On the stability of lumps and wave collapse in water waves

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg-de Vries solitary waves, the present study is concerned with a distinct class of gravity-capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney-Roskes-Davey-Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev-Petviashvili equation, a model for weakly nonlinear gravity-capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.     

Progressive Cross Waves due to the Horizontal Oscillations of a Vertical Cylinder in Water. Evolution Equations

The paper deals with the theoretical analysis of progressive cross waves excited due to the horizontal oscillations of a vertical, surface-piercing circular cylinder in water of constant depth. Although cross waves are a phenomenon well known in laboratory wave tanks, it seems that they have not been observed around horizontally oscillating structures in fluid up to now. Such observations have recently been carried out by the authors on various models of offshore gravity platforms subjected to earthquake-like horizontal excitation in a water tank. The theoretical analysis of the problem is based on a method developed by Becker and Miles (1992) for the radial cross waves due to the motion of an axisymmetric cylindrical wavemaker. Whitham's average-Lagrangian approach is applied. It is shown that the energy transfer to the cross wave is described by the functional which is quadratic, both in the forced basic wave and in the cross wave. Therefore, the solution to second-order problems is necessary for the derivation of the evolution equations. The evolution of the cross wave is found to be described by two complex nonlinear partial differential equations with coefficients depending on a slow radial variable both in linear and nonlinear terms. The evolution equations are coupled through the nonlinear terms and through the boundary conditions as well.     

Transformation of flexural gravity waves by heterogeneous boundaries

The transformation of flexural gravity waves due to wave scattering by heterogeneous boundaries is investigated under the assumption of the linearized water-wave theory. The heterogeneous boundaries include step-type bottom topography as well as heterogeneity in the material property of a floating ice-sheet. By applying the generalized expansion formulae along with the corresponding orthogonal mode-coupling relations, the boundary-value problem (BVP) is reduced to linear system of algebraic equations. The system of equations is solved numerically to determine the full solution of the problem under consideration. Energy relations are derived and used to check the accuracy of the computational results of the scattering problem. Explicit relations for the shoaling and scattering coefficients due to the change in water depth and heterogeneous ice-sheet are derived. These derivations are based on the law of conservation of energy flux under the assumptions of the linearized shallow-water theory. The change in water depth and the structural characteristics of the medium significantly contribute to the change in the scattering and shoaling coefficients and the deflection of the structure. The present results are likely to play a significant role in the analysis of flexural gravity-wave propagation in problems of variable topography for which a direct computational approach is being utilized.     

Long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves

The long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves is verified from the weakly nonlinear cubic-order truncation model derived from the free-surface boundary conditions of the Euler equations in the water-wave problem. The linearized operators corresponding to the cubic-order truncation model feature a skew-symmetric structure, consistent to the associated property of the Euler equations. From this, the leading-order initial long-wave transverse instability growth rate of the weakly nonlinear gravity–capillary solitary waves is estimated to be quantitatively identical, in the weakly nonlinear limit, to the earlier result obtained from the full Euler equations, through an equivalent perturbation procedure.     

Periodic shock waves in spherical resonators (survey)

This article surveys the literature on the problem of shock waves in spherical resonators. The published data is used to examine the feasibility of exciting shock waves in such resonators by means of a source of low-amplitude harmonic oscillations. A nonlinear wave equation is obtained to describe the propagation of unidimensional spherical waves in solids, liquids, and gases, as well as in bubbly liquids. A solution to the equation is constructed by the small-parameter method with the use of traveling-wave functions. Then, in solving boundary-value problems, linearized equations are integrated and the resonance frequencies at which the amplitudes of the oscillation increase without limit according to the linear solution are determined. Near these frequencies, the linear analysis is then refined by allowing for the nonlinear terms in the boundary-value problems. It is shown that an increase in the amplitude of the oscillations at resonance frequencies may lead to the formation of spherical periodic shock waves in the given resonators. An analogy is made between these waves and resonance shock waves excited in long unidimensional resonators.Translated from Problemy Prochnosti, No. 10, pp. 49–73, October, 1995.     

Water-wave scattering by two symmetric circular-arc-shaped thin plates

The problem of surface water-wave scattering by two symmetric circular-arc-shaped thin plates submerged in deep water is investigated in this paper assuming linear theory. The problem is formulated in terms of hypersingular integral equations which are solved approximately using finite series involving Chebyshev polynomials of the second kind. The coefficients of the finite series are obtained numerically by a collocation method. Very accurate numerical estimates for the reflection and the transmission coefficients are then obtained. The numerical results are depicted graphically against the wave number for different arc lengths of the plates, the depth of their submergence and the separation length. Known results for a circular cylinder and horizontal straight plate are recovered.     

Nonlinear axisymmetric waves in compressible hyperelastic rods: long finite amplitude waves

Summary This paper investigates nonlinear axisymmetric waves in compressible hyperelastic circular cylindrical rods. We consider first a compressible Mooney-Rivlin material to obtain exact governing equations. To further study the problem, we introduce the notion of long finite amplitude waves and derive the corresponding simplified model equations, which gives the framework for studying problems like wave-interactions arising through collision or reflection. The asymptotically valid far-field equation is consequently deduced from the simplified model equations. Then, using a strained-coordinate method, we obtain the second-order solitary wave solution. The result is not only of interest itself, but also provides a suitable initial condition for wave interaction problems. Finally, the results for a general hyperelastic rod are presented.     

A Cauchy integral-equation method for the numerical solution of the steady water-wave problem

Summary In this paper a method is developed for the numerical solution of singular integral equations related to the water-wave problem. Periodic gravity waves of constant form in water of finite depth have been studied. The problem has been programmed and run on a computer, and the computed results plotted and compared with those of other authors. Some difficulties of the computing and some checking of the solution are discussed.     

Photo-elastic waves

The basic equations of the dynamical theory of a hyperelastic dielectric are presented in spatial form and include Gibbs equation and the conservative form of energy equation here for the first time. The study of weak waves and shock waves, based on singular surface theory, is used to bring out the structure of linear and nonlinear theories of photo-elastic waves. An equation governing the lowest order nonlinear wave is derived for a mode and its solution exhibited to bring out amplitude dependence of frequency; further, though it can be considered isentropic, it is necessarily accompanied with elastic strain.     

A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves

The existence of a line solitary-wave solution to the water-wave problem with strong surface-tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg and G. de Vries and rigorously confirmed a century later by C. J. Amick and K. Kirchg?ssner in 1989. A model equation derived by B. B. Kadomtsev and V. I. Petviashvili in 1970 suggests that the Korteweg-de Vries line solitary wave belongs to a family of periodically modulated solitary waves which have a solitary-wave profile in the direction of motion and are periodic in the transverse direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. It is shown that the Korteweg-de Vries solitary wave undergoes a dimension-breaking bifurcation that generates a family of periodically modulated solitary waves. The term dimension-breaking phenomenon describes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.     

On the quantization of the electromagnetic field in conducting media

In this work we investigate the quantization of electromagnetic waves propagating through homogeneous conducting linear media with no charge density. We use Coulomb's gauge to reduce the problem to that of a time-dependent harmonic oscillator, which is described by the Caldirola–Kanai Hamiltonian. Furthermore, we obtain the corresponding exact wave functions with the help of quadratic invariants and of the dynamic invariant method. These wave functions are written in terms of a particular solution of the Milne–Pinney equation. We also construct coherent and squeezed states for the quantized electromagnetic waves and evaluate the quantum fluctuations in coordinates and momentum as well as the uncertainty product for each mode of the electromagnetic field.     

非线性尘埃等离子体孤立子波变分迭代解

在全球气候变暖的极端反常的情形下，大气尘埃的扩散现象会带来巨大的灾害．本文研究了大气尘埃等离子体扩散的一类广义非线性孤立子波模型．首先对非扰动情形下利用待定系数法得到孤立子波解的解析表示式．其次用广义变分迭代的方法求出对应的变分乘子并构造变分迭代式，依次求出孤子波的各次迭代解．然后用行波变换得到广义非线性尘埃等离子体扰动模型的孤立子波的各次近似解．最后，由得到解的近似函数序列据变分理论知，在自变量的一定区域内此序列为一致收敛的．因此便证明了迭代解的极限函数是尘埃等离子体低频振动非线性方程的精确解．本文得到的近似解是尘埃等离子体的低频振动孤立子波的近似解析解，据它可用解析运算来求出相关量的物理性态，如孤立子波的波峰值．可以根据本文理论采取相应措施，避免出现电荷超高密度的聚集而导致放电击穿现象等．     

Propagation of thickness-twist waves in an inhomogeneous piezoelectric plate with an imperfectly bonded interface

The propagation of thickness-twist waves in an inhomogeneous piezoelectric plate with an imperfectly bonded interface is investigated. Based on the spring-type relation, the imperfectly bonded interface is dealt with, and the exact solution is obtained from the equations of the linear theory of piezoelectricity. The amplitude ratio between the incident wave and the reflected wave, the displacement component and the stress component are all obtained and plotted. Both theoretical analysis and numerical examples show that the effect of the mechanical imperfection on the wave propagation is more evident than that of the electrical imperfection. When the incident wave frequency and the mechanical imperfect parameter meet some particular relation, no reflected waves can appear in the piezoelectric plate. The results are of fundamental importance to the design of resonators and other devices when imperfect joints are considered.     

An alternative analytical solution for water-wave motion over a submerged horizontal porous plate

This study gives an alternative analytical solution for water-wave motion over an offshore submerged horizontal porous-plate breakwater in the context of linear potential theory. The matched-eigenfunction-expansions method is used to obtain the solution. The solution consists of a symmetric part and an antisymmetric part. The symmetric part is also the solution of wave reflection by a vertical solid wall with a submerged horizontal porous plate attached to it. In comparison with previous analytical solutions with respect to finite submerged horizontal porous plates, no complex water-wave dispersion relations are included in the present solution. Thus, the present solution is easier for numerical implementation. Numerical examples show that the convergence of the present solution is satisfactory. The results of the present solution also agree well with previous results by different analytical approaches, as well as previous numerical results by different boundary-element methods. The present solution can be easily extended to the case of multi-layer submerged horizontal porous plates, which may be more significant in practice for meeting different tide levels.     

Davey-Stewartson Ⅰ方程新的精确周期解

Davey-Stewartson方程描述了有限深度的水中水波的运动,它的第一种类型称为(Davey-Stewartson I)是椭圆一双曲型方程。在物理学中,微分方程的精确解对考察非线性现象起着非常重要的作用,为了揭示Davey-Stewartson I方程的运动性质,本文研究它的精确周期解。应用F-代数方法并通过一个高阶辅助微分方程,获得了Davey-Stewartson I方程的一系列新的精确周期解,包括三角函数周期解,Jacobi椭圆函数周期解。     

Transmission line interpretation of the finite element mesh for a wave propagation problem

A one-dimensional exterior electromagnetic scattering problem is formulated using a differential equation approach followed by a finite element discretization. By interpreting the resulting linear algebraic equations as node voltage equations for a transmission line, a boundary element is obtained which satisfies the requirement of no wave reflection at the edge of the finite element region. Numerical results which show the elimination of non-physical standing waves from the scattered field are presented and discussed.     

一种由横波激发的特殊非线性声现象

目前对非线性超声的研究多集中在纵波激发的谐波性质以及对材料微观结构变化的实验检测上,横波激发的非线性声波性质少有研究。对横波激发的一维非线性声波方程入手,利用摄动法求解该方程,并改写为一阶偏微分方程,然后利用交错网格的有限差分形式进行数值求解。结果表明:采用横波激发,能产生线性横波和非线性纵波,且纵波的高次谐波内有两个信号,分别以纵波和横波两种速度传播。若采用较长的激发信号,纵波谐波能形成"拍"现象,成为一种奇特的声传播现象。     

Motion trapping structures in the three-dimensional water-wave problem

Trapped modes in the linearized water-wave problem are free oscillations of finite energy in an unbounded fluid with a free surface. It has been known for some time that such modes are supported by certain structures when held fixed, but recently it has been demonstrated that in two dimensions trapped modes are also possible for freely floating structures that are able to respond to the hydrodynamic forces acting upon them. For a freely floating structure such a mode is a coupled oscillation of the fluid and the structure that, in the absence of viscosity, persists for all time. Here previous work on the two-dimensional problem is extended to give motion trapping structures in the three-dimensional water-wave problem that have a vertical axis of symmetry. Nick Newman has made many important contributions to the theory of the interaction between water waves and structures, and his panel code WAMIT has been adopted as one of the industry standards for the calculation of wave loading on offshore structures. It gives us great pleasure to celebrate Nick’s achievements through the presentation of this work on a new type of structure that both draws on his theoretical work and uses WAMIT to perform relevant computations.     

Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution.