Self-similar solutions describing the propagation of solitary waves above underwater ridges and troughs

The nonlinear ray method [1] is used to investigate the propagation of solitary waves over an uneven bottom. In the process of nonlinear evolution of the wave front, singular points develop in it; these are treated in the given model as discontinuities [2, 3]. In contrast to earlier studies, it is not assumed here that the intensity of the discontinuity is weak. Boundary conditions at the discontinuities are introduced on the basis of the results of Miles and Bakholdin [4–6], and this makes it possible to take into account the energy loss at a discontinuity and the effects of wave reflection and construct a number of new self-similar solutions for the propagation of a wave above a ridge and trough. The main attention is devoted to considering how the type of solution depends on the parameters of the wave and the relief. For certain values of the parameters, the self-similar solution of the encounter of a homogeneous wave with a ridge is not unique. The reason for this is the singularity of the relief at the end of the ridge. A numerical investigation has therefore also been made of the encounter of a wave with a ridge having a smooth relief at its end. For an under-water trough and a ridge—trough system, self-similar solutions with complete or partial reflection or transmission of the wave energy into the trough are found. A reflected wave can also arise from an encounter with a ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 137–144, September–October, 1985.I thank A. G. Kulikovskii and A. A. Barmin for their interest in the work and for valuable comments made as the paper was being prepared for press.     

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.     

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 nonlinear waves in a porous medium with two-phase saturation by a liquid and a gas

The propagation of nonlinear waves through a porous medium saturated with a viscous liquid and a gas is investigated with allowance for the capillary pressure. Numerical solutions of the traveling-wave type are constructed for the generalized Korteweg-de Vries-Burgers equation for the wave amplitudes. Three types of regimes of longitudinal wave propagation, including soliton-like regimes, are found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 86–95, July–August, 1996.     

Structure of internal waves in a channel

One method of describing wave motion in a fluid with continuous stratification is to use normal waves (modes). The propagation of internal gravity waves in closed rectangular regions whose boundaries coincide with planes through which there is no normal motion is essentially different from wave motion in an unbounded medium [1, 2]. This paper describes a theoretical and experimental investigation of the propagation of internal waves in an exponentially stratified fluid in a horizontal channel of finite height.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–110, January–February, 1935.In conclusion the authors wish to express their gratitude to A. T. Onufriev for his interest in their work.     

Evolution of shock waves in polydisperse gas suspensions with a nonuniform particle concentration distribution

The results of a numerical investigation of the laws of shock wave propagation in polydisperse (two-fraction) gas suspensions with a non-uniform initial particle concentration distribution are presented. Examples of shock wave propagation in extended layers of a gas suspension with linearly increasing, linearly decreasing and sinusoidal laws of variation of the particle concentration are considered. It is shown that when shock waves pass through layers of a gas suspension with increasing and decreasing laws of variation of the particle concentration, respectively, amplification and attenuation of the waves are observed; when shock waves travel through gas suspensions with a periodic law of variation of the particle concentration the pressure distribution behind the wave fronts is nonmonotonic. The solutions corresponding to polydisperse and monodisperse gas suspensions with an effective particle size are examined. The nonequilibrium and thermodynamic-equilibrium solutions are compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 183–190, September–October, 1991.     

Interaction of rarefaction waves with wet foam

The interaction of rarefaction waves of different shapes with wet water foams is studied experimentally. It is found that the observed values of the pressure are greater, while the surface velocity is lower than the corresponding values predicted by the pseudogas model. The foam breakdown starts as the pressure decreases by 0.3 atm relative to the initial pressure. During downstream propagation of the rarefaction-wave leading edge the propagation velocity decreases.Using of water-based foams as effective screens for damping blast waves in different technological processes has caused considerable interest in studying wave propagation in such systems. The pressure wave dynamics in a foam have been investigated in much detail, both experimentally and theoretically [1–3]. However, the interaction of rarefaction waves with foam has practically never been studied, although it was mentioned in [4] that the unloading phase following the compression wave phase is one of the factors defining the damaging action of blast waves. Besides blast-wave damping, rarefaction wave propagation takes place if such waves are used to breakup foam in oil-producing wells [5].Below, the interaction of rarefaction waves of different shapes with wet water foams is studied. The vertical shock tube described in detail in [3] was used in these experiments.Brest. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 76–82, March–April, 1995.     

Transmission of waves in a liquid through absorbing layers and subsequent interaction of the waves with an obstacle

The propagation of waves in porous media is investigated both experimentally [1, 2] and by numerical simulation [3–5]. The influence of the relaxation properties of porous media on the propagation of waves has been investigated theoretically and compared with experiments [3, 4]. The interaction of a wave in air that passes through a layer of porous medium before interacting with an obstacle has been investigated with allowance for the relaxation properties [5]. In the present paper, in which the relaxation properties are also taken into account, a similar investigation is made into the interaction with an obstacle of a wave in a liquid that passes through a layer of a porous medium before encountering the obstacle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–53, March–April, 1983.     

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.     

Flow of a free jet of rheologically complex fluid from a vibrating capillary

The propagation of finite-amplitude waves along a vibrating capillary jet is studied. The standing wave equation is derived.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–10, May–June, 1985.     

Propagation of waves in a suspension of solid particles

A theory is developed for the propagation of waves in a suspension of elastic or rigid solid particles in a viscous or inviscid compressible fluid, using a homogenization process. We study the case where the characteristic length of the particles is small compared with the wave length. In the case of a viscous fluid, a law similar to Darcy's law for the average velocity of the suspension is established, and in the case of macroscopic homogeneity and isotropy, the propagation of a plane wave displays one dilatational, damped and dispersive wave. In the case of a barotropic inviscid fluid, the average acceleration of the suspension depends, in a linear way, on the mean pressure gradient and in the case of macroscopic homogeneity and isotropy, the propagation of a plane wave displays one dilatational, undamped and non dispersive wave.     

Soret effect on propagation of thermoconvective waves in a binary mixture

Effects of thermal diffusion (Soret effect) on the propagation of thermoconvective waves (TCW) in a layer of binary mixture are investigated. It is shown that undamped propagation of TCW is possible in a layer heated from below provided the Soret number is large and positive. A novel result of the analysis is that for a layer heated from below, TCW which are strongly damped in the absence of thermal diffusion may propagate almost undamped in the low frequency limit in the presence of thermal diffusion for some negative value of Soret number. It is further shown that for a layer heated from above such that the solute concentration increases vertically upward, weakly damped propagation of TCW in the low frequency limit is inhibited with increase in Soret number. Received on 14 July 1998     

Propagation of a soliton along a fluid-filled pipe

In [1, 2] a mathematical model of the motion of a fluid in a pipe whose axis is a curve in space was discussed and certain simplifications of the problem were studied. The propagation of linear and nonlinear waves in the framework of the model was studied. In the present paper we consider a simple wave flow in a pipe with elastic walls suing one of the models introduced in [1], which, unlike [2], takes into account axial displacements of the pipe. The basic equations describing the propagation of waves in the pipe are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 3, pp. 58–63, May–June, 1986.     

A spherical wave in a viscoelastic half-space

The propagation of spherical waves in an isotropie elastic medium has been studied sufficiently completely (see, e.g., [1–4]). it is proved [5, 6] that in imperfect solid media, the formation and propagation of waves similar to waves in elastic media are possible. With the use of asymptotic transform inversion methods in [7] a problem of an internal point source in a viscoelastic medium was investigated. The problem of an explosion in rocks in a half-space was considered in [8]. A numerical Laplace transform inversion, proposed by Bellman, is presented in [9] for the study of the action of an explosive pulse on the surface of a spherical cavity in a viscoelastic medium of Voigt type. In the present study we investigate the propagation of a spherical wave formed from the action of a pulsed load on the internal surface of a spherical cavity in a viscoelastic half-space. The potentials of the waves propagating in the medium are constructed in the form of series in special functions. In order to realize viscoelasticity we use a correspondence method [10]. The transform inversion is carried out by means of a representation of the potentials in integral form and subsequent use of asymptotic methods for their calculation. Thus, it becomes possible to investigate the behavior of a medium near the wave fronts. The radial stress is calculated on the surface of the cavity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 139–146, March–April, 1976.     

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 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.     

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 detonation burning of a fuel mixture resulting from a double explosion

A study is made of the propagation of a multifront detonation burning in a fuel mixture consisting of a gaseous fuel and an oxidant with additions of combustible solid or liquid particles arising as a result of a double point explosion. In such combustible media it is possible for there to be propagation of several detonation or burning fronts following one after the other. The easily igniting gaseous fuel burns in the first detonation wave, which propagates in the gaseous mixture with particles which are heated by the products of the explosion, ignite and burn in the second detonation wave or in the flame front. Self-similar regimes of propagation of such waves in an idealized formulation were studied in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–131, March–April, 1985.     

Internal waves in a two-layer electrically conducting fluid in the presence of a magnetic field

The propagation of linear and nonlinear internal waves along the interface between two weakly conducting media differing in density and electrical conductivity is investigated and the influence of MHD interaction effects on their characteristics is analyzed. It is shown that in this system the waves propagate with dispersion and dissipation, and for harmonic waves of infinitesimal amplitude there exists a range of wave numbers on which propagating modes do not exist. For waves of finite amplitude a nonlinear Schrödinger equation with a dissipative perturbation is obtained and its asymptotic solution is found. It is established that the presence of electrical conductivity and an applied magnetic field leads to a decrease in the amplitude and the frequency of the envelope of the wave train.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–108, September–October, 1990.     

Equations of hydrodynamics for wind wave circulation on a shelf

In order to predict the propagation of an impurity and water quality on a shelf it is necessary to know the water mass dynamics and the water exchange. However, the hydrodyamics of the shelf zone differ considerably from those of the open expanses of seas and lakes owing to the steepness of the bottom, the complex structure of the shoreline, the major role of wind waves, and their breaking [1]. In [2, 3] the importance of surface waves and their breaking for inshore flows was demonstrated and the equations of hydrodynamics, averaged over the depth, were derived. For regions of the shelf remote from the shoreline it is also necessary to take into account the interaction of waves with the bottom and with essentially three-dimensional flows. In this note the equations of hydrodynamics are derived for wind wave flows averaged over the wave period in the threedimensional formulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1-, pp. 174–176, January–February, 1987.