北极冰-水耦合系统中声波传播特性

利用固体和流体介质中波传播理论,导出了冰-水两层复合结构中导波频散方程。进一步,利用二分法对频散方程进行了数值求解,得到了ω-k频散曲线(ω与k分别为圆频率和波数),以及相速度和群速度频散曲线。结果表明:冰-水两层复合结构中导波由具有相同厚度水层和冰层中导波耦合而成,但与水层和冰层中导波频散曲线相比,复合结构中导波频散曲线除第1阶模式外,其余高阶模式均发生了很大变化。从原水层第1阶模式的截止频率开始,复合结构第2阶模式的相速度曲线被压低,各高阶(大于2阶)模式的相速度曲线出现一个跃变点,群速度曲线出现一个极大和一个极小值。水层越厚,复合结构各高阶模式的截止频率越低,相同频带内导波模式越丰富。水层厚度保持不变时,复合结构各阶模式的相速度和群速度曲线均随冰层厚度的增加而向低频方向移动。另外,还进一步分析了冰-水复合结构的导波波结构,发现第1阶导波模式的能量主要集中在冰层内和海表面附近,而2阶以上高阶导波模式的振动位移幅度随深度方向呈现周期性特征,并且模式阶数越高,振动越复杂。      

On the pressure of gravitational waves

A system of coupled point masses under the influence of gravitational waves is considered. By means of the geodesic deviation equation as the equation of motion it is shown, taking into account the second order small terms, that there exist forces which cause the acceleration of the system in the longitudinal direction. The longitudinal force is due to the fact that simultaneously with energy momentum is also absorbed from waves. It is proved directly on the basis of the equations of motion of the point masses that the energy and momentum absorbed by the test system obey the special relativistic relationship of a zero rest mass particle. The case when the Weber oscillator moves at a relativistic speed with respect to the source of gravitational waves is also examined. In this case, the absorption of energy and momentum by the Weber oscillator is much larger or smaller compared to the stationary situation.     

Universal geometric condition for the transverse instability of solitary waves

Transverse instabilities correspond to a class of perturbations traveling in a direction transverse to the direction of the basic solitary wave. Solitary waves traveling in one space direction generally come in one-parameter families. We embed them in a two-parameter family and deduce a new geometric condition for transverse instability of solitary waves. This condition is universal in the sense that it does not require explicit properties of the solitary wave-or the governing equation. In this paper the basic idea is presented and applied to the Zakharov-Kuznetsov equation for illustration. An indication of how the theory applies to a large class of equations in physics and oceanography is also discussed.     

Rogue wave observation in a water wave tank

The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schr?dinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments specifically designed for observation of breather states in the evolution of deep water waves have never been made in this double limit. In the present work, we present the first experimental results with observations of the Peregrine soliton in a water wave tank.     

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the combination of the third- and fifth-order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are close to solitons of the modified Korteweg-de Vries equation. However, the height of large-amplitude solutions has a limit approaching which solitary waves widen and acquire a table like shape similar to soluitons of the Gardner equation. Numerical calculations confirm that the collision of solitons of the derived equation is inelastic. Inelasticity is the most pronounced in the interaction of unipolar pulses. The direction of the shift of the phase of the higher-amplitude soliton owing to the interaction of solitons of different polarities depends on the amplitudes of the pulses.     

Existence of perturbed solitary wave solutions to a model equation for water waves

We prove the existence of travelling wave solutions to a fifth order partial differential equation, which is a formal asymptotic approximation for water waves with surface tension. These travelling waves are arbitrarily small perturbations of solitary waves, but are not solitary waves themselves, because they approach small amplitude oscillations for large values of the independent variable. This result suggests that for Bond numbers less than one third, there are branches of travelling wave solutions to the water wave equations, which are perturbations of supercritical elevation solitary waves, and which bifurcate from Froude number one and Bond number one third.     

大尺度浅水波方程中相互调制的非线性波

基于一般的浅水波方程, 根据大尺度正压大气的特点, 得到无量纲的控制大尺度大气的动力学非线性方程组. 利用多尺度法, 由无量纲的动力学方程组导出了扰动位势的非线性控制方程. 采用椭圆方程构造该扰动位势控制方程的解, 获得了扰动位势和速度的多周期波与冲击波(爆炸波) 并存的解析解. 扰动位势的解表明经向和纬向具有不同周期和波长的周期波, 且都受纬向孤波的调制; 速度的解表明大尺度大气流动存在气旋和反气旋周期性分布的现象. 关键词： 浅水波方程 大尺度正压大气 解析解 非线性波     

The wave equation with viscoelastic attenuation and its application in problems of shallow-sea acoustics

A suitable tool for the simulation of low frequency acoustic pulse signals propagating in a shallow sea is the numerical integration of the nonstationary wave equation. The main feature of such simulation problems is that in this case the sound waves propagate in the geoacoustic waveguide formed by the upper layers of the bottom and the water column. By this reason, the correct dependence of the attenuation of sound waves in the bottom on their frequency must be taken into account. In this paper we obtain an integro-differential equation for the sound waves in the viscoelastic fluid, which allows to simulate the arbitrary dependence of acoustic wave attenuation on frequency in the time domain computations. The procedure of numerical solution of this equation based on its approximation by a system of differential equations is then considered and the methods of artificial limitation of computational domain are described. We also construct a simple finite-difference scheme for the proposed equation suitable for the numerical solution of nonstationary problems arising in the shallow-sea acoustics.     

Darboux Transformation and Soliton Solutions for the (2+1)-Dimensional Generalization of Shallow Water Wave Equation with Symbolic Computation

In this paper, the (2+1)-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, we prove that the (2+1)-dimensional generalization of shallow water wave equation possesses the Painlevé property under a certain condition, and its Lax pair is constructed by applying the singular manifold method. Based on the obtained Lax representation, the Darboux transformation (DT) is constructed. The first iterated solution, second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT. Relevant properties are graphically illustrated, which might be helpful to understanding the propagation processes for ocean waves in shallow water.     

Rogue internal waves in the ocean: Long wave model

Rogue waves can be categorized as unexpectedly large waves, which are temporally and spatially localized. They have recently received much attention in the water wave context, and also been found in nonlinear optical fibers. In this paper, we examine the issue of whether rogue internal waves can be found in the ocean. Because large-amplitude internal waves are commonly observed in the coastal ocean, and are often modeled by weakly nonlinear long wave equations of the Korteweg-de Vries type, we focus our attention on this shallow-water context. Specifically, we examine the occurrence of rogue waves in the Gardner equation, which is an extended version of the Korteweg-de Vries equation with quadratic and cubic nonlinearity, and is commonly used for the modelling of internal solitary waves in the ocean. Importantly, we choose that version of the Gardner equation for which the coefficient of the cubic nonlinear term and the coefficient of the linear dispersive term have the same sign, as this allows for modulational instability. From numerical simulations of the evolution of a modulated narrow-band initial wave field, we identify several scenarios where rogue waves occur.     

A high-order nonlinear envelope equation for gravity waves in finite-depth water

A third-order nonlinear envelope equation is derived for surface waves in finite-depth water by assuming small wave steepness, narrow-band spectrum, and small depth as compared to the modulation length. A generalized Dysthe equation is derived for waves in relatively deep water. In the shallow-water limit, one of the nonlinear dispersive terms vanishes. This limit case is compared with the envelope equation for waves described by the Korteweg-de Vries equation. The critical regime of vanishing nonlinearity in the classical nonlinear Schrödinger equation for water waves (when kh ≈ 1.363) is analyzed. It is shown that the modulational instability threshold shifts toward the shallow-water (long-wavelength) limit with increasing wave intensity.     

Dispersion of doppleron-phonon modes in strong coupling regime

The dispersion equation for doppleron-phonon modes was constructed and solved analytically in the strong coupling regime. The Fermi surface model proposed previously for calculating the doppleron spectrum in an indium crystal was used. It was shown that in the vicinity of doppleron-phonon resonance, the dispersion curves of coupled modes form a gap qualitatively different from the one observed under helicon-phonon resonance: there is a frequency interval forbidden for existence of waves of definite circular polarization depending upon direction of the external DC magnetic field. The physical reason for it is interaction of the waves which have oppositely directed group velocities.     

Elastic surface waves guided by the edge of a slit

Surface waves propagating along the free surface of a homogeneous, isotropic, linearly elastic half-space, are shown to have the property that the normal displacement component at the free surface is governed by a reduced wave equation. This suggests a “membrane analogy”, and a corresponding family of surface waves. Of particular interest is a three-dimensional surface wave, whose displacement components in the sagittal plane vary linearly with the co-ordinate normal to that plane, while the displacement component in the direction normal to the sagittal plane is uniform in that direction. This new wave arises when surface waves propagate along the free surfaces of a semi-infinite slit, parallel to the edge of the slit, with the classical Rayleigh wave velocity. It is also shown that a semi-infinite slit cannot support surface waves which decay with the distance from the edge of the slit.     

Doppler effect of nonlinear waves and superspirals in oscillatory media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example in which waves originate from a source exhibiting a back-and-forth movement in a radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves ("superspiral"). Using direct simulations as well as numerical nonlinear analysis within the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonic growth or decay as well as saturation of these modulations depending on the perturbation frequency. Our findings elucidate recent experimental observations concerning superspirals and their decay to spatiotemporal chaos.     

Vibration analysis and sound field characteristics of a tubular ultrasonic radiator

A sort of tubular ultrasonic radiator used in ultrasonic liquid processing is studied. The frequency equation of the tubular radiator is derived, and its radiated sound field in cylindrical reactor is calculated using finite element method and recorded by means of aluminum foil erosion. The results indicate that sound field of tubular ultrasonic radiator in cylindrical reactor appears standing waves along both its radial direction and axial direction, and amplitudes of standing waves decrease gradually along its radial direction, and the numbers of standing waves along its axial direction are equal to the axial wave numbers of tubular radiator. The experimental results are in good agreement with calculated results.     

A feasibility study of the use of bounded beams resembling the shape of evanescent and inhomogeneous waves

Plane waves are solutions of the visco-elastic wave equation. Their wave vector can be real for homogeneous plane waves or complex for inhomogeneous and evanescent plane waves. Although interesting from a theoretical point of view, complex wave vectors normally only emerge naturally when propagation or scattering is studied of sound under the appearance of damping effects. Because of the particular behavior of inhomogeneous and evanescent waves and their estimated efficiency for surface wave generation, bounded beams, experimentally mimicking their infinite counterparts similar to (wide) Gaussian beams imitating infinite harmonic plane waves, are of special interest in this report. The study describes the behavior of bounded inhomogeneous and bounded evanescent waves in terms of amplitude and phase distribution as well as energy flow direction. The outcome is of importance to the applicability of bounded inhomogeneous ultrasonic waves for nondestructive testing.     

The generation of wind waves on clean water and in the presence of an oil film

The conditions of the resonant excitation of waves on a liquid surface by a horizontal air flow that has a decreasing velocity in the direction of motion were established, such that steady waves occurred when the period of the escape of a chain of eddies that is generated in a viscous layer of an air flow coincided with the period of natural oscillations, which is determined by the dispersion relationship for a group of waves. The dependence of the lengths of steady waves on the air-flow velocity over the surface of clean water and water with a light oil film was obtained. The resulting model was tested experimentally.     

Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos

In this paper, we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries (FKdV) framework and the polynomial chaos. The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third. When its steady symmetric solitary-wave-like solutions are randomly perturbed, the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations. By representing the perturbation with a random variable, the governing FKdV equation is interpreted as a stochastic equation. The polynomial chaos expansion of the random solution has been used for the study of stability in two ways. Firstly, it allows us to identify the stable solution of the stochastic governing equation. Secondly, it is used to construct upper and lower bounding surfaces for unstable solutions, which encompass the fluctuations of waves.     

Nonlinear gravitational waves on the surface of viscous fluid: Lagrangian approach

The problems of the asymptotic theory of weakly nonlinear surface waves in viscous fluid are discussed. For standing waves on deep water, the solutions obtained in the first- and second-order approximations in a small parameter—wave steepness—are analyzed. The evolution equation for the amplitude of wave packet envelope is obtained where the inverse Reynolds number is equal to the squared steepness. It is shown that this is a nonlinear Schrödinger equation with linear dissipation.