Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interactionInteraction de solitons enveloppes comme modèle pour la formation de « freak waves». Partie I : Interaction à long terme

We are concerned by a special mechanism that can explain the formation of freak waves. We study numerically the long time evolution of a surface gravity wave packet, comparing a fully nonlinear model with Schrödinger-like simplified equations. We observe that the interaction between envelope solitons generates large waves. This is predicted by both models. The fully nonlinear simulations show a qualitative behaviour that differs significantly from the ones preticted by Schrödinger models, however. Indeed, the occurence of freak waves is much more frequent with the fully nonlinear model. This is a consequence of the long-time interaction between envelope solitons, which, in the fully nonlinear model, is totally different from the Schrödinger scenario. The fundamental differences appear for times when the simplified equations cease to be valid. Possible statistical models, based on the latter, should hence under-estimate the probability of freak wave formation. To cite this article: D. Clamond, J. Grue, C. R. Mecanique 330 (2002) 575–580.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Freak waves as nonlinear stage of Stokes wave modulation instability

Numerical simulation of evolution of nonlinear gravity waves is presented. Simulation is done using two-dimensional code, based on conformal mapping of the fluid to the lower half-plane. We have considered two problems: (i) modulation instability of wave train and (ii) evolution of NLSE solitons with different steepness of carrier wave. In both cases we have observed formation of freak waves.     

无黏流体与固体界面的漏瑞利波

漏瑞利波存在于半无限无黏性流体和半无限固体媒质的界面处. 首先推导流固无限各向同性介质界面处漏瑞利波的特征方程和位移及应力的解析计算公式. 然后结合典型结构通过数值计算研究了漏瑞利波特性以及位移和应力在流体和固体中的分布规律. 数值计算结果表明漏瑞利波的相速度和衰减随流固密度比的增大而增大, 在流固界面上法向位移连续而切向位移不连续. 流固密度比对固体媒质中沿垂直于漏瑞利波的传播方向的位移、正应力和剪应力有比较大的影响,而对沿漏瑞利波的传播方向的正应力几乎没影响. 为利用漏瑞利波的无损检测与评价提供了理论基础.      

Complex surface rays associated with inhomogeneous skimming and Rayleigh waves

The Rayleigh wave, that propagates at the free surface of semi-infinite anisotropic medium, is composed of three inhomogeneous partial waves, each propagating along the surface with a different attenuation along the depth. Since this wave does not exhibit an attenuation on the surface, let us call it the homogeneous Rayleigh wave. The associated slowness corresponds to the real solution of the Rayleigh dispersion equation. Besides this classical solution, an infinite number of complex solutions of the Rayleigh dispersion equation exits. For such particular Rayleigh waves, the slowness vector, i.e. the identical component on the surface of the slowness of each partial waves, is taken to be complex. Thus, these Rayleigh waves are attenuated on the surface and as shown here, their attenuation is normal to the ray direction (or the energy velocity direction). Similarly to the infinite inhomogeneous plane waves which can be associated with complex rays, we call these waves, inhomogeneous Rayleigh waves. We use the inhomogeneous skimming waves, which are inhomogeneous plane waves, and the inhomogeneous Rayleigh waves to explain differently the usual diffraction phenomena on the free surface which cannot be explained by the real ray theory. For example, the arrival time of the wave packet observed beyond the cusp is in perfect accordance with the arrival time of some specific inhomogeneous Rayleigh waves. We show that these results are in agreement with the computation of the Green function. They apply to the theory of surface waves in linear elastodynamics with intrinsic anisotropy as well as to the theory of surface waves in linearised (incremental) elastodynamics with strain-induced anisotropy (also known as small-amplitude waves superimposed on the large static homogeneous deformation of a non-linear solid).     

Self-switching of displacement waves in elastic nonlinearly deformed materialsSelf-commutation d'ondes de déplacement dans les milieux élastiques non-linéaires

The problem of self-switching plane waves in elastic nonlinearly deformed materials is formulated. Reduced and evolution equations, which describe the interaction of two waves the power pumping wave and the faint signal wave are obtained. For the case of wave numbers matching the pumping and signal waves, a procedure of finding the exact solution of evolution equations is described. The solution is expressed by elliptic Jacobi functions. The existence of the power wave self-switching is shown and commented. To cite this article: J. Rushchitsky, C. R. Mecanique 330 (2002) 175–180.     

Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space

In this paper we consider the propagation of Rayleigh surface waves in an exponentially graded half-space made of an isotropic Kelvin-Voigt viscoelastic material. Here we take into account the effect of the viscoelastic dissipation energy upon the corresponding wave solutions. As a consequence we introduce the damped in time wave solutions and then we treat the Rayleigh surface wave problem in terms of such solutions. The explicit form of the secular equation is obtained in terms of the wave speed and the viscoelastic inhomogeneous profile. Furthermore, we use numerical methods and computations to solve the secular equation for some special homogeneous materials. The results sustain the idea, existent in literature on the argument, that there is possible to have more than one surface wave for the Rayleigh wave problem.     

Rayleigh waves in orthotropic fluid-saturated porous media

In this paper, we are interested in the propagation of Rayleigh waves in orthotropic fluid-saturated porous media. This problem was investigated by Liu and Liu (2004). The authors have derived the secular equation of the wave but that secular equation is still in implicit form. The main aim of this paper is to derive explicit secular equation of the wave. By employing the method of polarization vector, the secular equations of Rayleigh waves in explicit form is obtained. This equation recovers the dispersion equation of Rayleigh waves propagating in pure orthotropic elastic half-spaces. Remarkably, the secular equation obtained is not a complex equation as the one derived by Liu and Liu, it is a really real equation.     

Secular Equations for Rayleigh and Stoneley Waves in?Exponentially Graded Elastic Materials of General Anisotropy under the Influence of Gravity

The Stroh formalism is employed to study Rayleigh and Stoneley waves in exponentially graded elastic materials of general anisotropy under the influence of gravity. The 6×6 fundamental matrix N is no longer real. Nevertheless the coefficients of the sextic equation for the Stroh eigenvalue p are real. The orthogonality and closure relations are derived. Also derived are three Barnett-Lothe tensors. They are not necessarily real. Secular equations for Rayleigh and Stoneley wave speeds are presented. Explicit secular equations are obtained when the materials are orthotropic. In the literature, the secular equations for Stoneley waves in orthotropic materials are obtained without using the Stroh formalism. As a result, it requires computation of a 4×4 determinant. The secular equation presented here requires computation of a 2×2 determinant, and hence is fully explicit. A Rayleigh or Stoneley wave exists in the exponentially graded material under the influence of gravity if the wave can propagate in the homogeneous material without the influence of gravity. As the wave number k????, the Rayleigh or Stoneley wave speed approaches the speed for the homogeneous material.     

Formation de vagues géantes en eau peu profonde

A method, based on the spatio-temporal focusing phenomenon, is proposed to find the wave trains whose evolution leads to freak wave formation. The model is based on the classical Korteweg–de Vries equation. It is shown on the one hand that the Ursell number which measures nonlinearity with regards to dispersion is weak, that is to say freak waves are quasi-linear, and on the other hand that the phenomenon is a rare event with a short-lived character.     

Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero [Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2T, the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.     

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane _x3=0$x_3=0$. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.     

A simple and explicit algebraic expression for the Rayleigh wave velocity

In this note a factorization technique based on the theory of the Riemann problem is used to derive a compact algebraic formula for the velocity of Rayleigh waves. Unlike previous results based on rationalization and Cardan’s solution of a cubic, the present formulation leads to a formula for the velocity which is a continuous function of the Poission’s ratio and yet is simple enough to be of practical interest. The new formula also enables us to express the complex roots associated with the Rayleigh wave equation as simple functions of the Rayleigh wave velocity.     

小波变换在随机海浪及相关课题中的应用与前景

扼要介绍了小波变换的基本原理及其信号处理功能,综述了应用这一技术处理非平稳过程随机海浪,确定波群和异常波特性,判断波浪破碎,进行入反射分离,分析海洋和湖泊流场以及研究岸滩演变的时空变化过程与气象、海象的相关关系等方面的研究成果.讨论了这一技术在波浪动力特性认识、随机海浪的实验室模拟、波浪与建筑物相互作用研究方面的应用前景.      

Modeling nonlinear Rayleigh wave fields generated by angle beam wedge transducers—A theoretical study

Nonlinear Rayleigh wave fields generated by an angle beam wedge transducer are modeled in this study. The calculated area sound sources underneath the wedge are used to model the fundamental Rayleigh sound fields on the specimen surface, which are more accurate than the previously used line sources with uniform or Gaussian amplitude distributions. A general two-dimensional nonlinear Rayleigh wave equation without parabolic approximation is introduced and the solutions are obtained using the quasilinear theory. The second harmonic Rayleigh wave due to material nonlinearity is given in an integral expression with these fundamental Rayleigh waves radiated by the wedge transmitter acting as a forcing function. Multi-Gaussian beam (MGB) models are employed to simplify these integral solutions and to extract the diffraction and attenuation correction terms explicitly. The effect of nonlinearity of generating sources on the second harmonic Rayleigh wave fields is taken into consideration; simulation results show that it will affect the magnitude and diffraction correction of the second harmonic waves in the region close to the Rayleigh wave sound sources. This research provides a theoretical improvement to alleviate the experimental restriction on analyzing the effects of diffraction, attenuation and source nonlinearity when using angle beam wedge transducers as transmitters.     

Solitonic interactions,Darboux transformation and double Wronskian solutions for a variable-coefficient derivative nonlinear Schrödinger equation in the inhomogeneous plasmas

By symbolic computation we study a variable-coefficient derivative nonlinear Schrödinger (vc-DNLS) equation describing nonlinear Alfvén waves in inhomogeneous plasmas. Based on the Lax pair of the vc-DNLS equation, the N-fold Darboux transformation is constructed via a gauge transformation and the reduction technique. Multi-solitonic solutions in terms of the double Wronskian for the vc-DNLS equation are obtained. Two- and three-solitonic interactions are analyzed graphically, i.e., overtaking, head-on and parallel interactions. Plasma streaming and inhomogeneous magnetic field control the amplitudes and velocities of the solitonic waves, respectively. The nonuniform density affects the amplitudes of the solitonic waves. The effects of the spectral parameters on the dynamics of the two-solitonic waves are discussed. Our results might facilitate the analytic investigation on certain inhomogeneous systems in the Earth’s magnetosphere, solar winds, planetary bow shocks, dusty cometary tails and interplanetary shocks.     

The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett–Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett–Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i$i$ times the Stroh matrix.     

Internal gravity waves: parametric instability and deep ocean mixing

The Boussinesq approximation provides a convenient framework to describe the dynamics of stably-stratified fluids. A fundamental motion in these fluids consists of internal gravity waves, whatever the strength of the stratification. These waves may be unstable through parametric instability, which results in turbulence and mixing. After a brief review of the main properties of internal gravity waves, we show how the parametric instability of a monochromatic internal gravity wave organizes itself in space and time, using energetics arguments and a simple kinematic model. We provide an example, in the deep ocean, where such instability is likely to occur, as estimates of mixing from in situ measurements suggest. We eventually discuss the fundamental role of internal gravity wave mixing in the maintenance of the abyssal thermal stratification. To cite this article: C. Staquet, C. R. Mecanique 335 (2007).     

Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

In this paper, we study the existence and the role of solitary waves in the finite amplitude instability of a layer of a second-order fluid flowing down an inclined plane. The layer becomes unstable for disturbances of large wavelength for a critical value of Reynolds number which decreases with increase in the viscoelastic parameter M. The long-term evolution of a disturbance with an initial cosinusoidal profile as a result of this instability reveals the existence of a train of solitary waves propagating on the free surface. A novel result of this study is that the number of solitary waves decreases with in crease in M. When surface tension is large, we use dynamical system theory to describe solitary waves in a moving frame by homoclinic trajectories of an associated ordinary differential equation.     

曲面曲率对Rayleigh波传播特性的影响

对任意形状的均匀各向同性线弹性曲面物体,用 WKB～（1）方法求解了沿曲面传播的Rayleigh表面波的运动微分方程,同时考虑了波传播方向及其垂直方向曲面曲率对波的穿透性的影, 所获波动方程的势函数解答表明,在一般情况下垂直波传播方向的曲面曲率对波的穿透深度的影响是不容忽视的.进而以同种介质平面表面情况下的Rayleigh面波的传播特性为基准,给出了曲面曲率引起波数或波速变化的解析表达式.通过理论分析和数值算例,描述了曲面上Rayleigh面波传播行为的一些基本特征.