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.     

Surface wave diffraction at a crack in sheet ice

The time-periodic motions of a liquid layer of finite depth beneath an ice sheet with a straight infinite crack having a periodic dependence on the horizontal coordinate in the direction of the crack are considered. The ice sheet is simulated by a thin elastic plate. It is assumed that the thickness of the plate changes abruptly across the crack. The problems of plane-wave diffraction at a crack, plane-wave diffraction atN cracks in a uniform ice sheet, and plane-wave reflection from a rigid wall are solved. The effect of the pre-existing state of stress of the ice sheet on the properties of the reflected waves is investigated. The condition of nontransmission of fix-frequency waves beneath the edge of the ice is obtained.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 93–102, March–April, 1993.     

Behavior of a floating elastic plate during vibrations of a bottom segment

The Wiener-Hopf technique is used to obtain an analytical solution for the problem of vibrations of a floating semi-infinite elastic plate due to earthquake-induced vibrations of a bottom segment. An explicit solution is obtained ignoring the inertial term. The surface-wave amplitudes and ice-plate deflection are studied numerically as functions of the frequency and position of the vibrating bottom segment, ice thickness, and fluid depth.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 98–108, March–April, 2005.     

Motion of an External Load over a Semi-Infinite Ice Sheet in the Subcritical Regime

The three-dimensional problem of steady-state forced vibrations of fluid and semiinfinite ice sheet under the action of a local external load traveling along the rectilinear sheet edge at a constant velocity is considered. Two cases are analyzed. In the first case the fluid surface outside the ice sheet is free and in the second the fluid is confined by a rigid vertical wall and the ice sheet edge adjacent to the wall can be both clamped and free. The ice sheet is simulated by a thin elastic isotropic plate floating on the surface of fluid of finite depth. The load traveling velocity is assumed to be not higher than the minimum phase velocity of the flexural-gravity waves (subcritical regime). The solution to the linear problem is obtained by means of the integral Fourier transform and matching the expansions of the velocity potential in the vertical eigenfunctions. Examples of the numerical investigation of the ice sheet and fluid displacements are given.     

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.     

Theory of two-dimensional nonlinear waves in liquid covered by ice

The aim is to develop a method of Hamiltonian formalism for the waves in the liquid beneath an ice sheet and on that basis to construct a systematic nonlinear theory. Attention is concentrated on the investigation of the essentially two-dimensional effects whose properties depend to a large extent on the stresses in the ice.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–133, July–August, 1991.     

Plane Problem of Vibrations of an Elastic Floating Plate under Periodic External Loading

The Wiener–Hopf technique is used to construct an analytical solution of the problem of vibrations of a semiinfinite elastic floating plate under periodic external loading. The solution is obtained in explicit form ignoring draft. The dependences of the amplitudes of surface waves and iceplate deflection on the loading distribution and frequency, ice thickness, and liquid depth are studied numerically. It is established that for some types of acting load, no waves propagate in the plate and liquid and the plate vibrations are standing waves localized near the loading region. An example of such vibrations is given and a condition for the occurrence of localized vibrations is found.     

Vibrational equations of variable mass layers and membranes

Vibrations of layers and membranes, consisting of variable mass particles [1, 2], are considered. Adjacent and separated particles are classified. Differential equations and natural boundary conditions of vibrations of variable mass layers and membranes are derived. Two examples for vibrations are integrated by means of separation of variables.Translated from Zhurnal Prikladnoi Mekbaniki i Tekhnicheskoi Fiziki, No. 5, pp. 138–143, September–October, 1973.     

Free vibrations of two capillary liquids rotating in a cylindrical vessel

The problem of the small natural vibrations of two coaxially disposed ideal liquids rotating in a cylindrical vessel under conditions of complete weightlessness is considered. The set of normal vibrations of the system consists of internal wave motions and surface waves. Asymptotic formulas are derived for the vibration frequencies of the surface waves. The results of computer calculations are presented in the form of graphs and tables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 97–104, September–October, 1976.     

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.     

Impingement of surface waves on the edge of compressed ice

The impingement of small-amplitude surface waves on the edge of a solid compressed ice sheet in a basin of finite constant depth is considered. The influence of the cylindrical rigidity and the value of the compressing force on the dependence of the amplitude coefficients of reflection and transmission on the incident wave period is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 121–126, May–June, 1995.     

Hydroelastic vibrations of a beam floating on the surface of an incompressible liquid

The article considers the problem of determining the forms and the frequencies of the free vibrations of a floating beam of elliptical cross section, on the basis of simultaneous solution of the equations of mechanical bending vibrations and the equations of hydrodynamics. The difference between the forms of the vibrations of a beam in a liquid and in a void is evaluated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 128–132, January–February, 1973.     

Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Mathematical model of the dissociation of gas hydrates coexisting with ice in natural reservoirs

The dissociation of gas hydrate coexisting with ice in a low-temperature natural reservoir is investigated. A mathematical model of the process consisting of a generalization of the Stefan problem and containing two unknown moving phase transition boundaries — the hydrate dissociation and ice melting fronts — is constructed. It is shown that in high-permeability reservoirs the velocity of the dissociation surface is higher than that of the ice melting surface. As the permeability decreases, the fronts change places. The problem is solved in the self-similar approximation.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 84–92, March–April, 1993.     

Vibrations of an ice sheet with crack under a time-periodic load

The analytic solution of the problem of vibrations of an ice sheet with a rectilinear crack floating on the surface of an ideal incompressible fluid of shallow depth under the action of a local zone of the time-periodic pressure is obtained. The ice sheet is simulated by two thin viscoelastic semiinfinite plates of different thickness. Various conditions on the crack edges are considered. Far field asymptotics are investigated and it is revealed that the predominant directions of wave propagation at an angle to the crack can be distinguished in the far field in the case of contact of two plates of different thickness. In the case of contact of identical plates, a waveguide mode propagating along the crack is excited. It is shown that the waveguide mode is the same for the plates with the free edges and the free overlap since the part of the solution symmetric about the crack is the same while the difference between the solutions is caused by the antisymmetric part of the solution.     

A Comment on Governing Equations of Envelope Surface Created by a Two-Dimensional Directional Wavepacket Centered around a Propagation Direction on an Elastic Plate

Plates are common structural elements of most engineering structures, including aerospace, automotive, and civil engineering structures. The study of plates from theoretical perspective as well as experimental viewpoint is fundamental to understanding of the behavior of such structures. The dynamic characteristics of plates, such as natural vibrations, transient responses for the external forces and so on, are especially of importance in actual environments. In this paper, we consider natural vibrations of an elastic plate and the propagation of a wavepacket on it. We derive the two-dimensional equations that govern the spatial and temporal evolution of the amplitude of a wavepacket and discuss its features. We especially consider a directional wavepacket on an elastic plate, which propagation direction is centered around a main direction, but which wavenumber and frequency are fixed by a certain value. The fact that its envelope becomes time-invariant and is governed by the Schrödinger–Nohara type equation is shown.     

Calculation of wind currents in the edge zone of a sea ice sheet

The steady wind currents in the ocean near the edge of an ice sheet simulated by a rigid cover are studied. The characteristic horizontal scales of the problem correspond to several tens of kilometers. In this case, depending on the depth and viscosity of the fluid, the effect of the Coriolis force on its motion may be comparable with the effect of the wind shear stresses applied to the ice-free surface. Taking the Coriolis force into account leads to the formation of a spiral flow structure in the vertical direction.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 132–141, November–December, 1992.     

Vibrations of a system of fluid-filled cylindrical shells on an elastic base

The resonance synchronization of the vibrations of a system of shells is studied. It is established that the vibrations of isolated components excite the vibrations of the whole system, consisting of three cylindrical shells on a common elastic base.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 134–137, December 2004.     

Electric effects of vibrations of capillaries filled with droplets of mercury and an electrolyte solution

The appearance of a time-dependent difference of the electric potentials due to mechanical vibrations of capillaries filled with droplets of mercury and an electrolyte solution was studied in [1–5]. The phenomenon is used in electrokinetic transducers [6, 7]. Equivalent electric and mechanical circuits of the transducers were proposed in [1–4]. The experimental dependences of the amplitude of the potential difference that arises between the mercury droplets on the vibration frequency (the amplitude-frequency characteristics of the transducers) were investigated in [2, 4]. In the present paper, expressions are obtained for the difference of the electric potentials between mercury droplets that arises in the case of harmonic vibrations of a capillary with small amplitudes. The theoretical and experimental amplitudefrequency characteristics of an electrocapillary transducer are constructed. Good agreement between the results of the calculations and the experimental data is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika-Zhidkosti i Gaza, No. 2, pp. 131–136, March–April, 1988.     

Apparent masses of a lattice of plates in an incompressible fluid

The apparent masses of a plane lattice of plates in an incompressible fluid were determined by Gurevich (for the case of cophased translational vibrations [1]) and by Samoilovich (for cophased and antiphased translational and torsional vibrations of plates [2]). However, in a number of applications it is necessary to calculate apparent masses both for more complex forms of vibrations and also for different values of the phase shift between the vibrations of neighboring plates. In the present paper we present a calculation algorithm for the apparent masses of the lattice of plates that vibrate according to an arbitrary law with a constant phase shift between the vibrations of neighboring plates. The forms of the vibrations are approximated by trigonometric polynomials, which allows us to represent the apparent masses in the form of series in some standard coefficients. Results are given for the calculation of these coefficients over a wide range of variations of thickness, stagger, and phase shift.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–18, March–April, 1973.