The Effect of the Characteristic Dimension of a Microstructural Material and the Trough Length of a Solitary Wave on its Evolution

The profile of a solitary wave as a function of the trough size is studied. Profiles are specified in the form of the Chebyshov–Hermite and Whittaker functions. The convenience of introducing a new parameter in units of length is demonstrated     

Simulation of Traveling Solitary Plane Waves

The possibility of propagation of solitary plane waves with the Whittaker profile in materials with a microstructure (composites) is discussed. Solitary waves are defined as aperiodic smooth waves with an initial profile that is equal to zero everywhere except for some finite interval. Functions with indices 0.0, 0.1, –1/4, and 1/4 are chosen for computer simulation. It is observed that with some restrictions on the time or distance of propagation in the material, two modes of the traveling wave with the Whittaker profile and different phase-dependent phase velocities propagate simultaneously. The discussion section focuses attention on the conditions of blanking of the second mode for small values of the phase     

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.     

Plane Horizontal Transverse Waves in Elastic Piezopowders

A procedure and results of computer simulation of plane horizontal transverse waves are described. Three materials — gallium arsenide, bismuth germanate, and lead zirconate–titanate ceramics — are selected as the piezoelectric phase. The second phase of the powder is always lead. To describe waves in the powder, the microstructural theory of two-phase mixtures is used. Therefore, the computer simulation was intended to study the influence of the lead content by volume on the wave velocities and the microstructural wave-propagation pattern — decomposition of a wave into two modes, simultaneous propagation of both modes in each phase of the powder, etc. First, sets of physical constants (elastic, piezoelectric, and dielectric) of mixture theory were evaluated for three types of powders (with the piezoelectric phase as one of the above-mentioned materials) with the volume piezoelectric-phase content varying from 0.01 to 0.5 with step 0.005. Further, dispersion curves for both modes and 3D-graphs of amplitudes as functions of the wave propagation time and distance were plotted for 300 compositions of powders (three types, each of 100 modifications). Of the phenomena described, we should first of all point out that all the phase velocities increase twice upon changing the content of the powder in the piezoelectric phase from a very small amount to the maximum possible     

Features of the Interaction of Simple Waves Propagating In A Composite

Longitudinal plane simple waves having initial profiles in the form of the Chebyshov–Hermite function and propagating in a solid two-phase mixture are studied. The interaction between two simple waves, generation of the third wave because of this interaction, and the conditions for its occurrence are successively stated. The case where the third wave is not generated is analyzed numerically for the first time     

Shock wave in a gas-liquid medium

The propagation of weak shock waves and the conditions for their existence in a gas-liquid medium are studied in [1]. The article [2] is devoted to an examination of powerful shock waves in liquids containing gas bubbles. The possibility of the existence in such a medium of a shock wave having an oscillatory pressure profile at the front is demonstrated in [3] based on the general results of nonlinear wave dynamics. It is shown in [4, 5] that a shock wave in a gas-liquid mixture actually has a profile having an oscillating pressure. The drawback of [3–5] is the necessity of postulating the existence of the shock waves. This is connected with the absence of a direct calculation of the dissipative effects in the fundamental equations. The present article is devoted to the theoretical and experimental study of the structure of a shock wave in a gas-liquid medium. It is shown, within the framework of a homogeneous biphasic model, that the structure of the shock wave can be studied on the basis of the Burgers-Korteweg-de Vries equation. The results of piezoelectric measurements of the pressure profile along the shock wave front agree qualitatively with the theoretical representations of the structure of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 65–69, May–June, 1973.     

Discontinuities of the variables that characterize the propagation of solitary waves in a fluid layer

A nonlinear system of equations of hyperbolic type describing the propagation of solitary waves is considered [1]. A solitary wave is characterized in this approximation by two variables — the energy density per unit length measured along its crest, and the direction of the normal to the wave crest. The evolution of a wave described by the system may lead to the appearance of discontinuities, at which there are jumps in the energy density and the direction of the wave crest [2]. To establish the conditions at the discontinuities, a solution describing the interaction of nonparallel solitons [3, 4] is used. The obtained conditions are used to solve the problem of the decay of an arbitrary discontinuity in terms of soliton variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–93, May–June, 1984.I thank A. G. Kulikovskii and A. A. Barmin for helpful discussions and valuable comments in the preparation of the paper.     

Reflection of a Dam–Break Wave at a Vertical Wall. Numerical Modeling and Experiment

Results of the experimental study and numerical modeling of the reflection of a dam–break wave at the vertical end wall of a channel are given. A wave forms with distance from a partition creating the initial level difference of the liquid. It is shown that a numerical calculation based on the Zheleznyak—Pelinovskii nonlinear dispersion model satisfactorily describes the height of the splash–up, the amplitude of reflected waves, and the wave velocity in front of the wall for smooth and dam–break waves. It is also shown that, for smooth and weakly breaking (without significant entrainment of air) incoming waves, the experimental values of the height of the splash–up at the wall agree well with relevant experimental and calculated data for solitary waves.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

The Wave Energy in Nonlinearly Deformable Composites

The energy characteristics of waves propagating in composites are discussed. To describe the deformation of materials, two models are used — the classical model of an elastic body and the microstructural model of a two-phase elastic mixture. Both models take into account the quadratic nonlinearity of deformation based on the Murnaghan elastic potential. Analytical expressions for the velocity at which the energy of travelling plane longitudinal waves propagates are derived. It is shown that the nonlinearity of composite deformation decreases the velocities of energy propagation of both nondispersive and dispersive waves     

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.     

Solitary wave trains in a cold plasma

The existence of traveling solitary waves, the products of modulation instability in a cold quasi-neutral plasma, is considered. Solitary waves of this type (solitary wave trains) are formed as a result of bifurcation from a nonzero wave number of the linear wave spectrum. It is shown that the complete system of equations describing the wave process in a cold plasma has solutions of the solitary wave train type, at least when the undisturbed magnetic field is perpendicular to the wave front. Sufficient conditions of existence of solitary wave trains in weakly dispersive media are also formulated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 154–161, September–October, 1996.     

Solitary waves in a layer of viscous liquid

Solitary waves in a thin layer of viscous liquid which is running down a vertical surface under the action of gravity are investigated. The existence of such waves was demonstrated in the experiments of [1, 2]. The difficulties that must be faced in a theoretical computation were also noted in these studies. Below a solution of the problem of stationary waves is obtained by the method of expansion in the small parameter in two regions with subsequent matching and also by a numerical integration method. It is shown that in each case a solution of solitary wave type exists along with the single-parameter family of periodic solutions (parameter—the wave number ). On decreasing the wave number, the periodic waves go over into a succession of solitary waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 63–66, January–February, 1977.The authors thank L. N. Maurin for helpful discussions and A. M. Tereshchenko for assisting in the computations.     

Nonlinear wave effects in a fluid-saturated porous medium

Some one-dimensional nonlinear effects associated with wave propagation in weakly permeable fluid-saturated porous media are investigated. The effect of nonlinearity on the damping of monoharmonic waves is estimated and, moreover, the characteristics of the nonlinear parametric interaction of two waves excited in the medium by two monoharmonic sources of different frequencies are established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 74–77, January–February, 1992.     

The evolution of a solitary wave in a nonhomogeneous medium

Using the example of surface waves in a heavy liquid, the article discusses the propagation of a solitary wave in a nonhomogeneous medium. Ananalysis is made of processes of the decomposition of a wave into solitary waves, as a function of differences in the depth.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 80–85, November–December, 1971.The author thanks A. V. Gaponov, A. G. Litvak, and L. A. Ostrovskii for their evaluation of the results of the work.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Nonlinear surface waves in media with complex rheology

Weak nonlinear waves in a generalized viscoelastic medium with internal oscillators are considered. The rheological relations contain higher time derivatives of the stresses and strains as well as their tensor products. The method of expansion in a small parameter with the introduction of slow time and a running space coordinate is employed. The first approximation gives wave velocities and relations between the parameters equivalent to the results of an acoustic analysis at elastic wave fronts [1]. The second approximation leads to an evolution equation for the displacement velocity. For this a Fourier-Laplace double integral transformation is used. Reversion to the inverse transforms of the unknown functions leads to an integrodifferential evolution equation, which contains a Hubert transform and is a generalization of the Benjamin-Ono equation of deep water theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–103, September–October, 1990.     

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.     

Influence of an underwater ridge on a nonlinear wave expanding from a point on the surface of a liquid

A study is made of the influence of an underwater ridge on a solitary wave that prior to the interaction with the ridge has the form of a circle situated outside the ridge. It is shown that the nonlinear effects lead to a concentration of the wave energy above the ridge. As they move away from the source, the waves propagating above the ridge are not damped in the considered approximation but are damped everywhere away from the ridge. An analogy is pointed out between the propagation of the wave and two-dimensional steady flows of a fluid, and this makes it possible to use hydrodynamic intuition for qualitative predictions about the nature of the wave propagation in various cases. All the results of the paper can be extended to the case of waves that are periodic in time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 100–105, July–August, 1982.