Uniform approximation to surface gravity waves around a breakwater on water of variable depth

The linear surface gravity wavefield around a breakwater and on water of varying depth is described by a uniform asymptotic representation. The scattering of not necessarily irrotational wave packets generated by sources at arbitrary distance from the breakwater can be treated with the technique here expounded.     

Nonlinear surface waves on a ferromagnetic fluid of variable depth

A nonlinear ray method is used to study surface waves on a ferromagnetic fluid of variable depth subject to a horizontal magnetic field, and an equation of the KdV type with variable coefficients is derived. An approximate solution of the equation representing a three-dimensional soliton with varying amplitude and phase is constructed and numerical results are presented.     

Instability of capillary-gravity waves in water of arbitrary uniform depth

The Benjamin-Feir instability of periodic capillary-gravity waves on a liquid layer of arbitrary uniform depth is investigated. When surface tension is present, there is always instability for some wavenumber and liquid depth and bounds on the sideband frequencies for unbounded amplification are derived. The results are compared with the slow modulation theory using an averaged Lagrangian.     

Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis

We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale ε (weak nonlinearity), and on the scale, σ, of the depth variation. In our model, dispersion does not affect the evolution of the wave even over the large distances that tsunamis may travel. We allow a distribution of vorticity, in addition to variable depth. Our solution is not valid for depth=O(ε^4/5); the equations here are expressed in terms of the single parameter ε^2/5σ and matched to the solution in deep water. For a slow depth variation of the background state (consistent with our model), we prove that a constant-vorticity solution exists, from deep water to shoreline, and that regions of isolated vorticity can also exist, for appropriate bottom profiles. We describe how the wave properties are modified by the presence of vorticity. Some graphical examples of our various solutions are presented.     

New families of pure gravity waves in water of infinite depth

Nonlinear periodic gravity waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. It is known that there are both regular waves (for which all the crests are at the same height) and irregular waves (for which not all the crests are at the same height). We show numerically the existence of new branches of irregular waves which bifurcate from the branch of regular waves. Our results suggest there are an infinite number of such branches. In addition we found additional new branches of irregular waves which bifurcate from the previously calculated branches of irregular waves.     

A note on Hamiltonian for long water waves in varying depth

The Hamiltonian for two-dimensional long waves over a slowly varying depth is derived. The vertical variation of the velocity field is obtained by using a perturbation method in terms of velocity potential. Employing the canonical theorem, the conventional Boussinesq equations are recovered. The Hamiltonian becomes negative when the wavelength becomes short. A modified Hamiltonian is constructed so that it remains positive and finite for short waves. The corresponding Boussinesq-type equations are then given.     

On waves due to rolling of a ship in water of finite depth

Summary The problem of the generation of waves due to small rolling oscillations of a thin vertical plate partially immersed in uniform finite-depth water is investigated here by utilizing two mathematical methods assuming the linearised theory of water waves. In the first method, the use of eigenfunction expansion of the velocity potentials on the two sides of the plate produces the amplitude of wave motion at infinity in terms of an integral involving the unknown horizontal velocity across the gap, and also in terms of another integral involving the unknown difference of the potential across the plate. These unknown functions satisfy two integral equations. Any one of these, when solved numerically, can be used to compute the amplitude of the wave motion set up at either infinity on the two sides of the plate for various values of the wave number.In the second method, the problem is formulated in terms of a hypersingular integral equation involving the difference of the potential function across the plate. The hypersingular integral equation is solved numerically, and its numerical solution is used to compute the wave amplitude at infinity. The two methods produce almost the same numerical results. The results are illustrated graphically, and a comparison is made with the deep-water result. It is observed that the deep-water result effectively holds good if the plate is partially immersed to the order of one-tenth of the bottom depth.This work was initiated when the first Author was visiting Mathematics Department, Indian Institute of Science, Bangalore. It was partially supported by DST, and by CSIR. The authors take this opportunity to thank the Managing Editor for his suggestions to improve the paper in the present form.     

Particle trajectory and mass transport of finite-amplitude waves in water of uniform depth

A set of governing equations in Lagrangian form is derived for propagating gravity waves in water of uniform depth. The Lindstedt–Poincaré perturbation method is used to obtain approximations up to fifth order. Recognizing the Lagrangian frequency to be a position function for all particles is a key to find these higher-order approximations. The present solution has zero pressure at the free surface and satisfies exactly the dynamic boundary condition. Under the present approximations, the Lagrangian frequency is composed of two parts. The first part is constant for all particles and equivalent to the term in the fifth-order Stokes' wave theory [J.D. Fenton, A fifth-order Stokes theory for steady waves, J. Waterway, Port, Coastal Ocean Eng. 111 (1985) 216–234]. The second part is a function of the depth. All the particles move as open (nonclosed) loops and have mean drift displacements that decrease exponentially with the water depth. Thus, a new fourth-order mass transport velocity is found.     

On the simultaneous effect of viscosity and variable depth on the propagation of solitary waves

The Korteweg-de Vries equation modified by both the effect of viscosity and the effect of variable depth is derived and the evolution of a solitary wave in the presence of both of them is discussed by the method of multiple scales. The analysis has been focused on the eventual balance between both effects, which might allow a solitary wave to preserve its initial shape. It has been shown that cither the amplitude or the length or the speed of the wave can only be preserved and the corresponding forms of the channel have been found.     

The two-dimensional problem of steady waves on water of finite depth: regimes without waves of small amplitude

The two-dimensional problem of steady waves on water of finite depth is considered without assumptions about periodicity and symmetry of waves. A new form of Bernoulli's equation is derived, and it involves a new bifurcation parameter which is the product of the Froude number μ and the rate of flow ω. The main result obtained from this equation is the absence of waves, having sufficiently small amplitude, provided $|\mu \omega |>1$. To cite this article: V. Kozlov, N. Kuznetsov, C. R. Mecanique 333 (2005).     

Generation of two-dimensional steep water waves on finite depth with and without wind

This paper reports on a series of numerical simulations designed to investigate the action of wind on steep waves and breaking waves generated through the mechanism of dispersive focusing on finite depth. The dynamics of the wave packet propagating without wind at the free surface are compared to the dynamics of the packet propagating in the presence of wind. Wind is introduced in the numerical wave tank by means of a pressure term, corresponding to the modified Jeffreys' sheltering mechanism. The wind blowing over a strongly modulated wave group due to the dispersive focusing of an initial long wave packet increases the duration and maximal amplitude of the steep wave event. These results are coherent with those obtained within the framework of deep water. However, steep wave events are less unstable to wind perturbation in shallow water than in deep water.Furthermore, a comparison between experimental and numerical wave breaking is presented in the absence of wind. The numerical simulations show that the wind speeds up the wave breaking and amplifies slightly the wave height.The wall pressure during the runup of the steep wave event on a vertical wall is also investigated and a comparison between experimental and numerical results is provided.     

RANDOM VARIABLE WITH FUZZY PROBABILITY

IntroductionThematteroffuzzyrandomisusuallyclassifiedintothreeparts:fuzzyevent_exactitudeprobabilitymodel,crispevent_fuzzyprobabilitymodel,fuzzyevent_fuzzyprobabilitymodel.Uptonow ,themajorityofresearchesandachievementsareaboutthefirstmodel,buttherearefewresearchesandachievementsaboutthesecondmodel[1- 4].Whenresearchingthesecondmodel,thefuzzyprobabilityaboutcertainbasiceventsmustbegivenfirst,andthefuzzinessofvariablesiscausedbythefuzzyprobabilityaboutthesebasicevents.Theabove_mentionedrelation…     

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 constantdensity 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 so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic 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.Project Supported by National Natural Science Foundation of China.     

Fourth order evolution equations and stability analysis for stokes waves on arbitrary water depth

Fourth order evolution equations have been derived for three-dimensional Stokes waves on arbitrary water depth. In deep water the equations reduce to those of Dysthe, and on finite depth the third order terms agree with those of Benney and Roskes, Hasimoto and Ono and Davey and Stewartson. The results of the stability analysis for uniform waves based on the new arbitrary depth expressions are superior to those based on the finite depth approximation and they agree fairly well with the exact calculations of McLean. It is demonstrated that dimensionless water depth as well as wave steepness influences the applicability of the deep water stability expressions.     

A mixed FE-BE method for coupled vibration of gravity dam-reservoir system with variable water depth

To solve the coupled vibration of a gravity dam-reservoir system with variable water depth by using a hybrid element method, the fluid region with variable water depth needs to be discretized by FE meshes. However, such a method asks for a great computational cost owing to the excessive unknowns, especially when the fluid region with variable water depth is relatively large. To overcome the shortcoming, a refined boundary element method is proposed to analyze the fluid field, in which only the discretization for the boundary of the variable depth region is required. But as a basis of this approach, it is necessary to construct a new Green's function corresponding to an infinite strip region. The problem is solved as the first step in this paper by employing Fridman's operator function theory, and then a mixed FE-BE formulation for analyzing the free vibration of the gravity damreservoir system is derived by means of the coupling conditions on the dam-reservoir interface. Finally, a numerical example is provided to illustrate a great improvement of the method developed herein over the hybrid element method. The project supported by the National Key Research Plan of China.     

Harmonic components of viscous fluid waves in uniform depth

Summary The first harmonic of a sinusoidal progressing wave is considered here. Only gravity waves are dealt with, surface tension effects being ignored.

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Sommario Si esaminano in dettaglio le caratteristiche della prima armonica di un'onda sinusoidale progressiva. L'analisi riguarda le onde dominate dalla gravità, e prescinde quindi dalla tensione superficiale.

     

Unsteady waves in a viscous fluid of finite depth

Here we study the plane and three-dimensional problems of unsteady waves which arise on the surface of a viscous fluid of finite depth under the influence of a velocity pulse applied on the bottom of the basin.The problem is considered as the simplest scheme for studying, with account for the effect of viscosity, the propagation of waves of the tsunami type which result from an underwater shock.Similar problems on the propagation of waves which arise from initial surface disturbances are considered in [1–9].     

Hydrodynamic forces and moments due to interaction of linear water waves with truncated partial-porous cylinders in finite depth

An exact analytical method is employed for studying the diffraction problems in an ocean due to the presence of a specific type of cylinders. In this current work, two models are studied: (i) a floating surface-piercing truncated partial-porous cylinder, (ii) a surface-piercing truncated partial-porous cylinder placed at the bottom. In both cases, the configuration of the composite cylinder is such that it consists of an impermeable inner cylinder rising above the free surface and a coaxial truncated porous cylinder around the lower part of the inner cylinder with the top of the porous cylinder being impermeable. By using linear water wave theory, a three-dimensional representation of the problem is developed based on eigenfunction expansion method. The condition on the porous boundary is defined by applying Darcy’s law. Pressure and velocity satisfy continuity conditions across the linear interface between the adjacent fluid domains. Hydrodynamic force, moment and wave run-up are calculated by using the velocity potentials. Comparisons are carried out with results of wave diffraction by a floating and bottom-mounted compound cylinder, i.e., when the whole cylinder is non-porous. Handy agreements are observed from these comparisons. Through numerical tests, various experiments are carried out to investigate the impact of various parameters, such as porous coefficients, draft ratio, the ratio of inner and outer radii, the water depth etc., on hydrodynamic force, moment and wave run-up. The results clearly indicate that an appropriate optimal ratio for various parameters may be considered in designing practical ocean structures with minimum adverse hydrodynamic effect. The appearance of resonance in the results and role of porosity in mitigating resonance effect are explained. Proposal to select various appropriate parameters for the best possible effect is put forward.     

A new method for the calculation of steady periodic capillary-gravity waves on water of arbitrary uniform depth

This paper presents a method for the calculation of steady periodic capillary-gravity waves on water of arbitrary uniform depth. The method developed by Debiane and Kharif in 1997 for infinite depth is extended to finite depth. The water-wave problem is reduced to a system of nonlinear algebraic equations which is solved using Newton's method. For the resonant configurations, the method does not suffer from the Wilton's failures and is valid for all depths. In addition, it is shown that the method allows the computation of solitary waves and generalized solitary waves.