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.     

Correlation between theoretical and experimental solitary waves

Experimental data on surface solitary waves generated by five methods are given. These data and literature information show that at amplitudes 0.2<a/h<0.6 (h is the initial depth of the liquid), experimental solitary waves are in good agreement with their theoretical analogs obtained using the complete model of liquid potential flow. Some discrepancy is observed in the range of small amplitudes. The reasons why free solitary waves of theoretically limiting amplitude have not been realized in experiments are discussed, and an example of a forced wave of nearly limiting amplitude is given. The previously established fact that during evolution from the state of rest, undular waves break when the propagation speed of their leading front reaches the limiting speed of propagation of a solitary wave is confirmed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 44–52, May–June, 1999.     

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.     

Comparisons of CIP,compact and CIP‐CSL2 schemes for reproducing internal solitary waves

Six different models were evaluated for reproducing internal solitary waves which occur and propagate in a stratified flow field with a sharp interface. Three stages were used to compute internal solitary waves in a stratified field: (1) first‐phase computation of momentum equations, (2) second‐phase computation of momentum equations, which corresponds to computing the Poisson's equation, and (3) density computation. The six models discussed in this paper consisted of combinations of four different schemes, a three‐point combined compact difference scheme (CCD), a normal central difference scheme (CDS), a cubic‐polynomial interpolation (CIP), and an exactly conservative semi‐Lagrangian scheme (CIP‐CSL2). The residual cutting method was used to solve the Poisson's equation. Three tests were used to confirm the validity of the computations using KdV theory; i.e. the incremental wave speed and amplitude of internal solitary waves, the maximum horizontal velocity and amplitude, and the wave form. In terms of the shape of an internal solitary wave, using CIP for momentum equations was found to provide better performance than CCD. These results suggest one of the most appropriate scheme for reproducing internal solitary waves may be one in which CIP is used for momentum equations and CCD to solve the Poisson's equation. Copyright © 2005 John Wiley & Sons, Ltd.     

On the steady solitary-wave solution of the Green–Naghdi equations of different levels

The steady-state solitary wave solution of high-level Green–Naghdi (GN) equations is obtained by use of the Newton–Raphson method. Four aspects of solitary waves are studied: the wave speed, wave profile, velocity field and particle trajectory. A convergence study is performed for each individual case. Results of the converged model are compared with the existing laboratory experiments and other theoretical solutions for an inviscid and incompressible fluid, including the solutions of the Euler equations. Particle trajectories, predicted by the GN model, show close agreement with the laboratory measurements and provide a new approach to understanding the movement of the particles under a solitary wave. It is further shown that high-level GN equations can predict the solitary wave of the highest height.     

Modeling the interaction of solitary waves and semi-circular breakwaters by using unsteady reynolds equations

A vertical 2 -D numerical wave model was developed based on unsteady Reynolds equations. In this model, the k-epsilon models were used to close the Reynolds equations, and volume of fluid( VOF) method was used to reconstruct the free surface. The model was verified by experimental data. Then the model was used to simulate solitary wave interaction with submerged, alternative submerged and emerged semi-circular breakwaters. The process of velocity field, pressure field and the wave surface near the breakwaters was obtained. It is found that when the semi-circular breakwater is submerged, a large vortex will be generated at the bottom of the lee side wall of the breakwater ; when the still water depth is equal to the radius of the semi-circular breakwater, a pair of large vortices will be generated near the shoreward wall of the semi-circular breakwater due to wave impacting, but the velocity near the bottom of the lee side wall of the breakwater is always relatively small. When the semi-circular breakwater is emerged, and solitary wave cannot overtop it, the solitary wave surface will run up and down secondarily during reflecting from the breakwater. It can be further used to estate the diffusing and transportation of the contamination and transportation of suspended sediment.     

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.     

A study of shallow water waves with Gardner’s equation

In this paper the dynamics of solitary waves governed by Gardner’s equation for shallow water waves is studied. The mapping method is employed to carry out the integration of the equation. Subsequently, the perturbed Gardner equation is studied, and the fixed point of the soliton width is obtained. This fixed point is then classified. The integration of the perturbed Gardner equation is also carried out with the aid of He’s semi-inverse variational principle. Finally, Gardner’s equation with full nonlinearity is solved with the aid of the solitary wave ansatz method.     

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.     

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.     

高度非线性孤立波与弹性大板的耦合作用研究

一维颗粒链的一端受到一个有初速度颗粒的撞击,导致颗粒连中产生稳定传播的应力波——高度非线性孤立波,该应力波的波长、波速以及幅值都能保持很好的稳定性,且遇到边界才会反射. 孤立波是一种良好的信息载体,广泛应用于无损检测技术中. 基于孤立波的特性,研究高度非线性孤立波与弹性大板耦合作用,基于赫兹定律和板的内在非弹性理论,推导出晶体链与大板的耦合微分方程组. 用龙格库塔法求解该微分方程组,得到颗粒链中各颗粒的位移、速度曲线. 通过分析回弹波出现的时间、回弹波所携带的能量以及模量、厚度、重力等对孤立波的影响,发现反射孤立波对大板的弹性模量和厚度尤为敏感,此外,颗粒链的摆放对整个耦合过程也有影响. 研究的结果为孤立波对结构体的无损探伤提供了理论依据,该技术可实现对结构体的快速检查和可控性研究.     

Tsunamis and Barge Canals

Russell’s velocity formula was at the center of the controversy over the existence of the solitary wave; but today the topic is rarely mentioned. It is an immediate corollary of modern bifurcation theory; and it is fundamental to modelling waves in deep water. A tsunami 60 cm high in an ocean 4 km deep is 377 km long, travels with a velocity of 713 km/hr, and carries a transverse energy density of 2 × 10⁹ joules/meter, yet the maximum speed of the surface current is only 2.97 cm/s.     

On time-dependent,two-layer flow over topography: II. Evolution and propagation of solitary waves

The evolution of topographically generated interfacial motion is considered in a two-layer model. A system of two non-linear equations, similar to the Boussinesq equations for shallow water waves, is derived. The consequences of the cubic non-linearity of these equations on the nature of the solitary wave solutions are explored. A dispersion relation for solitary waves implies the existence of maxima for speed and displacement in a wave. The limiting values are shown to agree with other studies. The growth of solitary and/or cnoidal waves is studied for finite pulses of displacement and for internal bores.     

Uncertainty analysis of deflagration-to-detonation run-up distance

Three methods were adopted to estimate the deflagration-to-detonation run-up distance in a smooth tube, which comprise (1) the measurement of the propagation speed of pressure or combustion waves compared with the C–J detonation speed, (2) the time of the onset of detonation by the emission of visible light and the trajectory of pressure wave or combustion waves, and (3) the trajectory intersection with the presence of retonation wave. A nonstationary cross-correlation technique was applied to evaluate the uncertainty in estimating the run-up distance. Evaluation of the pressure wave (pressure wave speed or the pressure wave trajectory) appears to be more suitable to determine the deflagration-to-detonation run-up distance. PACS 47.40.-x Communicated by Z. Jiang     

A universal bifurcation mechanism arising from progressive hydroelastic waves

A unidirectional, weakly dispersive nonlinear model is proposed to describe the supercritical bifurcation arising from hydroelastic waves in deep water. This model equation, including quadratic, cubic, and quartic nonlinearities, is an extension of the famous Whitham equation. The coefficients of the nonlinear terms are chosen to match with the key properties of the full Euler equations, precisely, the associated cubic nonlinear Schr?dinger equation and the amplitude of the solitary wave at the bifurcation point. It is shown that the supercritical bifurcation, rich with Stokes, solitary, generalized solitary, and dark solitary waves in the vicinity of the phase speed minimum, is a universal bifurcation mechanism. The newly developed model can capture the essential features near the bifurcation point and easily be generalized to other nonlinear wave problems in hydrodynamics.     

Generation of stable solitary waves by a piston-type wave maker

The focus of present study is on how to generate solitary waves as pure as possible by using a piston type wave maker. A meshless numerical model, which can simulate the trajectories of fluid particles in a wave motion exerted by the wave paddle, is established for the purpose of present study. The present numerical model is verified by the comparison with experimental data before it is employed to the focused problem. Various wave paddle motions are considered. The results show that solitary waves generated by applying Fenton’s solitary solution to the paddle motion proposed by Goring are purer than those generated by other paddle motions.     

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.     

Numerical detection and generation of solitary waves for a nonlinear wave equation

This paper presents an automatic algorithm for detecting and generating solitary waves of nonlinear wave equations. With this purpose, dynamic simulations are carried out, the solution of which evolves into a main pulse along with smaller dispersive tails. The solitary waves are detected automatically by the algorithm by checking that they have constant amplitude and are symmetric respect to its maximum value. Once the main wave has been detected, the algorithm cleans the dispersive tails for time enough so that the solitary wave is obtained with the required precision.In order to use our algorithm, we need a spatial discretization with local basis. The numerical experiments are carried out for the BBM equation discretized in space with cubic finite elements along with periodic boundary conditions. Moreover, a geometric integrator in time is used in order to obtain good approximations of the solitary waves.     

Surface Water Waves and Tsunamis

Because of the enormous earthquake in Sumatra on December 26, 2004, and the devastating tsunami which followed, I have chosen the focus of my mini-course lectures at this year’s PASI to be on two topics which involve the dynamics of surface water waves. These topics are of interest to mathematicians interested in wave propagation, and particularly to Chilean scientists, I believe, because of Chile’s presence on the tectonically active Pacific Rim. My first lecture will describe the equations of fluid dynamics for the free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this, we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. In fact, the signal given by ocean waves generated by the Sumatra earthquake was felt globally; within 48 h distinguishable tsunami waves were measured by wave gages in Antarctica, Chile, Rio di Janeiro, the west coast of Mexico, the east coast of the United States, and at Halifax, Nova Scotia. To describe ocean waves, we will formulate the full nonlinear fluid dynamical equations as a Hamiltonian system [19], and we will introduce the Greens function and the Dirichlet-Neumann operator for the fluid domain along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations, we will derive the known Boussinesq-like systems and the KdV and KP equations, which govern the asymptotic behavior of tsunami waves over an idealized flat bottom. When the bottom is no longer assumed to be perfectly flat, a related theory [6, 13] gives a family of model equations taking this into account. My second lecture will describe a series of recent results in PDE, numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE), the Euler equations for a free surface do not admit clean (‘elastic’) interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical simulations which quantify the degree to which solitary waves lose energy during interactions [3, 4]. It is striking that this degree of ‘inelasticity’ is remarkably small. I will describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon [14, 18].     

Symmetry and Uniqueness of the Solitary-Wave Solution for the Ostrovsky Equation

We consider herein the Ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water, or the capillary waves in a plasma with the effects of rotation. Using the modified sliding method, we prove that the solitary wave moving to the left to the Ostrovsky equation is symmetric about the origin and unique up to translations. We also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed, weak rotation, and dispersive parameter.