Numerical investigation of oscillations induced by submerged sliding masses within a harbor of constant slope

Oscillations within a rectangular harbor of constant slope induced by submerged sliding masses are investigated numerically based on Boussinesq-type equations and results are used to reveal the characteristics of the generated oscillations. The numerical result of each transverse eigenfrequency is very close to the theoretical prediction and the spatial structure of each mode of the oscillations may also be well captured by the existing analytical solutions based on shallow water equations. The investigation shows that relatively small-scale sliding masses whose width is small compared with the harbor width may induce obvious transverse oscillations. The predominant transverse components are those with small mode numbers when the solid slides start moving from the backwall. In comparing the oscillations induced by the slides of constant velocity and those accelerated by gravity force with bottom friction, it is observed that the movements accelerated by gravity force may facilitate the development of very low transverse oscillation modes while those with constant velocity may also be in favor of the higher ones. The augmentation of the sliding velocity along the constant slope may shift the amplitudes of the oscillation components to smaller values, which corresponds to the physical understandings of the waves generated by underwater sliding masses or landslides. While the sliding masses may not act on an isolated point of the bottom but follow a certain trajectory along the harbor, the transverse oscillations induced by them are sensitive to their position of departure in both the cross-harbor direction and the offshore direction. Longitudinal oscillations may be induced by relatively large sliding masses of harbor width on a constant slope within the harbor. Although the longitudinal oscillations may not reach a steady state without forcing terms at the entrance of the harbor, some patterns of several low-mode ones occur and wavelet spectra are used to analyze their evolutions and comparisons are made with theoretical predictions. It is revealed that the longitudinal oscillations are also sensitive to the moving velocity and initial location of the sliding masses.     

Theoretical Analysis of Harbor Resonance in Harbor with An Exponential Bottom Profile

The general features of oscillations within a rectangular harbor of exponential bottom are investigated analytically. Based on the linear shallow water approximation, analytical solutions for longitudinal oscillations induced by the incident perpendicular wave are obtained by the method of matched asymptotics. The analytic results show that the resonant frequencies are shifted to larger values as the water depth increases and the oscillation amplitudes are enhanced due to the shoaling effect. Owing to the refraction effect, there could be several transverse oscillation modes existing in when the width of the harbor is on the order of the oscillation wavelength. These transverse oscillations are similar to standing edge waves, and there are m node lines in the longshore direction and n node lines running in the offshore direction corresponding to mode (n, m). Furthermore, the transverse eigen frequency is not only related to the width of the harbor, but also to the boundary condition at the backwall and the bottom shape.     

Numerical investigation of oscillations within a harbor of constant slope induced by seafloor movements

A numerical model, which can simulate wave generation and propagation is developed to simulate oscillations induced by seafloor movements inside a harbor of constant slope, and once verified and then validated through comparison with experimental results, the numerical results are used to examine the analytic solutions presented in Wang et al. (Wang, G., Dong, G., Perlin, M., Ma, X., Ma, Y., 2011. An analytic investigation of oscillations within a harbor of constant slope. Ocean Engineering 38, 479–486). Small-scale seafloor movement usually induces small longitudinal oscillations, but evident larger transverse oscillations. These transverse oscillations are sensitive to the location of the moveable seafloor. The numerical result of each transverse eigen frequency compares well with the theoretical solution; in addition the spatial structure of each mode is also well-captured by the theory. Furthermore, evident/larger longitudinal oscillations induced by large-scale seafloor movements are simulated, and the numerical resonant frequencies agree favorably with the analytical solutions. These longitudinal oscillations are sensitive to the horizontal location of the moveable seafloor.     

Study on Influences of Fringing Reef on Harbor Oscillations Triggered by N-Waves

Influences of topographic variations of the offshore fringing reef on the harbor oscillations excited by incident Nwaves with different amplitudes and waveform types are studied for the first time. Both the propagation of the Nwaves over the reef and the subsequently-induced harbor oscillations are simulated by a Boussinesq-type numerical model, FUNWAVE-TVD. The present study concentrates on revealing the influences of the plane reef-face slope,the reef-face profile shape and the lagoon width on the maximum runup, the wave energy distribution and the total wave energy within the harbor. It shows that both the wave energy distribution uniformity and the total wave energy gradually increase with decreasing reef-face slope. The profile shape of the reef face suffering leading-elevation Nwaves(LEN waves) has a negligible impact on the wave energy distribution uniformity, while for leading-depression N-waves(LDN waves), the latter gradually decreases with the mean water depth over the reef face. The total wave energy always first increases and then decreases with the mean water depth over the reef face. In general, the total wave energy first sharply decreases and then slightly increases with the lagoon width, regardless of the reef-face width and the incident waveform type. The maximum runup subjected to the LEN waves decreases monotonously with the lagoon width. However, for the LDN waves, its changing trend with the lagoon width relies on the incident wave amplitude.     

A composite numerical model for wave diffraction in a harbor with varying water depth

A composite numerical model is presented for computing the wave field in a harbor. The mild slope equation is discretized by a finite element method in the domain concerned. Out of the computational domain, the water depth is assumed to be constant. The boundary element method is applied to the outer boundary for dealing with the infinite boundary condition. Because the model satisfies strictly the infinite boundary condition, more accurate results can be obtained. The model is firstly applied to compute the wave diffraction in a narrow rectangular bay and the wave diffraction from a porous cylinder. The numerical results are compared with the analytical solution, experimental data and other numerical results. Good agreements are obtained. Then the model is applied to computing the wave diffraction in a square harbor with varying water depth. The effects of the water depth in the harbor and the incoming wave direction on the wave height distribution are discussed.     

A composite numerical model for wave diffraction in a harbor with varying water depth

A composite numerical model is presented for computing the wave field in a harbor. The mild slope equation is discretized by a finite element method in the domain concerned. Out of the computational domain, the water depth is assumed to be constant. The boundary element method is applied to the outer boundary for dealing with the infinite boundary condition. Because the model satisfies strictly the infinite boundary condition, more accurate results can be obtained. The model is firstly applied to compute the wave diffraction in a narrow rectangular bay and the wave diffraction from a porous cylinder. The numerical results are compared with the analytical solution, experimental data and other numerical results. Good agreements are obtained. Then the model is applied to computing the wave diffraction in a square harbor with varying water depth. The effects of the water depth in the harbor and the incoming wave direction on the wave height distribution are discussed.     

The applicability of the shallow water equations for modelling violent wave overtopping

The shallow water equations (SWE) have been used to model a series of experiments examining violent wave overtopping of a near-vertical sloping structure with impacting wave conditions. A finite volume scheme was used to solve the shallow water equations. A monotonic reconstruction method was applied to eliminate spurious oscillations and ensure proper treatment of bed slope terms. Both the numerical results and physical observations of the water surface closely followed the relevant Rayleigh probability distributions. However, the numerical model overestimated the wave heights and suffered from the lack of dispersion within the shallow water equations. Comparisons made on dimensionless parameters for the overtopping discharge and percentage of waves overtopping between the numerical model and the experimental observations indicated that for the lesser impacting waves, the shallow water equations perform satisfactorily and provide a good alternative to computationally more expensive methods.     

Bound waves and triad interactions in shallow water

Boussinesq type equations with improved linear dispersion characteristics are derived and applied to study wave-wave interaction in shallow water. Weakly nonlinear solutions are formulated in terms of Fourier series with constant or spatially varying coefficients for two purposes: to derive higher order boundary conditions for regular and irregular wave trains and to derive evolution equations on constant or variable water depth. Wave transformation of monochromatic, bichromatic and irregular waves is studied and comparison with measurements and direct time domain solutions shows good agreement. The improvement relative to classical models from the literature is discussed.     

Effects of depth discontinuity on harbor oscillations

The effects of water depth discontinuity near the harbor mouth on harbor oscillations are examined. Linear long-wave equations are used as the basis of the present study. For simplicity, only the normal incident waves are considered. Assuming that the harbor mouth is small in comparison with wavelength, the method of matched asymptotic expansion is employed to obtain the ocean impedance and the harbor responses. It is found that the incident waves can be trapped over the depth discontinuity which causes large oscillations near the harbor mouth. The radiation damping also decreases because of the appearance of the depth discontinuity, which leads to large amplifications at the lowest mode.     

Wave transformation by axisymmetric three-dimensional bathymetric anomalies with gradual transitions in depth

The development of an analytic model (Axisymmetric 3-D Step Model) for the propagation of linear water waves over an axisymmetric bathymetric anomaly in arbitrary water depth is presented. The Axisymmetric 3-D Step Model is valid in a region of uniform depth containing an axisymmetric bathymetric anomaly with gradual transitions in depth allowed as a series of steps approximating arbitrary slopes. The velocity potential is calculated by applying matching conditions at the interface between regions of constant depth. The velocity potential obtained determines the wave field in the domain for monochromatic incident waves of linear form. A second analytic model (3-D Shallow Water Exact Model) is developed for comparison within the shallow water limit.The Axisymmetric 3-D Step Model determines the wave transformation caused by the processes of wave refraction, diffraction and reflection. Wave transformation is demonstrated in plots of the relative amplitude for bathymetric anomalies in the form of pit or a shoal, highlighting areas of wave sheltering and wave focusing. Anomalies of constant volume, but variable cross-section are employed to isolate the effect of the transition slope on the wave transformation.Comparisons to a shallow water model, numerical models, and experimental data verify the results of the Axisymmetric 3-D Step Model for several bathymetries including both pits and shoals. Also included are estimates of the energy reflection induced by an axisymmetric depth anomaly. The 3-D Axisymmetric Step Model has been applied previously to account for nearshore transformation (sloping bathymetry) and associated shoreline changes [C.J. Bender, R.G. Dean, Coastal Engineering 51 (2004) 1143].     

一个改进的二阶Boussinesq方程模型在线性波浪反射问题中的适用性研究

基于改进型的二阶Boussinesq方程,在交错网络下建立数值模型.利用模型模拟波浪在常水深情况下的传播,波浪反射系数均低于2％.利用该模型模拟波浪在平斜坡前的反射,并将数值结果与解析解进行对比.结果表明,对于相对水深较大情况,坡度较陡时模拟结果明显偏大;对 于相对水深较小情况,坡度超过1:1时,数值结果仍与解析解有....     

A New Approach to High-Order Boussinesq-Type Equations with Ambient Currents

A new approach to high-order Boussinesq-type equations with ambient currents is presented. The current velocity is assumed to be uniform over depth and of the same magnitude as the shallow water wave celerity. The wave velocity field is expressed in terms of the horizontal and vertical wave velocity components at an arbitrary water depth level. Linear operators are introduced to improve the accuracy of the kinematic condition at the sea bottom. The dynamic and kinematic conditions at the free surface are expressed in terms of wave velocity variables defined directly on the free surface. The new equations provide high accuracy of linear properties as well as nonlinear properties from shallow to deep water, and extend the applicable range of relative water depth in the case of opposing currents.     

An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.     

Wave Run-up on A Vertical Seawall Protected by An Offshore Submerged Barrier

Submerged barriers are constructed in coastal zones for shoreline or harbor protection or to prevent the beach erosion. In the present study, the wave run-up on a vertical seawall protected by a submerged barrier is analyzed. The physical configurations include a rigid barrier and a long channel of finite depth. For linear water waves, by matching the velocity along the barrier and along the gap, the systems of linear equations about the velocity potentials are obtained. The wave run-up is further analyzed ...     

海啸波传播的线性和非线性特征及近海陆架效应影响的数值研究

浅水方程被广泛应用于海啸预警报业务及研究,而针对线性浅水方程与非线性浅水方程在不同海区水深地形条件下的适用范围、计算效率问题是海啸研究人员急需了解的。本文应用基于浅水方程的海啸数值预报模型就海啸波在南海、东海传播的线性、非线性特征以及陆架对其传播之影响进行了数值分析研究。海啸波在深水的传播表征为强线性特征,此时线性系统对海啸波幅的模拟计算具有较高的精度和效率,而弱的非线性特征及弱的色散特征对海啸波幅的预报影响甚微,可以忽略不计。海啸波传播至浅水大陆架后受海底坡度变化、海底粗糙度等因素影响,波动的非线性效应迅速传播、积累,与线性浅水方程计算的海啸波相比表现出较大差异,主要表现为:在南海区,水深小于100m时,海啸波首波以后的系列波动非线性特征比较明显,两者波幅差别较大,但首波波幅的区别不大,因此对于该区域在不考虑海啸爬高的情况下,应用线性系统计算得到的海啸波幅也可满足海啸预警报的要求;在东海区由于陆架影响,海啸波非线性特征明显增强,水深小于100m区域,首波及其后系列波波幅均差异较大,故在该区域必须考虑海啸波非线性作用。本文就底摩擦项对海啸波首波波幅的影响进行了数值对比分析,结果表明:底摩擦作用对海啸波首波波幅影响仅作用于小于100m水深。最后,该文通过敏感性试验,初步分析了陆架宽度及陆架边缘深度对海啸波波幅的影响,得出海啸波经陆架传播共振、变形后,海啸波幅的放大或减小与陆架的宽度及陆架边缘水深有关。     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

Numerical Simulation of Nonlinear Three Dimensional Waves in Water of Arbitrary Varying Topography

—The numerical simulation is based on the authors＇high-order models with a dissipative termfor nonlinear and dispersive wave in water of varying depth.Corresponding finite-difference equations andgeneral conditions for open and fixed natural boundaries with an arbitrary reflection coefficient and phaseshift are also given in this paper.The systematical tests of numerical simulation show that the theoreticalmodels,the finite-difference algorithms and the boundary conditions can give good calculation results forthe wave propagating in shallow and deep water with an arbitrary slope varying from gentle to steep.     

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.