一维非线性声波传播特性

针对一维非线性声波的传播问题进行了有限元仿真和实验研究. 首先推导了一维非线性声波方程的有限元形式, 含有高阶矩阵的非线性项导致声波具有波形畸变、谐波滋生、基频信号能量向高次谐波传递等非线性特性. 编制有限元程序对一维非线性声波进行了计算并对仿真得到的畸变非线性声波信号进行处理, 分析其传播性质和物理意义. 为验证有限元计算结果, 开展了水中的非线性声波传播的实验研究, 得到了不同输入信号幅度激励下和不同传播距离的畸变非线性声波信号. 然后对基波和二次谐波的传播性质进行详细讨论, 分析了二次谐波幅度与传播距离和输入信号幅度的变化关系及其意义, 拟合出二次谐波幅度随传播距离变化的方程并阐述了拟合方程的物理意义. 结果表明, 数值仿真信号及其频谱均与实验结果有较好的一致性, 证实计算方法和结果的正确性, 并提出了具有一定物理意义的二次谐波随传播距离变化的简单数学关系. 最后还对固体中的非线性声波传播性质进行了初步探讨. 本研究工作可为流体介质中的非线性声传播问题提供理论和实验依据.     

Laboratory and modeling study on modulated wave properties

Development of Benjamin?Feir instability is investigated under laboratory conditions and by analytical modeling. Nonlinear properties of the wave train with discrete spectrum are also investigated. The mechanically generated waves are composed of several discrete waves, while the newly generated harmonics are still combined into discrete spectra with the same frequency step. The technique proposed in this study allows us to study accurately the nonlinear variations in main properties of each harmonic with fixed frequency, such as amplitude, phase speed, and wavenumber along the wave tank together with velocities of wave packet crests, especially for the large transient waves. The phase speeds of short waves increase near large transient waves, and the velocities of longer waves are close to the values determined by the linear theory of waves. The relative long wave accompanied by short waves can dramatically change the local kurtosis and skewness of the wave field. They may play an important role for the generation of large transient wave and provide an opportunity for triggering of the freak waves.     

Resonant excitation and nonlinear evolution of waves in the equatorial waveguide in the presence of the mean current

We study interactions of planetary waves propagating across the equator with trapped Rossby or Yanai modes, and the mean flow. The equatorial waveguide with a mean current acts as a resonator and responds to planetary waves with certain wave numbers by making the trapped modes grow. Thus excited waves reach amplitudes greatly exceeding the amplitude of the incoming wave. Nonlinear saturation of the excited waves is described by an amplitude equation with one or two attracting equilibrium solutions. In the latter case spatial modulation leads to formation of characteristic defects in the wave field. The evolution of the envelopes of long trapped Rossby waves is governed by the driven complex Ginzburg-Landau equation, and by the damped-driven nonlinear Schr?dinger equation for short waves. The envelopes of the Yanai waves obey a simple wave equation with cubic nonlinearity.     

Reflection and refraction of surface acoustic waves by a periodic domain structure

Reflection and refraction of surface acoustic waves by a periodic domain structure formed in lithium niobate is studied. A second harmonic generation is observed. A mechanism underlying the linear and nonlinear interactions of acoustic waves with a periodic domain structure is proposed.     

Two-Mode Wave Solutions to the Degasperis--Procesi Equation

By introducing a new type of solutions, called the multiple-mode wave solutions which can be expressed in nonlinear superposition of single-mode waves with different speeds, we investigate the two-mode wave solutions in Degasperis-Procesi equation and two cases are derived. The explicit expressions for the two-mode waves as well as the existence conditions are presented. It is shown that the two-mode waves may be the nonlinear combinations of many types of single-mode waves, such as periodic waves, solJtons, compactons, etc., and more complicated multiple-mode waves can be obtained if higher order or more single-mode waves are taken into consideration. It is pointed out that the two-mode wave solutions can be employed to display the typical mechanism of the interactions between different single-mode waves.     

Electron plasma waves generation in suddenly created isotropicplasma

Electron plasma waves excitation in suddenly created isotropic plasma as a result of weak nonlinear interaction of linearly polarized plane electromagnetic (EM) wave and electrons has been considered. By the use of standard perturbation method the problem is solved in closed form for the case of a simple harmonic source EM wave. The appearance of the second harmonic and time independent modes have been demonstrated. The efficiency of excitation of these modes is possible to control by varying the frequency of the source wave     

Solution of the scalar wave equation over very long distances using nonlinear solitary waves: Relation to finite difference methods

The linear wave equation represents the basis of many linear electromagnetic and acoustic propagation problems. Features that a computational model must have, to capture large scale realistic effects (for over the horizon or “OTH” radar communication, for example), include propagation of short waves with scattering and partial absorption by complex topography. For these reasons, it is not feasible to use Green’s Function or any simple integral method, which neglects these intermediate effects and requires a known propagation function between source and observer. In this paper, we describe a new method for propagating such short waves over long distances, including intersecting scattered waves. The new method appears to be much simpler than conventional high frequency schemes: Lagrangian “particle” based approaches, such as “ray tracing” become very complex in 3-D, especially for waves that may be expanding, or even intersecting. The other high frequency scheme in common use, the Eikonal, also has difficulty with intersecting waves.Our approach, based on nonlinear solitary waves concentrated about centroid surfaces of physical wave features, is related to that of Whitham [1], which involves solving wave fronts propagating on characteristics. Then, the evolving electromagnetic (or acoustic) field can be approximated as a collection of propagating co-dimension one surfaces (for example, 2-D surfaces in three dimensions). This approach involves solving propagation equations discretely on an Eulerian grid to approximate the linear wave equation. However, to propagate short waves over long distances, conventional Eulerian numerical methods, which attempt to resolve the structure of each wave, require far too many grid cells and are not feasible on current or foreseeable computers. Instead, we employ an “extended” wave equation that captures the important features of the propagating waves. This method is first formulated at the partial differential equation (PDE) level, as a wave equation with an added “confining” term that involves both a positive and a negative dissipation. Once we have the stable PDE, the discrete formulation is simply a multidimensional PDE with (stable) perturbations caused by the discretization. The resulting discrete solution can then be low order and very simple and yet remain stable over arbitrarily long times. When discretized and solved on an Eulerian grid, this new method allows far coarser grids than required by conventional resolution considerations, while still accounting for the effects of varying atmospheric and topographic features. An important point is that the new method is in the same form as conventional discrete wave equation methods. However, the conventional solution eventually decays, and only the “intermediate asymptotic” solution can be used. Simply by adding an extra term, we show that a nontrivial true asymptotic solution can be obtained. A similar solitary wave based approach has been used successfully in a different problem (involving “Vorticity Confinement”), for a number of years.     

具微结构地壳中非线性地震波的演化

非线性地震波在地壳中传播及演化规律的研究具有重要实际意义.利用有限差分方法, 对非线性地震纵波和横波在具微结构地壳中的演化过程进行了详细的数值模拟.结果表明,非线性地震纵波和横波在具微结构地壳中可以逐渐演化成一个孤立波或两个孤立波或孤立波列或消失.地震波的初始幅度、频散系数和体力因子对演化过程和结果都有重要影响.研究结果揭示了非线性地震波在一种微结构地壳中的演化规律,这有助于在理论上解释一些特殊地震波现象.     

A two scale nonlinear fractal sea surface model in a one dimensional deep sea

Using the theory of nonlinear interactions between long and short waves,a nonlinear fractal sea surface model is presented for a one dimensional deep sea.Numerical simulation results show that spectra intensity changes at different locations(in both the wave number domain and temporal-frequency domain),and the system obeys the energy conservation principle.Finally,a method to limit the fractal parameters is also presented to ensure that the model system does not become ill-posed.     

Triggering rogue waves in opposing currents

We show that rogue waves can be triggered naturally when a stable wave train enters a region of an opposing current flow. We demonstrate that the maximum amplitude of the rogue wave depends on the ratio between the current velocity U(0) and the wave group velocity c(g). We also reveal that an opposing current can force the development of rogue waves in random wave fields, resulting in a substantial change of the statistical properties of the surface elevation. The present results can be directly adopted in any field of physics in which the focusing nonlinear Schr?dinger equation with nonconstant coefficient is applicable. In particular, nonlinear optics laboratory experiments are natural candidates for verifying experimentally our results.     

Periodic folded waves for a （2＋1）-dimensional modified dispersive water wave equation

A general solution, including three arbitrary functions, is obtained for a (2+1)-dimensional modified dispersive water-wave (MDWW) equation by means of the WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and the degenerated single folded solitary waves are investigated graphically and found to be completely elastic.     

有流存在时三层流体界面波的二阶Stokes波解

以小振幅波理论为基础，利用摄动方法研究了有背景流场存在时密度三层成层状态下的界面内波，得到了各层流体速度势的二阶渐近解及界面内波波面位移的二阶Stokes波解，并讨论了界面波的Kelvin-Helmholtz不稳定性.结果表明：有流存在的情况下三层密度成层流体界面内波的一阶渐近解(线性波解)、频散关系及二阶渐近解不仅依赖于各层流体的厚度和密度，也依赖于各层流体的背景流场；界面内波波面位移的二阶Stokes波解不仅描述了界面波之间的二阶非线性相互作用，也描述了背景流与界面波之间的二阶非线性相互作用；当每层流 关键词： 界面波 均匀流 二阶Stokes波解 Kelvin-Helmholtz不稳定性     

Vortex rings and solitary waves in trapped Bose–Einstein condensates

We discuss nonlinear excitations in an atomic Bose–Einstein condensate which is trapped in a harmonic potential. We focus on axially symmetric solitary waves propagating along a cylindrical condensate. A quasi one-dimensional dark soliton is the only nonlinear mode for a condensate with weak interactions. For sufficiently strong interactions of experimental interest solitary waves are hybrids of one-dimensional dark solitons and three-dimensional vortex rings. The energy-momentum dispersion of these solitary waves exhibits characteristics similar to a mode proposed sometime ago by Lieb in a strictly 1D model, as well as some rotonlike features. We subsequently discuss interactions between solitary waves. Head-on collisions between dark solitons are elastic. Slow vortex rings collide elastically but faster ones form intermediate structures during collisions before they lose energy to the background fluid. Solitary waves and their interactions have been observed in experiments. However, some of their intriguing features still remain to be experimentally identified.     

Nonlinear fast growth of water waves under wind forcing

In the wind-driven wave regime, the Miles mechanism gives an estimate of the growth rate of the waves under the effect of wind. We consider the case where this growth rate, normalised with respect to the frequency of the carrier wave, is of the order of the wave steepness. Using the method of multiple scales, we calculate the terms which appear in the nonlinear Schrödinger (NLS) equation in this regime of fast-growing waves. We define a coordinate transformation which maps the forced NLS equation into the standard NLS with constant coefficients, that has a number of known analytical soliton solutions. Among these solutions, the Peregrine and the Akhmediev solitons show an enhancement of both their lifetime and maximum amplitude which is in qualitative agreement with the results of tank experiments and numerical simulations of dispersive focusing under the action of wind.     

Detection of fatigue-induced micro-cracks in a pipe by using time-reversed nonlinear guided waves: a three-dimensional model study

Localization of fatigue-related micro-cracks in pipelines is of increasing importance in industrial applications. A three-dimensional (3D) fatigue-crack imaging technique combining nonlinear guided waves with time reversal is proposed in this paper for potential applications in pipeline inspections. By using this method, the non-classical nonlinear guided waves generated from micro-cracks with hysteretic behavior are recorded, and the third harmonic waves are used to reconstruct the fatigue-crack images in a pipe by using a time reversal (TR) process. The feasibility of this method is examined by the imaging simulations for a steel pipe with varied defect areas. A finite-difference time-domain (FDTD) code is programmed to solve the wave equations under cylindrical coordinates, and simulate the experimental process of wave propagation. The results show that: (1) the proposed technique has excellent spatial retrofocusing capability; (2) the accuracy of defect localization and sizing depends on the crack orientation and the adopted guided wave mode; and (3) different displacement/stress components have varied sensitivities to the crack orientation.     

Nonlinear equations for quasi-monochromatic waves near a caustic in a dispersive dissipative medium with cubic or quadratic nonlinearity

The behavior of a quasi-monochromatic nonlinear wave near a caustic is considered. Nonlinear ordinary differential equations for a dispersive dissipative medium with a cubic or quadratic nonlinearity are derived. For the latter medium, nonstationary equations describing it near the caustic are presented with allowance for the dissipative dispersive terms. These equations yield ordinary ones for quasi-monochromatic waves. The amplitude of the second harmonic is expressed in terms of the squared amplitude of the first harmonic. The amplitude of the second harmonic, as well as the solution as a whole, increases near the caustic.     

Freak waves in random oceanic sea states.

Freak waves are very large, rare events in a random ocean wave train. Here we study their generation in a random sea state characterized by the Joint North Sea Wave Project spectrum. We assume, to cubic order in nonlinearity, that the wave dynamics are governed by the nonlinear Schr?dinger (NLS) equation. We show from extensive numerical simulations of the NLS equation how freak waves in a random sea state are more likely to occur for large values of the Phillips parameter alpha and the enhancement coefficient gamma. Comparison with linear simulations is also reported.     

Rogue waves in shallow water

Most of the processes resulting in the formation of unexpectedly high surface waves in deep water (such as dispersive and geometrical focusing, interactions with currents and internal waves, reflection from caustic areas, etc.) are active also in shallow areas. Only the mechanism of modulational instability is not active in finite depth conditions. Instead, wave amplification along certain coastal profiles and the drastic dependence of the run-up height on the incident wave shape may substantially contribute to the formation of rogue waves in the nearshore. A unique source of long-living rogue waves (that has no analogues in the deep ocean) is the nonlinear interaction of obliquely propagating solitary shallow-water waves and an equivalent mechanism of Mach reflection of waves from the coast. The characteristic features of these processes are (i) extreme amplification of the steepness of the wave fronts, (ii) change in the orientation of the largest wave crests compared with that of the counterparts and (iii) rapid displacement of the location of the extreme wave humps along the crests of the interacting waves. The presence of coasts raises a number of related questions such as the possibility of conversion of rogue waves into sneaker waves with extremely high run-up. Also, the reaction of bottom sediments and the entire coastal zone to the rogue waves may be drastic.     

Fairly-long water waves

Water waves have fascinated artists and poets, fishermen and surfers, as well as mathematicians, scientists, and engineers. Recent developments in the understanding of some types of water waves are cast in terms sufficiently general that they can be applied to waves in media other than water, for example, in high temperature plasmas, in gases, and in solids, as well. Historically, many important advances in wave theory have been made with water waves motivating the research. For example, the work of Airy (1845) on long nonlinear water waves preceded the partially analogous work on nonlinear sound waves of Riemann (1858–9) and Earnshaw (1858). The research of Kelvin (Thomson, 1887) leading to the enunciation of the method of stationary phase, which has found wide application to problems of wave propagation in any dispersive media, is entitled, On the Waves Produced by a Single Impulse in Water of any Depth, or in a Dispersive Medium. More recently, the properties of resonant interactions among waves in dispersive systems are known best for water waves (Phillips, 1960; Longuet-Higgins, 1962; Longuet-Higgins and Phillips, 1962; Benney, 1962; Hasselman, 1962; 1963a, b; McGoldrick, 1965), where experiments can be performed relatively easily, (McGoldrick et al., 1966). Some contemporary works in resonant interactions among other types of waves include waves in plasmas (Fishman et al., 1960; Litvak, 1960), electromagnetic waves in solids (Armstrong et al., 1962), and many others. Solid state scientists should note that the fundamental paper on heat conduction in solids by Peierls (1929) was the direct ancestor to the work over the last decade on resonant wave-wave interactions. Finally, remarkable properties of the Korteweg deVries Equation are only now being brought to light, the equation having originally been derived for water waves (Korteweg and deVries, 1895), but applicable to a wide variety of physical situations. The hope of the author is that the methods and results of modern research on water waves can find application in other fields of science. Broadly speaking, if a problem in waves involves dispersion and/or nonlinearity, chances are better that a solution exists, or has been attempted, for water waves than for any other type.