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.

Empirical formula for wave length of ocean wave in finite depth water

In this paper, function characteristics of dispersion of ocean wave in finite depth water were analyzed systematically. The functional form of the fitting function is reasonably proposed, in which the parame- ters are optimally determined by the least square method (LSM). For infinitely deep and extremely shallow water, the fitting function fits strictly the dispersion to be fitted. A new technique is presented in application of LSM. An empirical formula with maximum error of less than 0.5% for computing wavelength in finite depth water is presented for practical applications.

EFFECT OF WATER DEPTH ON WIND-WAVE FREQUENCY SPECTRUM Ⅰ.SPECTRAL FORM

Wen et al.’s odhed dewiogh to obtain wind-wave frequency spectrum in deep was used toderive the spectrum in finite depth water. The spedrum S（ω） （ω bein angular frequency） when normalizedwith the zeroth moment m and peak frequercyω。 contains in adrition to the peakness factor P=ω。S（ω。）/m。, a twth parameter n=（2πm。）1/2d （d being water depPth）, so the spatrum behavior can bestudies for different ware growth stages and water depths.

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.

Depth inversion in coastal water based on SAR image of waves

波浪数字光谱技术被建议借助于波浪的合成的孔雷达(SAR ) 图象检索沿海的水深度。基于海浪的一般分散关系，在改变水深度上的一个表面波浪的波长变化能从 SAR 被导出。接近波浪并且用海浪的一般分散关系的 SAR 图象的分析，遥感深测术的这种间接技术在福建省被用于 Xiapu 的一个沿海的区域，中国。结果证明这种技术为有可变的水深度的近岸的区域特别对沿海的水合适。

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).

Exciting Forces for a Wave Energy Device Consisting of a Pair of Coaxial Cylinders in Water of Finite Depth

二同轴的垂直柱体一个是骑空柱体和其它在在一个透不过的水平底部上面的某距离的更大的半径的稳固的柱体，被考虑。由这二柱体的衍射的这个问题，被看作一个浮标和一个圆形的盘子的理想化，能被看作一台波浪精力设备。被这台设备创造并且转的波浪能量能适当地代替常规能量在许多应用被使用。变量的分离的方法被用来在四个清楚地识别的区域为使衍射的潜力获得分析表情。由沿着在这些区域之间的三条虚拟边界使用适当匹配条件，线性方程的一个系统被获得，它为未知系数被解决。潜力允许我们获得对两柱体起作用的令人激动的力量。令人激动的力量的集合为柱体的不同半径并且为在柱体之间的不同差距被获得。在半径和差距的变化在力量上有重要效果，这被观察。它主要被发现那令人激动的力量在更低的频率仅仅是重要的。令人激动的力量几乎消失在更高的频率。这个问题也为没有板安排的基础案例被调查，即，有仅仅漂浮的柱体的案例拴住到海床。为两安排的力量的比较被执行。以便由于巨浪运动，相应的增加的团和抑制系数照顾柱体的放射因为两柱体也被计算。所有结果图形地并且与可得到的结果相比被描绘。

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.