Transverse instability of gravity–capillary solitary waves

Gravity–capillary solitary waves of depression that bifurcate at the minimum phase speed on water of finite or infinite depth, while stable to perturbations along the propagation direction, are found to be unstable to transverse perturbations on the basis of a long-wave stability analysis. This suggests a possible generation mechanism of the new class of gravity–capillary lumps recently shown to also bifurcate at the minimum phase speed.     

Alternating rolls in non-degenerate optical parametric oscillators

Abstract

Pattern formation near threshold in large-aspect-ratio optical parametric oscillators with flat end reflectors and uniform pumping is investigated in the non-degenerate case. By deriving amplitude equations describing interaction of four orthogonal traveling waves, it is shown that, contrary to the degenerate case, a pure standing-wave pattern is always unstable, whereas a superposition of two perpendicularly oriented out-of-phase rolls (alternating rolls) may be stable near threshold. This state gives rise to square intensity patterns.     

Evolution equation for nonlinear Scholte waves

The evolution equations for nonlinear Scholte waves (finite amplitude elastic waves propagating along liquid/solid interface), which account for the second order nonlinearity of a liquid, are derived for the first time. For mathematical simplicity the nonlinearity of the solid, which influence is expected to be weak in the case of weak localization of the Scholte wave, is not taken into consideration. The analysis of these equations demonstrates that the nonlinear processes contributing to the evolution of the Scholte wave can be divided into two groups. The first group includes nonlinear processes leading to wave spectrum broadening which are common to bulk pressure waves in liquids and gases. The second group includes the nonlinear processes which are active only in the frequency down-conversion (leading to wave spectrum conservation or narrowing), which are specific to the confined nature of the interface wave. It is demonstrated that the nonlinear parameters, which characterize the efficiency of various nonlinear processes in the interface wave, strongly depend on the relative properties of the contacting liquid and solid (or, in other words, on the deviation of the Scholte wave velocity from the velocities of sound in liquid and in solid). In particular, the sign of the nonlinear parameter responsible for the second harmonic generation can differ from the sign of the nonlinear acoustic parameter of the liquid. It is also verified that there are particular liquid/solid combinations where the nonlinear processes, which are inactive in the frequency up-conversion, dominate in the evolution of the Scholte wave. In this case distortionless propagation of the finite amplitude harmonic interface wave is possible. The proposed theory should find applications in nonlinear acoustics, geophysics, and nondestructive testing.     

Particle trajectories in linear periodic capillary and capillary-gravity water waves

Surface tension plays a significant role as a restoration force in the setting of small-amplitude waves, leading to pure capillary and gravity-capillary waves. We show that within the framework of linear theory, the particle paths in a periodic gravity-capillary or pure capillary wave propagating at the surface of water over a flat bed are not closed.     

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。     

Direct Numerical Simulation of Supersonic Jet Flow

A numerical method is given for direct numerical simulation of the nonlinear evolution of instability waves in supersonic round jets, with spatial discretisation based on high-order compact finite differences. The numerical properties of a class of symmetric and asymmetric schemes are analysed. Implementation for the Navier–Stokes equations in cylindrical polar coordinates is discussed with particular attention given to treatment of the origin to ensure stability and efficiency. Validation of the schemes is carried out by detailed comparison with linear stability theory. The computer code is applied to study the initial stages of nonlinear development of unstable modes in a Mach 3 jet. The modes of instability that are present include strongly unstable axisymmetric acoustic and helical vortical waves, as well as weakly unstable radiating acoustic and vortical modes. Three distinctive wave patterns are observed from the simulations including a cross-hatched internal shock structure. Nonlinear interactions between the vortical and acoustic modes are investigated.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Evolution equations which model the ripples arising from an adjacent mode interaction

The method of multiple scales, in both space and time, is used to derive a set of three nonlinear coupled partial differential equations which model the evolution of a train of capillary-gravity waves contained in a channel of finite depth and which arise from the resonant interaction of the Nth and (N + l)th harmonics of the fundamental mode. These equations are used to investigate the stability of the waves to plane-wave perturbations, which may be longitudinal, transverse or oblique to the direction of the propagation of the waves.     

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.     

A perturbation method for non-linear dispersive waves with an application to water waves

Summary A multiple scale perturbation method is developed to obtain asymptotic evolution equations for slowly varying wave train solutions to non-linear dispersive wave problems. The method appears to give results which are a generalization of Whitham's theory on one hand and a generalization of the ray theory on the other hand. First an application is given to a non-linear Klein-Gordon equation, then the method is applied to two-dimensional water waves on water of finite depth (Stokes waves).     

Nonlinear dynamics of offshore structures under sea wave and earthquake forces

A study of dynamic response of offshore structures in random seas to inputs of earthquake ground motions is presented. Emphasis is placed on the evaluation of nonlinear hydrodynamic damping effects due to sea waves for the earthquake response. The structure is discretized using the finite element method. Sea waves are represented by Bretschneider’s power spectrum and the Morison equation defines the wave forcing function. Tajimi-Kanai’s power spectrum is used for the horizontal ground acceleration due to earthquakes. The governing equations of motion are obtained by the substructure method. Response analysis is carried out using the frequency-domain random-vibration approach. It is found that the hydrodynamic damping forces are higher in random seas than in still water and sea waves generally reduce the seismic response of offshore structures. Studies on the first passage probabilities of response indicate that small sea waves enhance the reliability of offshore structures against earthquakes forces.     

The influence of variable surface tension on capillary-gravity waves

Periodic gravity-capillary waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The surface tension is assumed to vary along the free surface. A numerical procedure is presented to solve the problem with an arbitrary distribution of surface tension on the free surface. It is found that there are many different families of solutions. These solutions generalize the classical theory of gravity-capillary waves with constant surface tension. An asymptotic solution is presented for a particular distribution of variable surface tension.     

A variational principle for nonlinear water waves

Summary This paper is concerned with a variational formulation of non-axisymmetric water waves and of two-dimensional surface waves in a running stream of finite depth. The full set of equations of motions for the non-axisymmetric water wave problem in cylindrical polar coordinates and for the two-dimensional surface waves in the running stream in Cartesian coordinates is obtained from a Lagrangian function which is equal to the pressure.With 1 Figure     

Wave forces on a submerged vertical plate

Summary A vertical plate of finite length and depth is attacked by gravity waves in water of finite depth. The forces and moments acting on the plate are computed by using the theory of linearized waves. The forces depend on three dimensionless parameters combining the draft, length, water depth and wave length and on the angle of attack. The problem is reduced to the solution of two infinite linear systems of equations. Numerical solutions are presented for different particular combinations of the parameter values. In most of the cases the standing wave approximation yields sufficiently accurate results.     

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.     

Steady two-layer flows over an obstacle

Nonlinear waves in a forced channel flow of two contiguous homogeneous fluids of different densities are considered. Each fluid layer is of finite depth. The forcing is due to an obstruction lying on the bottom. The study is restricted to steady flows. First a weakly nonlinear analysis is performed. At leading order the problem reduces to a forced Korteweg-de Vries equation, except near a critical value of the ratio of layer depths which leads to the vanishing of the nonlinear term. The weakly nonlinear results obtained by integrating the forced Korteweg-de Vries equation are validated by comparison with numerical results obtained by solving the full governing equations. The numerical method is based on boundary integral equation techniques. Although the problem of two-layer flows over an obstacle is a classical problem, several branches of solutions which have never been computed before are obtained.     

Gravity wave interaction with porous structures in two-layer fluid

Oblique wave interaction with rectangular porous structures of various configurations in two-layer fluid are analyzed in finite water depth. Wave characteristics within the porous structure are analyzed based on plane wave approximation. Oblique wave scattering by a porous structure of finite width and wave trapping by a porous structure near a wall are studied under small amplitude wave theory. The effectiveness of three types of porous structures—a semi-infinite porous structure, a finite porous structure backed by a rigid wall, and a porous structure with perforated front and rigid back walls—in reflecting and dissipating wave energy are analyzed. The reflection and transmission coefficients for waves in surface and internal modes and the hydrodynamic forces on porous structures of the aforementioned configurations are computed for various physical parameters in two-layer fluid. The eigenfunction expansion method is used to deal with waves past the porous structure in two-layer fluid assuming the associated eigenvalues are distinct. An alternate procedure based on the Green’s function technique is highlighted to deal with cases where the roots of the dispersion relation in the porous medium coalesce. Long wave equations are derived and the dispersion relation is compared with that derived based on small amplitude wave theory. The present study will be of significant importance in the design of various types of coastal structures used in the marine environment for the reflection and dissipation of wave energy.     

Davies’ surface condition and singularities of deep water waves

Davies’ surface condition is an approximate free-surface condition on gravity waves progressing in permanent form on water of infinite depth. It is known that this condition preserves essential features of finite-amplitude waves including the highest one. This paper proposes a new surface condition that generalizes Davies’ idea of approximation and covers a fully nonlinear condition. Analytic continuation of the proposed surface condition allows us to explore singularities of solutions that dominate the flow. The results of singularity analysis elucidate the connection between Davies’ approximate solution and the fully nonlinear solution. In addition, it is shown that the nonmonotonic variation of wave speed with wave steepness can be predicted using a linear sum of a relatively small number of singularities. This suggests that a suitable choice of a parameter in the proposed surface condition can move singularities away from the flow field without changing their structure and may reduce numerical difficulties due to singularities for large-amplitude waves.     

Nonlinear oscillations in a thin ring — I. Three-wave resonant interactions

Summary Nonlinear resonant interactions between planar waves in a thin circular ring are investigated. It is found that a high-frequency azimuthal wave is unstable against a pair of secondary low-frequency waves. The secondary waves are of two types; either two bending or azimuthal and bending. These are in phase with the primary wave. All three together compose a resonant triad. Such kind of instability causes the stress amplification in the ring. The stress growth constant and the period of energy exchange between the waves are estimated based on analytical solutions to the evolution equations driving the triad. The lowest-order nonlinear approximation analysis predicts stability for bending waves. A good qualitative agreement of the obtained results with some known experimental data is observed.     

横浪中船舶的随机混沌运动

采用概率密度函数和数值模拟的方法研究随机横浪中船舶的混沌运动特性和发生混沌运动的临界参数条件。综合考虑非线性阻尼、非线性恢复力矩以及白噪声横浪激励,建立了船舶的横摇非线性随机微分方程。用随机Melnikov均方准则确定混沌运动的系统参数域后,应用路径积分法求解随机微分方程得到了响应的概率密度函数。研究发现:当噪声强度大于混沌临界值时,船舶出现随机混沌运动;对于高的白噪声激励强度,系统响应有两种较大可能的状态并在这两个状态间随机跳跃,这时船舶的运动不稳定并可能发生倾覆。