Breaking of gravity waves in a neighborhood of their second critical propagation speed

Experimental data on gravity shallow-water waves generated by a vertical plate moving in a predetermined manner are given. The plate completely covers the cross section of the channel. It is found that with when the wave speed exceeds the first critical value known in hydraulics, the wave retains smoothness. Breaking of the waves begins at the second critical speed (which is about 1.3 times as high), whose value coincides with the limiting propagation speed of a solitary wave. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 52–58, March–April, 1998.     

Solitary waves for a nonlinear dispersive long wave equation

All the possible traveling wave solutions of Whitham-Broer-Kaup （WBK） equation are investigated in the present paper. By employing phase plane analysis, transition boundaries are derived to divide the parameter space into several regions associated with different types of phase portraits corresponding to different forms of wave solutions. All the exact expressions of bounded wave solutions are obtained as well as their existence conditions. The mechanism of bifurcation between different waves with varying Hamiltonian value has been revealed. It is pointed out that as the periods of two coexisted periodic waves tend to infinity, they may evolve to two solitary waves. Furthermore, when their trajectories pass through the common saddle point, the two solitary waves may merge into a periodic wave, and its amplitude is nearly equal to the sum of the amplitudes of the two solitary wave solutions.     

Waves ahead of a vertical plate in a channel

The behavior of disturbances propagating with supercritical speed ahead of a plate in a channel is analyzed on the basis of the experimental results obtained by the authors and data taken from the literature. In particular, the transition from smooth to breaking waves has been found to occur at higher propagation speeds than follows from the first approximation of shallow water theory. It has also been found that for waves widely encountered in practice the value of the propagation speed agrees well with the limiting propagation speed of solitary waves obtained on the basis of the complete equations of potential fluid flow. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 82–90, January–February, 1999. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 95-01-01164) and by the Integration Program of the Siberian Branch of the Russian Academy of Sciences under grant No. 97-43.     

Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.     

Propagation of weakly nonlinear disturbances of the interface between two layers of fluid

The dynamics of internal waves of small but finite amplitude in a two-layer fluid system bounded by rigid horizontal surfaces at bottom and top is investigated theoretically. For linear disturbances of the fluid interface the authors propose a polynomial approximation of the dispersion relation which has the same asymptotics as the exact formula in the limiting situations of very long and short waves. In the case of three-dimensional, weakly nonlinear disturbances of slowly varying shape (in the coordinate system moving with the wave) an equation like the wave equation is derived. This equation has Stokes solutions coinciding with the well-known results for infinitely deep layers. For fairly long disturbances solitary solutions of the model wave equation which fit the experimental data are determined. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 125–131, January–February, 1994.     

Solitary waves in a peridynamic elastic solid

The propagation of large amplitude nonlinear waves in a peridynamic solid is analyzed. With an elastic material model that hardens in compression, sufficiently large wave pulses propagate as solitary waves whose velocity can far exceed the linear wave speed. In spite of their large velocity and amplitude, these waves leave the material they pass through with no net change in velocity and stress. They are nondissipative and nondispersive, and they travel unchanged over large distances. An approximate solution for solitary waves is derived that reproduces the main features of these waves observed in computational simulations. It is demonstrated by numerical studies that the waves interact only weakly with each other when they collide. Wavetrains composed of many non-interacting solitary waves are found to form and propagate under certain boundary and initial conditions.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.     

Propagation of solitary waves over underwater ridges

The propagation of solitary waves is investigated on the basis of a nonlinear system of equations of hyperbolic type describing the motion of the crest of a solitary wave over the surface of a liquid of variable depth [1]. The existence of solutions with discontinuities, the boundary conditions at which are introduced on the basis of [2, 3], is assumed. In the case of an infinite cylindrical ridge both solitary and periodic captured waves are found. Depending upon the height of the ridge and the parameters of the wave, the encounter between a uniform wave and a semi-infinite ridge yields qualitatively different solutions — continuous and discontinuous, where the primary wave is broken down by the ridge into several solitary waves. The amplitude of the wave may either increase or decrease over the ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–93, January–February, 1985.The author is grateful to A. G. Kulikovskii and A. A. Barmin for their interest in his work, useful discussions and valuable comments offered during the preparation of the article for the press.     

Wave Localization in Hydroelastic Systems

The phenomenon of wave localization in hydroelastic systems leads to the strength concentration of radiation fields. The linear method considers the process of localization to be the formation of nonpropagation waves (trapped modes phenomenon). The presence of such waves in the total wave packet points to the existence of mixed natural spectrum of differential operators describing the behaviour of hydroelastic systems. The problem of liquid and oscillating structure interaction caused by the trapped modes phenomenon has been solved (membranes, dies). The interaction of the liquid and elastic structures with inclusions can lead to localized mode formation. In the case of solitary wave motion in nonlinear elastic media, contacting with the liquid, these solitons can be interpreted as “moving inclusions”. The analytical solution for solitary waves has been found. If the soliton speed v0 is more than the velocity of sound c0 in the liquid, the solitary waves strongly slow down. If c0 is close to v0, then a resonance can be observed and solitons move without any resistance. If the soliton speed is less than c0, the solitary wave slow-down is negligible, compared to the case v0 > c0. This revised version was published online in July 2006 with corrections to the Cover Date.     

Structure of a turbulent bore in a homogeneous liquid

A mathematical model for the propagation of nonlinear long waves is constructed with allowance for nonhydrostatic pressure distribution and the development of a surface boundary layer due to wave breaking. The problem of the structure of a bore in a homogeneous liquid is solved. In particular, the transition of a wave bore to a turbulent bore as its amplitude increases is described within a single model. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 56–68, March–April, 1999.     

On solitary waves in plasma

From the equations of hydrodynamics and electrodynamics, a system of coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exists for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The interaction of the longitudinal solitary waves with each other is studied and an upper bound on the amplitudes of these waves is obtained. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.     

Finite-amplitude solitary waves in a two-layer fluid

Second-mode nonlinear internal waves at a thin interface between homogeneous layers of immiscible fluids of different densities have been studied theoretically and experimentally. A mathematical model is proposed to describe the generation, interaction, and decay of solitary internal waves which arise during intrusion of a fluid with intermediate density into the interlayer. An exact solution which specifies the shape of solitary waves symmetric about the unperturbed interface is constructed, and the limiting transition for finite-amplitude waves at the interlayer thickness vanishing is substantiated. The fine structure of the flow in the vicinity of a solitary wave and its effect on horizontal mass transfer during propagation of short intrusions have been studied experimentally. It is shown that, with friction at the interfaces taken into account, the mathematical model adequately describes the variation in the phase and amplitude characteristics of solitary waves during their propagation.     

细观结构固体层中孤立波的传播及相互作用特性

利用直接微扰方法.确定了孤立波的放大或衰减与孤立波的初始幅度以及介质的结构参数之间的关系.然后利用线性化技术构造出一种二阶精度的稳定差分格式,并对孤立波在细观结构固体层中传播特性进行了数值模拟,特别对细观结构固体层中传播的不同幅度的孤立波的相互作用进行了详细的数值模拟,从而得到在适当条件下细观结构固体层中孤立波传播时即可以衰减、放大又可以稳定传播,且相互作用不影响这种传播特性.     

The Benjamin–Lighthill Conjecture for Steady Water Waves (revisited)

The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin–Lighthill conjecture is proved for these waves provided Bernoulli’s constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoulli’s constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoulli’s constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433–488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

On a class of internal solitary waves in a two-layer fluid

Solitary waves on an interface between two fluids are considered. A uniform asymptotic expansion is constructed for internal solitary waves with flat crests (of the plateau type) that degenerate into a bore in the limit. It is shown that, in this case, in contrast to a Korteweg-de Vries wave, the wave amplitude is of the same order of smallness as the longwave approximation parameter. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 55–61, September–October, 1999.     

On the existence of three solutions for a supercritical steady flow of a heavy fluid over obstructions

The paper is devoted to the solution of the steady problem of ideal incompressible fluid flow over a semi-circular cylinder located at the bottom. Calculations showed that the problem has at least three solutions for the Froude number. In the absence of an obstruction at the bottom, the proposed algorithm allows one to construct solitary waves up to limiting waves. The paper reports the most important wave characteristics: circulation, mass, and potential and kinetic energy. Analysis of the calculation results leads to the conclusion that all maximum values of the solitary-wave characteristics are attained before the maximum amplitude and the maximum of the mass does not coincide with the maxima of the total energy and the Froude number. Kemerovo State University, Kemerovo 650043. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 27–35, January–February, 1999.     

High-amplitude elastic solitary wave propagation in 1-D granular chains with preconditioned beads: Experiments and theoretical analysis

Elastic solitary waves resulting from Hertzian contact in one-dimensional (1-D) granular chains have demonstrated promising properties for wave tailoring such as amplitude-dependent wave speed and acoustic band gap zones. However, as load increases, plasticity or other material nonlinearities significantly affect the contact behavior between particles and hence alter the elastic solitary wave formation. This restricts the possible exploitation of solitary wave properties to relatively low load levels (up to a few hundred Newtons). In this work, a method, which we term preconditioning, based on contact pre-yielding is implemented to increase the contact force elastic limit of metallic beads in contact and consequently enhance the ability of 1-D granular chains to sustain high-amplitude elastic solitary waves. Theoretical analyses of single particle deformation and of wave propagation in a 1-D chain under different preconditioning levels are presented, while a complementary experimental setup was developed to demonstrate such behavior in practice. The experimental results show that 1-D granular chains with preconditioned beads can sustain high amplitude (up to several kN peak force) solitary waves. The solitary wave speed is affected by both the wave amplitude and the preconditioning level, while the wave spatial wavelength is still close to 5 times the preconditioned bead size. Comparison between the theoretical and experimental results shows that the current theory can capture the effect of preconditioning level on the solitary wave speed.     

Evolution of a laser-generated shock wave in iron and its interaction with martensitic transformation and twinning

Effects of shock waves (generated by a nanosecond laser pulse in plates of Armco-iron) on structural changes are analysed. Localisation of processes of martensitic transformation and twinning – for various values of laser pulse duration – is studied both experimentally and numerically. A proposed model accounts for interaction of shock wave propagation and structure changes. Realisation of martensitic transformation and twin formation influences wave front modification. A stress amplitude decrease with increasing distance from a microcrater determines, together with the pulse duration, a character of spatial localisation of structural changes. Numerical results are compared with experimental data and serve as a basis for additional interpretation of phenomena. Received 9 August 1994 / Accepted 30 June 1997     

Experimental verification of methods for calculating partial dam-break waves

This paper presents results of an experimental study of waves generated by partial break of two model dams. The previously proposed calculation methods are extended and compared with the experimental data obtained. It is shown that the wave propagation speed in the tailwater is significantly influenced by the energy losses due to flow through the breach. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 61–69, September–October, 2008