周边区域地震灾害风险普查、场地土动力学性质研究和地震动参数确定等提供参考。

Correlation of the slope of the initial part of P-wave envelope,hypocenter depth and plate boundary

We investigated correlation of the slope of the initial part of P-wave envelope, hypocenter depth and plate boundaries by B-Δ method, which is used to determine epicentral distances in the Japan Meteorological Agency and Japan Railway Company earthquake early warning(EEW) systems. The Tohoku region was chosen as the study region. 19,899 strong motion data for 265 events with magnitudes in the range from 5.0 to 7.6 from KiK-net(Kiban Kyoshin network) had been collected. The coefficient c to investigate is obtained from the linear relation between log B and log Δ. Compared to the hypocenter depth, the coefficients c of events is more likely to decide by the spatial correlation of the plate boundaries. The differences are likely to be due to earthquake characteristics, since deeper events in the subducting slabs the structural effects are likely to be larger than or comparable to those for shallow crust events.

基于Hoek-Brown准则的底板破坏深度分析

为了研究煤层采动底板最大破坏深度,以1509工作面回风巷道为工程背景。利用半无限体理论建立力学模型,结合材料力学和Hoek-Brown准则,推导出最大破坏深度计算公式,并经过试验和现场监测进行验证。结果显示:底板最大破坏深度为20.3 m,且呈“水滴形”塑性区边界。底板破坏深度与岩体软硬程度呈负相关,与破碎程度呈正相关。现场监测结果显示,随底板埋深增加,采动应力影响减小;波速-垂深曲线表示底板最大破坏深度约为20 m,与理论计算基本吻合。研究成果对类似工作面有借鉴意义,有助于预测底板破坏深度,确保安全生产,并为相关领域研究提供参考。

钻芯-超声对测法判定混凝土构件烧伤深度的实践应用

准确判定混凝土构件的烧伤深度,是火作用后混凝土结构加固修复的前提。围绕某混凝土框架地下车库的火灾案例,阐述了火作用后混凝土构件的损伤特点,根据超声波测试混凝土构件表层损伤厚度的原理,提出了钻芯-超声对测法对混凝土构件烧伤深度进行测试,可为其他火灾后混凝土检测鉴定中的构件烧伤深度的测定提供借鉴。

裂纹对弹性波在混凝土中传播特性的影响研究

目的为研究混凝土中裂纹对弹性波传播特性的影响,方法采用COMSOL软件构建弹性波在不同形态混凝土裂纹中的传播数值模型,并开展带裂纹混凝土的弹性波传播试验,系统研究混凝土裂纹特征(裂纹角度、裂纹深度、裂纹距声发射源距离、裂纹数量等)与弹性波传播之间的关系。结果研究发现:数值模拟结果与试验现象吻合较好,验证了数值模型的可靠性。其中,裂纹数量和裂纹距声发射源距离的变化对回波峰值影响较大,距声发射源2 cm裂纹的回波峰值最大,为3.46×10^(6)Pa;对波速和信号损伤影响最大的因素为裂纹深度,60 mm深度裂纹损伤率高达62.14%、波速衰减至2590.674 m/s,其次为裂纹数量,信号损伤率达31.28%。基于不同裂纹特征的波速变化规律,对裂纹角度和深度的波速关系模型进行回归分析,回归曲线拟合优度较好,R^(2)均大于0.95。结论混凝土中存在的裂纹和裂纹特征对弹性波传播具有显著影响,本文研究结果为混凝土无损检测技术提供了理论支撑。