New Solitary Waves in Elastic Materials: Existence and Nonlinear Interaction

Waves mentioned in the title were revealed in composite materials that are described by the microstructural theory of the second order — the theory of two-phase mixtures. For harmonic periodic waves, a mixture is always a dispersive medium. This medium admits existence of other waves — waves with profiles described by functions of mathematical physics (the Chebyshov–Hermite, Whittaker, Mathieu, and Lamé functions). If the initial profile of a plane wave is chosen in the form of the Chebyshev–Hermite or Whittaker function, then the wave may be regarded as an aperiodic solitary wave. The dispersivity of a mixture as a nonlinear frequency dependence of phase velocities transforms for nonperiodic solitary waves into a nonlinear phase-dependence of wave velocities. This and some other properties of such waves permit us to state that these waves fall into a new class of waves in materials, which is intermediate between the classical simple waves and the classical dispersion traveling waves. The existence of these new waves is proved in a computer analysis of phase-velocity-versus-phase plots. One of the main results of the interaction study is proof of the existence of this interaction itself. Some features of the wave interaction — triplets and the concept of synchronization — are commented on     

Some features of flexural-gravity wave propagation in the presence of a crack in sheet ice

The effect of a crack in an ice sheet on the propagation of surface flexural-gravity waves in a basin of constant depth is analyzed. The ice sheet is simulated by two floating semi-infinite fragments of a thin elastic isotropic plate. As the boundary-contact conditions on the line of contact between the parts of the plate the conditions of continuity of displacements for arbitrary slopes simulating one ice-floe overlying on another and free-edge conditions (crack) are considered. The dependence of the amplitude characteristics of the perturbations on the thickness of the ice, its degree of compression, the incident wave frequency, the depth of the basin, and the form of the boundary-contact conditions is investigated. Problems of wave diffraction on inhomogeneities of an elastic plate were solved in [1, 2], and on a crack in the ice sheet in [3, 4].Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–150, March–April, 1996.     

An explicit expression for the surface-impedance matrix of a generally anisotropic incompressible elastic material in a state of plane strain

Surface and interfacial impedance matrices play an important role in the construction of Green's functions, the analysis of surface and interfacial waves and the stability assessment of pre-stressed half-spaces or joined half-spaces. This paper studies these matrices for generally anisotropic pre-stressed incompressible elastic materials. It is shown that the surface-impedance matrix satisfies a simple matrix equation which, for plane-strain deformations, can be solved exactly. As a result, explicit secular equations for surface and interfacial wave speeds and explicit wrinkling/buckling conditions for pre-stressed half-spaces and joined half-spaces are obtained. It is also shown that the plane-strain surface-wave problem is mathematically identical to the edge-wave problem for thin elastic plates. Thus, the uniqueness of surface-wave speed is settled by drawing upon a recent proof of the uniqueness of edge-wave speed. Examples are used to show that it is straightforward to solve the secular equations based on the given formulae either exactly (where possible) or numerically.     

Some Exactly Solvable Problems of the Radiation of Three–Dimensional Periodic Internal Waves

An exact solution of the problem of the generation of three–dimensional periodic internal waves in an exponentially stratified, viscous fluid is constructed in a linear approximation. The wave source is an arbitrary part of the surface of a vertical circular cylinder which moves in radial, azimuthal, and vertical directions. Solutions satisfying exact boundary conditions, describe both the beam of outgoing waves and wave boundary layers of two types: internal boundary layers, whose thickness depends on the buoyancy frequency and the geometry of the problem, and viscous boundary layers, which, as in a homogeneous fluid, are determined by kinematic viscosity and frequency. Asymptotic solutions are derived in explicit form for cylinders of large, intermediate, and small dimensions relative to the natural scales of the problem.     

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.     

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     

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.     

The ionization-rate constant at high temperatures high electron concentrations

The electron distribution function and the rate constant for ionization of atoms by electron impacts have been calculated as they apply to the conditions that are characteristic of a shock wave — namely, the energy distribution of the electrons and the ionization-rate constants are determined as functions of the temperature of the heavy particles. The energy dependence of the effective cross section for the excitation of an atom by electron impact is assumed to be linear. Equations of the Fokker-Planck type are used in the solution of the problem, and the range of temperatures and concentrations in which the deviation of the distribution from Maxwellian leads to a substantial change of the ionization-rate constant is determined.Translated from Zhurnal Prikladnoi Mekhaniki i Tecknicheskoi Fiziki, No. 2, pp. 32–40, March–April, 1971.     

Torsional surface waves in a gradient-elastic half-space

The present work deals with torsional wave propagation in a linear gradient-elastic half-space. More specifically, we prove that torsional surface waves (i.e. waves with amplitudes exponentially decaying with distance from the free surface) do exist in a homogeneous gradient-elastic half-space. This finding is in contrast with the well-known result of the classical theory of linear elasticity that torsional surface waves do not exist in a homogeneous half-space. The weakness of the classical theory, at this point, is only circumvented by modeling the half-space as having material properties variable with depth (E. Meissner, Elastische Oberflachenwellen mit Dispersion in einem inhomogenen Medium, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich 66 (1921) 181–195; I. Vardoulakis, Torsional surface waves in inhomogeneous elastic media, Internat. J. Numer. Anal. Methods Geomech. 8 (1984) 287–296; G.A. Maugin, Shear horizontal surface acoustic waves on solids, in: D.F. Parker, G.A. Maugin (Eds.), Recent Developments in Surface Acoustic Waves, Springer Series on Wave Phenomena, vol. 7, Springer, Berlin, 1988, pp. 158–172), as a layered structure (Maugin, 1988; E. Reissner, Freie und erzwungene Torsionsschwingungen des elastischen Halbraumes, Ingenieur-Archiv 8 (1937) 229–245) or by considering couplings with electric and magnetic fields for different types of materials (Maugin, 1988). The theory employed here is the simplest possible version of Mindlin’s (R.D. Mindlin, Micro-structure in linear elasticity, Arch. Rat. Mech. Anal. 16 (1964) 51–78) generalized linear elasticity. A simple wave-propagation analysis based on Hankel transforms and complex-variable theory was done in order to determine the conditions for the existence of the torsional surface motions and to derive dispersion curves and cut-off frequencies. Also, we notice that, up to date, no other generalized linear continuum theory (including the integral-type non-local theory) has successfully been proposed to predict torsional surface waves in a homogeneous half-space.     

Discrete Homogenization in Graphene Sheet Modeling

Graphene sheets can be considered as lattices consisting of atoms and of interatomic bonds. Their bond lengths are smaller than one nanometer. Simple models describe their behavior by an energy that takes into account both the interatomic lengths and the angles between bonds. We make use of their periodic structure and we construct an equivalent macroscopic model by means of a discrete homogenization technique. Large three-dimensional deformations of graphene sheets are thus governed by a membrane model whose constitutive law is implicit. By linearizing around a prestressed configuration, we obtain linear membrane models that are valid for small displacements and whose constitutive laws are explicit. When restricting to two-dimensional deformations, we can linearize around a rest configuration and we provide explicit macroscopical mechanical constants expressed in terms of the interatomic tension and bending stiffnesses.     

Swell wave propagation in an inhomogeneous ice sheet

The propagation of surface waves beneath a periodically inhomogeneous ice sheet is considered. Areas of broken ice and hummock ridges are considered as irregularities. It is shown that waves with frequencies corresponding to wind and swell waves are strongly scattered by the irregularities and are damped exponentially as they propagate beneath the ice.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 162–169, September–October, 1996.     

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.     

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.     

Elastic Anisotropic Material with Purely Longitudinal and Transverse Waves

The simplest form of the matrix of elasticity moduli of an anisotropic material conducting purely longitudinal and transverse waves with an arbitrary direction of the wave normal is obtained. A generic solution of equations in displacements is represented in terms of three functions satisfying independent wave equations. In the case of planar deformation, this solution yields a complex representation coinciding with the Kolosov–Muskhelishvili formulas for an isotropic material. The formulas in the present work also determine an anisotropic material with Young's modulus identical for all directions, as in an isotropic medium.     

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.     

Diffraction of Love waves by a stress-free crack of finite width in the plane interface of a layered composite

A rigorous theory of the diffraction of Love waves by a stress-free crack of finite width in the interface of a layered composite is presented. The incident wave is taken to be either a bulk wave or a Love-wave mode. The resulting boundary-value problem for the unknown jump in the particle displacement across the crack is solved by employing the integral equation method. The unknown quantity is expanded in terms of a complete sequence of expansion functions in which each separate term satisfies the edge condition. This leads to an infinite system of linear, algebraic equations for the coefficients of the expansion functions. This system is solved numerically. The scattering matrix of the crack, which relates the amplitudes of the outgoing waves to the amplitudes of the incident waves, is computed. Several reciprocity and power-flow relations are obtained. Numerical results are presented for a range of material constants and geometrical parameters.     

Effect of leading radiation on the stability of strong shock waves in gases with an arbitrary equation of state

The effect of leading radiation on the stability of a strong shock wave in an ideal gas with an arbitrary equation of state is investigated. The ionization ahead of and behind the shock front and the radiation are assumed to be in equilibrium. The investigation is carried out in the linear approximation with respect to amplitude for disturbances with a wavelength much greater than the width of the relaxation zones ahead of and behind the shock. The conditions under which the leading radiation has a destabilizing effect on the shock wave are established. It is shown, in particular, that neutrally stable shock waves become unstable. The conditions under which the onset of instability is of the threshold type with respect to the radiation intensity are determined. It is found that the radiation also has a destabilizing effect on stable shock waves, including shock waves in a perfect gas. However, in this case instability can develop only when the disturbances have a wavelength comparable with the width of the relaxation zone. A simple physical mechanism of the onset of instability under the influence of leading radiation is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 125–133, May–June, 1990.The authors are grateful to A. G. Kulikovskii and A. A. Barmin for their constant interest and useful discussions.     

Spatial Simple Waves on a Shear Flow

The paper studies simple waves of the shallowwater equations describing threedimensional wave motions of a rotational liquid in a freeboundary layer. Simple wave equations are derived for the general case. The existence of unsteady or steady simple waves adjacent continuously to a given steady shear flow along a characteristic surface is proved. Exact solutions of the equations describing steady simple waves were found. These solutions can be treated as extension of Prandtl–Mayer waves for sheared flows. For shearless flows, a general solution of the system of equations describing unsteady spatial simple waves was found.     

The influence of a uniformly compressed floating elastic plate on the development of three-dimensional waves in a homogeneous fluid

A study is made in the linear formulation of the influence of a uniformly compressed floating elastic plate on the unsteady three-dimensional wave motion of a homogeneous fluid of finite depth. Waves are excited by a region of normal stresses which moves on the surface of the plate. Three-dimensional flexural-gravity waves were studied in [1, 2] without allowance for compressing forces. Plane waves under conditions of longitudinal compression were considered in [3, 4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 78–83, November–December, 1984.