Hamiltonian Boussinesq formulation of wave–ship interactions

In this paper a new approach is described for the fully nonlinear treatment of the dynamic wave–ship interaction for potential flows. A major reduction of computational complexity is obtained by describing the fluid motion in horizontal variables only, the surface elevation and the potential at the surface. In such Boussinesq type of equations, the internal fluid motion is not calculated, but modeled in a consistent approximative way. The equations for the wave–ship interaction are based on a Lagrangian variational principle, leading to the formulation of the coupled system as a Hamiltonian system. With the ship position and orientation as canonical coordinates, the canonically conjugate momentum variables are the sum of the ship momemta and the fluid momenta. A beneficial consequence of this is that the momentum exchange between fluid and ship will be described without the need to calculate the pressure, which simplifies the numerical implementation of the equations considerably. Provided that the potentials with mixed Dirichlet–Neumann data can be calculated, the presented ship dynamics can be inserted in existing free surface flow solvers.     

Finite element modelling of surface waves in a channel

A variational formulation of the vertically-integrated differential equations for free surface wave motion is presented. A finite element model is derived for solving this nonlinear system of hydrodynamic equations. The time integration scheme employed is discussed and the results obtained demonstrate its good stability and accuracy.Several applications of the model are considered: the first problem is an open channel of uniform depth and the second an open channel of linearly varying depth. The ‘inflow’ boundary condition is prescribed in terms of the velocity which represents a wavemaker and/or a flow source, while the ‘outflow’ boundary condition is specified in terms of the water elevation. The outflow condition is adjusted for two cases, a reflecting boundary (finite channel) and a non-reflecting boundary (open-ended channel). The latter boundary condition is examined in some detail and the results obtained show that the numerical model can produce the non-reflecting boundary that is similar to the analytical radiation condition for waves. Computational results for a third problem, involving wave reflection from a submerged cylinder, are also presented and compared with both experimental data and analytical predictions.The simplicity and the performance of the computational model suggest that free surface waves can be simulated without excessively complicated numerical schemes. The ability of the model to simulate outflow boundary conditions properly is of special importance since these conditions present serious problems for many numerical algorithms.     

On the propagation of elastic waves in two-phase media

At the present time a number of papers has been already devoted to the dynamics of two-phase media. One may mention the papers by Frenkel' [1], Rakhmatulin [2], Biot [3,4], Zwikker and Kosten [5], and others. However, the basic problem of the setting up of the equations of motion in two-phase media still cannot be considered solved and requires additional study and experimental verification.

This paper is concerned with the study of the simplest case of motion, which is the propagation of elastic waves in a homogeneous isotropic medium consisting of a solid and a fluid phase. The problems of the reflection of plane waves and surface waves at the free boundary of the half-space are solved. It is shown that the stress-strain relations established by Frenkel' are equivalent to the analogous relations proposed by Biot and that the equations of motion of the latter are more general.     

一个二流体系统中非线性水波的Hamilton描述

讨论了一个二流体系统中非线性水波的Hamilton描述,该系统由水平固壁之上的两层常密度不可压无粘流体组成,上表面为自由面.文中将速度势函数展开成垂向坐标的幂级数,在浅水长波的假定下,取下层流体的“动厚度”与上层流体的“折合动厚度”为广义位移、界面上和自由面上的速度势为广义动量,根据Hamilton原理并运用Legendre变换导出该系统的Hamilton正则方程,从而将单层流体情形的结果推广到分层流体的情形.     

Hamiltonian long‐wave expansions for free surfaces and interfaces

The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature. From the formulation, we develop a Hamiltonian perturbation theory for the long‐wave limits, and we carry out a systematic analysis of the principal long‐wave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker (28), Benjamin (3, 4), Ono (26), and many others. Our considerations include the Boussinesq and Korteweg–de Vries (KdV) regimes over finite‐depth fluids, the Benjamin‐Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long‐wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface. Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa (6, 7), we also derive novel systems of nonlinear dispersive long‐wave equations in the large‐amplitude, small‐slope regime. Our formulation of the dynamical free‐surface, free‐interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations. © 2005 Wiley Periodicals, Inc.     

Reflection of plane waves from a rigid wall and a free surface in a transverse isotropic medium

For the general solution of two-dimensional equations of dynamics of a transverse isotropic medium with the Carrier–Gassmann condition, we give a representation in terms of two resolvent functions satisfying two separate wave equations. The problem of reflection of plane waves from a rigid wall and a free surface is solved. The coefficients of reflection and transformation of the plane waves are found. These formulas yield a solution for isotropic media too. Some special cases are consideredwhere the shapes (amplitudes) of the reflectedwaves are not uniquely determined, but linearly related with the shape of the incident wave.     

Stability of reflection and refraction of shocks on interface

In this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media. The attack will produce a reflected wave and a refracted wave, and also let the interface deflected. In this paper we will mainly study the case, when the reflected wave is a shock, and the flow between the reflected wave and the refracted shock is relatively subsonic. Our result indicates that the wave structure and the flow field for the reflection-refraction problem in this case is conditionally stable.To describe the motion of the fluid we use the inviscid Euler system as the mathematical model. The reflection-refraction problem can be reduced to a free boundary value problem, where the unknown reflected shock and refracted shock are free boundaries, and the deflected interface is also to be determined. In the proof of the existence and the stability of the corresponding wave structure we apply the Lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic-hyperbolic composite system in its principal part. Meanwhile, some efficient weighted Sobolev estimates are established to derive the existence for corresponding nonlinear problems.     

On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance

The problem of orbital stability of a periodic motion of an autonomous two-degreeof- freedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourth-order resonance. It is believed that this case is the critical one for the resonance, when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion.Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive.     

Exact Free Surface Flows for Shallow Water Equations II: The Compressible Case

Exact free surface flows with shear in a compressible barotropic medium are found, extending the authors’ earlier work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The “shallow water” equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double psi-series. Simple wave and time-dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping traveling wave in the horizontal direction. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, when γ = 1, we again find simple wave and time-dependent solutions.     

低重环境下液体非线性晃动的稳态响应

在俯仰激励作用下,圆柱贮箱中液体晃动存在平面运动、旋转运动和平面运动中的旋转运动等,而这些运动的稳定、不稳定区间的分界线与贮箱的半径、充液深度、重力强度、表面张力系数和晃动阻尼等基本系统参数有关．据此,首先建立了液体非线性晃动的微分方程组,并借助变分原理建立了液体压力体积分形式的Lagrange函数;然后将速度势函数在自由液面处作波高函数的级数展开,通过变分从而导出自由液面运动学和动力学边界条件非线性方程组;最后用多尺度法求解非线性方程组,就重力强度对圆柱形贮箱中液体非线性晃动的全局稳态响应的影响进行了详细的理论分析,并发现系统软硬特性的变化、跳跃和滞后等非线性现象．     

Equations and interface conditions for acoustic phenomena in porous media

Acoustic phenomena are considered in a porous medium made of a rigid matrix hollowed by a lattice of periodically distributed canals of arbitrary shape, the period of the lattice being small compared with the wave length. If dissipation phenomena are taken into account in the solution of the Navier-Stokes equations at the microscopic level, the averaged acoustic variables satisfy equations analogous to those of acoustics in a fictitious homogeneous dissipative (and possibly anisotropic) medium whose properties are described. Interface phenomena between such a medium and a free fluid are studied. As a result, the interface conditions for the averaged variables on the one hand and the free fluid variables on the other hand are the continuity of pressure and normal velocity. The acoustic impedance of the surface is not a constant.     

Waves in a Perfectly Conducting Fluid Filling a Half-Space

The constant, maximal, energy preserving boundary conditionsfor the equations of magnetohydrodynamics in a perfectly conductinghalf-space give rise to two essentially different selfadjointoperators in the case when the external magnetic field is orthogonalto the boundary and exactly one such operator when the externalfield is parallel to the boundary. Neither of these problemsadmits surface waves. For a normalized external field, the generalizedeigenfunction expansion is given below. It is shown that, inthe second case, the modes are not coupled by the boundary,while for only one boundary condition for the orthogonal fieldis the wave motion essentially that of free space (in the sensethat solutions are delivered by the group which determines solutionsfor the free space problem for special initial data). The Alvnwave in the parallel field case acts as a grazing wave. Asymptoticwave motion for perturbed problems (inhomogeneous media) isinvestigated as well as local decay of energy (this is not altogethertrivial, since the operators involved are never coercive evenoff their null spaces).     

横观各向同性饱和地基的三维动力响应

首先引入位移函数,将直角坐标系下横观各向同性饱和土Biot波动方程转化为2个解耦的六阶和二阶控制方程;然后基于双重Fourier变换,求解了Biot波动方程,得到以土骨架位移和孔隙水压力为基本未知量的积分形式的一般解,并用一般解给出了饱和土总应力分量的表达式．在此基础上系统研究了横观各向同性饱和半空间体的稳态动力响应问题,考虑表面排水和不排水两种情况,得到了半空间体在任意分布的表面谐振荷载作用下,表面位移的稳态动力响应,文末给出了算例．     

Vibrations of the surface of an elastic double-layered half-space with a periodic system of cracks

The two-dimensional problem of the normal incidence of a plane transverse wave from the far field on to the free surface of an elastic double-layered half-space, comprising a homogeneous layer attached to a semi-infinite base of a different elastic material, is considered. At the boundary between the two media there is a system of plane cracks, arranged periodically along the separation line, which models the fracture zone at the interface between dense solid rock and soft sedimentary rock. The effect of the fractures on the transmission of a transverse seismic wave generated by a deep-focus earthquake, and of the type of vibrations of the free surface of the ground that result, is studied. It is difficult to predict whether the seismic wave is strengthened or weakened by the fracture zone. The effect of the system of cracks on vibrations of the free surface largely depends on the physical and geometrical parameters and, primarily, on the vibration frequencies.     

Remarks on a Strongly Nonlinear Model for Two‐Layer Flows with a Top Free Surface

We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two‐layer flows with a free surface proposed by Choi and Camassa [1] and Barros et al. [2]. Its solitary‐wave solutions were the object of the work by Barros and Gavrilyuk [3], who proved that such solutions are governed by a Hamiltonian system with two degrees of freedom. A detailed analysis of the critical points of the system is presented here, leading to some new results. It is shown that conjugate states for the long wave model are the same as those predicted by the fully nonlinear Euler equations. Some emphasis will be given to the baroclinic mode, where interfacial waves are known to change polarity according to different values of density and depth ratios. A critical depth ratio separates these two regimes and its analytical expression is derived directly from the model. In addition, we prove that such waves cannot exist throughout the whole range of speeds.     

Numerical study of linear and nonlinear string vibrations by means of physical discretization

To solve partial differential equations numerically, discretization of the continuous model is required and may be achieved either mathematically or physically. This paper illustrates how physical discretization of a continuous string may be accomplished by employing discrete model theory which has as its essential substance Newtonian mechanics.Typical examples of wave motion in discretized ‘linear’ and ‘non-linear’ strings are discussed. They include the transverse vibrations of a string after having been subjected to a given initial displacement, reflection and superposition of wave pulses in the string, and resonance of the string when coupled to a harmonic vibrator. The equations that arise after application of discrete model theory to these problems, describe the subsequent motion of the string, and are solved numerically by computer. In all cases the results obtained for the discrete linear string agree remarkably well with those for the corresponding continuous physical string. The stability of the solutions obtained by discretization are also investigated.     

A system of coupled partial differential equations exhibiting both elevation and depression rogue wave modes

Analytical solutions are obtained for a coupled system of partial differential equations involving hyperbolic differential operators. Oscillatory states are calculated by the Hirota bilinear transformation. Algebraically localized modes are derived by taking a Taylor expansion. Physically these equations will model the dynamics of water waves, where the dependent variable (typically the displacement of the free surface) can exhibit a sudden deviation from an otherwise tranquil background. Such modes are termed ‘rogue waves’ and are associated with ‘extreme and rare events in physics’. Furthermore, elevations, depressions and ‘four-petal’ rogue waves can all be obtained by modifying the input parameters.     

求解Hamilton约束的新算法

利用吴方法对多项式类型带约束的Hamilton系统作了研究.给出了判断系统是否正则的一个新算法.对于正则系统,可以得到Hamilton函数和运动方程,而对退化的系统给出了两个求解约束的新算法,得到带约束的Hamilton函数和运动方程.利用符号计算软件,这几个算法都可以在计算机上实现.     

Gravity waves on the surface of the sphere

Summary We propose a Hamiltonian model for gravity waves on the surface of a fluid layer surrounding a gravitating sphere. The general equations of motion are nonlocal and can be used as a starting point for simpler models, which can be derived systematically by expanding the Hamiltonian in dimensionless parameters. In this paper, we focus on the small wave amplitude regime. The first-order nonlinear terms can be eliminated by a formal canonical transformation. Similarly, many of the second order terms can be eliminated. The resulting model has the feature that it leaves invariant several finite-dimensional subspaces on which the motion is integrable. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.